{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mathematics, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Moradifam, Amir","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-08-25T18:02:30Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2010","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigenvalue problems, and simultaneous\npreconditioning and symmetrization of linear systems.\n\nIn the first part that consists of three research papers we study improved Hardy and Hardy-Rellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n \u2265 1, so that the following inequalities hold for all u \\in C_{0}^{\\infty}(B):\n\\begin{equation*} \\label{one}\n\\hbox{$\\int_{B}V(x)|\\nabla u\u00b2dx \\geq \\int_{B} W(x)u\u00b2dx,}\n\\end{equation*}\n\n\\begin{equation*} \\label{two}\n\\hbox{$\\int_{B}V(x)|\\Delta u|\u00b2dx \\geq \\int_{B} W(x)|\\nabla %%@\nu|^\u00b2dx+(n-1)\\int_{B}(\\frac{V(x)}{|x|\u00b2}-\\frac{V_r(|x|)}{|x|})|\\nabla\nu|\u00b2dx.\n\\end{equation*}\nThis characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities. In section 4, with a similar approach, we present\nvarious classes of Hardy-Rellich inequalities on H\u00b2\\cap H\u00b9\u2080\n\nThe second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6\nwe study the extremal solution u_{\\lambda^*}$ of the semilinear biharmonic equation $\\Delta\u00b2 u=\\frac{\\lambda}{(1-u)\u00b2, which models a simple Micro-Electromechanical System (MEMS) device on a\nball B\\subset R^N, under Dirichlet or Navier boundary conditions. We show that u* is regular provided N \u2264 8 while u_{\\lambda^*} is singular for N \u2265 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N \u2265 13.\n\nIn the third part, motivated by the theory of self-duality we propose new templates for solving non-symmetric linear systems. Our approach is efficient when dealing with certain ill-conditioned and highly non-symmetric systems.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/27775?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"Hardy-Rellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems by Amir Moradifam A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August, 2010 c\u00a9 Amir Moradifam 2010 Abstract This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigen- value problems, and simultaneous preconditioning and symmetrization of linear systems. In the first part that consists of three research papers we study improved Hardy and Hardy-Rellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n \u2265 1, so that the following inequalities hold for all u \u2208 C\u221e0 (B): \u222b B V (x)|\u2207u|2dx \u2265 \u222b BW (x)u 2dx, \u222b B V (x)|\u2206u|2dx \u2265 \u222b BW (x)|\u2207u|2dx+ (n\u2212 1) \u222b B( V (x) |x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equa- tions. This allows us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities. In section 4, with a similar ap- proach, we present various classes of Hardy-Rellich inequalities on H2\u2229H10 . The second part of the thesis studies the regularity of the extremal solu- tion of fourth order semilinear equations. In sections 5 and 6 we study the extremal solution u\u03bb\u2217 of the semilinear biharmonic equation \u22062u = \u03bb(1\u2212u)2 , which models a simple Micro-Electromechanical System (MEMS) device on a ball B \u2282 RN , under Dirichlet or Navier boundary conditions. We show that u\u2217 is regular provided N \u2264 8 while u\u03bb\u2217 is singular for N \u2265 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N \u2265 13. In the third part, motivated by the theory of self-duality we propose new templates for solving non-symmetric linear systems. Our approach is efficient when dealing with certain ill-conditioned and highly non-symmetric systems. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Co-authorship Statement . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Optimal improved Hardy and Hardy-Rellich inequalities . . . 1 1.2 Fourth order nonlinear eigenvalue problems . . . . . . . . . . 8 1.3 Preconditioning and symmetrization of non-symmetric linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 I Hardy and Hardy-Rellich Inequalities 13 2 On the best possible remaining term in the Hardy inequality 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Two dimensional inequalities . . . . . . . . . . . . . . . . . . 18 2.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . 20 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 iii Table of Contents 3 Bessel pairs and optimal Hardy and Hardy-Rellich inequal- ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 General Hardy Inequalities . . . . . . . . . . . . . . . . . . . 39 3.2.1 Integral criteria for Bessel pairs . . . . . . . . . . . . 44 3.2.2 New weighted Hardy inequalities . . . . . . . . . . . . 47 3.2.3 Improved Hardy and Caffarelli-Kohn-Nirenberg Inequal- ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 General Hardy-Rellich inequalities . . . . . . . . . . . . . . . 55 3.3.1 The non-radial case . . . . . . . . . . . . . . . . . . . 56 3.3.2 The case of power potentials |x|m . . . . . . . . . . . 62 3.4 Higher order Rellich inequalities . . . . . . . . . . . . . . . . 70 3.5 The class of Bessel potentials . . . . . . . . . . . . . . . . . . 72 3.6 The evaluation of an,m . . . . . . . . . . . . . . . . . . . . . 76 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 Optimal weighted Hardy-Rellich inequalities on H2 \u2229H10 . 84 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 General Hardy Inequalities . . . . . . . . . . . . . . . . . . . 88 4.3 General Hardy-Rellich inequalities . . . . . . . . . . . . . . . 91 4.3.1 The non-radial case . . . . . . . . . . . . . . . . . . . 93 4.3.2 The case of power potentials |x|m . . . . . . . . . . . 96 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 II Fourth Order Nonlinear Eigenvalue Problems 107 5 The critical dimension for a fourth order elliptic problem 108 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2 The effect of boundary conditions on the pull-in voltage . . . 112 5.2.1 Stability of the minimal branch of solutions . . . . . . 115 5.3 Regularity of the extremal solution for 1 \u2264 N \u2264 8 . . . . . . 119 5.4 The extremal solution is singular for N \u2265 9 . . . . . . . . . . 122 5.5 Improved Hardy-Rellich Inequalities . . . . . . . . . . . . . . 128 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 iv Table of Contents 6 On the critical dimension for a fourth order elliptic problem 133 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2 The pull-in voltage . . . . . . . . . . . . . . . . . . . . . . . 135 6.3 Stability of the minimal solutions . . . . . . . . . . . . . . . 138 6.4 Regularity of the extremal solutions in dimensions N \u2264 8 . . 142 6.5 The extremal solution is singular in dimensions N \u2265 9 . . . . 144 6.6 Improved Hardy-Rellich Inequalities . . . . . . . . . . . . . . 149 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7 The singular extremal solutions of the bilaplacian . . . . 158 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.2 An improved Hardy-Rellich inequality . . . . . . . . . . . . . 159 7.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 III Preconditioning of Nonsymmetric Linear Systems 166 8 Simultaneous preconditioning and symmetrization . . . . 167 8.1 Introduction and main results . . . . . . . . . . . . . . . . . 167 8.2 Selfdual methods for non-symmetric systems . . . . . . . . . 172 8.2.1 Exact methods . . . . . . . . . . . . . . . . . . . . . . 172 8.2.2 Inexact methods . . . . . . . . . . . . . . . . . . . . . 173 8.2.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . 174 8.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 177 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 v List of Tables 5.1 Summary 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.1 Summary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.1 Summary 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.1 GCGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 8.2 Number of iterations for (8.41) with the solution y = x sin(pix).177 8.3 Number of iterations for equation (8.41) with the solution y = x(1\u2212x)cos(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.4 Number of iterations for the backward scheme method (Ex- ample 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.5 Number of iterations for the centered difference scheme method (Example 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.6 Number of iterations for SD-CGN with different values of \u03b1. . 181 8.7 Number of iterations for (8.43) with different values of \u03b1. . . 182 vi Acknowledgements I am hugely grateful to my Ph.D. supervisor Nassif Ghoussoub for his help, constant support, and encouragement. He introduced me to the problems and provided me with helpful comments and suggestions throughout. I also wish to thank Anthony Pierce and Chen Greif for several discus- sions on simultaneous preconditioning and symmetrization of non-symmetric linear systems. I am grateful for their expert guidance and generous support. A special thanks goes to the administration of my department that made my life easier in many ways. Finally, a big thanks goes to all the people I met during my studies at UBC, my parents and my other friends for providing me with help and encouragement throughout my studies. vii Dedication This thesis is dedicated to my wonderful parents, Saed and Iran, who have raised me to be the person I am today. It is also dedicated to my sister, Aram, who has supported me in all of my life. You have been with me every step of the way, through good times and bad. Thank you for everything. viii Co-authorship Statement \u2022 Chapters 2, 3, and 8 were jointly authored by Nassif Ghoussoub and Amir Moradifam. \u2022 Chapter 5 was jointly authored by Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub and Amir Moradifam. In all of the joint papers of this thesis, all authors contributed equally to the identification and design of the research problem, performing the research, data analysis, and manuscript preparation. ix Chapter 1 Introduction The main focus of this thesis is improved Hardy and Hardy-Rellich inequal- ities, fourth order nonlinear eigenvalue problems, and the preconditioning issue of non-symmetric sparse linear systems. In the first part of the thesis we prove various classes of improved Hardy and Hardy-Rellich inequalities. In the second part we study the regularity of the extremal solutions of non- linear fourth order eigenvalue problems. Although the first two parts do not seem related at the first glance, we shall see in the second part that our improved Hardy-Rellich inequalities are crucial to show the singular nature of the extremal solutions in large dimensions close to the critical dimen- sion. In the last part, motivated by the theory of self-duality, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. 1.1 Optimal improved Hardy and Hardy-Rellich inequalities Let \u2126 be smooth bounded domain in Rn and 0 \u2208 \u2126. Hardy and Hardy- Rellich inequalities assert that\u222b \u2126 |\u2207u|2dx \u2265 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx, (1.1) for all u \u2208 H10 (\u2126) and n \u2265 3, and\u222b \u2126 |\u2206u|2dx \u2265 n 2(n\u22124)2 16 \u222b \u2126 u2 |x|4dx, (1.2) for u \u2208 H20 (\u2126) and n \u2265 5, respectively. These inequalities and their various improvements are used in many contexts, such as in the study of the stability of solutions of semi-linear elliptic and parabolic equations, in the analysis of the asymptotic behavior of the heat equation with singular potentials, as well as in the study of the stability of eigenvalues in elliptic problems such 1 1.1. Optimal improved Hardy and Hardy-Rellich inequalities as Schro\u0308dinger operators. It is well known that the constants appearing in the above inequalities are the best constants and they never achieved. So, one could anticipate improving these inequalities. Indeed, ever since Bre\u0301zis- Vazquez [4] showed that Hardy\u2019s inequality can be improved once restricted to a smooth bounded domain \u2126 in Rn, there was a flurry of activity about possible improvements of the following type: If n \u2265 3 then \u222b\u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b\u2126 |u|2|x|2 dx \u2265 \u222b\u2126 V (x)|u|2dx, (1.3) for all u \u2208 H10 (\u2126), as well as its fourth order counterpart If n \u2265 5 then \u222b\u2126 |\u2206u|2dx\u2212 n2(n\u22124)216 \u222b\u2126 u2|x|4dx \u2265 \u222b\u2126W (x)u2dx (1.4) for u \u2208 H20 (\u2126), where V,W are certain explicit radially symmetric potentials of order lower than 1 r2 (for V ) and 1 r4 (for W ) (see [1],[3], [4], [8], [12], [19]). In chapter 2 and 3, we provide an approach that completes, simplifies and improves most related results to-date regarding the Laplacian on Euclidean space as well as its powers. We also establish new inequalities some of which cover critical dimensions such as n = 2 for inequality (1.3) and n = 4 for (1.4). We start by giving necessary and sufficient conditions on positive radial functions V and W on a ball B in Rn, so that the following inequality holds for some c > 0:\u222b B V (x)|\u2207u|2dx \u2265 c \u222b BW (x)u 2dx for all u \u2208 C\u221e0 (B). (1.5) Assuming that the ball B has radius R and that \u222b R 0 1 rn\u22121V (r)dr = +\u221e, the condition is simply that the ordinary differential equation (BV,cW ) y\u2032\u2032(r) + (n\u22121r + Vr(r) V (r) )y \u2032(r) + cW (r)V (r) y(r) = 0 has a positive solution on the interval (0, R). We shall call such a couple (V,W ) a Bessel pair on (0, R). The weight of such a pair is then defined as \u03b2(V,W ;R) = sup { c; (BV,cW ) has a positive solution on (0, R) } . (1.6) This characterization makes an important connection between Hardy-type inequalities and the oscillatory behavior of the above equations. For exam- ple, by using recent results on ordinary differential equations, we can then infer that an integral condition on V,W of the form lim sup r\u21920 r2(n\u22121)V (r)W (r) ( \u222b R r d\u03c4 \u03c4n\u22121V (\u03c4) )2 < 1 4 (1.7) 2 1.1. Optimal improved Hardy and Hardy-Rellich inequalities is sufficient (and \u201calmost necessary\u201d) for (V,W ) to be a Bessel pair on a ball of sufficiently small radius \u03c1. Applied in particular, to a pair (V, 1 r2 V ) where the function rV \u2032(r) V (r) is assumed to decrease to \u2212\u03bb on (0, R), we obtain the following extension of Hardy\u2019s inequality: If \u03bb \u2264 n\u2212 2, then\u222b B V (x)|\u2207u|2dx \u2265 (n\u2212\u03bb\u221222 )2 \u222b B V (x) u2 |x|2dx for all u \u2208 C\u221e0 (B) (1.8) and (n\u2212\u03bb\u221222 ) 2 is the best constant. The case where V (x) \u2261 1 is obviously the classical Hardy inequality and when V (x) = |x|\u22122a for \u2212\u221e < a < n\u221222 , this is a particular case of the Caffarelli-Kohn-Nirenberg inequality. One can however apply the above criterion to obtain new inequalities such as the following: For a, b > 0 \u2022 If \u03b1\u03b2 > 0 and m \u2264 n\u221222 , then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m\u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (1.9) and (n\u22122m\u221222 ) 2 is the best constant in the inequality. \u2022 If \u03b1\u03b2 < 0 and 2m\u2212 \u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m+ \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (1.10) and (n\u22122m+\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. We can also extend some of the recent results of Blanchet-Bonforte-Dolbeault- Grillo-Vasquez [3]. \u2022 If \u03b1\u03b2 < 0 and \u2212\u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |\u2207u|2dx \u2265 b 2\u03b1 (n\u2212 \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx, (1.11) and b 2 \u03b1 (n\u2212\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. \u2022 If \u03b1\u03b2 > 0, and n \u2265 2, then there exists a constant C > 0 such that for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2|\u2207u|2dx \u2265 C \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx. (1.12) Moreover, b 2 \u03b1 (n\u221222 ) 2 \u2264 C \u2264 b 2\u03b1 (n+\u03b1\u03b2\u221222 )2. 3 1.1. Optimal improved Hardy and Hardy-Rellich inequalities On the other hand, by considering the pair V (x) = |x|\u22122a and Wa,c(x) = (n\u22122a\u221222 )2|x|\u22122a\u22122 + c|x|\u22122aW (x) we get the following improvement of the Caffarelli-Kohn-Nirenberg inequal- ities:\u222b B |x|\u22122a|\u2207u|2dx\u2212 (n\u2212 2a\u2212 2 2 )2 \u222b B |x|\u22122a\u22122u2dx \u2265 c \u222b B |x|\u22122aW (x)u2dx, (1.13) for all u \u2208 C\u221e0 (B), if and only if the following ODE (BcW ) y\u2032\u2032 + 1ry \u2032 + cW (r)y = 0 has a positive solution on (0, R). Such a function W will be called a Bessel potential on (0, R). More importantly, we establish that Bessel pairs lead to a myriad of op- timal Hardy-Rellich inequalities of arbitrary high order, therefore extending and completing a series of new results by Adimurthi et al. [2], Tertikas- Zographopoulos [19] and others. They are mostly based on the following theorem. Let V andW be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in Rn (n \u2265 1) such that \u222b R0 1rn\u22121V (r)dr = +\u221e and \u222b R 0 r n\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R) and \u03b2(V,W ;R) \u2265 1. 2. \u222b B V (x)|\u2207u|2dx \u2265 \u222b BW (x)u 2dx for all u \u2208 C\u221e0 (B). 3. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n\u22122, then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx, for all radial u \u2208 C\u221e0,r(B). 4. If in addition, W (r) \u2212 2V (r) r2 + 2Vr(r)r \u2212 Vrr(r) \u2265 0 on (0, R), then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx, for all u \u2208 C\u221e0 (B). 4 1.1. Optimal improved Hardy and Hardy-Rellich inequalities In other words, one can obtain as many Hardy and Hardy-Rellich type inequalities as one can construct Bessel pairs on (0, R). The relevance of the above result stems from the fact that there are plenty of such pairs that are easily identifiable. Indeed, even the class of Bessel potentials \u2013 equivalently thoseW such that ( 1, (n\u221222 ) 2|x|\u22122 + cW (x)) is a Bessel pair\u2013 is quite rich and contains several important potentials. Here are some of the most relevant properties of the class of C1 Bessel potentials W on (0, R), that we shall denote by B(0, R). First, the class is a closed convex solid subset of C1(0, R), that is if W \u2208 B(0, R) and 0 \u2264 V \u2264 W , then V \u2208 B(0, R). The \u201dweight\u201d of each W \u2208 B(R), that is \u03b2(W ;R) = sup { c > 0; (BcW ) has a positive solution on (0, R) } , (1.14) will be an important ingredient for computing the best constants in corre- sponding functional inequalities. Here are some basic examples of Bessel potentials and their corresponding weights. \u2022 W \u2261 0 is a Bessel potential on (0, R) for any R > 0. \u2022 W \u2261 1 is a Bessel potential on (0, R) for any R > 0, and \u03b2(1;R) = z20 R2 where z0 = 2.4048... is the first zero of the Bessel function J0. \u2022 If a < 2, then there exists Ra > 0 such that W (r) = r\u2212a is a Bessel potential on (0, Ra). \u2022 For k \u2265 1, R > 0 and \u03c1 = R(eee. .e((k\u22121)\u2212times) ), let Wk,\u03c1(r) = \u03a3kj=1 1 r2 ( j\u220f i=1 log(i) \u03c1 r )\u22122 , where the functions log(i) are defined iteratively as follows: log(1)(.) = log(.) and for k \u2265 2, log(k)(.) = log(log(k\u22121)(.)). Wk,\u03c1 is then a Bessel potential on (0, R) with \u03b2(Wk,\u03c1;R) = 14 . \u2022 For k \u2265 1, R > 0 and \u03c1 \u2265 R, define W\u0303k;\u03c1(r) = \u03a3kj=1 1 r2 X21 ( r \u03c1 )X22 ( r \u03c1 ) . . . X2j\u22121( r \u03c1 )X2j ( r \u03c1 ), where the functions Xi are defined iteratively as follows: X1(t) = (1 \u2212 log(t))\u22121 and for k \u2265 2, Xk(t) = X1(Xk\u22121(t)). Then again W\u0303k,\u03c1 is a Bessel potential on (0, R) with \u03b2(W\u0303k,\u03c1;R) = 14 . 5 1.1. Optimal improved Hardy and Hardy-Rellich inequalities \u2022 More generally, if W is any positive function on R such that lim inf r\u21920 ln(r) \u222b r 0 sW (s)ds > \u2212\u221e, then for every R > 0, there exists \u03b1 := \u03b1(R) > 0 such that W\u03b1(x) := \u03b12W (\u03b1x) is a Bessel potential on (0, R). What is remarkable is that the class of Bessel potentials W is also the one that leads to optimal improvements for fourth order inequalities (in dimension n \u2265 3) of the following type: \u222b B |\u2206u|2dx\u2212 C(n) \u222b B |\u2207u|2 |x|2 dx \u2265 c(W,R) \u222b BW (x)|\u2207u|2dx , (1.15) for all u \u2208 H20 (B), where C(3) = 2536 , C(4) = 3 and C(n) = n 2 4 for n \u2265 5. The case when W \u2261 W\u0303k,\u03c1 and n \u2265 5 was recently established by Tertikas- Zographopoulos [19]. Note that W can be chosen to be any one of the examples of Bessel potentials listed above. Moreover, both C(n) and the weight \u03b2(W ;R) are the best constants in the above inequality. Appropriate combinations of (1.5) and (1.15) then lead to a myriad of Hardy-Rellich inequalities in dimension n \u2265 4. For example, if W is a Bessel potential on (0, R) such that the function rWr(r)W (r) decreases to \u2212\u03bb, and if \u03bb \u2264 n\u2212 2, then we have for all u \u2208 C\u221e0 (BR)\u222b B |\u2206u|2dx \u2212 n 2(n\u2212 4)2 16 \u222b B u2 |x|4dx (1.16) \u2265 (n2 4 + (n\u2212 \u03bb\u2212 2)2 4 ) \u03b2(W ;R) \u222b B W (x) |x|2 u 2dx. By applying (1.16) to the various examples of Bessel functions listed above, one improves in many ways the recent results of Adimurthi et al. [2] and those by Tertikas-Zographopoulos [19]. Moreover, besides covering the critical dimension n = 4, we also establish that the best constant is (1 + n(n\u22124)8 ) for all the potentials Wk and W\u0303k defined above. For example we have for n \u2265 4,\u222b B |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4dx (1.17) + (1 + n(n\u2212 4) 8 ) k\u2211 j=1 \u222b B u2 |x|4 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx. 6 1.1. Optimal improved Hardy and Hardy-Rellich inequalities More generally, we show that for any m < n\u221222 , and any W Bessel potential on a ball BR \u2282 Rn of radius R, the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx, (1.18) where am,n and \u03b2(W ;R) are best constants that we compute in chapter 3 for all m and n and for many Bessel potentials W . We also establish a more general version of equation (1.16). Assuming again that rW \u2032(r) W (r) decreases to \u2212\u03bb on (0, R), and provided m \u2264 n\u221242 and \u03bb \u2264 n\u2212 2m\u2212 2, we then have for all u \u2208 C\u221e0 (BR),\u222b BR |\u2206u|2 |x|2m dx \u2265 \u03b2n,m \u222b BR u2 |x|2m+4dx (1.19) + \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b BR W (x) |x|2m+2u 2dx, where again the best constants \u03b2n,m are computed in chapter 3. This com- pletes the results in Theorem 1.6 of [19], where the inequality is established for n \u2265 5, 0 \u2264 m < n\u221242 , and the particular potential W\u0303k,\u03c1. Another inequality that relates the Hessian integral to the Dirichlet en- ergy is the following: Assuming \u22121 < m \u2264 n\u221242 and W is a Bessel potential on a ball B of radius R in Rn, then for all u \u2208 C\u221e0 (B),\u222b B |\u2206u|2 |x|2m dx\u2212 (n+ 2m)2(n\u2212 2m\u2212 4)2 16 \u222b B u2 |x|2m+4dx \u2265 (1.20) \u03b2(W ;R) (n+ 2m)2 4 \u222b B W (x) |x|2m+2u 2dx+ \u03b2(|x|2m;R)||u||H10 . This improves considerably Theorem A.2. in [2] where it is established \u2013 for m = 0 and without best constants \u2013 with the potential W1,\u03c1 in dimension n \u2265 5, and the potential W2,\u03c1 when n = 4. Finally, we establish several higher order Rellich inequalities for integrals of the form \u222b BR |\u2206mu|2 |x|2k dx, improving in many ways several recent results in [19]. Hardy-Rellich inequalities on H2 \u2229 H10 are important in the study of fourth order elliptic equations with Navier boundary condition and systems of second order elliptic equations. In [16], I developed a general approach to prove optimal weighted Hardy-Rellich inequalities on H2 \u2229H10 which leads to various new Hardy-Rellich inequalities. 7 1.2. Fourth order nonlinear eigenvalue problems The approach developed in [11], [12], and [18] basically finishes the prob- lem of improving Hardy and Hardy-Rellich inequalities in Rn. 1.2 Fourth order nonlinear eigenvalue problems Consider the fourth order elliptic problem \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bbf(u) in \u2126, . (G\u03bb) with either Dirichlet boundary condition u = \u2202\u03bdu = 0 or Navier boundary condition u = \u2206u = 0 on \u2202\u2126. Here \u03bb > 0 is a parameter, \u03c4 > 0, \u03b2 > 0 are fixed constants, and \u2126 \u2282 RN (N \u2265 2) is a bounded smooth domain. The case \u03b2 = 0, \u03c4 = 1, and f(u) = eu is the well known Gelfand problem. Under some technical assumptions on f one can show that there exists \u03bb\u2217 > 0 such that for every 0 < \u03bb < \u03bb\u2217, there exists a smooth minimal (smallest) solution of (S)\u03bb,f , while for \u03bb > \u03bb\u2217 there is no solution even in a weak sense. Moreover, the branch \u03bb 7\u2192 u\u03bb(x) is increasing for each x \u2208 \u2126, and there- fore the function u\u2217(x) := lim\u03bb\u2197\u03bb\u2217 u\u03bb(x) can be considered as a generalized solution that corresponds to the pull-in voltage \u03bb\u2217. Now the issue of the regularity of this extremal solution is an important question for many rea- sons, not the least of which being the fact that it decides whether the set of solutions stops there, or whether a new branch of solutions emanates from a bifurcation state (u\u2217, \u03bb\u2217). One of the main obstacles for this problem is the well-known difficulty of extracting energy estimates for solutions of fourth order problems from their stability properties which means that the methods used to analyze the regularity of the extremal solution of the second order problem could not carry to the corresponding fourth order problem. In the second part of this thesis I study the above problem with various nonlinear- ities f with both Dirichlet and Navier boundary conditions. Consider the fourth order elliptic problem\uf8f1\uf8f2\uf8f3 \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bb (1\u2212u)2 in \u2126, 0 < u \u2264 1 in \u2126, u = \u2206u = 0 on \u2202\u2126, (G\u03bb) where \u03bb > 0 is a parameter, \u03c4 > 0, \u03b2 > 0 are fixed constants, and \u2126 \u2282 RN (N \u2265 2) is a bounded smooth domain. This equation is derived in the study of the charged plates in electrostatic actuators. In chapter 4, we study the regularity of the extremal solution of the 8 1.2. Fourth order nonlinear eigenvalue problems above equation. \uf8f1\uf8f2\uf8f3 \u22062u = \u03bb (1\u2212u)2 in B 0 < u < 1 in B (P )\u03bb u = \u2202\u03bdu = 0 on \u2202B, where B is the unit ball in RN . This problem models a simple electrostatic Micro-Electromechanical Systems (MEMS) device. We proved that the ex- tremal solution u\u03bb\u2217 is regular (supB u\u03bb\u2217 < 1) provided N \u2264 8 while u\u03bb\u2217 is singular (supB u\u03bb\u2217 = 1) for N \u2265 9, in which case 1 \u2212 C0|x|4\/3 \u2264 u\u03bb\u2217(x) \u2264 1\u2212 |x|4\/3 on the unit ball, where C0 := ( \u03bb\u2217 \u03bb ) 1 3 and \u03bb\u0304 := 89(N \u2212 23)(N \u2212 83). This completes the results of F.H. Lin and Y.S. Yang [15]. In chapter 5, we study the problem (G\u03bb) on the unit ball in RN and showed that the critical dimension for (P\u03bb) is N = 9. Indeed I proved that the extremal solution of (P\u03bb) is regular (supB u\u2217 < 1) for N \u2264 8 and \u03b2, \u03c4 > 0 and it is singular (supB u\u2217 = 1) for N \u2265 9, \u03b2 > 0, and \u03c4 > 0 with \u03c4\u03b2 small. Our proof for the regularity of the extremal solutions is based on a blow up analysis and certain energy estimates which was recently introduced by Da\u0301vila et al. [7]. To show the singularity of the exterimal solutions in dimensions N \u2265 9 we use various improved Hardy-Rellich inequalities that are consequences of our main results in the first part of the thesis. Recently Da\u0301vila et al. [7] studied the fourth order counter part of the Gelfand problem { \u22062u = \u03bbeu in B u = \u2202u\u2202n = 0 on \u2202B, (1.21) They developed a new method to prove the regularity of the extremal solutions in low dimensions and showed that for N \u2264 12, u\u2217 is regular. They used a computer assisted proof to show that the extremal solution is singular in dimensions 13 \u2264 N \u2264 31 while they gave a mathematical proof in dimensions N \u2265 32. In [17], I introduced a unified mathematical approach to deal with this problem and showed that for N \u2265 13, the extremal solution is singular. The following lemma plays an essential role in our proof. Lemma 1.22. Suppose there exist \u03bb\u2032 > 0 and a radial function u \u2208 H2(B)\u2229 W 4,\u221eloc (B \\ {0}) such that \u22062u \u2264 \u03bb\u2032eu for all 0 < r < 1, (1.23) 9 1.3. Preconditioning and symmetrization of non-symmetric linear systems u(1) = 0, \u2202u \u2202n (1) = 0, (1.24) u \/\u2208 L\u221e(B), (1.25) and \u03b2 \u222b B eu\u03d52 \u2264 \u222b B (\u2206\u03d5)2 for all \u03d5 \u2208 C\u221e0 (B), (1.26) for some \u03b2 > \u03bb\u2032. Then u\u2217 is singular and \u03bb\u2217 \u2264 \u03bb\u2032 (1.27) The proof is based on the above lemma combined with certain improved Hardy-Rellich inequalities obtained in chapter 3. 1.3 Preconditioning and symmetrization of non-symmetric linear systems Many problems in scientific computing lead to systems of linear equations of the form, Ax = b, (1.28) where A \u2208 Rn\u00d7n is a nonsingular but sparse matrix, and b is a given vector in Rn, and various iterative methods have been developed for a fast and efficient resolution of such systems. In [13], motivated by the theory of self- duality ([10]) we propose new templates for solving non-symmetric linear systems. Our approach consists of symmetrizing the problem so as to be able to apply CG, MINRES, or SYMMLQ. We argue that for a large class of non-symmetric, ill-conditionned matrices, it is beneficial to replace problem (8.1) by one of the form ATMAx = ATMb, (1.29) where M is a symmetric and positive definite matrix that can be chosen properly so as to obtain good convergence behavior for CG when it is applied to the resulting symmetric ATMA. This reformulation should not only be seen as a symmetrization, but also as preconditioning procedure. While it is difficult to obtain general conditions on M that ensure higher efficiency of our approach, we show theoretically and numerically that by choosing M to be either the inverse of the symmetric part of A, or its resolvent, one can get surprisingly good numerical schemes to solve (8.1). 10 Bibliography [1] Adimurthi, N. Chaudhuri, and N. Ramaswamy, An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505. [2] Adimurthi, M. Grossi, and S. Santra, Optimal Hardy-Rellich inequal- ities, maximum principles and related eigenvalue problems, J. Funct. Anal., 240 (2006), 36-83. [3] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vasquez, Hardy- Poincare\u0301 inequalities and applications to nonlinear diffusions, C. R. Acad. Sci. Paris, Ser. I, 344 (2007), 431-436. [4] H. Brezis and J. L. Va\u0301zquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469. [5] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequal- ities with weights, Compositio Mathematica, 53 (1984), 259-275. [6] C. Cowan, P. Esposito, N. Ghoussoub, and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., In press (2009) 19 pp. [7] J. Davila, L. Dupaigne, I. Guerra, and M. Montenegro, Stable Solu- tions for the Bilaplacian with Exponential Nonlinearity, SIAM J. Math. Anal., 39 (2007) 565-592. [8] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), no. 1, 186-233. [9] I. Kombe and M. O\u0308zaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191- 6203. [10] N. Ghoussoub, Selfdual partial differential systems and their variational principles, Springer New York (2008). 11 Chapter 1. Bibliography [11] N. Ghoussoub, A. Moradifam, Bessel pairs and optimal improved Hardy and Hardy-Rellich inequalities, submitted to Math. Ann., 36 pages (2008). [12] N. Ghoussoub, A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Nat. Acad. Sci., 105, no. 37 (2008) p. 13746-13751. [13] N. Ghoussoub, A. Moradifam, Simultaneous preconditioning and sym- metrization of non-symmetric linear systems, submitted to Numer. Lin- ear Algebra Appl., 14 pages. [14] Z. Gui, J. Wei, On a fourth order nonlinear elliptic equation with neg- ative exponent, SIAM J. Math. Anal., 40 (2009), 2034-2054. [15] F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A, 463 (2007), 1323-1337. [16] A. Moradifam, On the critical dimension of a fourth order elliptic prob- lem with negative exponent, Journal of Differential Equations, 248 (2010), 594-616. [17] A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., In press (2009). [18] Optimal Weighted Hardy-Rellich inequalities on H2\u2229H10 , submitted to Trans. Amer. Math. Soc., 16 p (2009). [19] A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Advances in Mathematics, 209 (2007) 407-459. [20] Z-Q. Wang, M. Willem, Caffarelli-Kohn-Nirenberg inequalities with re- mainder terms, J. Funct. Anal., 203 (2003), 550-568. 12 Part I Hardy and Hardy-Rellich Inequalities 13 Chapter 2 On the best possible remaining term in the Hardy inequality 1 2.1 Introduction Let \u2126 be a bounded domain in Rn, n \u2265 3, with 0 \u2208 \u2126. The classical Hardy inequality asserts that\u222b \u2126 |\u2207u|2dx \u2265 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx for all u \u2208 H10 (\u2126). (2.1) This inequality and its various improvements are used in many contexts, such as in the study of the stability of solutions of semi-linear elliptic and parabolic equations [6, 7, 21], in the analysis of the asymptotic behavior of the heat equation with singular potentials [8, 22], as well as in the study of the stability of eigenvalues in elliptic problems such as Schro\u0308dinger operators [10, 12]. Now it is well known that (n\u221222 ) 2 is the best constant for inequality (2.1), and that this constant is however not attained in H10 (\u2126). So, one could anticipate improving this inequality by adding a non-negative correction term to the right hand side of (2.1) and indeed, several sharpened Hardy inequalities have been established in recent years [4, 5, 11, 12, 22], mostly triggered by the following improvement of Brezis and Va\u0301zquez [6].\u222b \u2126 |\u2207u|2dx \u2265 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx+ \u03bb\u2126 \u222b \u2126 |u|2dx for every u \u2208 H10 (\u2126). (2.2) The constant \u03bb\u2126 in (2.2) is given by \u03bb\u2126 = z20\u03c9 2\/n n |\u2126|\u2212 2 n , (2.3) 1A version of this chapter has been accepted for publication. N. Ghoussoub, A. Morad- ifam, On the best possible remaining term in the Hardy inequality, Proc. Nat. Acad. Sci. U.S.A., 105 (2008), 13746-13751. 14 2.1. Introduction where \u03c9n and |\u2126| denote the volume of the unit ball and \u2126 respectively, and z0 is the first zero of the bessel function J0(z). Moreover, \u03bb\u2126 is optimal when \u2126 is a ball, but is \u2013again\u2013 not achieved in H10 (\u2126). This led to one of the open problems mentioned in [6] (Problem 2), which is whether the two terms on the RHS of inequality (2.2) (i.e., the coefficients of |u|2) are just the first two terms of an infinite series of correcting terms. This question was addressed by several authors. In particular, Adimurthi et all [1] proved that for every integer k, there exists a constant c depending on n, k and \u2126 such that\u222b \u2126 |\u2207u|2dx \u2265 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx+ c \u2211k j=1 \u222b \u2126 |u|2 |x|2 (\u220fj i=1 log (i) \u03c1 |x| )\u22122 dx, (2.4) for u \u2208 H10 (\u2126), where \u03c1 = (supx\u2208\u2126 |x|)(ee e. .e(k\u2212times) ). Here we have used the notations log(1)(.) = log(.) and log(k)(.) = log(log(k\u22121)(.)) for k \u2265 2. Also motivated by the question of Brezis and Va\u0301zquez, Filippas and Tertikas proved in [11] that the inequality can be repeatedly improved by adding to the right hand side specific potentials which lead to an infinite series expansion of Hardy\u2019s inequality. More precisely, by defining iteratively the following functions, X1(t) = (1\u2212 log(t))\u22121, Xk(t) = X1(Xk\u22121(t)) k = 2, 3, ..., they prove that for any D \u2265 supx\u2208\u2126 |x|, the following inequality holds for any u \u2208 H10 (\u2126):\u222b \u2126 |\u2207u|2dx \u2265 (n\u2212 2 2 )2 \u222b \u2126 |u|2 |x|2dx (2.5) + 1 4 \u221e\u2211 i=1 \u222b \u2126 1 |x|2X 2 1 ( |x| D )X22 ( |x| D )...X2i ( |x| D )|u|2dx. Moreover, they proved that the constant 14 is the best constant for the corresponding k\u2212improved Hardy inequality which is again not attained in H10 (\u2126). In this paper, we show that all the above results \u2013and more\u2013 follow from a specific characterization of those potentials V that yield an improved Hardy inequality. Here is our main result. Let V be a radial function on a smooth bounded radial domain \u2126 in n with radius R, in such a way that V (x) = v(|x|) for some non-negative function v on (0, R). The following properties are then equivalent: 15 2.1. Introduction 1. The ordinary differential equation (DV ) y\u2032\u2032(r) + y\u2032(r) r + v(r)y(r) = 0 has a positive solution on the interval (0, R). 2. The following improved Hardy inequality holds (HV ) \u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx \u2265 \u222b \u2126 V (|x|)|u|2dx, for u \u2208 H10 (\u2126). Moreover, the best constant c(V ) := sup { c; (HcV ) holds } is the largest c so that y\u2032\u2032(r) + y \u2032(r) r + cv(r)y(r) = 0 has a positive solution on the interval (0, R). We note that the implication 1) implies 2) holds for any smooth bounded domain \u2126 in n containing 0, provided v(r) + (n\u221222 ) 2 1 r2 is non-increasing on (0, supx\u2208\u2126 |x|) and R is the radius of the ball which has the same volume as \u2126 (i.e. R = ( |\u2126|\u03c9n ) 1 n ). It is therefore clear from the above discussion that in order to find what potentials are candidates for an improved Hardy inequality, one needs to investigate the ordinary differential equation y\u2032\u2032 + y \u2032 r + v(r)y(r) = 0. We shall see that the results of Brezis-Va\u0301zquez, Adimurthi et al, and Filippas- Tertikas mentioned above can be easily deduced by simply checking that the potentials V they consider, correspond to equations (DV ) where an explicit positive solution can be found. Our approach turned out to be also useful for determining the best con- stants in the above mentioned improvements. Indeed, the case when V \u2261 1 will follow immediately from Theorem 2.1. A slightly more involved reason- ing \u2013 but also based of the above characterization \u2013 will allow us to find the best constant in the improvement of Adimurthi et al, and to recover the best one established by Filippas-Tertikas. Since the existence of positive solutions for ODEs of the form (DV ) is closely related to the oscillatory properties of second order equations of the form z\u2032\u2032(s)+a(s)z(s) = 0, Theorem 2.1 also allows for the use of the extensive literature on the oscillatory properties of such equations to deduce various interesting results such as the following corollary. Let V be a positive radial function on a smooth bounded radial domain \u2126 in Rn. 1. If lim infr\u21920 ln(r) \u222b r 0 sV (s)ds > \u2212\u221e, then there exists \u03b1 := \u03b1(\u2126) > 0 such that an improved Hardy inequality (HV\u03b1) holds for the scaled potential V\u03b1(x) := \u03b12V (\u03b1x). 16 2.1. Introduction 2. If limr\u21920 ln(r) \u222b r 0 sV (s)ds = \u2212\u221e, then there are no \u03b2, c > 0, for which (HV\u03b2,c) holds with V\u03b2,c = cV (\u03b2x). The following is a consequence of the two results above. For any \u03b1 < 2, inequality (HcV ) holds on a bounded domain \u2126 for V\u03b1(x) = 1|x|\u03b1 and some c > 0. Moreover, the best constant c( 1 |x|\u03b1 ) is equal to the largest c such that the equation y\u2032\u2032(r) + 1 r y\u2032(r) + c 1 |x|\u03b1 = 0, has a positive solution on (0, R), where R is the radius of the ball wich has the same volume as \u2126. Moreover, if \u03b1 \u2265 2 inequality (HV ) does not hold for V\u03b1,c(x) = c 1|x|\u03b1 for any c > 0. Note that the above corollary gives another proof of the fact that (n\u221222 ) 2 is the best constant for the classical Hardy inequality. Define now the class A\u2126 = {v : R\u2192 R+; v is non-increasing on(0, sup x\u2208\u2126 |x|), (Dv) has a positive solution on(0, ( |\u2126| \u03c9n ) 1 n )}. An immediate application of Theorem 2.1 coupled with Ho\u0308lder\u2019s inequality gives the following duality statement, which should be compared to inequal- ities dual to those of Sobolev, recently obtained via the theory of mass transport [2, 9]. Suppose that \u2126 is a smooth bounded domain in Rn containing 0. Then for any 0 < p \u2264 2, we have inf {\u222b \u2126 |\u2207u|2dx\u2212 (n\u2212 2 2 )2 \u222b \u2126 |u|2 |x|2dx; u \u2208 H 1 0 (\u2126), ||u||p = 1 } \u2265 sup \uf8f1\uf8f2\uf8f3 1||V \u22121(|x|)|| L p p\u22122 (\u2126) . ; V \u2208 A\u2126 \uf8fc\uf8fd\uf8fe . (2.6) Finally, consider the following classes of radial potentials: X = {V : \u2126\u2192+; V \u2208 L\u221eloc(\u2126 \\ {0}), lim inf r\u21920 ln(r) \u222b r 0 sV (s)ds > \u2212\u221e}, (2.7) 17 2.2. Two dimensional inequalities and Y = {V : \u2126\u2192+; V \u2208 L\u221eloc(\u2126 \\ {0}), lim r\u21920 ln(r) \u222b r 0 sV (s)ds = \u2212\u221e}. (2.8) For any 0 < \u00b5 < \u00b5n := (n\u22122) 2 ) 2 we consider the following weighted eigenvalue problem, (EV,\u00b5) { \u2212\u2206u\u2212 \u00b5|x|2u = \u03bbV u in \u2126, u = 0 on \u2126. (2.9) Our results above combine with standard arguments to yield the follow- ing. For any 0 < \u00b5 < \u00b5n, and V : \u2126 \u2192+ with V \u2208 L\u221eloc(\u2126 \\ {0}) and lim|x|\u21920 |x|2V (x) = 0, the problem (EV,\u00b5) admits a positive weak solution u\u00b5 \u2208 H10 (\u2126) corresponding to the first eigenvalue \u03bb = \u03bb1\u00b5(V ). Moreover, by letting \u03bb1(V ) = lim\u00b5\u2191\u00b5n \u03bb1\u00b5(V ), we have \u2022 If V \u2208 X, then there exists c > o such that \u03bb1(Vc) > 0. \u2022 If V \u2208 Y , then \u03bb1(Vc) = 0 for all c > 0, where Vc(x) := V (cx). 2.2 Two dimensional inequalities In this section, we start by establishing the following improvements of \u201ctwo- dimensional\u201d Poincare\u0301 and Poincare\u0301-Wirtinger inequalities. Let a < b, k > 0 is a differentiable function on (a, b), and \u03d5 be a strictly positive real valued differentiable function on (a, b). Then, every h \u2208 C1([a, b]) with \u2212\u221e < lim r\u2192a k(r)|h(r)| 2\u03d5 \u2032(r) \u03d5(r) = lim r\u2192b k(r)|h(r)|2\u03d5 \u2032(r) \u03d5(r) <\u221e, (2.10) satisfies the following inequality:\u222b b a |h\u2032(r)|2k(r)dr \u2265 \u222b b a \u2212|h(r)|2(k \u2032(r)\u03d5\u2032(r) + k(r)\u03d5\u2032\u2032(r) \u03d5(r) )dr. (2.11) 18 2.2. Two dimensional inequalities Moreover, assuming (2.10), the equality holds if and only if h(r) = \u03d5(r) for all r \u2208 (a, b). Proof. Define \u03c8(r) = h(r)\/\u03d5(r), r \u2208 [a, b]. Then\u222b b a |h\u2032(r)|2k(r)dr = \u222b b a |\u03c8(r)|2|\u03d5\u2032(r)|2k(r)dr + \u222b b a 2\u03d5(r)\u03d5\u2032(r)\u03c8(r)\u03c8\u2032(r)k(r)dr + \u222b b a |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr = \u222b b a |\u03c8(r)|2|\u03d5\u2032(r)|2k(r)dr \u2212 \u222b b a |\u03c8(r)|2(k\u03d5\u03d5\u2032)\u2032(r)dr + \u222b b a |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr = \u222b b a |\u03c8(r)|2(|\u03d5\u2032(r)|2k(r)\u2212 (k\u03d5\u03d5\u2032)\u2032(r)dr + \u222b b a |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr. Hence, we have\u222b b a |h\u2032(r)|2k(r)dr = \u222b b a \u2212|h(r)|2(k \u2032(r)\u03d5\u2032(r) + k(r)\u03d5\u2032\u2032(r) \u03d5 )dr + \u222b b a |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr \u2265 \u222b b a \u2212|h(r)|2(k \u2032(r)\u03d5\u2032(r) + k(r)\u03d5\u2032\u2032(r) \u03d5(r) )dr. Hence (2.11) holds. Note that the last inequality is obviously an idendity if and only if h(r) = \u03d5(r) for all r \u2208 (a, b). The proof is complete. \u0003 By applying Theorem 2.2 to the weight k(r) = r, we obtain the follow- ing generalization of the 2-dimensional Poincare\u0301 inequality. (Generalized 2-dimensional Poincare\u0301 inequality) Let 0 \u2264 a < b and \u03d5 be a strictly pos- itive real valued differentiable function on (a, b). Then every h \u2208 C1([a, b]) with \u2212\u221e < lim r\u2192a r|h(r)| 2\u03d5 \u2032(r) \u03d5(r) = lim r\u2192b r|h(r)|2\u03d5 \u2032(r) \u03d5(r) <\u221e, (2.12) satisfies the following inequality:\u222b b a |h\u2032(r)|2rdr \u2265 \u222b b a \u2212|h(r)|2(\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) )dr. (2.13) 19 2.3. Proof of the main theorem Moreover, under the assumption (2.12) the equality holds if and only if h(r) = \u03d5(r) for all r \u2208 (a, b). By applying Theorem 2.2 to the weight k(r) = 1, we obtain the following generalization of the 2-dimensional Poincare\u0301- Wirtinger inequality. (Generalized Poincare\u0301-Wirtinger inequality) Let a < b and \u03d5 be a strictly positive real valued differentiable function on (a, b). Then, every h \u2208 C1([a, b]) with \u2212\u221e < lim r\u2192a |h(r)| 2\u03d5 \u2032(r) \u03d5(r) = lim r\u2192b |h(r)|2\u03d5 \u2032(r) \u03d5(r) <\u221e, (2.14) satisfies the following inequality:\u222b b a |h\u2032(r)|2dr \u2265 \u222b b a \u2212|h(r)|2\u03d5 \u2032\u2032(r) \u03d5(r) dr. (2.15) Moreover, under assumption (2.14), the equality holds if and only if h(r) = \u03d5(r) for all r \u2208 (a, b). Remark 2.2.1. Note that all of inequalities presented in the above theo- rems hold when we replace the condtions (2.10), (2.12), and (2.14) with the following weaker conditions lim inf r\u2192b k(r)|h(r)|2\u03d5 \u2032(r) \u03d5(r) \u2265 lim sup r\u2192a k(r)|h(r)|2\u03d5 \u2032(r) \u03d5(r) , lim inf r\u2192b r|h(r)|2\u03d5 \u2032(r) \u03d5(r) \u2265 lim sup r\u2192a r|h(r)|2\u03d5 \u2032(r) \u03d5(r) , lim inf r\u2192b |h(r)|2\u03d5 \u2032(r) \u03d5(r) \u2265 lim sup r\u2192a |h(r)|2\u03d5 \u2032(r) \u03d5(r) , respectively, provided both sides in the above inequalities are not equal to \u2212\u221e or +\u221e. 2.3 Proof of the main theorem We start with the sufficient condition of Theorem 2.1 by establishing the following. Proposition 2.16. (Improved Hardy Inequality) Let \u2126 be a bounded smooth domain in Rn with 0 \u2208 \u2126, and set R = (|\u2126|\/\u03c9n)1\/n. Suppose V is a radially symmetric function on \u2126 and \u03d5 is a C2-function on (0, R) such that 0 \u2264 V (|x|) \u2264 \u2212\u03d5\u2032(|x|)+r\u03d5\u2032\u2032(|x|)|x|\u03d5(|x|) for all x \u2208 \u2126, 0 < |x| < R, (2.17) 20 2.3. Proof of the main theorem lim inf r\u21920 r\u03d5 \u2032(r) \u03d5(r) \u2265 0 and lim sup r\u2192R \u03d5\u2032(r) \u03d5(r) <\u221e, (2.18) (n\u221222 ) 2 1 |x|2 + V (|x|) is a decreasing function of |x|. (2.19) Then for any u \u2208 H10 (\u2126), we have\u222b \u2126 |\u2207u|2dx \u2265 (n\u2212 2 2 )2 \u222b \u2126 |u|2 |x|2dx+ \u222b \u2126 V (|x|)|u|2dx. (2.20) Moreover, if limr\u21920 r\u03d5(r)\u03d5\u2032(r) = limr\u2192R \u03d5(r)\u03d5\u2032(r) = 0, then equality holds if and only if u is a radial function on \u2126 such that u(x) = \u03d5(|x|) for all x \u2208 \u2126. Proof: We first prove the inequality for smooth radial positive functions on the ball \u2126 = BR. For such u \u2208 C20 (BR), we define v(r) = u(r)r(n\u22122)\/2, r = |x|. In view of Corollary 2.2, we can write\u222b \u2126 |\u2207u(x)|2dx \u2212 (n\u2212 2 2 )2 \u222b \u2126 u2(x) |x|2 dx = \u03c9n \u222b R 0 |n\u2212 2 2 r\u2212n\/2v(r)\u2212 r1\u2212n\/2v\u2032(r)|2rn\u22121dr \u2212 (n\u2212 2 2 )2\u03c9n \u222b R 0 v2(r) r dr = \u03c9n( n\u2212 2 2 )2 \u222b R 0 v2((1\u2212 2v \u2032(r)r (n\u2212 2)v(r)) 2 \u2212 1)dr r = \u03c9n \u222b R 0 (v\u2032(r))2r \u2212 \u03c9n(n\u2212 22 ) \u222b R 0 v(r)v\u2032(r)dr = \u03c9n \u222b R 0 (v\u2032(r))2r \u2265 \u03c9n \u222b R 0 \u2212v2(r)(\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) )dr = \u03c9n \u222b R 0 \u2212u2(r)(\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) )rn\u22122dr = \u2212 \u222b \u2126 u2(x)( \u03d5\u2032(|x|) + |x|\u03d5\u2032\u2032(|x|) |x|\u03d5(|x|) )dx. Hence, the inequality (2.20) holds for radial smooth positive functions. By density arguments, inequality (2.20) is valid for any u \u2208 H10 , u \u2265 0. For 21 2.3. Proof of the main theorem u \u2208 H10 which is not positve and general domain \u2126, we use symmetriza- tion arguments. Let BR be a ball having the same volume as \u2126 with R = (|\u2126|\/\u03c9n)1\/n and let |u|\u2217 be the symmetric decreasing rearrangement of the function |u|. Now note that for any u \u2208 H10 (\u2126), |u|\u2217 \u2208 H10 (BR) and |u|\u2217 > 0. It is well known that the symmetrization does not change the Lp-norm, and that it decreases the Dirichlet energy, while increasing the integrals \u222b \u2126(( n\u22122 2 ) 2 1 |x|2 + V (|x|)|u|2dx, since the weight (n\u221222 )2 1|x|2 + V (|x|) is a decreasing function of |x|. Hence, (2.20) holds for any u \u2208 H10 (\u2126). We shall need the following lemmas. Lemma 2.21. Let x(r) be a function in C1(0, R] that is a solution of rx\u2032(r) + x2(r) \u2264 \u2212F (r), 0 < r \u2264 R, (2.22) where F is a nonnegative continuous function. Then lim r\u21930 x(r) = 0. (2.23) Proof: Divide equation (2.22) by r and integrate once. Then we have x(r) \u2265 \u222b R r |x(s)|2 s ds+ x(1) + \u222b R r F (s) s ds. (2.24) It follows that limr\u21930 x(r) exists. In order to prove that this limit is zero, we claim that \u222b R r x2(s) s ds <\u221e. (2.25) Indeed, otherwise we have G(r) := \u222b R r x2(s) s ds \u2192 \u221e as r \u2192 0. From (2.22) we have (\u2212rG\u2032(r)) 12 \u2265 G(r) + x(1) + \u222b R r F (s) s ds. Note that F \u2265 0, and G goes to infinity as r goes to zero. Thus, for r sufficiently small we have \u2212rG\u2032(r) \u2265 12G2(r) hence, ( 1G(r))\u2032 \u2265 12 ln(r), which contradicts the fact that G(r) goes to infinity as r tends to zero. Thus, our claim is true and the limit in (2.23) is indeed zero. \u0003 Lemma 2.26. If the equation \u03c6\u2032\u2032 + \u03c6 \u2032 r + v(r)\u03c6 = 0 has a positive solution on some interval (0, R), then we have necessarily, lim inf r\u21920 r\u03d5 \u2032(r) \u03d5(r) \u2265 0 and lim sup r\u2192R \u03d5\u2032(r) \u03d5(r) <\u221e. (2.27) 22 2.3. Proof of the main theorem Proof: Since \u03d5(\u03b4) \u2265 0 and \u03d5(r) > 0 for 0 < r < \u03b4, it is obvious that \u03d5 satisfies the second condition. To obtain the first condition, set x(r) = r\u03d5 \u2032(r) \u03d5(r) . one may verify that x(r) satisfies the ODE: rx\u2032(r) + x2(r) = \u2212F (r), for 0 < r \u2264 \u03b4, where F (r) = r2v(r) \u2265 0. By Lemma 2.21 we conclude that limr\u21930 r\u03d5 \u2032(r) \u03d5(r) = limr\u21930 x(t) = 0. \u0003 Lemma 2.28. Let V be positive radial potential on the ball \u2126 of radius R in Rn (n \u2265 3). Assume that\u222b \u2126 ( |\u2207u|2 \u2212 (n\u221222 )2 |u| 2 |x|2 \u2212 V (|x|)|u|2 ) dx \u2265 0 for all u \u2208 H10 (\u2126). Then there exists a C2-supersolution to the equation \u2212\u2206u\u2212 ( n\u2212 2 2 )2 u |x|2 \u2212 V (|x|)u = 0, in \u2126, (2.29) u > 0 in \u2126 \\ {0}, (2.30) u = 0 in \u2202\u2126. (2.31) Proof: Define \u03bb1(V ) := inf{ \u222b \u2126 |\u2207\u03c8|2 \u2212 (n\u221222 )2|\u03c8|2 \u2212 V |\u03c8|2\u222b \u2126 |\u03c8|2 ; \u03c8 \u2208 C\u221e0 (\u2126 \\ {0})}. By our assumption \u03bb(V ) \u2265 0. Let (\u03c6n, \u03bbn1 ) be the first eigenpair for the problem (L\u2212 \u03bb1(V )\u2212 \u03bbr1)\u03c6r = 0 on \u2126 \\BR n \u03c6(r) = 0 on \u2202(\u2126 \\BR n ), where L = \u2212\u2206 \u2212 (n\u221222 )2 1|x|2 \u2212 V , and BRn is a ball of radius R n , n \u2265 2 . The eigenfunctions can be chosen in such a way that \u03c6n > 0 on \u2126 \\BR n and \u03d5n(b) = 1, for some b \u2208 \u2126 with R2 < |b| < R. Note that \u03bbn1 \u2193 0 as n \u2192 \u221e. Harnak\u2019s inequality yields that for any compact subset K, maxK\u03c6nmaxK\u03c6n \u2264 C(K) with the later constant being indepen- dant of \u03c6n. Also standard elliptic estimates also yields that the family (\u03c6n) have also uniformly bounded derivatives on compact sets \u2126\u2212BR n . Therefore, there exists a subsequence (\u03d5nl2 )l2 of (\u03d5n)n such that (\u03d5nl2 )l2 23 2.3. Proof of the main theorem converges to some \u03d52 \u2208 C2(\u2126 \\B(R2 )). Now consider (\u03d5nl2 )l2 on \u2126 \\B(R3 ). Again there exists a subsequence (\u03d5nl3 )l3 of (\u03d5nl2 )l2 which converges to \u03d53 \u2208 C2(\u2126 \\ B(R3 )), and \u03d53(x) = \u03d52(x) for all x \u2208 \u2126 \\ B(R2 ). By repeating this argument we get a supersolution \u03d5 \u2208 C2(\u2126\\{0}) i.e. L\u03d5 \u2265 0, such that \u03d5 > 0 on \u2126 \\ {0}. \u0003 Lemma 2.32. Let a be a locally integrable function on , then the following statements are equivalent. 1. z\u2032\u2032(s) + a(s)z(s) = 0, has a strictly positive solution on (b,\u221e). 2. There exists a function \u03c8 \u2208 C1(b,\u221e) such that \u03c8\u2032(r) + \u03c82(r) + a(t) \u2264 0, for r > b. Consequently, the equation y\u2032\u2032+ 1ry \u2032+ v(r)y = 0 has a positive supersolution on (0, \u03b4) if and only if it has a positive solution on (0, \u03b4). Proof: That 1) and 2) are equivalent follows from the work of Wintner [23, 24], a proof of which may be found in [14]. To prove the rest, we note that the change of variable z(s) = y(e\u2212s) maps the equation y\u2032\u2032 + 1ry \u2032 + v(r)y = 0 into z\u2032\u2032 + e\u22122sv(e\u2212s)z(s) = 0. On the other hand, the change of variables \u03c8(t) = \u2212e \u2212ty\u2032(e\u2212t) y(e\u2212t) maps y \u2032\u2032+ 1ry \u2032+ v(r)y into \u03c8\u2032(t) + \u03c82(t) + e\u22122tv(e\u2212t). This proves the lemma. \u0003 Proof of Theorem 2.1: The implication 1) implies 2) follows imme- diately from Proposition 2.16 and Lemma 2.26. It is valid for any smooth bounded domain provided v is assumed to be non-decreasing on (0, R). this condition is not needed if the domain is a ball of radius R. To show that 2) implies 1), we assume that inequality (HV ) holds on a ball \u2126 of radiusR, and then apply Lemma (2.28) to obtain a C2-supersolution for the equation (2.29). Now take the surface average of u, that is w(r) = 1 n\u03c9wrn\u22121 \u222b \u2202Br u(x)dS = 1 n\u03c9n \u222b |\u03c9|=1 u(r\u03c9)d\u03c9 > 0, (2.33) where \u03c9n denotes the volume of the unit ball in Rn. We may assume that the unit ball is contained in \u2126 (otherwise we just use a smaller ball). By a standard calculation we get w\u2032\u2032(r) + n\u2212 1 r w\u2032(r) \u2264 1 n\u03c9nrn\u22121 \u222b \u2202Br \u2206u(x)dS. (2.34) 24 2.4. Applications Since u(x) is a supersolution of (2.29), w satisfies the inequality: w\u2032\u2032(r)+ n\u2212 1 r w\u2032(r)+( n\u2212 2 2 )2 w(r) r2 \u2264 \u2212v(r)w(r), for 0 < r < R. (2.35) Now define \u03d5(r) = r n\u22122 2 w(r), in 0 < r < R. (2.36) Using (2.35), a straightforward calculation shows that \u03d5 satisfies the follow- ing inequality \u03d5\u2032\u2032(r) + \u03d5\u2032(r) r \u2264 \u2212\u03d5(r)v(r), for 0 < r < R. (2.37) By Lemma 2.32 we may conclude that the equation y\u2032\u2032(r) + 1ry \u2032+ v(r)y = 0 has actually a positive solution \u03c6 on (0, R). It is clear that by the sufficient condition c(V ) \u2265 c whenever y\u2032\u2032(r) + 1 ry \u2032 + cv(r)y = 0 has a positive solution on (0, R). On the other hand, the necessary condition yields that y\u2032\u2032(r) + 1ry \u2032 + c(V )v(r)y = 0 has a positive solution on (0, R). The proof is now complete. \u0003 2.4 Applications In this section we start by applying Theorem 2.1 to recover in a relatively simple and unified way, all previously known improvements of Hardy\u2019s in- equality. For that we need to investigate whether the ordinary differential equation y\u2032\u2032 + y\u2032 r + v(r)y(r) = 0, (2.38) corresponding to a potential v has a positive solution \u03c6 on (0, \u03b4) for some \u03b4 > 0. In this case, \u03c8(r) = \u03c6( \u03b4rR ) is a solution for y \u2032\u2032(r)+ 1ry \u2032+ \u03b4 2 R2 v( \u03b4Rr)y = 0 on (0, R), which means that the scaled potential V\u03b4(x) = \u03b4 2 R2 V ( \u03b4Rx) yields an improved Hardy formula (HV\u03b4) on a ball of radius R, with constant larger than one. Here is an immediate application of this criterium. 1) The Brezis-Va\u0301zquez improvement [6]: Here we need to show that we can have an improved inequality with a constant potential. In this case, the best constant for which the equation y\u2032\u2032 + y\u2032 r + cy(r) = 0, (2.39) 25 2.4. Applications has a positive solution on (0, R), with R = (|\u2126|\/\u03c9n) 1n is z20\u03c92\/nn |\u2126|\u22122\/n. Indeed, if z0 is the first root of the solution of the Bessel equation y\u2032\u2032 + y\u2032 r + y(r) = 0, then the solution of (2.39) in this case is the Bessel function \u03d5(r) = J0( rz0R ). This readily gives the result of Brezis-Va\u0301zquez mentioned in the introduction. 2) The Adimurthi et al. improvement [1]: In this case, one easily sees that the functions \u03d5j(r) = ( \u220fj i=1 log (i) \u03c1 r ) 1 2 is a solution of the equation \u2212\u03d5 \u2032 j(r) + r\u03d5 \u2032\u2032 j (r) r\u03d5j(r) = 1 4r2 ( n\u220f i=1 log(i) \u03c1 r )\u22122, on (0, R), which means that the inequality (HV ) holds for the potential V (x) = 1 4|x|2 ( \u220fn i=1 log (i) \u03c1 |x|) \u22122 which yields the result of Adimurthi et al. In the following, we use our characterization to show that the constant appear- ing in the above improvement is indeed the best constant in the following sense: 1 4 = inf u\u2208H10 (\u2126)\\{0} \u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx\u2212 14 \u2211m\u22121 j=1 \u222b \u2126 |u|2 |x|2 (\u220fj i=1 log (i) \u03c1 |x| )\u22122\u222b \u2126 |u|2 |x|2 (\u220fm i=1 log (i) R |x| )\u22122 , for all 1 \u2264 m \u2264 k. We proceed by contradiction, and assume that 1 4 + \u03bb = inf u\u2208H10 (\u2126)\\{0} \u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx\u2212 14 \u2211m\u22121 j=1 \u222b \u2126 |u|2 |x|2 (\u220fj i=1 log (i) \u03c1 |x| )\u22122\u222b \u2126 |u|2 |x|2 (\u220fm i=1 log (i) \u03c1 |x| )\u22122 , and \u03bb > 0. From Theorem 2.1 we deduce that there exists a positive function \u03d5 such that \u2212\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = 1 4 m\u22121\u2211 j=1 1 r ( j\u220f i=1 log(i) \u03c1 r )\u22122 + (1 4 + \u03bb) 1 r ( m\u220f i=1 log(i) \u03c1 r )\u22122 . Now define f(r) = \u03d5(r)\u03d5m(r) > 0, and calculate, \u03d5\u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = \u03d5\u2032m(r) + r\u03d5 \u2032\u2032 m(r) \u03d5m(r) + f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) m\u2211 i=1 1\u220fi j=1 log j(\u03c1r ) . 26 2.4. Applications Thus, f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) m\u2211 i=1 1\u220fi j=1 log j(\u03c1r ) = \u2212\u03bb1 r ( m\u220f i=1 log(i) \u03c1 r )\u22122 . (2.40) If now f \u2032(\u03b1n) = 0 for some sequence {\u03b1n}\u221en=1 that converges to zero, then there exists a sequence {\u03b2n}\u221en=1 that also converges to zero, such that f \u2032\u2032(\u03b2n) = 0, and f \u2032(\u03b2n) > 0. But this contradicts (2.40), which means that f is eventually monotone for r small enough. We consider the two cases according to whether f is increasing or decreasing: Case I: Assume f \u2032(r) > 0 for r > 0 sufficiently small. Then we will have (rf \u2032(r))\u2032 rf \u2032(r) \u2264 m\u2211 i=1 1 r \u220fi j=1 log j(\u03c1r ) . Integrating once we get f \u2032(r) \u2265 c r \u220fm j=1 log j(\u03c1r ) , for some c > 0. Hence, limr\u21920 f(r) = \u2212\u221e which is a contradiction. Case II: Assume f \u2032(r) < 0 for r > 0 sufficiently small. Then (rf \u2032(r))\u2032 rf \u2032(r) \u2265 m\u2211 i=1 1 r \u220fi j=1 log j(\u03c1r ) . Thus, f \u2032(r) \u2265 \u2212 c r \u220fm j=1 log j(\u03c1r ) , (2.41) for some c > 0 and r > 0 sufficiently small. On the other hand f \u2032(r) + rf \u2032\u2032(r) f(r) \u2264 \u2212\u03bb m\u2211 j=1 1 r ( j\u220f i=1 log(i) R r )\u22122 \u2264 \u2212\u03bb( 1\u220fm j=1 log j(\u03c1r ) )\u2032. Since f \u2032(r) < 0, there exists l such that f(r) > l > 0 for r > 0 sufficiently small. From the above inequality we then have bf \u2032(b)\u2212 af \u2032(a) < \u2212\u03bbl( 1\u220fm j=1 log j(\u03c1b ) \u2212 1\u220fm j=1 log j( \u03c1a ) ). From (2.41) we have lima\u21920 af \u2032(a) = 0. Hence, bf \u2032(b) < \u2212 \u03bbl\u220fm j=1 log j(\u03c1b ) , 27 2.4. Applications for every b > 0, and f \u2032(r) < \u2212 \u03bbl r \u220fm j=1 log j(\u03c1r ) , for r > 0 sufficiently small. Therefore, lim r\u21920 f(r) = +\u221e, and by choosing l large enouph (e.g., l > c\u03bb ) we get to contradict (2.41) and the proof is now complete. 3) The Filippas and Tertikas improvement [11]: Let D \u2265 supx\u2208\u2126 |x|, and define \u03d5k(r) = (X1( r D )X2( r D ) . . . Xi\u22121( r D )Xi( r D ))\u2212 1 2 , i = 1, 2, . . . . Using the fact that X \u2032k(r) = 1 rX1(r)X2(r) . . . Xk\u22121(r)X 2 k(r) for k = 1, 2, . . ., we get \u2212\u03d5 \u2032 k(r) + r\u03d5 \u2032\u2032 k(r) \u03d5k(r) = 1 4r X21 ( r D )X22 ( r D ) . . . X2k\u22121( r D )X2k( r D ). This means that the inequality (HV ) holds for the potential V (x) = 1 4|x|2X 2 1 ( |x| D )X22 ( |x| D ) . . . X2k\u22121( |x| D )X2k( |x| D ), which yields the result of Filippas and Tertikas [11]. We now identify the best constant by showing that: 1 4 = inf u\u2208H10 (\u2126)\\{0}\u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx\u2212 14 \u2211m\u22121 j=1 \u222b \u2126 |u|2 |x|2X 2 1 ( |x| D )X 2 2 ( |x| D ) . . . X 2 j\u22121( |x| D )X 2 j ( |x| D )\u222b \u2126 |u|2 |x|2X 2 1 ( |x| D )X 2 2 ( |x| D ) . . . X 2 m\u22121( |x| D )X 2 m( |x| D ) , for all 1 \u2264 m \u2264 k. We proceed again by contradiction and in a way very similar to the above case. Indeed, assuming that 1 4 + \u03bb = inf u\u2208H10 (\u2126)\\{0}\u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx\u2212 14 \u2211m\u22121 j=1 \u222b \u2126 |u|2 |x|2X 2 1 ( |x| D )X 2 2 ( |x| D ) . . . X 2 j\u22121( |x| D )X 2 j ( |x| D )\u222b \u2126 |u|2 |x|2X 2 1 ( |x| D )X 2 2 ( |x| D ) . . . X 2 m\u22121( |x| D )X 2 m( |x| D ) , and \u03bb > 0, we use again Theorem 2.1 to find a positive function \u03d5 such that \u2212\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = 1 4 m\u22121\u2211 j=1 1 r X21 ( r D )X22 ( r D ) . . . X2j\u22121( r D )X2j ( r D ) + ( 1 4 + \u03bb) 1 r X21 ( r D )X22 ( r D ) . . . X2m\u22121( r D )X2m( r D ). 28 2.4. Applications Setting f(r) = \u03d5(r)\u03d5m(r) > 0, we have \u03d5\u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = \u03d5\u2032m(r) + r\u03d5 \u2032\u2032 m(r) \u03d5m(r) + f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) m\u2211 i=1 i\u220f j=1 Xj( r D ). Thus, f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) m\u2211 i=1 i\u220f j=1 Xj( r D ) = \u2212\u03bb1 r m\u220f j=1 X2j ( r D ). (2.42) Arguing as before, we deduce that f is eventually monotone for r small enough, and we consider two cases: Case I: If f \u2032(r) > 0 for r > 0 sufficiently small, then we will have (rf \u2032(r))\u2032 rf \u2032(r) \u2264 m\u2211 i=1 1 r i\u220f j=1 Xj( r D ). Integrating once we get f \u2032(r) \u2265 c r m\u220f j=1 Xj( r D ), for some c > 0, and therefore limr\u21920 f(r) = \u2212\u221e which is a contradiction. Case II: Assume f \u2032(r) < 0 for r > 0 sufficiently small. Then (rf \u2032(r))\u2032 rf \u2032(r) \u2265 m\u2211 i=1 1 r i\u220f j=1 Xj( r D ) Thus, f \u2032(r) \u2265 \u2212 c r m\u220f j=1 Xj( r D ), (2.43) for some c > 0 and r > 0 sufficiently small. On the other hand f \u2032(r) + rf \u2032\u2032(r) f(r) \u2264 \u2212\u03bb m\u2211 j=1 1 r j\u220f i=1 X2j \u2264 \u2212\u03bb( m\u220f j=1 Xj( r D ))\u2032. Since f \u2032(r) < 0, we may assume f(r) > l > 0 for r > 0 sufficiently small, and from the above inequality we have bf \u2032(b)\u2212 af \u2032(a) < \u2212\u03bbl( m\u220f j=1 Xj( b D )\u2212 m\u220f j=1 Xj( a D )). 29 2.4. Applications From (2.43) we have lima\u21920 af \u2032(a) = 0. Hence, f \u2032(r)) < \u2212\u03bbl r m\u220f j=1 Xj( r D ), for r > 0 sufficiently small. Therefore, lim r\u21920 f(r) = +\u221e, and by choosing l large enouph (i.e. l > c\u03bb ) we contradict (2.43) and the proof is complete. \u0003 We shall now make the connection between improved Hardy inequalities and the existence of non-oscillatory solutions (i.e., those z(s) such that z(s) > 0 for s > 0 sufficiently large) for the second order linear differential equations z\u2032\u2032(s) + a(s)z(s) = 0. (2.44) Interesting results in this direction were established by many authors (see [14, 15, 23, 24, 25]). Here is a typical criterium about the oscillatory properties of equation (2.44): 1. If lim supt\u2192\u221e t \u222b\u221e t a(s)ds < 14 , then Eq. (2.44) is non-oscillatory. 2. If lim inft\u2192\u221e t \u222b\u221e t a(s)ds > 14 , then Eq. (2.44) is oscillatory. This result combined with Theorem 2.1 and Lemma 2.32 clearly yields Corollary 2.1. Proof of Corollary 2.1: It follows from Ho\u0308lder\u2019s inequality that ( \u222b \u2126 V (|x|)u2(x)dx) 1s \u2265 \u222b \u2126 u 2 s (x)dx ( \u222b \u2126 V \u2212 r s (|x|)dx) 1r , where s \u2265 1 and 1s + 1r = 1. Letting p = 2s , we get\u222b \u2126 V (|x|)u2(x)dx \u2265 ( \u222b \u2126 up(x)dx) 2 p 1 ||V \u22121(|x|)|| L p 2\u2212p (\u2126) . Inequality (2.1) now follows from Theorem 2.1. \u0003 Proof of Corollary 2.1: Define the functional F\u00b5(u) = \u222b \u2126 |\u2207u(x)|2dx\u2212 \u00b5 \u222b \u2126 u2(x) |x|2 dx, (2.45) which is continuous, Gateaux differentiable and coercive on H10 (\u2126). Let u\u00b5 > 0 be a minimizer of F\u00b5 over the manifold M = {u \u2208 H10 (\u2126)| \u222b \u2126 u2(x)V (x) = 1} and 30 2.4. Applications assume \u03bb1\u00b5 is the infimum. It is clear that \u03bb 1 \u00b5 > 0. By standard arguments we can conclude that u\u00b5 is a weak solution of (EV,\u00b5). The rest of the proof follows from Corollary 2.1 and the fact that \u03bb1(V ) = lim \u00b5\u2192\u00b5n \u03bb1\u00b5 = inf u\u2208H10 (\u2126)\\{0} \u222b \u2126 (|\u2207u|2 \u2212 \u00b5n u 2(x) |x|2 )dx\u222b \u2126 |u(x)|2V (x)dx . 31 Bibliography [1] Adimurthi, N. Chaudhuri, and N. Ramaswamy, An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc. 130 (2002), 489-505. [2] M. Agueh, N. Ghoussoub, X. S. Kang: Geometric inequalities via a general comparison principle for interacting gases, Geom. And Funct. Anal., Vol 14, 1 (2004) p. 215-244 [3] H. Brezis, E. H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86. [4] H. 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Soc. 144 (1969) 197-215. 33 Chapter 3 Bessel pairs and optimal Hardy and Hardy-Rellich inequalities 2 3.1 Introduction Ever since Bre\u0301zis-Vazquez [10] showed that Hardy\u2019s inequality can be improved once restricted to a smooth bounded domain \u2126 in Rn, there was a flurry of activity about possible improvements of the following type: If n \u2265 3 then \u222b \u2126 |\u2207u|2dx\u2212 (n\u221222 )2 \u222b \u2126 |u|2 |x|2 dx \u2265 \u222b \u2126 V (x)|u|2dx, (3.1) for all u \u2208 H10 (\u2126), as well as its fourth order counterpart If n \u2265 5 then \u222b \u2126 |\u2206u|2dx\u2212 n2(n\u22124)216 \u222b \u2126 u2 |x|4 dx \u2265 \u222b \u2126 W (x)u2dx (3.2) for u \u2208 H20 (\u2126), where V,W are certain explicit radially symmetric potentials of order lower than 1r2 (for V ) and 1 r4 (for W ) (see [1], [3], [6], [4], [5], [8], [9], [10], [15], [16], [17], [19], [31]). In this section, we provide an approach that completes, simplifies and improves most related results to-date regarding the Laplacian on Euclidean space as well as its powers. We also establish new inequalities some of which cover critical dimensions such as n = 2 for inequality (3.1) and n = 4 for (3.2). We start by giving necessary and sufficient conditions on positive radial func- tions V and W on a ball B in Rn, so that the following inequality holds for some c > 0: \u222b B V (x)|\u2207u|2dx \u2265 c \u222b B W (x)u2dx for all u \u2208 C\u221e0 (B). (3.3) Assuming that the ball B has radius R and that \u222b R 0 1 rn\u22121V (r)dr = +\u221e, the condition is simply that the ordinary differential equation (BV,cW ) y\u2032\u2032(r) + (n\u22121r + Vr(r) V (r) )y \u2032(r) + cW (r)V (r) y(r) = 0 2A version of this chapter has been accepted for publication. N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. An- nalen, Published Online (2010). 34 3.1. Introduction has a positive solution on the interval (0, R). We shall call such a couple (V,W ) a Bessel pair on (0, R). The weight of such a pair is then defined as \u03b2(V,W ;R) = sup { c; (BV,cW ) has a positive solution on (0, R) } . (3.4) This characterization makes an important connection between Hardy-type inequal- ities and the oscillatory behavior of the above equations. For example, by using recent results on ordinary differential equations, we can then infer that an integral condition on V,W of the form lim sup r\u21920 r2(n\u22121)V (r)W (r) ( \u222b R r d\u03c4 \u03c4n\u22121V (\u03c4) )2 < 1 4 (3.5) is sufficient (and \u201calmost necessary\u201d) for (V,W ) to be a Bessel pair on a ball of sufficiently small radius \u03c1. Applied in particular, to a pair (V, 1r2V ) where the function rV \u2032(r) V (r) is assumed to decrease to \u2212\u03bb on (0, R), we obtain the following extension of Hardy\u2019s inequality: If \u03bb \u2264 n\u2212 2, then\u222b B V (x)|\u2207u|2dx \u2265 (n\u2212\u03bb\u221222 )2 \u222b B V (x) u 2 |x|2 dx for all u \u2208 C\u221e0 (B) (3.6) and (n\u2212\u03bb\u221222 ) 2 is the best constant. The case where V (x) \u2261 1 is obviously the classical Hardy inequality and when V (x) = |x|\u22122a for \u2212\u221e < a < n\u221222 , this is a particular case of the Caffarelli-Kohn-Nirenberg inequality. One can however apply the above criterium to obtain new inequalities such as the following: For a, b > 0 \u2022 If \u03b1\u03b2 > 0 and m \u2264 n\u221222 , then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m\u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (3.7) and (n\u22122m\u221222 ) 2 is the best constant in the inequality. \u2022 If \u03b1\u03b2 < 0 and 2m\u2212 \u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m+ \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (3.8) and (n\u22122m+\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. We can also extend some of the recent results of Blanchet-Bonforte-Dolbeault- Grillo-Vasquez [4]. \u2022 If \u03b1\u03b2 < 0 and \u2212\u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |\u2207u|2dx \u2265 b 2\u03b1 (n\u2212 \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx, (3.9) and b 2 \u03b1 (n\u2212\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. 35 3.1. Introduction \u2022 If \u03b1\u03b2 > 0, and n \u2265 2, then there exists a constant C > 0 such that for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |\u2207u|2dx \u2265 C \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx. (3.10) Moreover, b 2 \u03b1 (n\u221222 ) 2 \u2264 C \u2264 b 2\u03b1 (n+\u03b1\u03b2\u221222 )2. On the other hand, by considering the pair V (x) = |x|\u22122a and Wa,c(x) = (n\u22122a\u221222 )2|x|\u22122a\u22122 + c|x|\u22122aW (x) we get the following improvement of the Caffarelli-Kohn-Nirenberg inequalities:\u222b B |x|\u22122a|\u2207u|2dx\u2212 (n\u2212 2a\u2212 2 2 )2 \u222b B |x|\u22122a\u22122u2dx \u2265 c \u222b B |x|\u22122aW (x)u2dx, (3.11) for all u \u2208 C\u221e0 (B), if and only if the following ODE (BcW ) y\u2032\u2032 + 1ry \u2032 + cW (r)y = 0 has a positive solution on (0, R). Such a functionW will be called a Bessel potential on (0, R). This type of characterization was established recently by the authors [19] in the case where a = 0, yielding in particular the recent improvements of Hardy\u2019s inequalities (on bounded domains) established by Brezis-Va\u0301zquez [10], Adimurthi et al. [1], and Filippas-Tertikas [17]. Our results here include in addition those proved by Wang-Willem [34] in the case where a < n\u221222 and W (r) = 1 r2(ln Rr ) 2 , but also cover the previously unknown limiting case corresponding to a = n\u221222 as well as the critical dimension n = 2. More importantly, we establish here that Bessel pairs lead to a myriad of op- timal Hardy-Rellich inequalities of arbitrary high order, therefore extending and completing a series of new results by Adimurthi et al. [2], Tertikas-Zographopoulos [31] and others. They are mostly based on the following theorem which summarizes the main thrust of this chapter. Let V and W be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in Rn (n \u2265 1) such that \u222b R 0 1 rn\u22121V (r)dr = +\u221e and\u222b R 0 rn\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R) and \u03b2(V,W ;R) \u2265 1. 2. \u222b B V (x)|\u2207u|2dx \u2265 \u222b B W (x)u2dx for all u \u2208 C\u221e0 (B). 3. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n\u2212 2, then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx, for all radial u \u2208 C\u221e0,r(B). 36 3.1. Introduction 4. If in addition, W (r)\u2212 2V (r)r2 + 2Vr(r)r \u2212 Vrr(r) \u2265 0 on (0, R), then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx, for all u \u2208 C\u221e0 (B). In other words, one can obtain as many Hardy and Hardy-Rellich type inequali- ties as one can construct Bessel pairs on (0, R). The relevance of the above result stems from the fact that there are plenty of such pairs that are easily identifi- able. Indeed, even the class of Bessel potentials \u2013equivalently those W such that( 1, (n\u221222 ) 2|x|\u22122 + cW (x)) is a Bessel pair\u2013 is quite rich and contains several impor- tant potentials. Here are some of the most relevant properties of the class of C1 Bessel potentials W on (0, R), that we shall denote by B(0, R). First, the class is a closed convex solid subset of C1(0, R), that is ifW \u2208 B(0, R) and 0 \u2264 V \u2264W , then V \u2208 B(0, R). The \u201dweight\u201d of each W \u2208 B(R), that is \u03b2(W ;R) = sup { c > 0; (BcW ) has a positive solution on (0, R) } , (3.12) will be an important ingredient for computing the best constants in corresponding functional inequalities. Here are some basic examples of Bessel potentials and their corresponding weights. \u2022 W \u2261 0 is a Bessel potential on (0, R) for any R > 0. \u2022 W \u2261 1 is a Bessel potential on (0, R) for any R > 0, and \u03b2(1;R) = z20R2 where z0 = 2.4048... is the first zero of the Bessel function J0. \u2022 If a < 2, then there exists Ra > 0 such that W (r) = r\u2212a is a Bessel potential on (0, Ra). \u2022 For k \u2265 1, R > 0 and \u03c1 = R(eee. .e((k\u22121)\u2212times) ), let Wk,\u03c1(r) = \u03a3kj=1 1 r2 ( j\u220f i=1 log(i) \u03c1 r )\u22122 , where the functions log(i) are defined iteratively as follows: log(1)(.) = log(.) and for k \u2265 2, log(k)(.) = log(log(k\u22121)(.)). Wk,\u03c1 is then a Bessel potential on (0, R) with \u03b2(Wk,\u03c1;R) = 14 . \u2022 For k \u2265 1, R > 0 and \u03c1 \u2265 R, define W\u0303k;\u03c1(r) = \u03a3kj=1 1 r2 X21 ( r \u03c1 )X22 ( r \u03c1 ) . . . X2j\u22121( r \u03c1 )X2j ( r \u03c1 ), where the functions Xi are defined iteratively as follows: X1(t) = (1 \u2212 log(t))\u22121 and for k \u2265 2, Xk(t) = X1(Xk\u22121(t)). Then again W\u0303k,\u03c1 is a Bessel potential on (0, R) with \u03b2(W\u0303k,\u03c1;R) = 14 . 37 3.1. Introduction \u2022 More generally, if W is any positive function on R such that lim inf r\u21920 ln(r) \u222b r 0 sW (s)ds > \u2212\u221e, then for every R > 0, there exists \u03b1 := \u03b1(R) > 0 such that W\u03b1(x) := \u03b12W (\u03b1x) is a Bessel potential on (0, R). What is remarkable is that the class of Bessel potentials W is also the one that leads to optimal improvements for fourth order inequalities (in dimension n \u2265 3) of the following type:\u222b B |\u2206u|2dx\u2212 C(n) \u222b B |\u2207u|2 |x|2 dx \u2265 c(W,R) \u222b B W (x)|\u2207u|2dx , (3.13) for all u \u2208 H20 (B), where C(3) = 2536 , C(4) = 3 and C(n) = n 2 4 for n \u2265 5. The case when W \u2261 W\u0303k,\u03c1 and n \u2265 5 was recently established by Tertikas-Zographopoulos [31]. Note that W can be chosen to be any one of the examples of Bessel potentials listed above. Moreover, both C(n) and the weight \u03b2(W ;R) are the best constants in the above inequality. Appropriate combinations of (3.3) and (3.13) then lead to a myriad of Hardy- Rellich inequalities in dimension n \u2265 4. The For example, ifW is a Bessel potential on (0, R) such that the function rWr(r)W (r) decreases to \u2212\u03bb, and if \u03bb \u2264 n\u2212 2, then we have for all u \u2208 C\u221e0 (BR)\u222b B |\u2206u|2dx \u2212 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx (3.14) \u2265 (n2 4 + (n\u2212 \u03bb\u2212 2)2 4 ) \u03b2(W ;R) \u222b B W (x) |x|2 u 2dx. By applying (3.14) to the various examples of Bessel functions listed above, one improves in many ways the recent results of Adimurthi et al. [2] and those by Tertikas-Zographopoulos [31]. Moreover, besides covering the critical dimension n = 4, we also establish that the best constant is (1+ n(n\u22124)8 ) for all the potentials Wk and W\u0303k defined above. For example we have for n \u2265 4,\u222b B |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx (3.15) + (1 + n(n\u2212 4) 8 ) k\u2211 j=1 \u222b B u2 |x|4 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx. More generally, we show that for any m < n\u221222 , and any W Bessel potential on a ball BR \u2282 Rn of radius R, the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx, (3.16) 38 3.2. General Hardy Inequalities where am,n and \u03b2(W ;R) are best constants that we compute in the appendices for all m and n and for many Bessel potentialsW . Worth noting is Corollary 3.3 where we show that inequality (3.16) restricted to radial functions in C\u221e0 (BR) holds with a best constant equal to (n+2m2 ) 2, but that an,m can however be strictly smaller than (n+2m2 ) 2 in the non-radial case. These results improve considerably Theorem 1.7, Theorem 1.8, and Theorem 6.4 in [31]. We also establish a more general version of equation (3.14). Assuming again that rW \u2032(r) W (r) decreases to \u2212\u03bb on (0, R), and provided m \u2264 n\u221242 and \u03bb \u2264 n\u2212 2m\u2212 2, we then have for all u \u2208 C\u221e0 (BR),\u222b BR |\u2206u|2 |x|2m dx \u2265 \u03b2n,m \u222b BR u2 |x|2m+4 dx (3.17) + \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b BR W (x) |x|2m+2u 2dx, where again the best constants \u03b2n,m are computed in section 3. This completes the results in Theorem 1.6 of [31], where the inequality is established for n \u2265 5, 0 \u2264 m < n\u221242 , and the particular potential W\u0303k,\u03c1. Another inequality that relates the Hessian integral to the Dirichlet energy is the following: Assuming \u22121 < m \u2264 n\u221242 and W is a Bessel potential on a ball B of radius R in Rn, then for all u \u2208 C\u221e0 (B),\u222b B |\u2206u|2 |x|2m dx\u2212 (n+ 2m)2(n\u2212 2m\u2212 4)2 16 \u222b B u2 |x|2m+4 dx \u2265 (3.18) \u03b2(W ;R) (n+ 2m)2 4 \u222b B W (x) |x|2m+2u 2dx+ \u03b2(|x|2m;R)||u||H10 . This improves considerably Theorem A.2. in [2] where it is established \u2013 for m = 0 and without best constants \u2013 with the potential W1,\u03c1 in dimension n \u2265 5, and the potential W2,\u03c1 when n = 4. Finally, we establish several higher order Rellich inequalities for integrals of the form \u222b BR |\u2206mu|2 |x|2k dx, improving in many ways several recent results in [31]. Theorem 3.1 also leads to certain improved Hardy-Rellich inequalities that are crucial to show the singular nature of the external solutions in various fourth order nonlinear eigenvalue problems [12, 23, 24]. 3.2 General Hardy Inequalities Here is the main result of this section. Let V andW be positive radial C1-functions on BR\\{0}, where BR is a ball centered at zero with radius R (0 < T \u2264 +\u221e) in Rn (n \u2265 1). Assume that \u222b a 0 1 rn\u22121V (r)dr = +\u221e and \u222b a 0 rn\u22121V (r)dr < \u221e for some 0 < a < R. Then the following two statements are equivalent: 1. The ordinary differential equation (BV,W ) y\u2032\u2032(r) + (n\u22121r + Vr(r) V (r) )y \u2032(r) + W (r)V (r) y(r) = 0 39 3.2. General Hardy Inequalities has a positive solution on the interval (0, R] (possibly with \u03d5(R) = 0). 2. For all u \u2208 C\u221e0 (BR) (HV,W ) \u222b BR V (x)|\u2207u(x)|2dx \u2265 \u222b BR W (x)u2dx. Before proceeding with the proofs, we note the following immediate but useful corollary. Let V and W be positive radial C1-functions on B\\{0}, where B is a ball with radius R in Rn (n \u2265 1) and centered at zero, such that \u222b R 0 1 rn\u22121V (r)dr = +\u221e and\u222b R 0 rn\u22121V (r)dr < \u221e. Then (V,W ) is a Bessel pair on (0, R) if and only if for all u \u2208 C\u221e0 (BR), we have\u222b BR V (x)|\u2207u|2dx \u2265 \u03b2(V,W ;R) \u222b BR W (x)u2dx, with \u03b2(V,W ;R) being the best constant. For the proof of Theorem 3.2, we shall need the following lemmas. Lemma 3.19. Let \u2126 be a smooth bounded domain in Rn with n \u2265 1 and let \u03d5 \u2208 C1(0, R := supx\u2208\u2202\u2126 |x|) be a positive solution of the ordinary differential equation y\u2032\u2032 + ( n\u2212 1 r + Vr(r) V (r) )y\u2032 + W (r) V (r) y = 0, (3.20) on (0, R) for some V (r),W (r) \u2265 0 where \u222b R 0 1 rn\u22121V (r)dr = +\u221e and \u222b R 0 rn\u22121V (r)dr < \u221e. Setting \u03c8(x) = u(x)\u03d5(|x|) for any u \u2208 C\u221e0 (\u2126), we then have the following properties: 1. \u222b R 0 rn\u22121V (r)(\u03d5 \u2032(r) \u03d5(r) ) 2dr <\u221e and limr\u21920 rn\u22121V (r)\u03d5 \u2032(r) \u03d5(r) = 0. 2. \u222b \u2126 V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx <\u221e. 3. \u222b \u2126 V (|x|)\u03d52(|x|)|\u2207\u03c8|2(x)dx <\u221e. 4. | \u222b \u2126 V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8(x)dx| <\u221e. 5. limr\u21920 | \u222b \u2202Br V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c82(x)ds| = 0, where Br \u2282 \u2126 is a ball of radius r centered at 0. Proof: 1) Setting x(r) = rn\u22121V (r)\u03d5 \u2032(r) \u03d5(r) , we have rn\u22121V (r)x\u2032(r) + x2(r) = r2(n\u22121)V 2(r) \u03d5 (\u03d5\u2032\u2032(r) + ( n\u2212 1 r + Vr(r) V (r) )\u03d5\u2032(r)) = \u2212r 2(n\u22121)V (r)W (r) \u03d5(r) \u2264 0, 0 < r < R. 40 3.2. General Hardy Inequalities Dividing by rn\u22121V (r) and integrating once, we obtain x(r) \u2265 \u222b R r |x(s)|2 sn\u22121V (s) ds+ x(R). (3.21) To prove that limr\u21920G(r) < \u221e, where G(r) := \u222b R r x2(s) sn\u22121V (s)ds, we assume the contrary and use (3.21) to write that (\u2212rn\u22121V (r))G\u2032(r)) 12 \u2265 G(r) + x(R). Thus, for r sufficiently small we have\u2212rn\u22121V (r)G\u2032(r) \u2265 12G2(r) and hence, ( 1G(r) )\u2032 \u2265 1 2rn\u22121V (r) , which contradicts the fact that G(r) goes to infinity as r tends to zero. Also in view of (3.21), we have that x0 := limr\u21920 x(r) exists, and since limr\u21920G(r) < \u221e, we necessarily have x0 = 0 and 1) is proved. For assertion 2), we use 1) to see that\u222b \u2126 V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx \u2264 ||u||2\u221e \u222b \u2126 V (|x|) (\u03d5 \u2032(|x|))2 \u03d52(|x|) dx <\u221e. 3) Note that |\u2207\u03c8(x)| \u2264 |\u2207u(x)|\u03d5(|x|) + |u(x)| |\u03d5 \u2032(|x|)| \u03d52(|x|) \u2264 C1\u03d5(|x|) + C2 |\u03d5 \u2032(|x|)| \u03d52(|x|) , for all x \u2208 \u2126, where C1 = maxx\u2208\u2126 |\u2207u| and C2 = maxx\u2208\u2126 |u|. Hence we have\u222b \u2126 V (|x|)\u03d52(|x|)|\u2207\u03c8|2(x)dx \u2264 \u222b \u03c9 V (|x|) (C1\u03d5(|x|) + C2\u03d5 \u2032(|x|))2 \u03d52(|x|) dx = \u222b \u2126 C21V (|x|)dx+ \u222b \u2126 2C1C2 |\u03d5\u2032(|x|)| \u03d5(|x|) V (|x|)dx+ \u222b \u2126 C22 ( \u03d5\u2032(|x|) \u03d5(|x|) ) 2V (|x|)dx \u2264 L1 + 2C1C2 ( \u222b \u2126 V (|x|)(\u03d5 \u2032(|x|) \u03d5(|x|) ) 2dx ) 1 2 ( \u222b \u2126 V (|x|)dx) 12 + L2 < \u221e, which proves 3). 4) now follows from 2) and 3) since V (|x|)|\u2207u|2 = V (|x|)(\u03d5\u2032(|x|))2\u03c82(x) + 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8(x) + V (|x|)\u03d5 2(|x|)|\u2207\u03c8|2. Finally, 5) follows from 1) since | \u222b \u2202Br V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c82(x)ds| < ||u||2\u221e| \u222b \u2202Br V (|x|)\u03d5 \u2032(|x|) \u03d5(|x|) ds = ||u||2\u221eV (r) |\u03d5\u2032(r)| \u03d5(r) \u222b \u2202Br 1ds = n\u03c9n||u||2\u221ern\u22121V (r) |\u03d5\u2032(r)| \u03d5(r) . 41 3.2. General Hardy Inequalities Lemma 3.22. Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in Rn (n \u2265 1) and centered at zero. Assuming\u222b B ( V (x)|\u2207u|2 \u2212W (x)|u|2) dx \u2265 0 for all u \u2208 C\u221e0 (B), then there exists a C2-supersolution to the following linear elliptic equation \u2212div(V (x)\u2207u)\u2212W (x)u = 0, in B, (3.23) u > 0 in B \\ {0}, (3.24) u = 0 in \u2202B. (3.25) (3.26) Proof: Define \u03bb1(V ) := inf{ \u222b B V (x)|\u2207\u03c8|2 \u2212W (x)|\u03c8|2\u222b B |\u03c8|2 ; \u03c8 \u2208 C \u221e 0 (B \\ {0})}. By our assumption \u03bb1(V ) \u2265 0. Let (\u03c6n, \u03bbn1 ) be the first eigenpair for the problem (L\u2212 \u03bb1(V )\u2212 \u03bbn1 )\u03c6n = 0 on B \\BR n \u03c6n = 0 on \u2202(B \\BR n ), where Lu = \u2212div(V (x)\u2207u) \u2212W (x)u, and BR n is a ball of radius Rn , n \u2265 2 . The eigenfunctions can be chosen in such a way that \u03c6n > 0 on B \\BR n and \u03d5n(b) = 1, for some b \u2208 B with R2 < |b| < R. Note that \u03bbn1 \u2193 0 as n \u2192 \u221e. Harnak\u2019s inequality yields that for any compact subset K, maxK\u03c6nminK\u03c6n \u2264 C(K) with the later constant being independant of \u03c6n. Also standard elliptic estimates also yields that the family (\u03c6n) have also uniformly bounded derivatives on the compact sets B \u2212BR n . Therefore, there exists a subsequence (\u03d5nl2 )l2 of (\u03d5n)n such that (\u03d5nl2 )l2 converges to some \u03d52 \u2208 C2(B \\ B(R2 )). Now consider (\u03d5nl2 )l2 on B \\ B(R3 ). Again there exists a subsequence (\u03d5nl3 )l3 of (\u03d5nl2 )l2 which converges to \u03d53 \u2208 C2(B \\ B(R3 )), and \u03d53(x) = \u03d52(x) for all x \u2208 B \\ B(R2 ). By repeating this argument we get a supersolution \u03d5 \u2208 C2(B \\ {0}) i.e. L\u03d5 \u2265 0, such that \u03d5 > 0 on B \\ {0}. \u0003 Proof of Theorem 3.2: First we prove that 1) implies 2). Let \u03c6 \u2208 C1(0, R] be a solution of (BV,W ) such that \u03c6(x) > 0 for all x \u2208 (0, R). Define u(x)\u03d5(|x|) = \u03c8(x). Then |\u2207u|2 = (\u03d5\u2032(|x|))2\u03c82(x) + 2\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8 + \u03d5 2(|x|)|\u2207\u03c8|2. Hence, V (|x|)|\u2207u|2 \u2265 V (|x|)(\u03d5\u2032(|x|))2\u03c82(x) + 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8(x). 42 3.2. General Hardy Inequalities Thus, we have\u222b B V (|x|)|\u2207u|2dx \u2265 \u222b B V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx + \u222b B 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx. Let B\u000f be a ball of radius \u000f centered at the origin. Integrate by parts to get\u222b B V (|x|)|\u2207u|2dx \u2265 \u222b B V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx+ \u222b B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx + \u222b B\\B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx = \u222b B\u000f V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx+ \u222b B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx \u2212 \u222b B\\B\u000f (V (|x|)\u03d5\u2032\u2032(|x|)\u03d5(|x|) + ((n\u2212 1)V (|x|) r + Vr(|x|))\u03d5\u2032(|x|)\u03d5(|x|) ) \u03c82(x)dx + \u222b \u2202(B\\B\u000f) V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c82(x)ds Let \u000f\u2192 0 and use Lemma 3.19 and the fact that \u03c6 is a solution of (Dv,w) to get\u222b B V (|x|)|\u2207u|2dx \u2265 \u2212 \u222b B [V (|x|)\u03d5\u2032\u2032(|x|) + ( (n\u2212 1)V (|x|) r + Vr(|x|))\u03d5\u2032(|x|)] u 2(x) \u03d5(|x|)dx = \u222b B W (|x|)u2(x)dx. To show that 2) implies 1), we assume that inequality (HV,W ) holds on a ball B of radius R, and then apply Lemma 3.22 to obtain a C2-supersolution for the equation (3.23). Now take the surface average of u, that is y(r) = 1 n\u03c9wrn\u22121 \u222b \u2202Br u(x)dS = 1 n\u03c9n \u222b |\u03c9|=1 u(r\u03c9)d\u03c9 > 0, (3.27) where \u03c9n denotes the volume of the unit ball in Rn. We may assume that the unit ball is contained in B (otherwise we just use a smaller ball). We clearly have y\u2032\u2032(r) + n\u2212 1 r y\u2032(r) = 1 n\u03c9nrn\u22121 \u222b \u2202Br \u2206u(x)dS. (3.28) Since u(x) is a supersolution of (3.23), we have\u222b \u2202Br div(V (|x|)\u2207u)ds\u2212 \u222b \u2202B W (|x|)udx \u2265 0, 43 3.2. General Hardy Inequalities and therefore, V (r) \u222b \u2202Br \u2206udS \u2212 Vr(r) \u222b \u2202Br \u2207u.xds\u2212W (r) \u222b \u2202Br u(x)ds \u2265 0. It follows that V (r) \u222b \u2202Br \u2206udS \u2212 Vr(r)y\u2032(r)\u2212W (r)y(r) \u2265 0, (3.29) and in view of (3.27), we see that y satisfies the inequality V (r)y\u2032\u2032(r) + ( (n\u2212 1)V (r) r + Vr(r))y\u2032(r) \u2264 \u2212W (r)y(r), for 0 < r < R, (3.30) that is it is a positive supersolution for (BV,W ). Standard results in ODE now allow us to conclude that (BV,W ) has actually a positive solution on (0, R), and the proof of theorem 3.2 is now complete. 3.2.1 Integral criteria for Bessel pairs In order to obtain criteria on V and W so that inequality (HV,W ) holds, we clearly need to investigate whether the ordinary differential equation (BV,W ) has positive solutions. For that, we rewrite (BV,W ) as (rn\u22121V (r)y\u2032)\u2032 + rn\u22121W (r)y = 0, and then by setting s = 1r and x(s) = y(r), we see that y is a solution of (BV,W ) on an interval (0, \u03b4) if and only if x is a positive solution for the equation (s\u2212(n\u22123)V ( 1s )x \u2032(s))\u2032 + s\u2212(n+1)W ( 1s )x(s) = 0 on ( 1 \u03b4 ,\u221e). (3.31) Now recall that a solution x(s) of the equation (3.31) is said to be oscillatory if there exists a sequence {an}\u221en=1 such that an \u2192 +\u221e and x(an) = 0. Otherwise we call the solution non-oscillatory. It follows from Sturm comparison theorem that all solutions of (3.31) are either all oscillatory or all non-oscillatory. Hence, the fact that (V,W ) is a Bessel pair or not is closely related to the oscillatory behavior of the equation (3.31). The following theorem is therefore a consequence of Theorem 3.2, combined with a relatively recent result of Sugie et al. in [29] about the oscillatory behavior of the equation (3.31). Let V and W be positive radial C1-functions on BR\\{0}, where BR is a ball centered at 0 with radius R in Rn (n \u2265 1). Assume \u222b R 0 1 \u03c4n\u22121V (\u03c4)d\u03c4 = +\u221e and\u222b R 0 rn\u22121v(r)dr <\u221e. \u2022 Assume lim sup r\u21920 r2(n\u22121)V (r)W (r) ( \u222b R r 1 \u03c4n\u22121V (\u03c4) d\u03c4 )2 < 1 4 (3.32) then (V,W ) is a Bessel pair on (0, \u03c1) for some \u03c1 > 0 and consequently, inequality (HV,W ) holds for all u \u2208 C\u221e0 (B\u03c1), where B\u03c1 is a ball of radius \u03c1. 44 3.2. General Hardy Inequalities \u2022 On the other hand, if lim inf r\u21920 r2(n\u22121)V (r)W (r) ( \u222b R r 1 \u03c4n\u22121V (\u03c4) d\u03c4 )2 > 1 4 (3.33) then there is no interval (0, \u03c1) on which (V,W ) is a Bessel pair and conse- quently, there is no smooth domain \u2126 on which inequality (HV,W ) holds. A typical Bessel pair is (|x|\u2212\u03bb, |x|\u2212\u03bb\u22122) for \u03bb \u2264 n \u2212 2. It is also easy to see by a simple change of variables in the corresponding ODEs that W is a Bessel potential if and only if (|x|\u2212\u03bb, |x|\u2212\u03bb(|x|\u22122 +W (|x|)) is a Bessel pair. (3.34) More generally, the above integral criterium allows to show the following. Let V be an strictly positive C1-function on (0, R) such that for some \u03bb \u2208 R rVr(r) V (r) + \u03bb \u2265 0 on (0, R) and limr\u21920 rVr(r) V (r) + \u03bb = 0. (3.35) If \u03bb \u2264 n\u2212 2, then for any Bessel potential W on (0, R), and any c \u2264 \u03b2(W ;R), the couple (V,W\u03bb,c) is a Bessel pair, where W\u03bb,c(r) = V (r)(( n\u2212 \u03bb\u2212 2 2 )2r\u22122 + cW (r)). (3.36) Moreover, \u03b2 ( V,W\u03bb,c;R ) = 1 for all c \u2264 \u03b2(W ;R). We need the following easy lemma. Lemma 3.37. Assume the equation y\u2032\u2032 + a r y\u2032 + V (r)y = 0, has a positive solution on (0, R), where a \u2265 1 and V (r) > 0. Then y is strictly decreasing on (0, R). Proof: First observe that y can not have a local minimum, hence it is either increasing or decreasing on (0, \u03b4), for \u03b4 sufficiently small. Assume y is increasing. Under this assumption if y\u2032(a) = 0 for some a > 0, then y\u2032\u2032(a) = 0 which contradicts the fact that y is a positive solution of the above ODE. So we have y \u2032\u2032 y\u2032 \u2264 \u2212ar , thus, y\u2032 \u2265 c ra . Therefore, x(r)\u2192 \u2212\u221e as r \u2192 0 which is a contradiction. Since, y can not have a local minimum it should be strictly decreasing on (0, R). \u0003 Proof of Theorem 3.2.1: Write Vr(r)V (r) = \u2212\u03bbr + f(r) where f(r) \u2265 0 on (0, R) 45 3.2. General Hardy Inequalities and lim r\u21920 rf(r) = 0. In order to prove that ( V (r), V (r)((n\u2212\u03bb\u221222 ) 2r\u22122 + cW (r)) ) is a Bessel pair, we need to show that the equation y\u2032\u2032 + ( n\u2212 \u03bb\u2212 1 r + f(r))y\u2032 + (( n\u2212 \u03bb\u2212 2 2 )2r\u22122 + cW (r))y(r) = 0, (3.38) has a positive solution on (0, R). But first we note that the equation x\u2032\u2032 + ( n\u2212 \u03bb\u2212 1 r )x\u2032 + (( n\u2212 \u03bb\u2212 2 2 )2r\u22122 + cW (r))x(r) = 0, has a positive solution on (0, R), whenever c \u2264 \u03b2(W ;R). Since now f(r) \u2265 0 and since, by the proceeding lemma, x\u2032(r) \u2264 0, we get that x is a positive subsolution for the equation (3.38) on (0, R), and thus it has a positive solution of (0, R). Note that this means that \u03b2(V,W\u03bb,c;R) \u2265 1. For the reverse inequality, we shall use the criterium in Theorem 3.2.1. Indeed apply criteria (3.32) to V (r) and W1(r) = C V (r) r2 to get lim r\u21920 r2(n\u22121)V (r)W1(r) ( \u222b R r 1 \u03c4n\u22121V (\u03c4) d\u03c4 )2 = C lim r\u21920 r2(n\u22122)V 2(r) ( \u222b R r 1 \u03c4n\u22121V (\u03c4) d\u03c4 )2 = C ( lim r\u21920 r(n\u22122)V (r) \u222b R r 1 \u03c4n\u22121V (\u03c4) d\u03c4 )2 = C ( lim r\u21920 1 rn\u22121V (r) (n\u22122)rn\u22123V (r)+rn\u22122Vr(r) r2(n\u22122)V 2(r) )2 = C ( lim r\u21920 1 (n\u2212 2) + r Vr(r)V (r) )2 = C (n\u2212 \u03bb\u2212 2)2 . For ( V,CV (r\u22122 + cW ) ) to be a Bessel pair, it is necessary that C(n\u2212\u03bb\u22122)2 \u2264 14 , and the proof for the best constant is complete. \u0003 With a similar argument one can also prove the following. Let V and W be positive radial C1-functions on BR\\{0}, where BR is a ball centered at zero with radius R in Rn (n \u2265 1). Assume that lim r\u21920 r Vr(r)V (r) = \u2212\u03bb and \u03bb \u2264 n\u2212 2. (3.39) \u2022 If lim sup r\u21920 r2W (r)V (r) < ( n\u2212\u03bb\u22122 2 ) 2, then (V,W ) is a Bessel pair on some interval (0, \u03c1), and consequently there exists a ball B\u03c1 \u2282 Rn such that inequality (HV,W ) holds for all u \u2208 C\u221e0 (B\u03c1). \u2022 On the other hand, if lim inf r\u21920 r2W (r)V (r) > ( n\u2212\u03bb\u22122 2 ) 2, then there is no smooth domain \u2126 \u2282 Rn such that inequality (HV,W ) holds on \u2126. 46 3.2. General Hardy Inequalities 3.2.2 New weighted Hardy inequalities An immediate application of Theorem 3.2.1 and Theorem 3.2 is the following very general Hardy inequality. Let V (x) = V (|x|) be a strictly positive radial function on a smooth domain \u2126 containing 0 such that R = supx\u2208\u2126 |x|. Assume that for some \u03bb \u2208 R rVr(r) V (r) + \u03bb \u2265 0 on (0, R) and limr\u21920 rVr(r) V (r) + \u03bb = 0. (3.40) 1. If \u03bb \u2264 n\u2212 2, then the following inequality holds for any Bessel potential W on (0, R): \u222b \u2126 V (x)|\u2207u|2dx \u2265 (n\u2212 \u03bb\u2212 2 2 )2 \u222b \u2126 V (x) |x|2 u 2dx + \u03b2(W ;R) \u222b \u2126 V (x)W (x)u2dx, (3.41) for all u \u2208 C\u221e0 (\u2126) and both (n\u2212\u03bb\u221222 )2 and \u03b2(W ;R) are the best constants. 2. In particular, \u03b2(V, r\u22122V ;R) = (n\u2212\u03bb\u221222 ) 2 is the best constant in the following inequality\u222b \u2126 V (x)|\u2207u|2dx \u2265 (n\u2212\u03bb\u221222 )2 \u222b \u2126 V (x) |x|2 u 2dx for all u \u2208 C\u221e0 (\u2126). (3.42) Applied to V1(r) = r\u2212mWk,\u03c1(r) and V2(r) = r\u2212mW\u0303k,\u03c1(r) where Wk,\u03c1(r) = \u03a3kj=1 1 r2 ( j\u220f i=1 log(i) \u03c1 r )\u22122 and W\u0303k;\u03c1(r) = \u03a3kj=1 1 r2X 2 1 ( r \u03c1 )X 2 2 ( r \u03c1 ) . . . X 2 j\u22121( r \u03c1 )X 2 j ( r \u03c1 ) are the iterated logs intro- duced in the introduction, and noting that in both cases the corresponding \u03bb is equal to 2m+ 2, we get the following new Hardy inequalities. Let \u2126 be a smooth bounded domain in Rn (n \u2265 1) and m \u2264 n\u221242 . Then the following inequalities hold.\u222b \u2126 Wk,\u03c1(x) |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m\u2212 4 2 )2 \u222b \u2126 Wk,\u03c1(x) |x|2m+2 u 2dx (3.43)\u222b \u2126 W\u0303k,\u03c1(x) |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m\u2212 4 2 )2 \u222b \u2126 W\u0303k,\u03c1(x) |x|2m+2 u 2dx. (3.44) Moreover, the constant (n\u22122m\u221242 ) 2 is the best constant in both inequalities. Remark 3.2.1. The two following theorems deal with Hardy-type inequalities on the whole of Rn. Theorem 3.2 already yields that inequality (HV,W ) holds for all u \u2208 C\u221e0 (Rn) if and only if the ODE (BV,W ) has a positive solution on (0,\u221e). The 47 3.2. General Hardy Inequalities latter equation is therefore non-oscillatory, which will again be a very useful fact for computing best constants, in view of the following criterium at infinity (Theorem 2.1 in [29]) applied to the equation (a(r)y\u2032)\u2032 + b(r)y(r) = 0, (3.45) where a(r) and b(r) are positive real valued functions. Assuming that \u222b\u221e d 1 a(\u03c4)d\u03c4 < \u221e for some d > 0, and that the following limit L := lim r\u2192\u221e a(r)b(r) (\u222b \u221e r 1 a(r) dr )2 , exists. Then for the equation (3.45) equation to be non-oscillatory, it is necessary that L \u2264 14 . Let a, b > 0, and \u03b1, \u03b2,m be real numbers. \u2022 If \u03b1\u03b2 > 0, and m \u2264 n\u221222 , then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m\u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (3.46) and (n\u22122m\u221222 ) 2 is the best constant in the inequality. \u2022 If \u03b1\u03b2 < 0, and 2m\u2212 \u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m |\u2207u| 2dx \u2265 (n\u2212 2m+ \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2 |x|2m+2 u 2dx, (3.47) and (n\u22122m+\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. Proof: Letting V (r) = (a+br \u03b1)\u03b2 r2m , then r V \u2032(r) V (r) = \u22122m+ b\u03b1\u03b2r \u03b1 a+ br\u03b1 = \u22122m+ \u03b1\u03b2 \u2212 a\u03b1\u03b2 a+ br\u03b1 . Hence, in the case \u03b1, \u03b2 > 0 and 2m \u2264 n\u2212 2, (3.46) follows directly from Theorem 3.2.2. The same holds for (3.47) since it also follows directly from Theorem 3.2.2 in the case where \u03b1 < 0, \u03b2 > 0 and 2m\u2212 \u03b1\u03b2 \u2264 n\u2212 2. For the remaining two other cases, we will use Theorem 3.2. Indeed, in this case the equation (BV,W ) becomes y\u2032\u2032 + ( n\u2212 2m\u2212 1 r + b\u03b1\u03b2r\u03b1\u22121 a+ br\u03b1 )y\u2032 + c r2 y = 0, (3.48) and the best constant in inequalities (3.46) and (3.47) is the largest c such that the above equation has a positive solution on (0,+\u221e). Note that by Lemma 3.37, we 48 3.2. General Hardy Inequalities have that y\u2032 < 0 on (0,+\u221e). Hence, if \u03b1 < 0 and \u03b2 < 0, then the positive solution of the equation y\u2032\u2032 + n\u2212 2m\u2212 1 r y\u2032 + (n\u22122m\u221222 ) 2 r2 y = 0 is a positive super-solution for (3.48) and therefore the latter ODE has a positive solution on (0,+\u221e), from which we conclude that (3.46) holds. To prove now that (n\u22122m\u221222 ) 2 is the best constant in (3.46), we use the fact that if the equation (3.48) has a positive solution on (0,+\u221e), then the equation is necessarily non-oscillatory. By rewriting (3.48) as( rn\u22122m\u22121(a+ br\u03b1)\u03b2y\u2032 )\u2032 + crn\u22122m\u22123(a+ br\u03b1)\u03b2y = 0, (3.49) and by noting that \u222b \u221e d 1 rn\u22122m\u22121(a+ br\u03b1)\u03b2 <\u221e, and lim r\u2192\u221e cr 2(n\u22122m\u22122)(a+ br\u03b1)2\u03b2 (\u222b \u221e r 1 rn\u22122m\u22121(a+ br\u03b1)\u03b2 dr )2 = c (n\u2212 2m\u2212 2)2 , we can use Theorem 2.1 in [29] to conclude that for equation (3.49) to be non- oscillatory it is necessary that c (n\u2212 2m\u2212 2)2 \u2264 1 4 . Thus, (n\u22122m\u22122) 2 4 is the best constant in the inequality (3.46). A very similar argument applies in the case where \u03b1 > 0, \u03b2 < 0, and 2m < n\u22122, to obtain that inequality (3.47) holds for all u \u2208 C\u221e0 (Rn) and that (n\u22122m+\u03b1\u03b2\u221222 )2 is indeed the best constant. \u0003 Note that the above two inequalities can be improved on smooth bounded domains by using Theorem 3.2.2. We shall now extend the recent results of Blanchet-Bonforte-Dolbeault-Grillo- Vasquez [4] and address some of their questions regarding best constants. Let a, b > 0, and \u03b1, \u03b2 be real numbers. \u2022 If \u03b1\u03b2 < 0 and \u2212\u03b1\u03b2 \u2264 n\u2212 2, then for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |\u2207u|2dx \u2265 b 2\u03b1 (n\u2212 \u03b1\u03b2 \u2212 2 2 )2 \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx, (3.50) and b 2 \u03b1 (n\u2212\u03b1\u03b2\u221222 ) 2 is the best constant in the inequality. \u2022 If \u03b1\u03b2 > 0 and n \u2265 2, then there exists a constant C > 0 such that for all u \u2208 C\u221e0 (Rn)\u222b Rn (a+ b|x|\u03b1)\u03b2 |\u2207u|2dx \u2265 C \u222b Rn (a+ b|x|\u03b1)\u03b2\u2212 2\u03b1u2dx. (3.51) 49 3.2. General Hardy Inequalities Moreover, b 2 \u03b1 (n\u221222 ) 2 \u2264 C \u2264 b 2\u03b1 (n+\u03b1\u03b2\u221222 )2. Proof: Letting V (r) = (a+ br\u03b1)\u03b2 , then we have r V \u2032(r) V (r) = b\u03b1\u03b2r\u03b1 a+ br\u03b1 = \u03b1\u03b2 \u2212 a\u03b1\u03b2 a+ br\u03b1 . Inequality (3.50) and its best constant in the case when \u03b1 < 0 and \u03b2 > 0, then follow immediately from Theorem 3.2.2 with \u03bb = \u2212\u03b1\u03b2. The proof of the remaining cases will use Theorem 3.2 as well as the integral criteria for the oscillatory behavior of solutions for ODEs of the form (BV,W ). Assuming still that \u03b1\u03b2 < 0, then with an argument similar to that of The- orem 3.2.2 above, one can show that the positive solution of the equation y\u2032\u2032 + (n+\u03b1\u03b2\u22121r )y \u2032 + (n+\u03b1\u03b2\u22122) 2 4r2 y = 0 on (0,+\u221e) is a positive supersolution for the equa- tion y\u2032\u2032 + ( n\u2212 1 r + V \u2032(r) V (r) )y\u2032 + b 2 \u03b1 (n+ \u03b1\u03b2 \u2212 2)2 4(a+ br\u03b1) 2 \u03b1 y = 0. Theorem 3.2 then yields that the inequality (3.50) holds for all u \u2208 C\u221e0 (Rn). To prove now that b 2 \u03b1 (n+\u03b1\u03b2\u221222 ) 2 is the best constant in (3.50) it is enough to show that if the following equation( rn\u22121(a+ br\u03b1)\u03b2y\u2032 )\u2032 + crn\u22121(a+ br\u03b1)\u03b2\u2212 2 \u03b1 y = 0 (3.52) has a positive solution on (0,+\u221e), then c \u2264 b 2\u03b1 (n+\u03b1\u03b2\u221222 )2. If now \u03b1 > 0 and \u03b2 < 0, then we have lim r\u2192\u221e cr 2(n\u22121)(a+ br\u03b1)2\u03b2\u2212 2 \u03b1 (\u222b \u221e r 1 rn\u22121(a+ br\u03b1)\u03b2 dr )2 = c b 2 \u03b1 (n+ \u03b1\u03b2 \u2212 2)2 . Hence, by Theorem 2.1 in [29] again, the non-oscillatory aspect of the equation holds for c \u2264 b 2 \u03b1 (n+\u03b1\u03b2\u22122)2 4 which completes the proof of the first part. A similar argument applies in the case where \u03b1\u03b2 > 0 to prove that (3.51) holds for all u \u2208 C\u221e0 (Rn) and b 2 \u03b1 (n\u221222 ) 2 \u2264 C \u2264 b 2\u03b1 (n+\u03b1\u03b2\u221222 )2. The best constants are estimated by carefully studying the existence of positive solutions for the ODE (3.52). Remark 3.2.2. Recently, Blanchet et al. in [4] studied a special case of inequality (3.50) (a = b = 1, and \u03b1 = 2) under the additional condition:\u222b Rn (1 + |x|2)\u03b2\u22121u(x)dx = 0, for \u03b2 < n\u2212 2 2 . (3.53) Note that we do not assume (3.53) in Theorem 3.2.2, and that we have found the best constants for \u03b2 \u2264 0, a case that was left open in [4]. 50 3.2. General Hardy Inequalities 3.2.3 Improved Hardy and Caffarelli-Kohn-Nirenberg Inequalities In [11] Caffarelli-Kohn-Nirenberg established a set inequalities of the following form:( \u222b Rn |x|\u2212bp|u|pdx) 2p \u2264 Ca,b \u222bRn |x|\u22122a|\u2207u|2dx for all u \u2208 C\u221e0 (Rn), (3.54) where for n \u2265 3, \u2212\u221e < a < n\u221222 , a \u2264 b \u2264 a+ 1, and p = 2nn\u22122+2(b\u2212a) . (3.55) For the cases n = 2 and n = 1 the conditions are slightly different. For n = 2 \u2212\u221e < a < 0, a < b \u2264 a+ 1, and p = 2b\u2212a , (3.56) and for n = 1 \u2212\u221e < a < \u2212 12 , a+ 12 < b \u2264 a+ 1, and p = 2\u22121+2(b\u2212a) . (3.57) Let D1,2a be the completion of C \u221e 0 (R n) for the inner product (u, v) = \u222b Rn |x|\u22122a\u2207u.\u2207vdx and let S(a, b) = inf u\u2208D1,2a \\{0} \u222b Rn |x|\u22122a|\u2207u|2dx ( \u222b Rn |x|\u2212bp|u|pdx)2\/p (3.58) denote the best embedding constant. We are concerned here with the \u201cHardy critical\u201d case of the above inequalities, that is when b = a + 1. In this direction, Catrina and Wang [14] showed that for n \u2265 3 we have S(a, a+1) = (n\u22122a\u221222 )2 and that S(a, a+ 1) is not achieved while S(a, b) is always achieved for a < b < a+ 1. For the case n = 2 they also showed that S(a, a + 1) = a2, and that S(a, a + 1) is not achieved, while for a < b < a + 1, S(a, b) is again achieved. For n = 1, S(a, a+ 1) = ( 1+2a2 ) 2 is also not achieved. In this section we give a necessary and sufficient condition for improvement of (3.54) with b = a + 1 and n \u2265 1. Our results cover also the critical case when a = n\u221222 which is not allowed by the methods of [11]. Let W be a positive radial function on the ball B in Rn (n \u2265 1) with radius R and centered at zero. Assume a \u2264 n\u221222 . The following two statements are then equivalent: 1. W is a Bessel potential on (0, R). 2. There exists c > 0 such that the following inequality holds for all u \u2208 C\u221e0 (B) (Ha,cW ) \u222b B |x|\u22122a|\u2207u(x)|2dx \u2265 (n\u2212 2a\u2212 2 2 )2 \u222b B |x|\u22122a\u22122u2dx + c \u222b B |x|\u22122aW (x)u2dx, 51 3.2. General Hardy Inequalities Moreover, (n\u22122a\u221222 ) 2 is the best constant and \u03b2(W ;R) = sup{c; (Ha,cW )holds}, where \u03b2(W ;R) is the weight of W on (0, R). On the other hand, there is no strictly positive W \u2208 C1(0,\u221e), such that the following inequality holds for all u \u2208 C\u221e0 (Rn),\u222b Rn |x|\u22122a|\u2207u(x)|2dx \u2265 (n\u2212 2a\u2212 2 2 )2 \u222b Rn |x|\u22122a\u22122u2dx + c \u222b Rn W (|x|)u2dx. (3.59) (3.60) Proof: It suffices to use Theorems 3.2 and 3.2.2 with V (r) = r\u22122a to get that W is a Bessel function if and only if the pair ( r\u22122a,Wa,c(r) ) is a Bessel pair on (0, R) for some c > 0, where Wa,c(r) = ( n\u2212 2a\u2212 2 2 )2r\u22122\u22122a + cr\u22122aW (r). For the last part, assume that (3.59) holds for some W . Then it follows from Theorem 3.2.3 that for V = cr2aW (r) the equation y\u2032\u2032(r) + 1ry \u2032 + v(r)y = 0 has a positive solution on (0,\u221e). From Lemma 3.37 we know that y is strictly decreasing on (0,+\u221e). Hence, y\u2032\u2032(r)y\u2032(r) \u2265 \u2212 1r which yields y\u2032(r) \u2264 br , for some b > 0. Thus y(r)\u2192 \u2212\u221e as r \u2192 +\u221e. This is a contradiction and the proof is complete. \u0003 Remark 3.2.3. Theorem 3.2.3 characterizes the best constant only when \u2126 is a ball, while for general domain \u2126, it just gives a lower and upper bounds for the best constant corresponding to a given Bessel potential W . It is indeed clear that CBR(W ) \u2264 C\u2126(W ) \u2264 CB\u03c1(W ), where BR is the smallest ball containing \u2126 and B\u03c1 is the largest ball contained in it. If now W is a Bessel potential such that \u03b2(W,R) is independent of R, then clearly \u03b2(W,R) is also the best constant in inequality (Ha,cW ) for any smooth bounded domain. This is clearly the case for the potentials Wk,\u03c1 and W\u0303k,\u03c1 where \u03b2(W,R) = 14 for all R, while for W \u2261 1 the best constant is still not known for general domains even for the simplest case a = 0. Using the integral criteria for Bessel potentials, we can also deduce immediately the following. Let \u2126 be a bounded smooth domain in Rn with n \u2265 1, and let W be a non- negative function in C1(0, R =: supx\u2208\u2202\u2126 |x|] and a \u2264 n\u221222 . 1. If lim inf r\u21920 ln(r) \u222b r 0 sW (s)ds > \u2212\u221e, then there exists \u03b1 := \u03b1(\u2126) > 0 such that an improved Hardy inequality (Ha,W\u03b1) holds for the scaled potential W\u03b1(x) := \u03b12W (\u03b1|x|). 2. If lim r\u21920 ln(r) \u222b r 0 sW (s)ds = \u2212\u221e, then there are no \u03b1, c > 0, for which (Ha,W\u03b1,c) holds with W\u03b1,c = cW (\u03b1|x|). 52 3.2. General Hardy Inequalities By applying the above to various examples of Bessel potentials, we can now deduce several old and new inequalities. The first is an extension of a result estab- lished by Brezis and Va\u0301zquez [10] in the case where a = 0, and b = 0. Let \u2126 be a bounded smooth domain in Rn with n \u2265 1 and a \u2264 n\u221222 . Then, for any b < 2a+ 2 there exists c > 0 such that for all u \u2208 C\u221e0 (\u2126)\u222b \u2126 |x|\u22122a|\u2207u|2dx \u2265 (n\u22122a\u221222 )2 \u222b \u2126 |x|\u22122a\u22122u2dx+ c \u222b \u2126 |x|\u2212bu2dx. (3.61) Moreover, when \u2126 is a ball B of radius R the best constant c for which (3.61) holds is equal to the weight \u03b2(r2a\u2212b;R) of the Bessel potential W (r) = r2a\u2212b on (0, R]. In particular,\u222b B |x|\u22122a|\u2207u|2dx \u2265 (n\u22122a\u221222 )2 \u222b B |x|\u22122a\u22122u2dx+ \u03bbB \u222b B |x|\u22122au2dx, (3.62) where the best constant \u03bbB is equal to z0\u03c9 2\/n n |\u2126|\u22122\/n, where \u03c9n and |\u2126| denote the volume of the unit ball and \u2126 respectively, and z0 = 2.4048... is the first zero of the Bessel function J0(z). Proof: It suffices to apply Theorem 3.2.3 with the function W (r) = rb+2a which is a Bessel potential whenever b > \u22122a\u2212 2 since then lim inf r\u21920 ln(r) \u222b r 0 s2a+1W (s)ds > \u2212\u221e . In the case where b = \u22122a and thereforeW \u2261 1, we use the fact that \u03b2(1;R) = z20R2 (established in section 3.5) to deduce that the best constant is then equal to z0\u03c9 2\/n n |\u2126|\u22122\/n. \u0003 The following corollary is an extension of a recent result by Adimurthi et all [1] established in the case where a = 0, and of another result by Wang and Willem in [34] (Theorem 2) in the case k = 1. We also provide here the value of the best constant. Let B be a bounded smooth domain in Rn with n \u2265 1 and a \u2264 n\u221222 . Then for every integer k, and \u03c1 = (supx\u2208\u2126 |x|)(ee e. .e((k\u22121)\u2212times) ), we have for any u \u2208 H10 (\u2126),\u222b \u2126 |x|\u22122a|\u2207u|2dx \u2265 (n\u2212 2a\u2212 2 2 )2 \u222b \u2126 u2 |x|2a+2 dx + 1 4 k\u2211 j=1 \u222b \u2126 |u|2 |x|2a+2 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx. (3.63) Moreover, 14 is the best constant which is not attained in H 1 0 (\u2126). Proof: As seen in section 3.5, Wk,\u03c1(r) = \u2211k j=1 1 r2 (\u220fj i=1 log (i) \u03c1 |x| )\u22122 dx is a Bessel potential on (0, R) where R = supx\u2208\u2126 |x|, and \u03b2(Wk,\u03c1;R) = 14 . \u0003 53 3.2. General Hardy Inequalities The very same reasoning leads to the following extension of a result established by Filippas and Tertikas [17] in the case where a = 0. Let \u2126 be a bounded smooth domain in Rn with n \u2265 1 and a \u2264 n\u221222 . Then for every integer k, and any D \u2265 supx\u2208\u2126 |x|, we have for u \u2208 H10 (\u2126),\u222b \u2126 |\u2207u|2 |x|2a dx \u2265 ( n\u2212 2a\u2212 2 2 )2 \u222b \u2126 u2 |x|2a+2 dx + 1 4 \u221e\u2211 i=1 \u222b \u2126 1 |x|2a+2X 2 1 ( |x| D )X22 ( |x| D )...X2i ( |x| D )|u|2dx, (3.64) and 14 is the best constant which is not attained in H 1 0 (\u2126). The classical Hardy inequality is valid for dimensions n \u2265 3. We now present optimal Hardy type inequalities for dimension two in bounded domains, as well as the corresponding best constants. Let \u2126 be a smooth domain in R2 and 0 \u2208 \u2126. Then we have the following inequalities. \u2022 Let D \u2265 supx\u2208\u2126 |x|, then for all u \u2208 H10 (\u2126),\u222b \u2126 |\u2207u|2dx \u2265 14 \u2211\u221e i=1 \u222b \u2126 1 |x|2X 2 1 ( |x| D )X 2 2 ( |x| D )...X 2 i ( |x| D )|u|2dx (3.65) and 14 is the best constant. \u2022 Let \u03c1 = (supx\u2208\u2126 |x|)(ee e. .e((k\u22121)\u2212times) ), then for all u \u2208 H10 (\u2126)\u222b \u2126 |\u2207u|2dx \u2265 14 \u2211k j=1 \u222b \u2126 |u|2 |x|2 (\u220fj i=1 log (i) \u03c1 |x| )\u22122 dx, (3.66) and 14 is the best constant for all k \u2265 1. \u2022 If \u03b1 < 2, then there exists c > 0 such that for all u \u2208 H10 (\u2126),\u222b \u2126 |\u2207u|2dx \u2265 c \u222b \u2126 u2 |x|\u03b1 dx, (3.67) and the best constant is larger or equal to \u03b2(r\u03b1; sup x\u2208\u2126 |x|). An immediate application of Theorem 3.2 coupled with Ho\u0308lder\u2019s inequality gives the following duality statement, which should be compared to inequalities dual to those of Sobolev\u2019s, recently obtained via the theory of mass transport [3, 13]. Suppose that \u2126 is a smooth bounded domain containing 0 in Rn (n \u2265 1) with R := supx\u2208\u2126 |x|. Then, for any a \u2264 n\u221222 and 0 < p \u2264 2, we have the following dual inequalities: inf {\u222b \u2126 |x|\u22122a|\u2207u|2dx\u2212 (n\u2212 2a\u2212 2 2 )2 \u222b \u2126 |x|\u22122a\u22122|u|2dx; u \u2208 C\u221e0 (\u2126), ||u||p = 1 } \u2265 sup {(\u222b \u2126 ( |x|\u22122a W (x) ) p p\u22122 dx ) 2\u2212p p ; W \u2208 B(0, R) } . 54 3.3. General Hardy-Rellich inequalities 3.3 General Hardy-Rellich inequalities Let 0 \u2208 \u2126 \u2282 Rn be a smooth domain, and denote Ck0,r(\u2126) = {v \u2208 Ck0 (\u2126) : v is radial and supp v \u2282 \u2126}, Hm0,r(\u2126) = {u \u2208 Hm0 (\u2126) : u is radial}. We start by considering a general inequality for radial functions. Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in Rn (n \u2265 1) and centered at zero. Assume \u222b R 0 1 rn\u22121V (r)dr =\u221e and limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n \u2212 2. Then the following statements are equivalent: 1. (V,W ) is a Bessel pair on (0, R). 2. There exists c > 0 such that the following inequality holds for all radial functions u \u2208 C\u221e0,r(B) (HRV,cW ) \u222b B V (x)|\u2206u|2dx \u2265 c \u222b B W (x)|\u2207u|2dx + (n\u2212 1) \u222b B ( V (x) |x|2 \u2212 Vr(|x|) |x| )|\u2207u| 2dx. Moreover, the best constant is given by \u03b2(V,W ;R) = sup { c; (HRV,cW ) holds for radial functions } . (3.68) Proof: Assume u \u2208 C\u221e0,r(B) and observe that\u222b B V (x)|\u2206u|2dx = n\u03c9n{ \u222b R 0 V (r)u2rrr n\u22121dr + (n\u2212 1)2 \u222b R 0 V (r) u2r r2 rn\u22121dr + 2(n\u2212 1) \u222b R 0 V (r)uurrn\u22122dr}. Setting \u03bd = ur, we then have\u222b B V (x)|\u2206u|2dx = \u222b B V (x)|\u2207\u03bd|2dx+ (n\u2212 1) \u222b B ( V (|x|) |x|2 \u2212 Vr(|x|) |x| )|\u03bd| 2dx. Thus, (HRV,W ) for radial functions is equivalent to\u222b B V (x)|\u2207\u03bd|2dx \u2265 \u222b B W (x)\u03bd2dx. Letting x(r) = \u03bd(x) where |x| = r, we then have\u222b R 0 V (r)(x\u2032(r))2rn\u22121dr \u2265 \u222b R 0 W (r)x2(r)rn\u22121dr. (3.69) 55 3.3. General Hardy-Rellich inequalities It therefore follows from Theorem 3.2 that 1) and 2) are equivalent. \u0003 By applying the above theorem to the Bessel pair V (x) = |x|\u22122m and Wm(x) = V (x) [ (n\u22122m\u221222 ) 2|x|\u22122 +W (x)] where W is a Bessel potential, and by using Theorem 3.2.2, we get the following result in the case of radial functions. Suppose n \u2265 1 and m < n\u221222 . Let BR \u2282 Rn be a ball of radius R > 0 and centered at zero. Let W be a Bessel potential on (0, R). Then we have for all u \u2208 C\u221e0,r(BR)\u222b BR |\u2206u|2 |x|2m \u2265 ( n+ 2m 2 )2 \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx. (3.70) Moreover, (n+2m2 ) 2 and \u03b2(W ;R) are the best constants. 3.3.1 The non-radial case The decomposition of a function into its spherical harmonics will be one of our tools to prove the corresponding result in the non-radial case. This idea has also been used in [31]. Any function u \u2208 C\u221e0 (\u2126) could be extended by zero outside \u2126, and could therefore be considered as a function in C\u221e0 (R n). By decomposing u into spherical harmonics we get u = \u03a3\u221ek=0uk where uk = fk(|x|)\u03d5k(x) and (\u03d5k(x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(n+ k \u2212 2), k \u2265 0. The functions fk belong to C\u221e0 (\u2126) and satisfy fk(r) = O(r k) and f \u2032(r) = O(rk\u22121) as r \u2192 0. In particular, \u03d50 = 1 and f0 = 1n\u03c9nrn\u22121 \u222b \u2202Br uds = 1n\u03c9n \u222b |x|=1 u(rx)ds. (3.71) We also have for any k \u2265 0, and any continuous real valued functions v and w on (0,\u221e), \u222b Rn V (|x|)|\u2206uk|2dx = \u222b Rn V (|x|)(\u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2dx, (3.72) and\u222b Rn W (|x|)|\u2207uk|2dx = \u222b Rn W (|x|)|\u2207fk|2dx+ ck \u222b Rn W (|x|)|x|\u22122f2kdx. (3.73) Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in Rn (n \u2265 1) and centered at zero. Assume \u222b R 0 1 rn\u22121V (r)dr =\u221e and limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < (n\u2212 2). If W (r)\u2212 2V (r) r2 + 2Vr(r) r \u2212 Vrr(r) \u2265 0 for 0 \u2264 r \u2264 R, (3.74) then the following statements are equivalent. 56 3.3. General Hardy-Rellich inequalities 1. (V,W ) is a Bessel pair with \u03b2(V,W ;R) \u2265 1. 2. The following inequality holds for all u \u2208 C\u221e0 (B), (HRV,W ) \u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx + (n\u2212 1) \u222b B ( V (x) |x|2 \u2212 Vr(|x|) |x| )|\u2207u| 2dx. Moreover, if \u03b2(V,W ;R) \u2265 1, then the best constant is given by \u03b2(V,W ;R) = sup { c; (HRV,cW ) holds } . (3.75) Proof: That 2) implies 1) follows from Theorem 3.3 and does not require condition (3.74). To prove that 1) implies 2) assume that the equation (BV,W ) has a positive solution on (0, R]. We prove that the inequality (HRV,W ) holds for all u \u2208 C\u221e0 (B) by frequently using that \u222b R 0 V (r)|x\u2032(r)|2rn\u22121dr \u2265 \u222b R 0 W (r)x2(r)rn\u22121dr for all x \u2208 C1(0, R]. (3.76) Indeed, for all n \u2265 1 and k \u2265 0 we have 1 nwn \u222b Rn V (x)|\u2206uk|2dx = 1 nwn \u222b Rn V (x) ( \u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2 dx = \u222b R 0 V (r) ( f \u2032\u2032k (r) + n\u2212 1 r f \u2032k(r)\u2212 ck fk(r) r2 )2 rn\u22121dr = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1)2 \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + c2k \u222b R 0 V (r)f2k (r)r n\u22125 + 2(n\u2212 1) \u222b R 0 V (r)f \u2032\u2032k (r)f \u2032 k(r)r n\u22122 \u2212 2ck \u222b R 0 V (r)f \u2032\u2032k (r)fk(r)r n\u22123dr \u2212 2ck(n\u2212 1) \u222b R 0 V (r)f \u2032k(r)fk(r)r n\u22124dr. Integrate by parts and use (3.71) for k = 0 to get 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx (3.77) = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1 + 2ck) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + (2ck(n\u2212 4) + c2k) \u222b R 0 V (r)rn\u22125f2k (r)dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 5) \u222b R 0 Vr(r)f2k (r)r n\u22124dr \u2212 ck \u222b R 0 Vrr(r)f2k (r)r n\u22123dr. 57 3.3. General Hardy-Rellich inequalities Now define gk(r) = fk(r) r and note that gk(r) = O(r k\u22121) for all k \u2265 1. We have\u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 = \u222b R 0 V (r)(g\u2032k(r)) 2rn\u22121dr + \u222b R 0 2V (r)gk(r)g\u2032k(r)r n\u22122dr + \u222b R 0 V (r)g2k(r)r n\u22123dr = \u222b R 0 V (r)(g\u2032k(r)) 2rn\u22121dr \u2212 (n\u2212 3) \u222b R 0 V (r)g2k(r)r n\u22123dr \u2212 \u222b R 0 Vr(r)g2k(r)r n\u22122dr Thus, \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 \u2265 \u222b R 0 W (r)f2k (r)r n\u22123dr \u2212 (n\u2212 3) \u222b R 0 V (r)f2k (r)r n\u22125dr \u2212 \u222b R 0 Vr(r)f2k (r)r n\u22124dr. (3.78) Substituting 2ck \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 in (3.78) by its lower estimate in the last inequality (3.78), we get 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx \u2265 \u222b R 0 W (r)(f \u2032k(r)) 2rn\u22121dr + \u222b R 0 W (r)(fk(r))2rn\u22123dr + (n\u2212 1) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + ck(n\u2212 1) \u222b R 0 V (r)(fk(r))2rn\u22125dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 1) \u222b R 0 Vr(r)rn\u22124(fk)2(r)dr + ck(ck \u2212 (n\u2212 1)) \u222b R 0 V (r)rn\u22125f2k (r)dr + ck \u222b R 0 (W (r)\u2212 2V (r) r2 + 2Vr(r) r \u2212 Vrr(r))f2k (r)rn\u22123dr. The proof is now complete since the last term is non-negative by condition (3.74). Note also that because of this condition, the formula for the best constant requires that \u03b2(V,W ;R) \u2265 1, since if W satisfies (3.74) then cW satisfies it for any c \u2265 1. \u0003 Remark 3.3.1. In order to apply the above theorem to the Bessel pair V (x) = |x|\u22122m and Wm,c(x) = V (x) [ (n\u22122m\u221222 ) 2|x|\u22122 + cW (x)] where W is a Bessel potential, we see that even in the simplest case V \u2261 1 and Wm,c(x) = (n\u221222 ) 2|x|\u22122+W (x), condition (3.74) reduces to (n\u221222 )2|x|\u22122+W (x) \u2265 2|x|\u22122, which is then guaranteed only if n \u2265 5. 58 3.3. General Hardy-Rellich inequalities More generally, if V (x) = |x|\u22122m, then in order to satisfy (3.74) we need to have \u2212(n+ 4)\u2212 2\u221an2 \u2212 n+ 1 6 \u2264 m \u2264 \u2212(n+ 4) + 2 \u221a n2 \u2212 n+ 1 6 , (3.79) and in this case, we have for m < n\u221222 and any Bessel potential W on BR, that for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206u|2 |x|2m \u2265 ( n+ 2m 2 )2 \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx. (3.80) Moreover, (n+2m2 ) 2 and \u03b2(W ;R) are the best constant. Therefore, inequality (3.80) in the case wherem = 0 and n \u2265 5, already includes Theorem 1.5 in [31] as a special case. It also extends Theorem 1.8 in [31] where it is established under the condition 0 \u2264 m \u2264 \u2212(n+ 4) + 2 \u221a n2 \u2212 n+ 1 6 (3.81) which is more restrictive than (3.79). We shall see however that this inequality remains true without condition (3.79), but with a constant that is sometimes dif- ferent from (n+2m2 ) 2 in the cases where (3.79) is not valid. For example, if m = 0, then the best constant is 3 in dimension 4 and 2536 in dimension 3. We shall now give a few immediate applications of the above in the case where m = 0 and n \u2265 5. Actually the results are true in lower dimensions, and will be stated as such, but the proofs for n < 5 will require additional work and will be postponed to the next section. Assume W is a Bessel potential on BR \u2282 Rn with n \u2265 3, then for all u \u2208 C\u221e0 (BR) we have\u222b BR |\u2206u|2dx \u2265 C(n) \u222b BR |\u2207u|2 |x|2 dx+ \u03b2(W ;R) \u222b BR W (x)|\u2207u|2dx, (3.82) where C(3) = 2536 , C(4) = 3 and C(n) = n2 4 for all n \u2265 5. Moreover, C(n) and \u03b2(W ;R) are the best constants. In particular, the following holds for any smooth bounded domain \u2126 in Rn with R = supx\u2208\u2126 |x|, and any u \u2208 H20 (\u2126). \u2022 For any \u03b1 < 2,\u222b \u2126 |\u2206u|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx+ \u03b2(|x| \u03b1;R) \u222b \u2126 |\u2207u|2 |x|\u03b1 dx, (3.83) and for \u03b1 = 0,\u222b \u2126 |\u2206u|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx+ z20 R2 \u222b \u2126 |\u2207u|2dx, (3.84) the constants being optimal when \u2126 is a ball. 59 3.3. General Hardy-Rellich inequalities \u2022 For any k \u2265 1, and \u03c1 = R(eee. .e(k\u2212times) ), we have\u222b \u2126 |\u2206u(x)|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx + 1 4 k\u2211 j=1 \u222b \u2126 |\u2207u|2 |x|2 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx, (3.85) \u2022 For D \u2265 R, and Xi is defined as (3.113) we have\u222b \u2126 |\u2206u(x)|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx + 1 4 \u221e\u2211 i=1 \u222b \u2126 |\u2207u| |x|2 X 2 1 ( |x| D )X22 ( |x| D )...X2i ( |x| D )dx, (3.86) Moreover, all constants appearing in the above two inequality are optimal. Let W (x) = W (|x|) be radial Bessel potential on a ball B of radius R in Rn with n \u2265 4, and such that Wr(r)W (r) = \u03bbr + f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. If \u03bb < n\u2212 2, then the following Hardy-Rellich inequality holds:\u222b B |\u2206u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx + ( n2 4 + (n\u2212 \u03bb\u2212 2)2 4 )\u03b2(W ;R) \u222b B W (x) |x|2 u 2dx, (3.87) Proof: Use first Theorem 3.3.1 with the Bessel potential W , then Theorem 3.2.3 with the Bessel pair (|x|\u22122, |x|\u22122( (n\u22124)24 |x|\u22122 +W ), then Theorem 3.2.2 with the Bessel pair (W, (n\u2212\u03bb\u22122) 2 4 )|x|\u22122W ) to obtain\u222b B |\u2206u|2dx \u2265 C(n) \u222b B |\u2207u|2 |x|2 dx+ \u03b2(W,R) \u222b B W (x)|\u2207u|2dx \u2265 C(n) (n\u2212 4) 2 4 \u222b B u2 |x|4 dx + C(n)\u03b2(W,R) \u222b B W (x) |x|2 u 2 + \u03b2(W,R) \u222b W (x)|\u2207u|2dx \u2265 C(n) (n\u2212 4) 2 4 \u222b B u2 |x|4 dx+ (C(n) + (n\u2212 \u03bb\u2212 2)2 4 )\u03b2(W,R) \u222b B W (x) |x|2 u 2dx. Recall that C(n) = n 2 4 for n \u2265 5, giving the claimed result in these dimensions. This is however not the case when n = 4, and therefore another proof will be given in the next section to cover these cases. 60 3.3. General Hardy-Rellich inequalities The following is immediate from Theorem 3.3.1 and from the fact that \u03bb = 2 for the Bessel potential under consideration. Remark 3.3.2. After we wrote this paper, we learnt that Beckner [5] has also computed the values of the constants C(3), C(4), and C(n). Let \u2126 be a smooth bounded domain in Rn, n \u2265 4 and R = supx\u2208\u2126 |x|. Then the following holds for all u \u2208 H20 (\u2126) 1. If \u03c1 = R(ee e. .e(k\u2212times) ) and log(i)(.) is defined as (3.112), then\u222b \u2126 |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b \u2126 u2 |x|4 dx + (1 + n(n\u2212 4) 8 ) k\u2211 j=1 \u222b \u2126 u2 |x|4 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx. (3.88) 2. If D \u2265 R and Xi is defined as (3.113), then\u222b \u2126 |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b \u2126 u2 |x|4 dx + (1 + n(n\u2212 4) 8 ) \u221e\u2211 i=1 \u222b \u2126 u2 |x|4X 2 1 ( |x| D )X22 ( |x| D )...X2i ( |x| D )dx. (3.89) Let W1(x) and W2(x) be two radial Bessel potentials on a ball B of radius R in Rn with n \u2265 4. If a < 1, then there exists c(a,R) > 0 such that for all u \u2208 H20 (B)\u222b B |\u2206u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx+ n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx +c( n\u2212 2a\u2212 2 2 )2 \u222b B u2 |x|2a+2 dx+ c\u03b2(W2;R) \u222b B W2(x) u2 |x|2a dx, Proof: Here again we shall give a proof when n \u2265 5. The case n = 4 will be handled in the next section. We again first use Theorem 3.3.1 (for n \u2265 5) with the Bessel potential |x|\u22122a where a < 1, then Theorem 3.2.3 with the Bessel pair (|x|\u22122, |x|\u22122( (n\u22124)24 |x|\u22122+W )), then again Theorem 3.2.3 with the Bessel pair 61 3.3. General Hardy-Rellich inequalities (|x|\u22122a, |x|\u22122a((n\u22122a\u221222 )2|x|\u22122 +W ) to obtain\u222b B |\u2206u|2dx \u2265 n 2 4 \u222b B |\u2207u|2 |x|2 dx+ \u03b2(|x| \u22122a;R) \u222b B |\u2207u|2 |x|\u22122a dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx + n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx+ \u03b2(|x| \u22122a;R) \u222b B |\u2207u|2 |x|\u22122a dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx+ n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx + \u03b2(|x|\u22122a;R)(n\u2212 2a\u2212 2 2 )2 \u222b B u2 |x|2a+2 dx + \u03b2(|x|\u22122a;R)\u03b2(W2;R) \u222b B W2(x) u2 |x|2a dx. The following theorem will be established in full generality (i.e with V (r) = r\u2212m) in the next section. Let W (x) = W (|x|) be a radial Bessel potential on a smooth bounded domain \u2126 in Rn, n \u2265 4. Then,\u222b \u2126 |\u2206u(x)|2dx\u2212 n2(n\u22124)216 \u222b \u2126 u2 |x|4 dx\u2212 n 2 4 \u222b \u2126 W (x)u2dx \u2265 z20R2 ||u||2W 1,20 (\u2126), u \u2208 H20 (\u2126). 3.3.2 The case of power potentials |x|m The general Theorem 3.3.1 allowed us to deduce inequality (3.90) below for a re- stricted interval of powers m. We shall now prove that the same holds for all m < n\u221222 . The following theorem improves considerably Theorem 1.7, Theorem 1.8, and Theorem 6.4 in [31]. Suppose n \u2265 1 and m < n\u221222 , and let W be a Bessel potential on a ball BR \u2282 Rn of radius R. Then for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx, (3.90) where an,m = inf \uf8f1\uf8f2\uf8f3 \u222b BR |\u2206u|2 |x|2m dx\u222b BR |\u2207u|2 |x|2m+2 dx ; u \u2208 C\u221e0 (BR) \\ {0} \uf8fc\uf8fd\uf8fe . Moreover, \u03b2(W ;R) and am,n are the best constants to be computed in section 3.6. Proof: Assuming the inequality\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx, 62 3.3. General Hardy-Rellich inequalities holds for all u \u2208 C\u221e0 (BR), we shall prove that it can be improved by any Bessel potential W . We will use the following inequality frequently in the proof which follows directly from Theorem 3.2.3 with n=1.\u222b R 0 r\u03b1(f \u2032(r))2dr \u2265 (\u03b1\u2212 1 2 )2 \u222b R 0 r\u03b1\u22122f2(r)dr + \u03b2(W ;R) \u222b R 0 r\u03b1W (r)f2(r)dr, \u03b1 \u2265 1, (3.91) for all f \u2208 C\u221e(0, R), where both (\u03b1\u221212 )2 and \u03b2(W ;R) are best constants. Decompose u \u2208 C\u221e0 (BR) into its spherical harmonics \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x). We evaluate Ik = 1nwn \u222b Rn |\u2206uk|2 |x|2m dx in the following way Ik = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 \u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + [( n+ 2m 2 )2 + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 \u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +\u03b2(W )[( n+ 2m 2 )2 + 2ck \u2212 an,m] \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr + ( ( n\u2212 2m\u2212 4 2 )2[( n+ 2m 2 )2 + 2ck \u2212 an,m] + ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] ) \u222b R 0 rn\u22122m\u22125(fk(r))2dr. Now by (3.122) we have( ( n\u2212 2m\u2212 4 2 )2[( n+ 2m 2 )2+2ck\u2212an,m]+ ck[ck+(n\u2212 2m\u2212 4)(2m+2)] \u2265 ckan,m, 63 3.3. General Hardy-Rellich inequalities for all k \u2265 0. Hence, we have Ik \u2265 an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + an,mck \u222b R 0 rn\u22122m\u22125(fk(r))2dr + \u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + \u03b2(W )[( n+ 2m 2 )2 + 2ck \u2212 an,m] \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr \u2265 an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + an,mck \u222b R 0 rn\u22122m\u22125(fk(r))2dr +\u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + \u03b2(W )ck \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr = an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ) \u222b BR W (x) |\u2207u|2 |x|2m dx. Moreover, it is easy to see from Theorem 3.2 and the above calculation that \u03b2(W ;R) is the best constant. Let \u2126 be a smooth domain in Rn with n \u2265 1 and let V \u2208 C2(0, R =: supx\u2208\u2126 |x|) be a non-negative function that satisfies the following conditions: Vr(r) \u2264 0 and \u222b R 0 1 rn\u22123V (r)dr = \u2212 \u222b R 0 1 rn\u22124Vr(r) dr = +\u221e. (3.92) There exists \u03bb1, \u03bb2 \u2208 R such that rVr(r) V (r) + \u03bb1 \u2265 0 on (0, R) and limr\u21920 rVr(r) V (r) + \u03bb1 = 0, (3.93) rVrr(r) Vr(r) + \u03bb2 \u2265 0 on (0, R) and lim r\u21920 rVrr(r) Vr(r) + \u03bb2 = 0, (3.94) and ( 1 2 (n\u2212 \u03bb1 \u2212 2)2 + 3(n\u2212 3) ) V (r)\u2212 (n\u2212 5)rVr(r)\u2212 r2Vrr(r) \u2265 0, (3.95) for all r \u2208 (0, R). Then the following inequality holds:\u222b \u2126 V (|x|)|\u2206u|2dx \u2265 ( (n\u2212 \u03bb1 \u2212 2) 2 4 + (n\u2212 1))(n\u2212 \u03bb1 \u2212 4) 2 4 \u222b \u2126 V (|x|) |x|4 u 2dx \u2212 (n\u2212 1)(n\u2212 \u03bb2 \u2212 2) 2 4 \u222b \u2126 Vr(|x|) |x|3 u 2dx. (3.96) 64 3.3. General Hardy-Rellich inequalities Proof: We have by Theorem 3.2.2 and condition (3.95), 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1 + 2ck) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + (2ck(n\u2212 4) + c2k) \u222b R 0 V (r)rn\u22125f2k (r)dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 5) \u222b R 0 Vr(r)f2k (r)r n\u22124dr \u2212 ck \u222b R 0 Vrr(r)f2k (r)r n\u22123dr \u2265 \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr + ck \u222b R 0 (( 1 2 (n\u2212 \u03bb1 \u2212 2)2 + 3(n\u2212 3) ) V (r)\u2212 (n\u2212 5)rVr(r)\u2212 r2Vrr(r) ) f2k (r)r n\u22125 The rest of the proof follows from the above inequality combined with Theorem 3.2.2. \u0003 Remark 3.3.3. Let V (r) = r\u22122m withm \u2264 n\u221242 . Then in order to satisfy condition (3.95) we must have \u22121 \u2212 \u221a 1+(n\u22121)2 2 \u2264 m \u2264 n\u221242 . Under this assumption the inequality (3.96) gives the following weighted second order Rellich inequality:\u222b B |\u2206u|2 |x|2m dx \u2265 ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2 \u222b B u2 |x|2m+4 dx. In the following theorem we will show that the constant appearing in the above inequality is optimal. Moreover, we will see that if m < \u22121\u2212 \u221a 1+(n\u22121)2 2 , then the best constant is strictly less than ( (n+2m)(n\u22124\u22122m)4 ) 2. This shows that inequality (3.96) is actually sharp. Let m \u2264 n\u221242 and define \u03b2n,m = inf u\u2208C\u221e0 (B)\\{0} \u222b B |\u2206u|2 |x|2m dx\u222b B u2 |x|2m+4 dx . (3.97) Then \u03b2n,m = ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2 + min k=0,1,2,... {k(n+ k \u2212 2)[k(n+ k \u2212 2) + (n+ 2m)(n\u2212 2m\u2212 4) 2 ]}. Consequently the values of \u03b2n,m are as follows. 65 3.3. General Hardy-Rellich inequalities 1. If \u22121\u2212 \u221a 1+(n\u22121)2 2 \u2264 m \u2264 n\u221242 , then \u03b2n,m = ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2. 2. If n2 \u2212 3 \u2264 m \u2264 \u22121\u2212 \u221a 1+(n\u22121)2 2 , then \u03b2n,m = ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2+(n\u22121)[(n\u22121)+ (n+ 2m)(n\u2212 2m\u2212 4) 2 ]. 3. If k := n\u22122m\u221242 \u2208 N , then \u03b2n,m = ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2+k(n+k\u22122)[k(n+k\u22122)+(n+ 2m)(n\u2212 2m\u2212 4) 2 ]. 4. If k < n\u22122m\u221242 < k + 1 for some k \u2208 N , then \u03b2n,m = (n+ 2m)2(n\u2212 2m\u2212 4)2 16 + a(m,n, k) where a(m,n, k) = min{k(n+ k \u2212 2)[k(n+ k \u2212 2) + (n+ 2m)(n\u2212 2m\u2212 4) 2 ], (k + 1)(n+ k \u2212 1)[(k + 1)(n+ k \u2212 1) + (n+ 2m)(n\u2212 2m\u2212 4) 2 ]}. Proof: Decompose u \u2208 C\u221e0 (BR) into spherical harmonics \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x). we have 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 (( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2 + ck[ck + (n+ 2m)(n\u2212 2m\u2212 4) 2 ] ) \u222b R 0 rn\u22122m\u22125(fk(r))2dr, by Hardy inequality. Hence, \u03b2n,m \u2265 B(n,m, k) := ((n+ 2m)(n\u2212 4\u2212 2m)4 ) 2 + min k=0,1,2,... {k(n+ k \u2212 2)[k(n+ k \u2212 2) + (n+ 2m)(n\u2212 2m\u2212 4) 2 ]}. 66 3.3. General Hardy-Rellich inequalities To prove that \u03b2n,m is the best constant, let k be such that \u03b2n,m = (n+ 2m)(n\u2212 4\u2212 2m) 4 )2 + k(n+ k \u2212 2)[k(n+ k \u2212 2) + (n+ 2m)(n\u2212 2m\u2212 4) 2 ]. (3.98) Set u = |x|\u2212n\u221242 +m+\u000f\u03d5k(x)\u03d5(|x|), where \u03d5k(x) is an eigenfunction corresponding to the eigenvalue ck and \u03d5(r) is a smooth cutoff function, such that 0 \u2264 \u03d5 \u2264 1, with \u03d5 \u2261 1 in [0, 12 ]. We have\u222b BR |\u2206u|2 |x|2m dx\u222b BR u2 |x|2m+4 dx = (\u2212 (n+ 2m)(n\u2212 4\u2212 2m) 4 \u2212 ck + \u000f(2 + 2m+ \u000f))2 +O(1). Let now \u000f\u2192 0 to obtain the result. Thus the inequality\u222b BR |\u2206u|2 |x|2m \u2265 \u03b2n,m \u222b BR u2 |x|2m+4 dx, holds for all u \u2208 C\u221e0 (BR). To calculate explicit values of \u03b2n,m we need to find the minimum point of the function f(x) = x(x+ (n+ 2m)(n\u2212 2m\u2212 4) 2 ), x \u2265 0. Observe that f \u2032(\u2212 (n+ 2m)(n\u2212 2m\u2212 4) 4 ) = 0. To find minimizer k \u2208 N we should solve the equation k2 + (n\u2212 2)k + (n+ 2m)(n\u2212 2m\u2212 4) 4 = 0. The roots of the above equation are x1 = n+2m2 and x2 = n\u22122m\u22124 2 . 1) follows from Theorem 3.3.2. It is easy to see that if m \u2264 \u22121\u2212 \u221a 1+(n\u22121)2 2 , then x1 < 0. Hence, for m \u2264 \u22121 \u2212 \u221a 1+(n\u22121)2 2 the minimum of the function f is attained in x2. Note that if m \u2264 \u22121\u2212 \u221a 1+(n\u22121)2 2 , then B(n,m1) \u2264 B(n,m, 0). Therefore claims 2), 3), and 4) follow. \u0003 The following theorem extends Theorem 1.6 of [31] in many ways. First, we do not assume that n \u2265 5 or m \u2265 0, as was assumed there. Moreover, inequality (3.99) below includes inequalities (1.17) and (1.22) of [31] as special cases. 67 3.3. General Hardy-Rellich inequalities Let m \u2264 n\u221242 and let W (x) be a Bessel potential on a ball B of radius R in Rn with radius R. Assume W (r)Wr(r) = \u2212\u03bbr + f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. Then the following inequality holds for all u \u2208 C\u221e0 (B)\u222b B |\u2206u|2 |x|2m dx \u2265 \u03b2n,m \u222b B u2 |x|2m+4 dx (3.99) + \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b B W (x) |x|2m+2u 2dx. Proof: Again we will frequently use inequality (3.91) in the proof. Decomposing u \u2208 C\u221e0 (BR) into spherical harmonics \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x), we can write 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) +2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 (n+ 2m 2 )2 \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + \u03b2(W ;R) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr +ck[ck + 2( n\u2212 \u03bb\u2212 4 2 )2 + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr, 68 3.3. General Hardy-Rellich inequalities where we have used the fact that ck \u2265 0 to get the above inequality. We have 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx \u2265 \u03b2n,m \u222b R 0 rn\u22122m\u22125(fk)2dr +\u03b2(W ;R) (n+ 2m)2 4 \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr +\u03b2(W ;R) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr \u2265 \u03b2n,m \u222b R 0 rn\u22122m\u22125(fk)2dr +\u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr \u2265 \u03b2n,m n\u03c9n \u222b B u2k |x|2m+4 dx + \u03b2(W ;R) n\u03c9n ( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b B W (x) |x|2m+2u 2 kdx, by Theorem 3.2.2. Hence, (3.99) holds and the proof is complete. \u0003 Assume \u22121 < m \u2264 n\u221242 and letW (x) be a Bessel potential on a ball B of radius R and centered at zero in Rn (n \u2265 1). Then the following holds for all u \u2208 C\u221e0 (B):\u222b B |\u2206u|2 |x|2m dx \u2265 (n+ 2m)2(n\u2212 2m\u2212 4)2 16 \u222b B u2 |x|2m+4 dx (3.100) + \u03b2(W ;R) (n+ 2m)2 4 \u222b B W (x) |x|2m+2u 2dx+ \u03b2(|x|2m;R)||u||H10 . Proof: Decomposing again u \u2208 C\u221e0 (BR) into its spherical harmonics \u03a3\u221ek=0uk 69 3.4. Higher order Rellich inequalities where uk = fk(|x|)\u03d5k(x), we calculate 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 (n+ 2m 2 )2 \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + \u03b2(|x|2m;R) \u222b R 0 rn\u22121(f \u2032k) 2dr + ck \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr \u2265 (n+ 2m) 2(n\u2212 2m\u2212 4)2 16 \u222b R 0 rn\u22122m\u22125(fk)2dr +\u03b2(W ;R) (n+ 2m)2 4 \u222b R 0 W (r)rn\u22122m\u22123(fk)2dr + \u03b2(|x|2m;R) \u222b R 0 rn\u22121(f \u2032k) 2dr + ck\u03b2(|x|2m;R) \u222b R 0 rn\u22123(fk)2dr = (n+ 2m)2(n\u2212 2m\u2212 4)2 16n\u03c9n \u222b Rn u2k |x|2m+4 dx + \u03b2(W ;R) n\u03c9n ( (n+ 2m)2 4 ) \u222b Rn W (x) |x|2m+2u 2 kdx+ \u03b2(|x|2m;R)||uk||W 1,20 . Hence (3.100) holds. \u0003 We note that even for m = 0 and n \u2265 4, Theorem 3.3.2 improves considerably Theorem A.2. in [2]. 3.4 Higher order Rellich inequalities In this section we will repeat the results obtained in the previous section to derive higher order Rellich inequalities with corresponding improvements. Let W be a Bessel potential, \u03b2n,m be defined as in Theorem 3.3.2 and \u03c3n,m = \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ). For the sake of convenience we make the following convention: 0\u220f i=1 ai = 1. Let BR be a ball of radius R andW be a Bessel potential on BR such that W (r) Wr(r) = \u2212\u03bbr +f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. Assumem \u2208 N , 1 \u2264 l \u2264 m, and 2k+4m \u2264 n. 70 3.4. Higher order Rellich inequalities Then the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206mu|2 |x|2k dx \u2265 l\u22121\u220f i=0 \u03b2n,k+2i \u222b BR |\u2206m\u2212lu|2 |x|2k+4l dx (3.101) + l\u22121\u2211 i=0 \u03c3n,k+2i l\u22121\u220f j=1 \u03b2n,k+2j\u22122 \u222b BR W (x)|\u2206m\u2212i\u22121u|2 |x|2k+4i+2 dx Proof: Follows directly from theorem 3.3.2. \u0003 Let BR be a ball of radius R and W be a Bessel potential on BR such that W (r) Wr(r) = \u2212\u03bbr + f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. Assume m \u2208 N , 1 \u2264 l \u2264 m, and 2k + 4m + 2 \u2264 n. Then the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2207\u2206mu|2 |x|2k dx \u2265 ( n\u2212 2k \u2212 2 2 )2 l\u22121\u220f i=0 \u03b2n,k+2i+1 \u222b BR |\u2206m\u2212lu|2 |x|2k+4l+2 dx + ( n\u2212 2k \u2212 2 2 )2 l\u22121\u2211 i=0 \u03c3n,k+2i+1 l\u22121\u220f j=1 \u03b2n,k+2j\u22121 \u222b BR W (x)|\u2206m\u2212i\u22121u|2 |x|2k+4i+4 dx + \u03b2(W ;R) \u222b BR W (x) |\u2206mu|2 |x|2k dx (3.102) Proof: Follows directly from Theorem 3.2.3 and the previous theorem. \u0003 Remark 3.4.1. For k = 0 Theorems 3.4 and 3.4 include Theorem 1.9 in [31] as a special case. Let BR be a ball of radius R and W be a Bessel potential on BR such that W (r) Wr(r) = \u2212\u03bbr + f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. Assume m \u2208 N , 1 \u2264 l \u2264 m \u2212 1, and 2k + 4m \u2264 n. Then the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206mu|2 |x|2k dx \u2265 an,k( n\u2212 2k \u2212 4 2 )2 l\u22121\u220f i=0 \u03b2n,k+2i+2 \u222b BR |\u2206m\u2212l\u22121u|2 |x|2k+4l+4 dx + an,k( n\u2212 2k \u2212 4 2 )2 l\u22121\u2211 i=0 \u03c3n,k+2i+2 l\u22121\u220f j=1 \u03b2n,k+2j \u222b BR W (x)|\u2206m\u2212i\u22122u|2 |x|2k+4i+6 dx + \u03b2(W ;R)an,k \u222b BR W (x) |\u2206m\u22121u|2 |x|2k+2 dx + \u03b2(W ;R) \u222b BR W (x) |\u2207\u2206m\u22121u|2 |x|2k dx, (3.103) where an,m is defined in Theorem 3.3.2. Proof: Follows directly from Theorem 3.3.2 and the previous theorem. \u0003 71 3.5. The class of Bessel potentials The following improves Theorem 1.10 in [31] in many ways, since it is assumed there that l \u2264 \u2212n+8+2 \u221a n2\u2212n+1 12 and 4m < n. Even for k = 0, Theorem 3.4 below shows that we can drop the first condition and replace the second one by 4m \u2264 n. Let BR be a ball of radius R and W be a Bessel potential on BR such that . Assume m \u2208 N , 1 \u2264 l \u2264 m, and 2k + 4m \u2264 n. Then the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206mu|2 |x|2k dx (3.104) \u2265 l\u220f i=1 a n,k+2i\u22122(n\u2212 2k \u2212 4i)2 4 \u222b BR |\u2206m\u2212lu|2 |x|2k+4l dx (3.105) + \u03b2(W ;R) l\u2211 i=1 l\u22121\u220f j=1 a n,k+2j\u22122(n\u2212 2k \u2212 4j)2 4 \u222b BR W (x) |\u2207\u2206m\u2212iu|2 |x|2k+4i\u22124 dx + \u03b2(W ;R) l\u2211 i=1 a n,k+2i\u22122 l\u22121\u220f j=1 a n,k+2j\u22122(n\u2212 2k \u2212 4j)2 4 \u222b BR W (x) |\u2206m\u2212iu|2 |x|2k+4i\u22122 dx, where an,m are the best constants in inequality (3.90). Proof: Follows directly from Theorem 3.3.2. \u0003 3.5 The class of Bessel potentials The Bessel equation associated to a potential W (BW ) y\u2032\u2032 + 1ry \u2032 +W (r)y = 0 is central to all results revolving around the inequalities of Hardy and Hardy-Rellich type. We summarize in this section the various properties of these equations that were used throughout this chapter. We say that a non-negative real valued C1- function is a Bessel potential on (0, R) if there exists c > 0 such that the equation (BcW ) has a positive solution on (0, R). The class of Bessel potentials on (0, R) will be denoted by B(0, R). Note that the change of variable z(s) = y(e\u2212s) maps the equation y\u2032\u2032 + 1ry \u2032 +W (r)y = 0 into (B\u2032W ) z \u2032\u2032 + e\u22122sW (e\u2212s)z(s) = 0. (3.106) On the other hand, the change of variables \u03c8(t) = \u2212e \u2212ty\u2032(e\u2212t) y(e\u2212t) maps it into the nonlinear equation (B\u2032\u2032W ) \u03c8 \u2032(t) + \u03c82(t) + e\u22122tW (e\u2212t) = 0. (3.107) This will allow us to relate the existence of positive solutions of (BW ) to the non- oscillatory behaviour of equations (B\u2032W ) and (B \u2032\u2032 W ). 72 3.5. The class of Bessel potentials The theory of sub\/supersolutions \u2013applied to (B\u2032\u2032W ) (See Wintner [35, 36, 20])\u2013 already yields, that if (BW ) has a positive solution on an interval (0, R) for some non-negative potential W \u2265 0, then for any W such that 0 \u2264 V \u2264W , the equation (BV ) has also a positive solution on (0, R). This leads to the definition of the weight of a potential W \u2208 B(0, R) as: \u03b2(W ;R) = sup{c > 0; (BcW ) has a positive solution on (0, R)}. (3.108) The following is now straightforward. Proposition 3.109. 1) The class B(0, R) is a closed convex and solid subset of C1(0, R). 2) For every W \u2208 B(0, R), the equation (BW ) y\u2032\u2032 + 1ry \u2032 + \u03b2(W ;R)W (r)y = 0 has a positive solution on (0, R). The following gives an integral criteria for Bessel potentials. Proposition 3.110. Let W be a positive locally integrable function on . 1. If lim inf r\u21920 ln(r) \u222b r 0 sW (s)ds > \u2212\u221e, then for every R > 0, there exists \u03b1 := \u03b1(R) > 0 such that the scaled function W\u03b1(x) := \u03b12W (\u03b1x) is a Bessel potential on (0, R). 2. If lim r\u21920 ln(r) \u222b r 0 sW (s)ds = \u2212\u221e, then there are no \u03b1, c > 0, for which W\u03b1,c = cW (\u03b1|x|) is a Bessel potential on (0, R). Proof: This relies on well known results concerning the existence of non- oscillatory solutions (i.e., those z(s) such that z(s) > 0 for s > 0 sufficiently large) for the second order linear differential equations z\u2032\u2032(s) + a(s)z(s) = 0, (3.111) where a is a locally integrable function on . For these equations, the following integral criteria are available. We refer to [20, 21, 35, 36, 37]) among others for proofs and related results. i) If lim supt\u2192\u221e t \u222b\u221e t a(s)ds < 14 , then Eq. (3.111) is non-oscillatory. ii) If lim inft\u2192\u221e t \u222b\u221e t a(s)ds > 14 , then Eq. (3.111) is oscillatory. It follows that if lim inf r\u21920 ln(r) \u222b r 0 sW (s)ds > \u2212\u221e holds, then there exists \u03b4 > 0 such that (BW ) has a positive solution on (0, \u03b4). An easy scaling argument then shows that there exists \u03b1 > 0 such thatW\u03b1(x) := \u03b12W (\u03b1x) is a Bessel potential on (0, R). The rest of the proof is similar. \u0003 73 3.5. The class of Bessel potentials We now exhibit a few explicit Bessel potentials and compute their weights. We use the following notation. log(1)(.) = log(.) and log(k)(.) = log(log(k\u22121)(.)) for k \u2265 2. (3.112) and X1(t) = (1\u2212 log(t))\u22121, Xk(t) = X1(Xk\u22121(t)) k = 2, 3, ..., (3.113) Explicit Bessel potentials 1. W \u2261 0 is a Bessel potential on (0, R) for any R > 0. 2. The Bessel function J0 is a positive solution for equation (BW ) with W \u2261 1, on (0, z0), where z0 = 2.4048... is the first zero of J0. Moreover, z0 is larger than the first root of any other solution for (B1). In other words, for every R > 0, \u03b2(1;R) = z20 R2 . (3.114) 3. If a < 2, then there exists Ra > 0 such that W (r) = r\u2212a is a Bessel potential on (0, Ra). 4. For each k \u2265 1 and \u03c1 > R(eee. .e(k\u2212times) ), the equation (B 1 4Wk,\u03c1 ) correspond- ing to the potential Wk,\u03c1(r) = \u03a3kj=1Uj where Uj(r) = 1 r2 (\u220fj i=1 log (i) \u03c1 r )\u22122 has a positive solution on (0, R) that is explicitly given by \u03d5k,\u03c1(r) = ( \u220fk i=1 log (i) \u03c1 r ) 1 2 . On the other hand, the equation (B 1 4Wk,\u03c1+\u03bbUk ) corresponding to the potential 1 4Wk,\u03c1 + \u03bbUk has no positive solution for any \u03bb > 0. In other words, Wk,\u03c1 is a Bessel potential on (0, R) with \u03b2(Wk;\u03c1, R) = 14 for any k \u2265 1. (3.115) 5. For each k \u2265 1 and R > 0, the equation (B 1 4 W\u0303k,R ) corresponding to the potential W\u0303k,R(r) = \u03a3kj=1U\u0303j where U\u0303j(r) = 1 r2X 2 1 ( r R )X 2 2 ( r R ) . . . X 2 j\u22121( r R )X 2 j ( r R ) has a positive solution on (0, R) that is explicitly given by \u03d5k(r) = (X1( r R )X2( r R ) . . . Xk\u22121( r R )Xk( r R ))\u2212 1 2 . On the other hand, the equation (B 1 4 W\u0303k,R+\u03bbU\u0303k ) corresponding to the poten- tial 14W\u0303k,R + \u03bbU\u0303k has no positive solution for any \u03bb > 0. In other words, W\u0303k,R is a Bessel potential on (0, R) with \u03b2(W\u0303k,R;R) = 14 for any k \u2265 1. (3.116) 74 3.5. The class of Bessel potentials Proof: 1) It is clear that \u03c6(r) = \u2212log( eRr) is a positive solution of (B0) on (0, R) for any R > 0. 2)The best constant for which the equation y\u2032\u2032 + 1ry \u2032 + cy = 0 has a positive solution on (0, R) is z 2 0 R2 , where z0 = 2.4048... is the first zero of Bessel function J0(z). Indeed if \u03b1 is the first root of the an arbitrary solution of the Bessel equation y\u2032\u2032+ y \u2032 r +y(r) = 0, then we have \u03b1 \u2264 z0. To see this let x(t) = aJ0(t)+bY0(t), where J0 and Y0 are the two standard linearly independent solutions of Bessel equation, and a and b are constants. Assume the first zero of x(t) is larger than z0. Since the first zero of Y0 is smaller than z0, we have a \u2265 0. Also b \u2264 0, because Y0(t)\u2192 \u2212\u221e as t \u2192 0. Finally note that Y0(z0) > 0, so if b < 0, then x(z0 + \u000f) < 0 for \u000f sufficiently small. Therefore, b = 0 which is a contradiction. 3) follows directly from the integral criteria. 4) That \u03c6k is an explicit solution of the equation (B 1 4Wk ) is straightforward. Assume now that there exists a positive function \u03d5 such that \u2212\u03d5 \u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = 1 4 k\u22121\u2211 j=1 1 r ( j\u220f i=1 log(i) \u03c1 r )\u22122 + (1 4 + \u03bb) 1 r ( k\u220f i=1 log(i) \u03c1 r )\u22122 . Define f(r) = \u03d5(r)\u03d5k(r) > 0, and calculate, \u03d5\u2032(r) + r\u03d5\u2032\u2032(r) \u03d5(r) = \u03d5\u2032k(r) + r\u03d5 \u2032\u2032 k(r) \u03d5k(r) + f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) k\u2211 i=1 1\u220fi j=1 log j(\u03c1r ) . Thus, f \u2032(r) + rf \u2032\u2032(r) f(r) \u2212 f \u2032(r) f(r) k\u2211 i=1 1\u220fi j=1 log j(\u03c1r ) = \u2212\u03bb1 r ( k\u220f i=1 log(i) \u03c1 r )\u22122 . (3.117) If now f \u2032(\u03b1n) = 0 for some sequence {\u03b1n}\u221en=1 that converges to zero, then there exists a sequence {\u03b2n}\u221en=1 that also converges to zero, such that f \u2032\u2032(\u03b2n) = 0, and f \u2032(\u03b2n) > 0. But this contradicts (3.117), which means that f is eventually monotone for r small enough. We consider the two cases according to whether f is increasing or decreasing: Case I: Assume f \u2032(r) > 0 for r > 0 sufficiently small. Then we will have (rf \u2032(r))\u2032 rf \u2032(r) \u2264 k\u2211 i=1 1 r \u220fi j=1 log j(\u03c1r ) . Integrating once we get f \u2032(r) \u2265 c r \u220fk j=1 log j(\u03c1r ) , 75 3.6. The evaluation of an,m for some c > 0. Hence, limr\u21920 f(r) = \u2212\u221e which is a contradiction. Case II: Assume f \u2032(r) < 0 for r > 0 sufficiently small. Then (rf \u2032(r))\u2032 rf \u2032(r) \u2265 k\u2211 i=1 1 r \u220fi j=1 log j(\u03c1r ) . Thus, f \u2032(r) \u2265 \u2212 c r \u220fk j=1 log j(\u03c1r ) , (3.118) for some c > 0 and r > 0 sufficiently small. On the other hand f \u2032(r) + rf \u2032\u2032(r) f(r) \u2264 \u2212\u03bb k\u2211 j=1 1 r ( j\u220f i=1 log(i) R r )\u22122 \u2264 \u2212\u03bb( 1\u220fk j=1 log j(\u03c1r ) )\u2032. Since f \u2032(r) < 0, there exists l such that f(r) > l > 0 for r > 0 sufficiently small. From the above inequality we then have bf \u2032(b)\u2212 af \u2032(a) < \u2212\u03bbl( 1\u220fk j=1 log j(\u03c1b ) \u2212 1\u220fk j=1 log j( \u03c1a ) ). From (3.118) we have lima\u21920 af \u2032(a) = 0. Hence, bf \u2032(b) < \u2212 \u03bbl\u220fk j=1 log j(\u03c1b ) , for every b > 0, and f \u2032(r) < \u2212 \u03bbl r \u220fk j=1 log j(\u03c1r ) , for r > 0 sufficiently small. Therefore, lim r\u21920 f(r) = +\u221e, and by choosing l large enouph (e.g., l > c\u03bb ) we get to contradict (3.118). The proof of 5) is similar and is left to the interested reader. \u0003 3.6 The evaluation of an,m Here we evaluate the best constants an,m which appear in Theorem 3.3.2. Suppose n \u2265 1 and m \u2264 n\u221222 . Then for any R > 0, the constants an,m = inf \uf8f1\uf8f2\uf8f3 \u222b BR |\u2206u|2 |x|2m dx\u222b BR |\u2207u|2 |x|2m+2 dx ; u \u2208 C\u221e0 (BR) \\ {0} \uf8fc\uf8fd\uf8fe are given by the following expressions. 76 3.6. The evaluation of an,m 1. For n = 1 \u2022 if m \u2208 (\u2212\u221e,\u2212 32 ) \u222a [\u2212 76 ,\u2212 12 ], then a1,m = ( 1 + 2m 2 )2 \u2022 if \u2212 32 < m < \u2212 76 , then a1,m = min{(n+ 2m2 ) 2, ( (n\u22124\u22122m)(n+2m)4 + 2) 2 (n\u22124\u22122m2 ) 2 + 2 }. 2. If m = n\u221242 , then am,n = min{(n\u2212 2)2, n\u2212 1}. 3. If n \u2265 2 and m \u2264 \u2212(n+4)+2 \u221a n2\u2212n+1 6 , then an,m = ( n+2m 2 ) 2. 4. If 2 \u2264 n \u2264 3 and \u2212(n+4)+2 \u221a n2\u2212n+1 6 < m \u2264 n\u221222 , or n \u2265 4 and n\u221242 < m \u2264 n\u22122 2 , then an,m = ( (n\u22124\u22122m)(n+2m)4 + n\u2212 1)2 (n\u22124\u22122m2 ) 2 + n\u2212 1 . 5. For n \u2265 4 and \u2212(n+4)+2 \u221a n2\u2212n+1 6 < m < n\u22124 2 , define k \u2217 = [( \u221a 3 3 \u2212 12 )(n\u2212 2)]. \u2022 If k\u2217 \u2264 1, then an,m = ( (n\u22124\u22122m)(n+2m)4 + n\u2212 1)2 (n\u22124\u22122m2 ) 2 + n\u2212 1 . \u2022 For k\u2217 > 1 the interval (m10 := \u2212(n+4)+2 \u221a n2\u2212n+1 6 ,m 2 0 := n\u22124 2 ) can be divided in 2k\u2217 \u2212 1 subintervals. For 1 \u2264 k \u2264 k\u2217 define m1k := 2(n\u2212 5)\u2212\u221a(n\u2212 2)2 \u2212 12k(k + n\u2212 2) 6 , m2k := 2(n\u2212 5) +\u221a(n\u2212 2)2 \u2212 12k(k + n\u2212 2) 6 . If m \u2208 (m10,m11] \u222a [m21,m20)], then an,m = ( (n\u22124\u22122m)(n+2m)4 + n\u2212 1)2 (n\u22124\u22122m2 ) 2 + n\u2212 1 . 77 3.6. The evaluation of an,m \u2022 For k \u2265 1 and m \u2208 (m1k,m1k+1] \u222a [m2k+1,m2k), then an,m = min{ ( (n\u22124\u22122m)(n+2m)4 + k(n+ k \u2212 2))2 (n\u22124\u22122m2 ) 2 + k(n+ k \u2212 2) , ( (n\u22124\u22122m)(n+2m)4 + (k + 1)(n+ k \u2212 1))2 (n\u22124\u22122m2 ) 2 + (k + 1)(n+ k \u2212 1) }. For m \u2208 (m1k\u2217 ,m2k\u2217), then an,m = min{ ( (n\u22124\u22122m)(n+2m)4 + k \u2217(n+ k\u2217 \u2212 2))2 (n\u22124\u22122m2 ) 2 + k\u2217(n+ k\u2217 \u2212 2) , ( (n\u22124\u22122m)(n+2m)4 + (k \u2217 + 1)(n+ k\u2217 \u2212 1))2 (n\u22124\u22122m2 ) 2 + (k\u2217 + 1)(n+ k\u2217 \u2212 1) }. Proof: Letting V (r) = r\u22122m then, W (r)\u2212 2V (r) r2 + 2Vr(r) r \u2212Vrr(r) = ((n\u2212 2m\u2212 22 ) 2\u22122\u22124m\u22122m(2m+1))r\u22122m\u22122. In order to satisfy condition (3.74) we should have \u2212(n+ 4) + 2\u221an2 \u2212 n+ 1 6 \u2264 m \u2264 \u2212(n+ 4) + 2 \u221a n2 \u2212 n+ 1 6 . (3.119) So, by Theorem 3.3.1 under the above condition we have an,m = (n+2m2 ) 2 as in the radial case. For the rest of the proof we will use an argument similar to that of Theorem 6.4 in [31] who computed an,m in the case where n \u2265 5 and for certain intervals of m. Decomposing again u \u2208 C\u221e0 (BR) into spherical harmonics; u = \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x), one has\u222b Rn |\u2206uk|2 |x|2m dx = \u222b Rn |x|\u22122m(f \u2032\u2032k (|x|))2dx (3.120) + ((n\u2212 1)(2m+ 1) + 2ck) \u222b Rn |x|\u22122m\u22122(f \u2032k)2dx + ck(ck + (n\u2212 4\u2212 2m)(2m+ 2)) \u222b Rn |x|\u22122m\u22124(fk)2dx, and \u222b Rn |\u2207uk|2 |x|2m+2 dx = \u222b Rn |x|\u22122m\u22122(f \u2032k)2dx+ ck \u222b Rn |x|\u22122m\u22124(fk)2dx. (3.121) One can then prove as in [31] that an,m = min {A(k,m, n); k \u2208} (3.122) 78 3.6. The evaluation of an,m where A(k,m, n) = ( (n\u22124\u22122m)(n+2m) 4 +ck) 2 (n\u22124\u22122m2 ) 2+ck if m = n\u221242 (3.123) and A(k,m, n) := ck if m = n\u221242 and n+ k > 2. (3.124) Note that when m = n\u221242 and n+ k > 2, then ck 6= 0. Actually, this also holds for n+ k \u2264 2, in which case one deduces that if m = n\u221242 , then an,m = min{(n\u2212 2)2 = (n+ 2m2 ) 2, (n\u2212 1) = c1} which is statement 2). The rest of the proof consists of computing the infimum especially in the cases not considered in [31]. For that we consider the function f(x) = ( (n\u22124\u22122m)(n+2m)4 + x) 2 (n\u22124\u22122m2 ) 2 + x . It is easy to check that f \u2032(x) = 0 at x1 and x2, where x1 = \u2212 (n\u2212 4\u2212 2m)(n+ 2m)4 (3.125) x2 = (n\u2212 4\u2212 2m)(\u2212n+ 6m+ 8) 4 . (3.126) Observe that for for n \u2265 2, n\u221286 \u2264 n\u221242 . Hence, for m \u2264 n\u221286 both x1 and x2 are negative and hence an,m = (n+2m2 ) 2. Also note that \u2212(n+ 4)\u2212 2\u221an2 \u2212 n+ 1 6 \u2264 n\u2212 8 6 for all n \u2265 1. Hence, under the condition in 3) we have an,m = (n+2m2 ) 2. Also for n = 1 if m \u2264 \u221232 both critical points are negative and we have a1,m \u2264 ( 1+2m2 ) 2. Comparing A(0,m, n) and A(1,m, n) we see that A(1,m, n) \u2265 A(0,m, n) if and only if (3.119) holds. For n = 1 and \u2212 32 < m < \u2212 76 both x1 and x2 are positive. Consider the equations x(x\u2212 1) = x1 = (2m+ 3)(2m+ 1)4 , and x(x\u2212 1) = x2 = \u2212 (2m+ 3)(6m+ 7)4 . By simple calculations we can see that all four solutions of the above two equations are less that two. Since, A(1,m, 1) < A(0,m, 1) for m < \u2212 76 , we have a1,m \u2264 min{A(1,m, 1), A(2,m, 1)} and 1) follows. 79 3.6. The evaluation of an,m For n \u2265 2 and n\u221242 < m < n\u221222 we have x1 > 0 and x2 < 0. Consider the equation x(x+ n\u2212 2) = x1 = \u2212 (n\u2212 4\u2212 2m)(n+ 2m)4 . 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Va\u0301zquez, Domain of existence and blowup for the exponential reaction diffusion equation, Indiana Univ. Math. J. 48 (1999), 677-709. [33] J. L. Va\u0301zquez and E. Zuazua, The Hardy inequality and the asymptotic behav- iour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), 103-153. [34] Z-Q. Wang, M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203 (2003), 550-568. [35] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951) 368-380. [36] A. Wintner, On the comparision theorem of Knese-Hille, Math. Scand. 5 (1957) 255-260. [37] James S. W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969) 197-215. 83 Chapter 4 Optimal weighted Hardy-Rellich inequalities on H2 \u2229H10 3 4.1 Introduction Let \u2126 be a smooth bounded domain in n and 0 \u2208 \u2126. Let us recall that the classical Hardy-Rellich inequality assets that\u222b \u2126 |\u2206u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b \u2126 u2 |x|4 dx, for u \u2208 H 2 0 (\u2126), (4.1) where the constant appearing in the above inequality is the best constant and it is never achieved in H20 . Recently there has been a flurry of activity about possible improvements of the following type If n \u2265 5 then \u222b \u2126 |\u2206u|2dx\u2212 n2(n\u22124)216 \u222b \u2126 u2 |x|4 dx \u2265 \u222b \u2126 W (x)u2dx, (4.2) for u \u2208 H20 (\u2126) as well as If n \u2265 3 then \u222b \u2126 |\u2206u|2dx\u2212 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx \u2265 \u222b \u2126 V (x)|\u2207u|2dx, (4.3) for all u \u2208 H20 (\u2126), where V,W are certain explicit radially symmetric potentials of order lower than 1r2 (for V ) and 1 r4 (for W ) (see [2], [3], [8], [10], [11], [15], and [18]. The inequality (4.1) was first proved by Rellich [17] for u \u2208 H20 (\u2126) and then it was extended to functions in H2(\u2126) \u2229H10 (\u2126) by Donal et al. in [11]. So far most of the results about improved Hardy-Rellich inequalities and the inequalities of the form (4.3) are proved for u \u2208 H20 (\u2126) (see [8], [15], and [18]). The goal of this paper is to provide a general approach to prove optimal weighted Hardy-Rellich inequalities on H2(\u2126) \u2229 H10 (\u2126) and inequalities of type (4.3) on H2(\u2126) which are important in the study of fourth order elliptic equations with Navier boudary condition and systems of second order elliptic equations (see [16]). 3A version of this chapter has been submitted for publication; A. Moradifam, Optimal weighted Hardy-Rellich inequalities on H2 \u2229H10 (2009). 84 4.1. Introduction We start \u2013 in section 2 \u2013 by giving necessary and sufficient conditions on positive radial functions V andW on a ball B in Rn, so that the following inequality holds for some c > 0 and b < 0:\u222b B V (x)|\u2207u|2dx \u2265 c \u222b B W (x)u2dx+ b \u222b \u2202B u2 for all u \u2208 H1(B). (4.4) Assuming that the ball B has radius R and that \u222b R 0 1 rn\u22121V (r)dr = +\u221e, the condition is simply that the ordinary differential equation (BV,cW ) y\u2032\u2032(r) + (n\u22121r + Vr(r) V (r) )y \u2032(r) + cW (r)V (r) y(r) = 0 has a positive solution \u03c6 on the interval (0, R) with V (R)\u03c6 \u2032(R) \u03c6(R) = b. As in [15], we shall call such a couple (V,W ) a Bessel pair on (0, R). The weight of such a pair is then defined as \u03b2(V,W ;R) = sup { c; (BV,cW ) has a positive solution on (0, R) } . (4.5) We call W a Bessel potential if (1,W ) is a Bessel pair. This characterization makes an important connection between Hardy-type inequalities and the oscillatory be- havior of the above equations. For a detailed analysis of Bessel pairs see [15]. The above theorem in the general form of improved Hardy-type inequalities which re- cently has been of interest for many authors (see [1], [4], [5], [6], [7], [9], [12], [13], [19], and [20]). Here is the main result of this paper. Let V and W be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in n (n \u2265 1) such that \u222b R 0 1 rn\u22121V (r)dr = +\u221e and\u222b R 0 rn\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R) with \u03b8 := V (R)\u03c6 \u2032(R) \u03c6(R) , where \u03c6 is the corre- sponding solution of (B(V,W )). 2. \u222b B V (x)|\u2207u|2dx \u2265 \u222b B W (x)u2dx+ \u03b8 \u222b \u2202B u2ds for all u \u2208 C\u221e(B\u0304). 3. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n\u2212 2, then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B ( V (x) |x|2 \u2212 Vr(|x|)|x| )|\u2207u| 2dx+ (\u03b8 + (n\u2212 1)V (R)) \u222b \u2202B |\u2207u|2, for all radial u \u2208 C\u221e(B\u0304). 85 4.1. Introduction 4. If in addition, W (r)\u2212 2V (r)r2 + 2Vr(r)r \u2212 Vrr(r) \u2265 0 on (0, R), then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B ( V (x) |x|2 \u2212 Vr(|x|)|x| )|\u2207u| 2dx+ (\u03b8 + (n\u2212 1)V (R)) \u222b \u2202B |\u2207u|2, for all u \u2208 C\u221e(B\u0304). Appropriate combinations of 4) and 2) in the above theorem and lead to a myriad of Hardy-Rellich type inequalities on H2(\u2126) \u2229H10 (\u2126). Remark 4.1.1. The condition W (r) \u2212 2V (r)r2 + 2Vr(r)r \u2212 Vrr(r) \u2265 0 in the above theorem guarantees that the minimizing sequences are radial functions. We shall see in section 3 that even with out this condition our approach is applicable, although the minimizing sequences are no longer radial functions. Remark 4.1.2. To see the importance and generality of the above theorem, notice that inequalities (7) and (8) in [16] which are the author\u2019s main tools to prove sin- gularity of the extremal solutions in dimensions n \u2265 9 (see [16]) are an immediate consequence of the above theorem combined with (4.4). This theorem will also allow us to extend most of the results about Hardy and Hardy-Rellich type inequalities on C\u221e0 (\u2126) to corresponding inequalities on C \u221e(\u2126\u0304) such as those in [15] and [18]. We shall show that for \u2212n2 \u2264 m \u2264 n\u221222 Hn,m = inf u\u2208H2(B)\\{0} \u222b B |\u2206u|2 |x|2m\u222b B |\u2207u|2 |x|2m+2 = inf u\u2208H20 (B)\\{0} \u222b B |\u2206u|2 |x|2m\u222b B |\u2207u|2 |x|2m+2 , (4.6) and for \u2212n2 \u2264 m \u2264 n\u221242 an,m = inf u\u2208H2(B)\u2229H10 (B)\\{0} \u222b B |\u2206u|2 |x|2m\u222b B u2 |x|2m+4 = \u222b B |\u2206u|2 |x|2m\u222b B u2 |x|2m+4 , (4.7) where the constants Hn,m and an,m have been computed in [18] and then more generally in [15]. For example an,0 = n 2 4 for n \u2265 5, a4,0 = 3, and a3,0 = 2536 . The above general theorem also allows us to obtain improved Hardy-Rellich inequalities on H2(B) \u2229 H10 (B). For instance, assume W is a Bessel potential on (0, R) and \u03c6 is the corresponding solution of (B(1,W )) with R \u03c6\u2032(R) \u03c6(R) \u2265 \u2212n2 . If rWr(r)W (r) decreases to \u2212\u03bb and \u03bb \u2264 n\u2212 2, then we have for all H2(B) \u2229H10 (B)\u222b B |\u2206u|2dx\u2212 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx \u2265 (n2 4 + (n\u2212 \u03bb\u2212 2)2 4 ) \u03b2(W ;R) \u222b B W (x) |x|2 u 2dx. (4.8) 86 4.1. Introduction By applying (4.8) to the various examples of Bessel functions, we can vari- ous improved Hardy-Rellich inequalities on H2(B) \u2229H10 (B). Here are some basic examples of Bessel potentials, their corresponding solution \u03c6 of (B(1,W )). \u2022 W \u2261 0 is a Bessel potential on (0, R) for any R > 0 and \u03c6 = 1. \u2022 W \u2261 1 is a Bessel potential on (0, R) for any R > 0, \u03c6(r) = J0(\u00b5rR ), where J0 is the Bessel function and z0 = 2.4048... is the first zero of the Bessel function J0. Moreover R \u03c6\u2032(R) \u03c6(R) = \u2212n2 . \u2022 For k \u2265 1, R > 0, letWk,\u03c1(r) = \u03a3kj=1 1r2 (\u220fj i=1 log (i) \u03c1 r )\u22122 where the functions log(i) are defined iteratively as follows: log(1)(.) = log(.) and for k \u2265 2, log(k)(.) = log(log(k\u22121)(.)). Wk,\u03c1 is then a Bessel potential on (0, R) with the corresponding solution \u03c6k = ( j\u220f i=1 log(i) \u03c1 r )\u2212 12 . It is easy to see that for \u03c1 \u2265 R(eee. .e((k\u22121)\u2212times) ) large enough we have R \u03c6\u2032k(R) \u03c6k(R) \u2265 \u2212n2 . \u2022 For k \u2265 1, and R > 0, define W\u0303k;\u03c1(r) = \u03a3kj=1 1 r2 X21 ( r R )X22 ( r R ) . . . X2j\u22121( r R )X2j ( r R ) where the functions Xi are defined iteratively as follows: X1(t) = (1 \u2212 log(t))\u22121 and for k \u2265 2, Xk(t) = X1(Xk\u22121(t)). Then again W\u0303k,\u03c1 is a Bessel potential on (0, R) with \u03c6k = (X1( rR )X2( r R ) . . . Xj\u22121( r R )Xk( r R )) 1 2 . More- over, R\u03c6 \u2032 k(R) \u03c6k(R) = \u2212k2 . As an example, let k \u2265 1 and choose \u03c1 \u2265 R(eee. .e(k\u2212times) ) large enough so that R\u03c6 \u2032(R) \u03c6(R) \u2265 \u2212n2 , where \u03c6 = ( j\u220f i=1 log(i) \u03c1 |x| ) 1 2 . (4.9) Then we have\u222b B |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx (4.10) + (1 + n(n\u2212 4) 8 ) k\u2211 j=1 \u222b B u2 |x|4 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx, for all H2(B) \u2229H10 (B) which corresponds to the result od Adimurthi et al. [2]. 87 4.2. General Hardy Inequalities More generally, we show that for any \u2212n2 \u2264 m < n\u221222 , and any W Bessel potential on a ball BR \u2282 Rn of radius R, if for the corresponding solution \u03c6 of (B(1,W )) we have R \u03c6\u2032(R) \u03c6(R) \u2265 \u2212n2 \u2212 m, then the following inequality holds for all u \u2208 C\u221e0 (BR)\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx. (4.11) We also establish a more general version of equation (4.8). Assuming again that rW \u2032(r) W (r) decreases to \u2212\u03bb on (0, R), and provided m \u2264 n\u221242 and n2 +m \u2265 \u03bb \u2265 n\u2212 2m\u2212 4, we then have for all u \u2208 C\u221e0 (BR),\u222b BR |\u2206u|2 |x|2m dx \u2265 (n+ 2m)2(n\u2212 2m\u2212 4)2 16 \u222b BR u2 |x|2m+4 dx (4.12) + \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b BR W (x) |x|2m+2u 2dx. 4.2 General Hardy Inequalities Here is the main result of this section. Let V andW be positive radial C1-functions on BR\\{0}, where BR is a ball centered at zero with radius R (0 < R \u2264 +\u221e) in n (n \u2265 1). Assume that \u222b a 0 1 rn\u22121V (r)dr = +\u221e and \u222b a 0 rn\u22121V (r)dr < \u221e for some 0 < a < R. Then the following two statements are equivalent: 1. The ordinary differential equation (BV,W ) y\u2032\u2032(r) + (n\u22121r + Vr(r) V (r) )y \u2032(r) + W (r)V (r) y(r) = 0 has a positive solution on the interval (0, R] with \u03b8 := V (R)\u03c6 \u2032(R) \u03c6(R) . 2. For all u \u2208 H1(BR) (HV,W ) \u222b BR V (x)|\u2207u(x)|2dx \u2265 \u222b BR W (x)u2dx+\u03b8 \u222b \u2202B u2 ds. The above theorem allows to generalize all Hardy type inequalities on H10 (\u2126) to a corresponding inequality on H1(\u2126). For instance we can get the following general form of the Caffarelli-Kohn-Nirenberg inequalities. Assume B is the ball of radius R and and centered at zero in n. If a \u2264 n\u221222 , then\u222b B |x|\u22122a|\u2207u(x)|2dx \u2265 (n\u2212 2a\u2212 2 2 )2 \u222b B |x|\u22122a\u22122u2dx (4.13) \u2212 (n\u2212 2a\u2212 2)R \u22122a\u22121 2 \u222b \u2202B u2dx, for all u \u2208 H1(B). To prove Theorem 4.2 we shall need the following lemma. 88 4.2. General Hardy Inequalities Lemma 4.14. Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in n (n \u2265 1) and centered at zero. Assume\u222b B ( V (x)|\u2207u|2 \u2212W (x)|u|2) dx\u2212 \u03b8 \u222b \u2202B u2ds \u2265 0 for all u \u2208 H1(B), for some \u03b8 < 0. Then there exists a C2-supersolution to the following linear elliptic equation \u2212div(V (x)\u2207u)\u2212W (x)u = 0, in B, (4.15) u > 0 in B \\ {0}, (4.16) V\u2207u.\u03bd = \u03b8u in \u2202B. (4.17) Proof: Define \u03bb1(V ) := inf{ \u222b B V (x)|\u2207\u03c8|2 \u2212W (x)|\u03c8|2 \u2212 \u03b8 \u222b \u2202B u2\u222b B |\u03c8|2 ; \u03c8 \u2208 C \u221e 0 (B \\ {0})}. By our assumption \u03bb1(V ) \u2265 0. Let (\u03c6n, \u03bbn1 ) be the first eigenpair for the problem (L\u2212 \u03bb1(V )\u2212 \u03bbn1 )\u03c6n = 0 on B \\BR n \u03c6n = 0 on \u2202BR n V\u2207\u03c6n.\u03bd = \u03b8\u03c6n on \u2202B, where Lu = \u2212div(V (x)\u2207u) \u2212W (x)u, and BR n is a ball of radius Rn , n \u2265 2 . The eigenfunctions can be chosen in such a way that \u03c6n > 0 on B \\BR n and \u03d5n(b) = 1, for some b \u2208 B with R2 < |b| < R. Note that \u03bbn1 \u2193 0 as n \u2192 \u221e. Harnak\u2019s inequality yields that for any com- pact subset K, maxK\u03c6nminK\u03c6n \u2264 C(K) with the later constant being independent of \u03c6n. Also standard elliptic estimates also yields that the family (\u03c6n) have also uniformly bounded derivatives on the compact sets B \u2212BR n . Therefore, there exists a subsequence (\u03d5nl2 )l2 of (\u03d5n)n such that (\u03d5nl2 )l2 converges to some \u03d52 \u2208 C2(B \\ B(R2 )). Now consider (\u03d5nl2 )l2 on B \\ B(R3 ). Again there exists a subsequence (\u03d5nl3 )l3 of (\u03d5nl2 )l2 which converges to \u03d53 \u2208 C2(B \\ B(R3 )), and \u03d53(x) = \u03d52(x) for all x \u2208 B \\ B(R2 ). By repeating this argument we get a supersolution \u03d5 \u2208 C2(B \\ {0}) i.e. L\u03d5 \u2265 0, such that \u03d5 > 0 on B \\ {0} and V\u2207\u03c6.\u03bd = \u03b8\u03c6 on \u2202B. \u0003 Proof of Theorem 4.2: First we prove that 1) implies 2). Let \u03c6 \u2208 C1(0, R] be a solution of (BV,W ) such that \u03c6(x) > 0 for all x \u2208 (0, R). Define u(x)\u03d5(|x|) = \u03c8(x). Then |\u2207u|2 = (\u03d5\u2032(|x|))2\u03c82(x) + 2\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8 + \u03d5 2(|x|)|\u2207\u03c8|2. Hence, V (|x|)|\u2207u|2 \u2265 V (|x|)(\u03d5\u2032(|x|))2\u03c82(x) + 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8(x). 89 4.2. General Hardy Inequalities Thus, we have\u222b B V (|x|)|\u2207u|2dx \u2265 \u222b B V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx + \u222b B 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx. Let B\u000f be a ball of radius \u000f centered at the origin. Integrate by parts to get\u222b B V (|x|)|\u2207u|2dx \u2265 \u222b B V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx + \u222b B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx + \u222b B\\B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx = \u222b B\u000f V (|x|)(\u03d5\u2032(|x|))2\u03c82(x)dx+ \u222b B\u000f 2V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c8(x) x|x| .\u2207\u03c8dx \u2212 \u222b B\\B\u000f {( V (|x|)\u03d5\u2032\u2032(|x|)\u03d5(|x|) + ((n\u2212 1)V (|x|) r + Vr(|x|))\u03d5\u2032(|x|)\u03d5(|x|) ) \u03c82(x) } dx + \u222b \u2202(B\\B\u000f) V (|x|)\u03d5\u2032(|x|)\u03d5(|x|)\u03c82(x)ds Let \u000f\u2192 0 and use Lemma 2.3 in [15] and the fact that \u03c6 is a solution of (Dv,w) to get \u222b B V (|x|)|\u2207u|2dx \u2265 \u2212 \u222b B [V (|x|)\u03d5\u2032\u2032(|x|) + ((n\u2212 1)V (|x|) r + Vr(|x|))\u03d5\u2032(|x|)] u 2(x) \u03d5(|x|)dx = \u222b B W (|x|)u2(x)dx\u2212 \u03b8 \u222b \u2202B u2ds. To show that 2) implies 1), we assume that inequality (HV,W ) holds on a ball B of radius R, and then apply Lemma 4.14 to obtain a C2-supersolution for the equation (4.15). Now take the surface average of u, that is y(r) = 1 n\u03c9wrn\u22121 \u222b \u2202Br u(x)dS = 1 n\u03c9n \u222b |\u03c9|=1 u(r\u03c9)d\u03c9 > 0, (4.18) where \u03c9n denotes the volume of the unit ball in Rn. We may assume that the unit ball is contained in B (otherwise we just use a smaller ball). It is easy to see that V (R)y \u2032(R) y(R) = \u03b8. We clearly have y\u2032\u2032(r) + n\u2212 1 r y\u2032(r) = 1 n\u03c9nrn\u22121 \u222b \u2202Br \u2206u(x)dS. (4.19) 90 4.3. General Hardy-Rellich inequalities Since u(x) is a supersolution of (4.15), we have\u222b \u2202Br div(V (|x|)\u2207u)ds\u2212 \u222b \u2202B W (|x|)udx \u2265 0, and therefore, V (r) \u222b \u2202Br \u2206udS \u2212 Vr(r) \u222b \u2202Br \u2207u.xds\u2212W (r) \u222b \u2202Br u(x)ds \u2265 0. It follows that V (r) \u222b \u2202Br \u2206udS \u2212 Vr(r)y\u2032(r)\u2212W (r)y(r) \u2265 0, (4.20) and in view of (4.18), we see that y satisfies the inequality V (r)y\u2032\u2032(r) + ( (n\u2212 1)V (r) r + Vr(r))y\u2032(r) \u2264 \u2212W (r)y(r), for 0 < r < R, (4.21) that is it is a positive supersolution y for (BV,W ) with V (R) y\u2032(R) y(R) = \u03b8. Standard results in ODE now allow us to conclude that (BV,W ) has actually a positive solu- tion on (0, R), and the proof of theorem 4.2 is now complete. \u0003 An immediate application of Theorem 2.6 in [15] and Theorem 4.2 is the fol- lowing very general Hardy inequality. Let V (x) = V (|x|) be a strictly positive radial function on a smooth domain \u2126 containing 0 such that R = supx\u2208\u2126 |x|. Assume that for some \u03bb \u2208 rVr(r) V (r) + \u03bb \u2265 0 on (0, R) and limr\u21920 rVr(r) V (r) + \u03bb = 0. (4.22) If \u03bb \u2264 n\u2212 2, then the following inequality holds for any Bessel potential W on (0, R):\u222b \u2126 V (x)|\u2207u|2dx \u2265 (n\u2212 \u03bb\u2212 2 2 )2 \u222b \u2126 V (x) |x|2 u 2dx+ \u03b2(W ;R) \u222b \u2126 V (x)W (x)u2dx + V (R)( \u03c6\u2032(R) \u03c6(R) \u2212 n\u2212 \u03bb\u2212 2 2R ) \u222b \u2202B u2 for u \u2208 H1(\u2126), where \u03c6 is the corresponding solution of (B1,W ). Proof: Under our assumptions, it is easy to see that y = r n\u2212\u03bb\u22122 2 \u03c6(r) is a positive super-solution of B(V,V (n\u2212\u03bb\u221222 )2r\u22122+W ). Now apply Theorem 2.6 in [15] and Theorem 4.2 to complete the proof. \u0003 4.3 General Hardy-Rellich inequalities Let 0 \u2208 \u2126 \u2282 Rn be a smooth domain, and denote Ckr (\u2126\u0304) = {v \u2208 Ck(\u2126\u0304) : v is radial }. 91 4.3. General Hardy-Rellich inequalities We start by considering a general inequality for radial functions. Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in n (n \u2265 1) and centered at zero. Assume \u222b R 0 1 rn\u22121V (r)dr =\u221e and limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n \u2212 2. Then the following statements are equivalent: 1. (V,W ) is a Bessel pair on (0, R) with \u03b8 := V (R)\u03c6 \u2032(R) \u03c6(R) , where \u03c6 is the corre- sponding solution of (B(V,W )). 2. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < n\u2212 2, then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx + (n\u2212 1) \u222b B ( V (x) |x|2 \u2212 Vr(|x|) |x| )|\u2207u| 2dx + (\u03b8 + (n\u2212 1)V (R)) \u222b \u2202B |\u2207u|2, for all radial u \u2208 C\u221e(B\u0304). Proof: Assume u \u2208 C\u221er (B\u0304) and observe that\u222b B V (x)|\u2206u|2dx = n\u03c9n{ \u222b R 0 V (r)u2rrr n\u22121dr + (n\u2212 1)2 \u222b R 0 V (r) u2r r2 rn\u22121dr + 2(n\u2212 1) \u222b R 0 V (r)uurrn\u22122dr}. Setting \u03bd = ur, we then have\u222b B V (x)|\u2206u|2dx = \u222b B V (x)|\u2207\u03bd|2dx + (n\u2212 1) \u222b B ( V (|x|) |x|2 \u2212 Vr(|x|)|x| )|\u03bd| 2dx+ (n\u2212 1)V (R) \u222b \u2202B |\u03bd|2ds. Thus, (HRV,W ) for radial functions is equivalent to\u222b B V (x)|\u2207\u03bd|2dx \u2265 \u222b B W (x)\u03bd2dx. It therefore follows from Theorem 4.2 that 1) and 2) are equivalent. \u0003 92 4.3. General Hardy-Rellich inequalities 4.3.1 The non-radial case The decomposition of a function into its spherical harmonics will be one of our tools to prove our results. This idea has also been used in [18] and [15]. Let u \u2208 C\u221e(B\u0304). By decomposing u into spherical harmonics we get u = \u03a3\u221ek=0uk where uk = fk(|x|)\u03d5k(x) and (\u03d5k(x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(N + k \u2212 2), k \u2265 0. The functions fk belong to u \u2208 C\u221e([0, R]), fk(R) = 0, and satisfy fk(r) = O(rk) and f \u2032(r) = O(rk\u22121) as r \u2192 0. In particular, \u03d50 = 1 and f0 = 1n\u03c9nrn\u22121 \u222b \u2202Br uds = 1n\u03c9n \u222b |x|=1 u(rx)ds. (4.23) We also have for any k \u2265 0, and any continuous real valued functions v and w on (0,\u221e), \u222b Rn V (|x|)|\u2206uk|2dx = \u222b Rn V (|x|)(\u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2dx, (4.24) and\u222b Rn W (|x|)|\u2207uk|2dx = \u222b Rn W (|x|)|\u2207fk|2dx+ ck \u222b Rn W (|x|)|x|\u22122f2kdx. (4.25) Let V and W be positive radial C1-functions on a ball B\\{0}, where B is a ball with radius R in n (n \u2265 1) and centered at zero. Assume \u222b R 0 1 rn\u22121V (r)dr =\u221e and limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < (n\u2212 2). If W (r)\u2212 2V (r) r2 + 2Vr(r) r \u2212 Vrr(r) \u2265 0 for 0 \u2264 r \u2264 R, (4.26) and the ordinary differential equation (BV,W ) has a positive solution \u03c6 on the interval (0, R] such that (n\u2212 1 +R\u03c6 \u2032(R) \u03c6(R) )V (R) \u2265 0, (4.27) then the following inequality holds for all u \u2208 H2(B). (HRV,W ) \u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (n\u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2. Moreover, if \u03b2(V,W ;R) \u2265 1, then the best constant is given by \u03b2(V,W ;R) = sup { c; (HRV,cW ) holds } . (4.28) 93 4.3. General Hardy-Rellich inequalities Proof: Assume that the equation (BV,W ) has a positive solution on (0, R]. We prove that the inequality (HRV,W ) holds for all u \u2208 C\u221e0 (B) by frequently using that \u222b R 0 V (r)|x\u2032(r)|2rn\u22121dr \u2265 \u222b R 0 W (r)x2(r)rn\u22121dr + V (R) \u03c6\u2032(R) \u03c6(R) Rn\u22121(x(R))2, (4.29) for all x \u2208 C1(0, R]. Indeed, for all n \u2265 1 and k \u2265 0 we have 1 nwn \u222b Rn V (x)|\u2206uk|2dx = 1 nwn \u222b Rn V (x) ( \u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2 dx = \u222b R 0 V (r) ( f \u2032\u2032k (r) + n\u2212 1 r f \u2032k(r)\u2212 ck fk(r) r2 )2 rn\u22121dr = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1)2 \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr +c2k \u222b R 0 V (r)f2k (r)r n\u22125 + 2(n\u2212 1) \u222b R 0 V (r)f \u2032\u2032k (r)f \u2032 k(r)r n\u22122 \u22122ck \u222b R 0 V (r)f \u2032\u2032k (r)fk(r)r n\u22123dr \u2212 2ck(n\u2212 1) \u222b R 0 V (r)f \u2032k(r)fk(r)r n\u22124dr. Integrate by parts and use (4.23) for k = 0 to get 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1 + 2ck) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + (2ck(n\u2212 4) + c2k) \u222b R 0 V (r)rn\u22125f2k (r)dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 5) \u222b R 0 Vr(r)f2k (r)r n\u22124dr \u2212 ck \u222b R 0 Vrr(r)f2k (r)r n\u22123dr. + (n\u2212 1)V (R)(f \u2032k(R))2Rn\u22122 94 4.3. General Hardy-Rellich inequalities Now define gk(r) = fk(r) r and note that gk(r) = O(r k\u22121) for all k \u2265 1. We have\u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 = \u222b R 0 V (r)(g\u2032k(r)) 2rn\u22121dr + \u222b R 0 2V (r)gk(r)g\u2032k(r)r n\u22122dr + \u222b R 0 V (r)g2k(r)r n\u22123dr = \u222b R 0 V (r)(g\u2032k(r)) 2rn\u22121dr \u2212 (n\u2212 3) \u222b R 0 V (r)g2k(r)r n\u22123dr \u2212 \u222b R 0 Vr(r)g2k(r)r n\u22122dr Thus,\u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 \u2265 \u222b R 0 W (r)f2k (r)r n\u22123dr (4.30) \u2212 (n\u2212 3) \u222b R 0 V (r)f2k (r)r n\u22125dr \u2212 \u222b R 0 Vr(r)f2k (r)r n\u22124dr. Substituting 2ck \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123 in (4.30) by its lower estimate in the last inequality (4.30), we get 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx \u2265 \u222b R 0 W (r)(f \u2032k(r)) 2rn\u22121dr + \u222b R 0 W (r)(fk(r))2rn\u22123dr + (n\u2212 1) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + ck(n\u2212 1) \u222b R 0 V (r)(fk(r))2rn\u22125dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 1) \u222b R 0 Vr(r)rn\u22124(fk)2(r)dr + ck(ck \u2212 (n\u2212 1)) \u222b R 0 V (r)rn\u22125f2k (r)dr + ck \u222b R 0 (W (r)\u2212 2V (r) r2 + 2Vr(r) r \u2212 Vrr(r))f2k (r)rn\u22123dr + (n\u2212 1)V (R)(f \u2032k(R))2Rn\u22122 + V (R) \u03c6\u2032(R) \u03c6(R) Rn\u22121(f \u2032k(R)) 2 The proof is now complete since the last two terms are non-negative by our as- sumptions. \u0003 Remark 4.3.1. In order to apply the above theorem to V (x) = |x|\u22122m 95 4.3. General Hardy-Rellich inequalities we see that even in the simplest case V \u2261 1 condition (4.26) reduces to (n\u221222 )2|x|\u22122 \u2265 2|x|\u22122, which is then guaranteed only if n \u2265 5. More generally, if V (x) = |x|\u22122m, then in order to satisfy (4.26) we need to have \u2212(n+ 4)\u2212 2\u221an2 \u2212 n+ 1 6 \u2264 m \u2264 \u2212(n+ 4) + 2 \u221a n2 \u2212 n+ 1 6 . (4.31) Also to satisfy the condition (4.27) we need to have m > \u2212n2 . Thus for m satisfying (4.31) the inequality \u222b BR |\u2206u|2 |x|2m \u2265 ( n+ 2m 2 )2 \u222b BR |\u2207u|2 |x|2m+2 dx. (4.32) for all u \u2208 H2(BR). Moreover, (n+2m2 )2 is the best constant. We shall see however that this inequality remains true without condition (4.31), but with a constant that is sometimes different from (n+2m2 ) 2 in the cases where (4.31) is not valid. For example, if m = 0, then the best constant is 3 in dimension 4 and 2536 in dimension 3. 4.3.2 The case of power potentials |x|m The general Theorem 4.3.1 allowed us to deduce inequality (4.36) below for a re- stricted interval of powers m. We shall now prove that the same holds for all \u2212n2 \u2264 m < n\u221222 . We start with the following result. Assume \u2212n2 \u2264 m < n\u221222 and \u2126 be a smooth domain in n, n \u2265 1. Then an,m = inf \uf8f1\uf8f2\uf8f3 \u222b BR |\u2206u|2 |x|2m dx\u222b BR |\u2207u|2 |x|2m+2 dx ; H2(\u2126) \\ {0} \uf8fc\uf8fd\uf8fe = inf \uf8f1\uf8f2\uf8f3 \u222b BR |\u2206u|2 |x|2m dx\u222b BR |\u2207u|2 |x|2m+2 dx ; u \u2208 H20 (\u2126) \\ {0} \uf8fc\uf8fd\uf8fe Proof. Decomposing again u \u2208 C\u221e(B\u0304R) into spherical harmonics; u = \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x), one has\u222b n |\u2206uk|2 |x|2m dx = \u222b n |x|\u22122m(f \u2032\u2032k (|x|))2dx + ((n\u2212 1)(2m+ 1) + 2ck) \u222b n |x|\u22122m\u22122(f \u2032k)2dx + ck(ck + (n\u2212 4\u2212 2m)(2m+ 2)) \u222b n |x|\u22122m\u22124(fk)2dx + (n\u2212 1)Rn\u22122m\u22122(f \u2032k(R))2, (4.34) 96 4.3. General Hardy-Rellich inequalities and \u222b n |\u2207uk|2 |x|2m+2 dx = \u222b n |x|\u22122m\u22122(f \u2032k)2dx+ ck \u222b n |x|\u22122m\u22124(fk)2dx. (4.35) The rest of the proof follows from the inequality (4.13) and an argument similar to that of Theorem 6.1 in [15]. \u0003 Remark 4.3.2. The constant an,m has been computed explicitly in [15] (Theorem 6.1). Suppose n \u2265 1 and \u2212n2 \u2264 m < n\u221222 , and W is a Bessel potential on BR \u2282 Rn with n \u2265 3 and \u03c6 is the corresponding solution for the (B1,W ). If R \u03c6\u2032(R) \u03c6(R) \u2265 \u2212n 2 \u2212m, then for all u \u2208 H2(BR) we have\u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ;R) \u222b BR W (x) |\u2207u|2 |x|2m dx, (4.36) where an,m = inf \uf8f1\uf8f2\uf8f3 \u222b BR |\u2206u|2 |x|2m dx\u222b BR |\u2207u|2 |x|2m+2 dx ; u \u2208 H2(BR) \\ {0} \uf8fc\uf8fd\uf8fe . Moreover \u03b2(W ;R) and am,n are the best constants. Proof: Assuming the in- equality \u222b BR |\u2206u|2 |x|2m \u2265 an,m \u222b BR |\u2207u|2 |x|2m+2 dx, holds for all u \u2208 C\u221e(B\u0304R), we shall prove that it can be improved by any Bessel potentialW . We will use the following inequality in the proof which follows directly from the inequality (4.13) with n=1.\u222b R 0 r\u03b1(f \u2032(r))2dr \u2265 (\u03b1\u2212 1 2 )2 \u222b R 0 r\u03b1\u22122f2(r)dr (4.37) + \u03b2(W ;R) \u222b R 0 r\u03b1W (r)f2(r)dr + ( \u03c6\u2032(R) \u03c6(R) \u2212 \u03b1\u2212 1 2R )R\u03b1, for \u03b1 \u2265 1 and for all f \u2208 C\u221e(0, R], where both (\u03b1\u221212 )2 and \u03b2(W ;R) are best constants. Decompose u \u2208 C\u221e(B\u0304R) into its spherical harmonics \u03a3\u221ek=0uk, where 97 4.3. General Hardy-Rellich inequalities uk = fk(|x|)\u03d5k(x). We evaluate Ik = 1nwn \u222b Rn |\u2206uk|2 |x|2m dx in the following way Ik = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr + (n\u2212 1)Rn\u22122m\u22122(f \u2032k(R))2 \u2265 \u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + [( n+ 2m 2 )2 + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr \u2265 \u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr +\u03b2(W )[( n+ 2m 2 )2 + 2ck \u2212 an,m] \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr + ( ( n\u2212 2m\u2212 4 2 )2[( n+ 2m 2 )2 + 2ck \u2212 an,m] + ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] ) \u222b R 0 rn\u22122m\u22125(fk(r))2dr. Now by (115) in [15] we have( ( n\u2212 2m\u2212 4 2 )2[( n+ 2m 2 )2+2ck\u2212an,m]+ ck[ck+(n\u2212 2m\u2212 4)(2m+2)] \u2265 ckan,m, for all k \u2265 0. Hence, we have Ik \u2265 an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + an,mck \u222b R 0 rn\u22122m\u22125(fk(r))2dr +\u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr +\u03b2(W )[( n+ 2m 2 )2 + 2ck \u2212 an,m] \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr \u2265 an,m \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + an,mck \u222b R 0 rn\u22122m\u22125(fk(r))2dr +\u03b2(W ) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + \u03b2(W )ck \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr = an,m \u222b BR |\u2207u|2 |x|2m+2 dx+ \u03b2(W ) \u222b BR W (x) |\u2207u|2 |x|2m dx. \u0003 In the following theorem we prove a very general class of weighted Hardy-Rellich inequalities on H2(\u2126) \u2229H10 . 98 4.3. General Hardy-Rellich inequalities Let \u2126 be a smooth domain in Rn with n \u2265 1 and let V \u2208 C2(0, R =: supx\u2208\u2126 |x|) be a non-negative function that satisfies the following conditions: Vr(r) \u2264 0 and \u222b R 0 1 rn\u22123V (r)dr = \u2212 \u222b R 0 1 rn\u22124Vr(r) dr = +\u221e. (4.38) There exists \u03bb1, \u03bb2 \u2208 R such that rVr(r) V (r) + \u03bb1 \u2265 0 on (0, R) and limr\u21920 rVr(r) V (r) + \u03bb1 = 0, (4.39) rVrr(r) Vr(r) + \u03bb2 \u2265 0 on (0, R) and lim r\u21920 rVrr(r) Vr(r) + \u03bb2 = 0, (4.40) and( 1 2 (n\u2212 \u03bb1 \u2212 2)2 + 3(n\u2212 3) ) V (r)\u2212 (n\u2212 5)rVr(r)\u2212 r2Vrr(r) \u2265 0 for all r \u2208 (0, R). (4.41) If \u03bb1 \u2264 n, then the following inequality holds:\u222b \u2126 V (|x|)|\u2206u|2dx \u2265 ( (n\u2212 \u03bb1 \u2212 2) 2 4 + (n\u2212 1))(n\u2212 \u03bb1 \u2212 4) 2 4 \u222b \u2126 V (|x|) |x|4 u 2dx \u2212 (n\u2212 1)(n\u2212 \u03bb2 \u2212 2) 2 4 \u222b \u2126 Vr(|x|) |x|3 u 2dx. (4.42) Proof: We have by Theorem 4.2 and condition (4.41), 1 n\u03c9n \u222b Rn V (x)|\u2206uk|2dx = \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1 + 2ck) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr + (2ck(n\u2212 4) + c2k) \u222b R 0 V (r)rn\u22125f2k (r)dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr \u2212 ck(n\u2212 5) \u222b R 0 Vr(r)f2k (r)r n\u22124dr \u2212 ck \u222b R 0 Vrr(r)f2k (r)r n\u22123dr + (n\u2212 1)V (R)(f \u2032k(R))2Rn\u22122 \u2265 \u222b R 0 V (r)(f \u2032\u2032k (r)) 2rn\u22121dr + (n\u2212 1) \u222b R 0 V (r)(f \u2032k(r)) 2rn\u22123dr \u2212 (n\u2212 1) \u222b R 0 Vr(r)rn\u22122(f \u2032k) 2(r)dr + ck \u222b R 0 (( 1 2 (n\u2212 \u03bb1 \u2212 2)2 + 3(n\u2212 3) ) V (r)\u2212 (n\u2212 5)rVr(r)\u2212 r2Vrr(r) ) f2k (r)r n\u22125dr + (n\u2212 1)V (R)(f \u2032k(R))2Rn\u22122 The rest of the proof follows from the above inequality combined with Theorem 4.2. \u0003 99 4.3. General Hardy-Rellich inequalities Remark 4.3.3. Let V (r) = r\u22122m with \u2212n2 \u2264 m \u2264 n\u221242 . Then in order to satisfy condition (4.41) we must have\u22121\u2212 \u221a 1+(n\u22121)2 2 \u2264 m \u2264 n\u221242 . Since\u22121\u2212 \u221a 1+(n\u22121)2 2 \u2264\u2212n2 , if \u2212n2 \u2264 m \u2264 n\u221242 the inequality (4.42) gives the following weighted second order Rellich inequality:\u222b B |\u2206u|2 |x|2m dx \u2265 Hn,m \u222b B u2 |x|2m+4 dx u \u2208 H 2(\u2126) \u2229H10 (\u2126), where Hn,m := ( (n+ 2m)(n\u2212 4\u2212 2m) 4 )2. (4.43) The following theorem includes a large class of improved Hardy-Rellich inequal- ities as special cases. Let \u2212n2 \u2264 m \u2264 n\u221242 and letW (x) be a Bessel potential on a ball B of radius R in Rn with radius R. Assume W (r)Wr(r) = \u2212\u03bbr +f(r), where f(r) \u2265 0 and limr\u21920 rf(r) = 0. If \u03bb \u2264 n2 +m, then the following inequality holds for all u \u2208 H2 \u2229H10 (B)\u222b B |\u2206u|2 |x|2m dx \u2265 Hn,m \u222b B u2 |x|2m+4 dx (4.44) + \u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b B W (x) |x|2m+2u 2dx. Moreover, both constants are the best constants. Proof: Again we will frequently use inequality (4.37) in the proof. Decompos- ing u \u2208 C\u221e(B\u0304R) into spherical harmonics \u03a3\u221ek=0uk, where uk = fk(|x|)\u03d5k(x), we can write 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx = \u222b R 0 rn\u22122m\u22121(f \u2032\u2032k (r)) 2dr + [(n\u2212 1)(2m+ 1) + 2ck] \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + ck[ck + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr + (n\u2212 1)(f \u2032k(R))2Rn\u22122m\u22122 \u2265 (n+ 2m 2 )2 \u222b R 0 rn\u22122m\u22123(f \u2032k) 2dr + \u03b2(W ;R) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr + ck[ck + 2( n\u2212 \u03bb\u2212 4 2 )2 + (n\u2212 2m\u2212 4)(2m+ 2)] \u222b R 0 rn\u22122m\u22125(fk(r))2dr + (n\u2212 1)(f \u2032k(R))2Rn\u22122m\u22122, 100 4.3. General Hardy-Rellich inequalities where we have used the fact that ck \u2265 0 to get the above inequality. We have 1 n\u03c9n \u222b Rn |\u2206uk|2 |x|2m dx \u2265 \u03b2n,m \u222b R 0 rn\u22122m\u22125(fk)2dr +\u03b2(W ;R) (n+ 2m)2 4 \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr +\u03b2(W ;R) \u222b R 0 rn\u22122m\u22121W (x)(f \u2032k) 2dr \u2265 \u03b2n,m \u222b R 0 rn\u22122m\u22125(fk)2dr +\u03b2(W ;R)( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b R 0 rn\u22122m\u22123W (x)(fk)2dr \u2265 \u03b2n,m n\u03c9n \u222b B u2k |x|2m+4 dx + \u03b2(W ;R) n\u03c9n ( (n+ 2m)2 4 + (n\u2212 2m\u2212 \u03bb\u2212 2)2 4 ) \u222b B W (x) |x|2m+2u 2 kdx, by Theorem 4.2. Hence, (4.44) holds and the proof is complete. \u0003 We shall now give a few immediate applications of the above in the case where m = 0 and n \u2265 3. Assume W is a Bessel potential on BR \u2282 Rn with n \u2265 3 and \u03c6 is the corre- sponding solution for the (B1,W ). If R \u03c6\u2032(R) \u03c6(R) \u2265 \u2212n 2 , then for all u \u2208 H2(BR) we have\u222b BR |\u2206u|2dx \u2265 C(n) \u222b BR |\u2207u|2 |x|2 dx+ \u03b2(W ;R) \u222b BR W (x)|\u2207u|2dx, (4.45) where C(3) = 2536 , C(4) = 3 and C(n) = n2 4 for all n \u2265 5. Moreover, C(n) and \u03b2(W ;R) are best constants. The following holds for any smooth bounded domain \u2126 in Rn with R = supx\u2208\u2126 |x|, and any u \u2208 H2(\u2126). 1. Let z0 be the first zero of the Bessel function J0(z) and choose 0 < \u00b5 < z0 so that \u00b5 J \u20320(\u00b5) J0(\u00b5) = \u2212n 2 . (4.46) Then \u222b \u2126 |\u2206u|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx+ \u00b52 R2 \u222b \u2126 |\u2207u|2dx (4.47) 101 4.3. General Hardy-Rellich inequalities 2. For any k \u2265 1, choose \u03c1 \u2265 R(eee. .e(k\u2212times) ) large enough so that R\u03c6 \u2032(R) \u03c6(R) \u2265 \u2212n2 , where \u03c6 = ( j\u220f i=1 log(i) \u03c1 |x| ) 1 2 . (4.48) Then we have\u222b \u2126 |\u2206u(x)|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx (4.49) + 1 4 k\u2211 j=1 \u222b \u2126 |\u2207u|2 |x|2 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx, 3. We have\u222b \u2126 |\u2206u(x)|2dx \u2265 C(n) \u222b \u2126 |\u2207u|2 |x|2 dx (4.50) + 1 4 n\u2211 i=1 \u222b \u2126 |\u2207u| |x|2 X 2 1 ( |x| R )X22 ( |x| R )...X2i ( |x| R )dx. The following is immediate from Theorem 4.3.2 and from the fact that \u03bb = 2 for the Bessel potential under consideration. Let \u2126 be a smooth bounded domain in n, n \u2265 4 and R = supx\u2208\u2126 |x|. Then the following holds for all u \u2208 H2(\u2126) \u2229H10 (\u2126) 1. Choose \u03c1 \u2265 R(eee. .e(k\u2212times) ) so that R\u03c6 \u2032(R) \u03c6(R) \u2265 \u2212n2 . Then\u222b \u2126 |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b \u2126 u2 |x|4 dx (4.51) + (1 + n(n\u2212 4) 8 ) k\u2211 j=1 \u222b \u2126 u2 |x|4 ( j\u220f i=1 log(i) \u03c1 |x| )\u22122 dx. 2. Let Xi is defined as in the introduction, then\u222b \u2126 |\u2206u(x)|2dx \u2265 n 2(n\u2212 4)2 16 \u222b \u2126 u2 |x|4 dx (4.52) + (1 + n(n\u2212 4) 8 ) n\u2211 i=1 \u222b \u2126 u2 |x|4X 2 1 ( |x| R )X22 ( |x| R )...X2i ( |x| R )dx. Moreover, all constants in the above inequalities are best constants. 102 4.3. General Hardy-Rellich inequalities Let W1(x) and W2(x) be two radial Bessel potentials on a ball B of radius R in Rn with n \u2265 4. Then for all u \u2208 H2(B) \u2229H10 (B)\u222b B |\u2206u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx+ n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx +\u00b5( n\u2212 2 2 )2 \u222b B u2 |x|2 dx+ \u00b5\u03b2(W2;R) \u222b B W2(x)u2dx, where \u00b5 is defined by (4.46). Proof: Here again we shall give a proof when n \u2265 5. The case n = 4 will be handled in the next section. We again first use Theorem 4.3.2 (for n \u2265 5), then Theorem 2.15 in [15] with the Bessel pair (|x|\u22122, |x|\u22122( (n\u22124)24 |x|\u22122+W )), then again Theorem 4.2 with the Bessel pair (1, (n\u221222 )2|x|\u22122+ W ) to obtain\u222b B |\u2206u|2dx \u2265 n 2 4 \u222b B |\u2207u|2 |x|2 dx+ \u00b5 \u222b B |\u2207u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx+ n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx +\u00b5 \u222b B |\u2207u|2dxdx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx+ n2 4 \u03b2(W1;R) \u222b B W1(x) u2 |x|2 dx +\u00b5( n\u2212 2 2 )2 \u222b B u2 |x|2 dx+ \u00b5\u03b2(W2;R) \u222b B W2(x)u2dx. Assume n \u2265 4 and let W (x) be a Bessel potential on a ball B of radius R and centered at zero in Rn. Then the following holds for all u \u2208 H2(B) \u2229H10 (B):\u222b B |\u2206u|2dx \u2265 n 2(n\u2212 4)2 16 \u222b B u2 |x|4 dx (4.53) +\u03b2(W ;R) n2 4 \u222b B W (x) |x|2 u 2dx+ \u00b52 R2 ||u||H10 , (4.54) where \u00b5 2 R2 is defined by (4.46). Proof: Decomposing again u \u2208 C\u221e(B\u0304R) into its 103 4.3. General Hardy-Rellich inequalities spherical harmonics \u03a3\u221ek=0uk where uk = fk(|x|)\u03d5k(x), we calculate 1 n\u03c9n \u222b Rn |\u2206uk|2dx = \u222b R 0 rn\u22121(f \u2032\u2032k (r)) 2dr + [n\u2212 1 + 2ck] \u222b R 0 rn\u22123(f \u2032k) 2dr + ck[ck + n\u2212 4] \u222b R 0 rn\u22125(fk(r))2dr + (n\u2212 1)(f \u2032k(R))2Rn\u22122m\u22122 \u2265 n 2 4 \u222b R 0 rn\u22123(f \u2032k) 2dr + \u00b52 R2 \u222b R 0 rn\u22121(f \u2032k) 2dr + ck \u222b R 0 rn\u22123(f \u2032k) 2dr \u2265 n 2(n\u2212 4)2 16 \u222b R 0 rn\u22125(fk)2dr +\u03b2(W ;R) n2 4 \u222b R 0 W (r)rn\u22123(fk)2dr + \u00b52 R2 \u222b R 0 rn\u22121(f \u2032k) 2dr + ck \u00b52 R2 \u222b R 0 rn\u22123(fk)2dr = n2(n\u2212 4)2 16n\u03c9n \u222b Rn u2k |x|2m+4 dx + \u03b2(W ;R) n\u03c9n ( n2 4 ) \u222b Rn W (x) |x|2 u 2 kdx+ \u00b52 n\u03c9nR2 ||uk||W 1,20 . Hence (4.53) holds. \u0003 104 Bibliography [1] Adimurthi, N. Chaudhuri, and N. 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Wang, M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203 (2003), 550-568. 106 Part II Fourth Order Nonlinear Eigenvalue Problems 107 Chapter 5 The critical dimension for a fourth order elliptic problem with singular nonlinearity 4 5.1 Introduction The following model has been proposed for the description of the steady-state of a simple Electrostatic MEMS device:\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 \u03b1\u22062u = ( \u03b2 \u222b \u2126 |\u2207u|2dx+ \u03b3)\u2206u+ \u03bbf(x) (1\u2212u)2 \u0010 1+\u03c7 R \u2126 dx (1\u2212u)2 \u0011 in \u2126 0 < u < 1 in \u2126 u = \u03b1\u2202\u03bdu = 0 on \u2202\u2126, (5.1) where \u03b1, \u03b2, \u03b3, \u03c7 \u2265 0, f \u2208 C(\u2126, [0, 1]) are fixed, \u2126 is a bounded domain in RN and \u03bb \u2265 0 is a varying parameter (see for example Bernstein and Pelesko [19]). The function u(x) denotes the height above a point x \u2208 \u2126 \u2282 RN of a dielectric membrane clamped on \u2202\u2126, once it deflects torwards a ground plate fixed at height z = 1, whenever a positive voltage \u2013 proportional to \u03bb \u2013 is applied. In studying this problem, one typically makes various simplifying assumptions on the parameters \u03b1, \u03b2, \u03b3, \u03c7, and the first approximation of (5.1) that has been studied extensively so far is the equation\uf8f1\uf8f2\uf8f3 \u2212\u2206u = \u03bb f(x) (1\u2212u)2 in \u2126 0 < u < 1 in \u2126 (S)\u03bb,f u = 0 on \u2202\u2126, where we have set \u03b1 = \u03b2 = \u03c7 = 0 and \u03b3 = 1 (see for example [6], [8], [9], and the monograph [7]) . This simple model, which lends itself to the vast literature on second order semilinear eigenvalue problems, is already a rich source of interesting mathematical problems. The case when the \u201cpermittivity profile\u201d f is constant (f = 1) on a general domain was studied in [16], following the pioneering work of 4A version of this chapter has been accepted for publication. C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Archive, Arch. Ration. Mech. Anal., In press (2010). 108 5.1. Introduction Joseph and Lundgren [13] who had considered the radially symmetric case. The case for a non constant permittivity profile f was advocated by Pelesko [18], taken up by [11], and studied in depth in [6],[8], [9]. The starting point of the analysis is the existence of a pull-in voltage \u03bb\u2217(\u2126, f), defined as \u03bb\u2217(\u2126, f) := sup { \u03bb > 0 : there exists a classical solution of (S)\u03bb,f } . It is then shown that for every 0 < \u03bb < \u03bb\u2217, there exists a smooth minimal (smallest) solution of (S)\u03bb,f , while for \u03bb > \u03bb\u2217 there is no solution even in a weak sense. More- over, the branch \u03bb 7\u2192 u\u03bb(x) is increasing for each x \u2208 \u2126, and therefore the function u\u2217(x) := lim\u03bb\u2197\u03bb\u2217 u\u03bb(x) can be considered as a generalized solution that corre- sponds to the pull-in voltage \u03bb\u2217. Now the issue of the regularity of this extremal solution \u2013 which, by elliptic regularity theory, is equivalent to whether sup\u2126 u\u2217 < 1 \u2013 is an important question for many reasons, not the least of which being the fact that it decides whether the set of solutions stops there, or whether a new branch of solutions emanates from a bifurcation state (u\u2217, \u03bb\u2217). This issue turned out to depend closely on the dimension and on the permittivity profile f . Indeed, it was shown in [9] that u\u2217 is regular in dimensions 1 \u2264 N \u2264 7, while it is not necessarily the case for N \u2265 8. In other words, the dimension N = 7 is critical for equation (S)\u03bb (when f = 1, we simplify the notation (S)\u03bb,1 into (S)\u03bb). On the other hand, it is shown in [8] that the regularity of u\u2217 can be restored in any dimension, provided we allow for a power law profile |x|\u03b7 with \u03b7 large enough. The case where \u03b2 = \u03b3 = \u03c7 = 0 (and \u03b1 = 1) in the above model, that is when we are dealing with the following fourth order analog of (S)\u03bb\uf8f1\uf8f2\uf8f3 \u22062u = \u03bb(1\u2212u)2 in \u2126 0 < u < 1 in \u2126 (P )\u03bb u = \u2202\u03bdu = 0 on \u2202\u2126, was also considered by [4] and [14] but with limited success. One of the reasons is the lack of a \u201cmaximum principle\u201d which plays such a crucial role in developing the theory for the Laplacian. Indeed, it is a well known fact that such a principle does not normally hold for general domains \u2126 (at least for the clamped boundary conditions u = \u2202\u03bdu = 0 on \u2202\u2126) unless one restricts attention to the unit ball \u2126 = B in RN , where one can exploit a positivity preserving property of \u22062 due to T. Boggio [3]. This is precisely what was done in the references mentioned above, where a theory of the minimal branch associated with (P )\u03bb is developed along the same lines as for (S)\u03bb. The second obstacle is the well-known difficulty of extracting energy estimates for solutions of fourth order problems from their stability properties. This means that the methods used to analyze the regularity of the extremal solution for (S)\u03bb could not carry to the corresponding problem for (P )\u03bb. This is the question we address in this paper as we eventually show the following result. 109 5.1. Introduction Theorem 5.2. The unique extremal solution u\u2217 for (P )\u03bb\u2217 in B is regular in di- mension 1 \u2264 N \u2264 8, while it is singular (i.e, supB u\u2217 = 1) for N \u2265 9. In other words, the critical dimension for (P )\u03bb in B is N = 8, as opposed to being equal to 7 in (S)\u03bb. We add that our methods are heavily inspired by the recent paper of Davila et al. [5] where it is shown that N = 12 is the critical dimension for the fourth order nonlinear eigenvalue problem{ \u22062u = \u03bbeu in B u = \u2202\u03bdu = 0 on \u2202B, while the critical dimension for its second order counterpart (i.e., the Gelfand prob- lem) is N = 9. There is, however, one major difference between our approach and the one used by Davila et al. [5]. It is related to the most delicate dimensions \u2013 just above the critical one \u2013 where they use a computer assisted proof to establish the singularity of the extremal solution, while our method is more analytical and relies on improved and non standard Hardy-Rellich inequalities recently established by Ghoussoub-Moradifam [10] (See the lase section). Throughout this paper, we will always consider problem (P )\u03bb on the unit ball B. We start by recalling some of the results from [4] concerning (P )\u03bb, that will be needed in the sequel. We define \u03bb\u2217 := sup { \u03bb > 0 : there exists a classical solution of (P )\u03bb } , and note that we are not restricting our attention to radial solutions. We will deal also with weak solutions: Definition 5.3. We say that u is a weak solution of (P )\u03bb if 0 \u2264 u \u2264 1 a.e. in B, 1 (1\u2212u)2 \u2208 L1(B) and\u222b B u\u22062\u03c6 = \u03bb \u222b B \u03c6 (1\u2212 u)2 , \u2200\u03c6 \u2208 C 4(B\u0304) \u2229H20 (B). We say that u is a weak super-solution (resp. weak sub-solution) of (P )\u03bb if the equality is replaced with the inequality \u2265 (resp. \u2264) for all \u03c6 \u2208 C4(B\u0304)\u2229H20 (B) with \u03c6 \u2265 0. We also introduce notions of regularity and stability. Definition 5.4. Say that a weak solution u of (P )\u03bb is regular (resp. singular) if \u2016u\u2016\u221e < 1 (resp. =) and stable (resp. semi-stable) if \u00b51(u) = inf {\u222b B (\u2206\u03c6)2 \u2212 2\u03bb \u222b B \u03c62 (1\u2212 u)3 : \u03c6 \u2208 H 2 0 (B), \u2016\u03c6\u2016L2 = 1 } is positive (resp. non-negative). 110 5.1. Introduction The following extension of Boggio\u2019s principle will be frequently used in the sequel (see [2, Lemma 16] and [5, Lemma 2.4]): Lemma 5.5 (Boggio\u2019s Principle). Let u \u2208 L1(B). Then u \u2265 0 a.e. in B, provided one of the following conditions hold: 1. u \u2208 C4(B), \u22062u \u2265 0 on B, and u = \u2202u\u2202n = 0 on \u2202B. 2. \u222b B u\u22062\u03c6dx \u2265 0 for all 0 \u2264 \u03c6 \u2208 C4(B) \u2229H20 (B). 3. u \u2208 H2(B), u = 0 and \u2202u\u2202n \u2264 0 on \u2202B, and \u222b B \u2206u\u2206\u03c6 \u2265 0 for all 0 \u2264 \u03c6 \u2208 H20 (B). Moreover, either u \u2261 0 or u > 0 a.e. in B. The following theorem summarizes the main results in [4] that will be needed in the sequel: Theorem 5.6. The following assertions hold: 1. For each 0 < \u03bb < \u03bb\u2217 there exists a classical minimal solution u\u03bb of (P )\u03bb. Moreover u\u03bb is radial and radially decreasing. 2. For \u03bb > \u03bb\u2217, there are no weak solutions of (P )\u03bb. 3. For each x \u2208 B the map \u03bb 7\u2192 u\u03bb(x) is strictly increasing on (0, \u03bb\u2217). 4. The pull-in voltage \u03bb\u2217 satisfies the following bounds: max { 32(10N \u2212N2 \u2212 12) 27 , 8 9 (N \u2212 2 3 )(N \u2212 8 3 ) } \u2264 \u03bb\u2217 \u2264 4\u03bd1 27 where \u03bd1 denotes the first eigenvalue of \u22062 in H20 (B). 5. For each 0 < \u03bb < \u03bb\u2217, u\u03bb is a stable solution (i.e., \u00b51(u\u03bb) > 0). Using the stability of u\u03bb, it can be shown that u\u03bb is uniformly bounded in H20 (B) and that 11\u2212u\u03bb is uniformly bounded in L 3(B). Since now \u03bb 7\u2192 u\u03bb(x) is increasing, the function u\u2217(x) := lim\u03bb\u2197\u03bb\u2217 u\u03bb(x) is well defined (in the pointwise sense), u\u2217 \u2208 H20 (B), 1 1\u2212u\u2217 \u2208 L3(B) and u\u2217 is a weak solution of (P )\u03bb\u2217 . Moreover u\u2217 is the unique weak solution of (P )\u03bb\u2217 . The second result we list from [4] is critical in identifying the extremal solution. Theorem 5.7. If u \u2208 H20 (B) is a singular weak solution of (P )\u03bb, then u is semi- stable if and only if (u, \u03bb) = (u\u2217, \u03bb\u2217). 111 5.2. The effect of boundary conditions on the pull-in voltage 5.2 The effect of boundary conditions on the pull-in voltage As in [5], we are led to examine problem (P )\u03bb with non-homogeneous boundary conditions such as \uf8f1\uf8f2\uf8f3 \u22062u = \u03bb(1\u2212u)2 in B \u03b1 < u < 1 in B (P )\u03bb,\u03b1,\u03b2 u = \u03b1 , \u2202\u03bdu = \u03b2 on \u2202B, where \u03b1, \u03b2 are given. Notice first that some restrictions on \u03b1 and \u03b2 are necessary. Indeed, letting \u03a6(x) := (\u03b1\u2212 \u03b22 ) + \u03b22 |x|2 denote the unique solution of{ \u22062\u03a6 = 0 in B \u03a6 = \u03b1 , \u2202\u03bd\u03a6 = \u03b2 on \u2202B, (5.8) we infer immediately from Lemma 5.5 that the function u \u2212 \u03a6 is positive in B, which yields to sup B \u03a6 < sup B u \u2264 1. To insure that \u03a6 is a classical sub-solution of (P )\u03bb,\u03b1,\u03b2 , we impose \u03b1 6= 1 and \u03b2 \u2264 0, and condition sup B \u03a6 < 1 rewrites as \u03b1 \u2212 \u03b22 < 1. We will then say that the pair (\u03b1, \u03b2) is admissible if \u03b2 \u2264 0, and \u03b1\u2212 \u03b22 < 1. This section will be devoted to obtaining results for (P )\u03bb,\u03b1,\u03b2 when (\u03b1, \u03b2) is an admissible pair, which are analogous to those for (P )\u03bb. To cut down on notation, we shall sometimes drop \u03b1 and \u03b2 from our expressions whenever such an emphasis is not needed. For example in this section u\u03bb and u\u2217 will denote the minimal and extremal solution of (P )\u03bb,\u03b1,\u03b2 . We now introduce a notion of weak solution for (P )\u03bb,\u03b1,\u03b2 . Definition 5.9. We say that u is a weak solution of (P )\u03bb,\u03b1,\u03b2 if \u03b1 \u2264 u \u2264 1 a.e. in B, 1(1\u2212u)2 \u2208 L1(B) and if\u222b B (u\u2212 \u03a6)\u22062\u03c6 = \u03bb \u222b B \u03c6 (1\u2212 u)2 , \u2200\u03c6 \u2208 C 4(B\u0304) \u2229H20 (B), where \u03a6 is given in (5.8). We say that u is a weak super-solution (resp. weak sub-solution) of (P )\u03bb,\u03b1,\u03b2 if the equality is replaced with the inequality \u2265 (resp. \u2264) for \u03c6 \u2265 0. We now define as before \u03bb\u2217 := sup{\u03bb > 0 : (P )\u03bb,\u03b1,\u03b2 has a classical solution} 112 5.2. The effect of boundary conditions on the pull-in voltage and \u03bb\u2217 := sup{\u03bb > 0 : (P )\u03bb,\u03b1,\u03b2 has a weak solution}. Observe that by the Implicit Function Theorem, one can always solve (P )\u03bb,\u03b1,\u03b2 for small \u03bb\u2019s. Therefore, \u03bb\u2217 (and also \u03bb\u2217) is well defined. Let now U be a weak super-solution of (P )\u03bb,\u03b1,\u03b2 . Recall the following standard existence result. Theorem 5.10 ([2]). For every 0 \u2264 f \u2208 L1(B), there exists a unique 0 \u2264 u \u2208 L1(B) which satisfies \u222b B u\u22062\u03c6 = \u222b B f\u03c6 for all \u03c6 \u2208 C4(B\u0304) \u2229H20 (B). We can now introduce the following \u201cweak iterative scheme\u201d: Start with u0 = U and (inductively) let un, n \u2265 1, be the solution of\u222b B (un \u2212 \u03a6)\u22062\u03c6 = \u03bb \u222b B \u03c6 (1\u2212 un\u22121)2 \u2200 \u03c6 \u2208 C 4(B\u0304) \u2229H20 (B) given by Theorem 5.10. Since 0 is a sub-solution of (P )\u03bb,\u03b1,\u03b2 , one can easily show inductively by using Lemma 5.5 that \u03b1 \u2264 un+1 \u2264 un \u2264 U for every n \u2265 0. Since (1\u2212 un)\u22122 \u2264 (1\u2212 U)\u22122 \u2208 L1(B), we get by Lebesgue Theorem, that the function u = lim n\u2192+\u221eun is a weak solution of (P )\u03bb,\u03b1,\u03b2 such that \u03b1 \u2264 u \u2264 U . In other words, the following result holds. Proposition 5.2.1. Assume the existence of a weak super-solution U of (P )\u03bb,\u03b1,\u03b2. Then there exists a weak solution u of (P )\u03bb,\u03b1,\u03b2 so that \u03b1 \u2264 u \u2264 U a.e. in B. In particular, we can find a weak solution of (P )\u03bb,\u03b1,\u03b2 for every \u03bb \u2208 (0, \u03bb\u2217). Now we show that this is still true for regular weak solutions. Proposition 5.2.2. Let (\u03b1, \u03b2) be an admissible pair and let u be a weak solution of (P )\u03bb,\u03b1,\u03b2. Then for every 0 < \u00b5 < \u03bb, there is a regular solution for (P )\u00b5,\u03b1,\u03b2. Proof. Let \u03b5 \u2208 (0, 1) be given and let u\u0304 = (1\u2212 \u03b5)u+ \u03b5\u03a6, where \u03a6 is given in (5.8). We have that sup B u\u0304 \u2264 (1\u2212 \u03b5) + \u03b5 sup B \u03a6 < 1 , inf B u\u0304 \u2265 (1\u2212 \u03b5)\u03b1+ \u03b5 inf B \u03a6 = \u03b1, and for every 0 \u2264 \u03c6 \u2208 C4(B\u0304) \u2229H20 (B) there holds:\u222b B (u\u0304\u2212 \u03a6)\u22062\u03c6 = (1\u2212 \u03b5) \u222b B (u\u2212 \u03a6)\u22062\u03c6 = (1\u2212 \u03b5)\u03bb \u222b B \u03c6 (1\u2212 u)2 = (1\u2212 \u03b5)3\u03bb \u222b B \u03c6 (1\u2212 u\u0304+ \u03b5(\u03a6\u2212 1))2 \u2265 (1\u2212 \u03b5) 3\u03bb \u222b B \u03c6 (1\u2212 u\u0304)2 . 113 5.2. The effect of boundary conditions on the pull-in voltage Note that 0 \u2264 (1\u2212\u03b5)(1\u2212u) = 1\u2212 u\u0304+\u03b5(\u03a6\u22121) < 1\u2212 u\u0304. So u\u0304 is a weak super-solution of (P )(1\u2212\u03b5)3\u03bb,\u03b1,\u03b2 satisfying sup B u\u0304 < 1. From Proposition 5.2.1 we get the existence of a weak solution w of (P )(1\u2212\u03b5)3\u03bb,\u03b1,\u03b2 so that \u03b1 \u2264 w \u2264 u\u0304. In particular, sup B w < 1 and w is a regular weak solution. Since \u03b5 \u2208 (0, 1) is arbitrarily chosen, the proof is complete. Proposition 5.2.2 implies in particular the existence of a regular weak solution U\u03bb for every \u03bb \u2208 (0, \u03bb\u2217). Introduce now a \u201cclassical\u201d iterative scheme: u0 = 0 and (inductively) un = vn +\u03a6, n \u2265 1, where vn \u2208 H20 (B) is the (radial) solution of \u22062vn = \u22062(un \u2212 \u03a6) = \u03bb(1\u2212 un\u22121)2 in B. (5.11) Since vn \u2208 H20 (B), un is also a weak solution of (5.11), and by Lemma 5.5 we know that \u03b1 \u2264 un \u2264 un+1 \u2264 U\u03bb for every n \u2265 0. Since sup B un \u2264 sup B U\u03bb < 1 for n \u2265 0, we get that (1 \u2212 un\u22121)\u22122 \u2208 L2(B) and the existence of vn is guaranteed. Since vn is easily seen to be uniformly bounded in H20 (B), we have that u\u03bb := lim n\u2192+\u221eun does hold pointwise and weakly in H2(B). By Lebesgue Theorem, we have that u\u03bb is a radial weak solution of (P )\u03bb,\u03b1,\u03b2 so that sup B u\u03bb \u2264 sup B U\u03bb < 1. By elliptic regularity theory [1] u\u03bb \u2208 C\u221e(B\u0304) and u\u03bb \u2212 \u03a6 = \u2202\u03bd(u\u03bb \u2212 \u03a6) = 0 on \u2202B. So we can integrate by parts to get\u222b B \u22062u\u03bb\u03c6 = \u222b B \u22062(u\u03bb \u2212 \u03a6)\u03c6 = \u222b B (u\u03bb \u2212 \u03a6)\u22062\u03c6 = \u03bb \u222b B \u03c6 (1\u2212 u\u03bb)2 for every \u03c6 \u2208 C4(B\u0304) \u2229H20 (B). Hence, u\u03bb is a radial classical solution of (P )\u03bb,\u03b1,\u03b2 showing that \u03bb\u2217 = \u03bb\u2217. Moreover, since \u03a6 and v\u03bb := u\u03bb \u2212\u03a6 are radially decreasing in view of [20], we get that u\u03bb is radially decreasing too. Since the argument above shows that u\u03bb < U for any other classical solution U of (P )\u00b5,\u03b1,\u03b2 with \u00b5 \u2265 \u03bb, we have that u\u03bb is exactly the minimal solution and u\u03bb is strictly increasing as \u03bb \u2191 \u03bb\u2217. In particular, we can define u\u2217 in the usual way: u\u2217(x) = lim \u03bb\u2197\u03bb\u2217 u\u03bb(x). Finally, we show the finiteness of the pull-in voltage. Lemma 5.12. If (\u03b1, \u03b2) is an admissible pair, then \u03bb\u2217(\u03b1, \u03b2) < +\u221e. Proof. Let u be a classical solution of (P )\u03bb,\u03b1,\u03b2 and let (\u03c8, \u03bd1) denote the first eigenpair of \u22062 in H20 (B) with \u03c8 > 0. Now, let C be such that\u222b \u2202B (\u03b2\u2206\u03c8 \u2212 \u03b1\u2202\u03bd\u2206\u03c8) = C \u222b B \u03c8. Multiplying (P )\u03bb,\u03b1,\u03b2 by \u03c8 and then integrating by parts one arrives at\u222b B ( \u03bb (1\u2212 u)2 \u2212 \u03bd1u\u2212 C ) \u03c8 = 0. 114 5.2. The effect of boundary conditions on the pull-in voltage Since \u03c8 > 0 there must exist a point x\u0304 \u2208 B where \u03bb(1\u2212u(x\u0304))2 \u2212\u03bd1u(x\u0304)\u2212C \u2264 0. Since \u03b1 < u(x\u0304) < 1, one can conclude that \u03bb \u2264 sup\u03b1__ \u03bb\u2217 there are no weak solutions of (P )\u03bb,\u03b1,\u03b2. 5.2.1 Stability of the minimal branch of solutions This section is devoted to the proof of the following stability result for minimal solutions. We shall need yet another notion of H2(B)\u2212weak solutions, which is an intermediate class between classical and weak solutions. Definition 5.14. We say that u is a H2(B)\u2212weak solution of (P )\u03bb,\u03b1,\u03b2 if u\u2212\u03a6 \u2208 H20 (B), \u03b1 \u2264 u \u2264 1 a.e. in B, 1(1\u2212u)2 \u2208 L1(B) and if\u222b B \u2206u\u2206\u03c6 = \u03bb \u222b B \u03c6 (1\u2212 u)2 , \u2200\u03c6 \u2208 C 4(B\u0304) \u2229H20 (B), where \u03a6 is given in (5.8). We say that u is a H2(B)\u2212weak super-solution (resp. H2(B)\u2212weak sub-solution) of (P )\u03bb,\u03b1,\u03b2 if for \u03c6 \u2265 0 the equality is replaced with \u2265 (resp. \u2264) and u \u2265 \u03b1 (resp. \u2264), \u2202\u03bdu \u2264 \u03b2 (resp. \u2265) on \u2202B. Theorem 5.15. Suppose (\u03b1, \u03b2) is an admissible pair. 1. The minimal solution u\u03bb is then stable and is the unique semi-stable H2(B)\u2212weak solution of (P )\u03bb,\u03b1,\u03b2. 2. The function u\u2217 := lim \u03bb\u2197\u03bb\u2217 u\u03bb is a well-defined semi-stable H2(B)\u2212weak solution of (P )\u03bb\u2217,\u03b1,\u03b2. 3. When u\u2217 is classical solution, then \u00b51(u\u2217) = 0 and u\u2217 is the unique H2(B)\u2212weak solution of (P )\u03bb\u2217,\u03b1,\u03b2. 4. If v is a singular, semi-stable H2(B)\u2212weak solution of (P )\u03bb,\u03b1,\u03b2, then v = u\u2217 and \u03bb = \u03bb\u2217 The crucial tool is a comparison result which is valid exactly in this class of solu- tions. 115 5.2. The effect of boundary conditions on the pull-in voltage Lemma 5.16. Let (\u03b1, \u03b2) be an admissible pair and u be a semi-stable H2(B)\u2212weak solution of (P )\u03bb,\u03b1,\u03b2. Assume U is a H2(B)\u2212weak super-solution of (P )\u03bb,\u03b1,\u03b2 so that U \u2212 \u03a6 \u2208 H20 (B). Then 1. u \u2264 U a.e. in B; 2. If u is a classical solution and \u00b51(u) = 0 then U = u. Proof. (i) Define w := u \u2212 U . Then by the Moreau decomposition [17] for the biharmonic operator, there exist w1, w2 \u2208 H20 (B), with w = w1 + w2, w1 \u2265 0 a.e., \u22062w2 \u2264 0 in the H2(B)\u2212weak sense and \u222b B \u2206w1\u2206w2 = 0. By Lemma 5.5, we have that w2 \u2264 0 a.e. in B. Given now 0 \u2264 \u03c6 \u2208 C\u221ec (B), we have that\u222b B \u2206w\u2206\u03c6 \u2264 \u03bb \u222b B (f(u)\u2212 f(U))\u03c6, where f(u) := (1\u2212 u)\u22122. Since u is semi-stable, one has \u03bb \u222b B f \u2032(u)w21 \u2264 \u222b B (\u2206w1)2 = \u222b B \u2206w\u2206w1 \u2264 \u03bb \u222b B (f(u)\u2212 f(U))w1. Since w1 \u2265 w one also has\u222b B f \u2032(u)ww1 \u2264 \u222b B (f(u)\u2212 f(U))w1, which once re-arranged gives \u222b B f\u0303w1 \u2265 0, where f\u0303(u) = f(u) \u2212 f(U) \u2212 f \u2032(u)(u \u2212 U). The strict convexity of f gives f\u0303 \u2264 0 and f\u0303 < 0 whenever u 6= U . Since w1 \u2265 0 a.e. in B one sees that w \u2264 0 a.e. in B. The inequality u \u2264 U a.e. in B is then established. (ii) Since u is a classical solution, it is easy to see that the infimum in \u00b51(u) is attained at some \u03c6. The function \u03c6 is then the first eigenfunction of \u22062 \u2212 2\u03bb(1\u2212u)3 in H20 (B). Now we show that \u03c6 is of fixed sign. Using the above decomposition, one has \u03c6 = \u03c61 + \u03c62 where \u03c6i \u2208 H20 (B) for i = 1, 2, \u03c61 \u2265 0, \u222b B \u2206\u03c61\u2206\u03c62 = 0 and \u22062\u03c62 \u2264 0 in the H20 (B)\u2212weak sense. If \u03c6 changes sign, then \u03c61 6\u2261 0 and \u03c62 < 0 in B (recall that either \u03c62 < 0 or \u03c62 = 0 a.e. in B). We can write now: 0 = \u00b51(u) \u2264 \u222b B (\u2206(\u03c61 \u2212 \u03c62))2 \u2212 \u03bbf \u2032(u)(\u03c61 \u2212 \u03c62)2\u222b B (\u03c61 \u2212 \u03c62)2 < \u222b B (\u2206\u03c6)2 \u2212 \u03bbf \u2032(u)\u03c62\u222b B \u03c62 = \u00b51(u) 116 5.2. The effect of boundary conditions on the pull-in voltage in view of \u03c61\u03c62 < \u2212\u03c61\u03c62 in a set of positive measure, leading to a contradiction. So we can assume \u03c6 \u2265 0, and by the Boggi\u2019s principle we have \u03c6 > 0 in B. For 0 \u2264 t \u2264 1 define g(t) = \u222b B \u2206 [tU + (1\u2212 t)u]\u2206\u03c6\u2212 \u03bb \u222b B f(tU + (1\u2212 t)u)\u03c6, where \u03c6 is the above first eigenfunction. Since f is convex one sees that g(t) \u2265 \u03bb \u222b B [tf(U) + (1\u2212 t)f(u)\u2212 f(tU + (1\u2212 t)u)]\u03c6 \u2265 0 for every t \u2265 0. Since g(0) = 0 and g\u2032(0) = \u222b B \u2206(U \u2212 u)\u2206\u03c6\u2212 \u03bbf \u2032(u)(U \u2212 u)\u03c6 = 0, we get that g\u2032\u2032(0) = \u2212\u03bb \u222b B f \u2032\u2032(u)(U \u2212 u)2\u03c6 \u2265 0. Since f \u2032\u2032(u)\u03c6 > 0 in B, we finally get that U = u a.e. in B. Based again on Lemma 5.5(3), we can show a more general version of the above Lemma 5.16. Lemma 5.17. Let (\u03b1, \u03b2) be an admissible pair and \u03b2\u2032 \u2264 0. Let u be a semi-stable H2(B)\u2212weak sub-solution of (P )\u03bb,\u03b1,\u03b2 with u = \u03b1, \u2202\u03bdu = \u03b2\u2032 \u2265 \u03b2 on \u2202B. Assume that U is a H2(B)\u2212weak super-solution of (P )\u03bb,\u03b1,\u03b2 with U = \u03b1, \u2202\u03bdU = \u03b2 on \u2202B. Then U \u2265 u a.e. in B. Proof. Let u\u0303 \u2208 H20 (B) denote a weak solution to \u22062u\u0303 = \u22062(u \u2212 U) in B. Since u\u0303\u2212u+U = 0 and \u2202\u03bd(u\u0303\u2212u+U) \u2264 0 on \u2202B, by Lemma 5.5 one has that u\u0303 \u2265 u\u2212U a.e. in B. Again by the Moreau decomposition [17], we may write u\u0303 as u\u0303 = w + v, where w, v \u2208 H20 (B), w \u2265 0 a.e. in B, \u22062v \u2264 0 in a H2(B)\u2212weak sense and\u222b B \u2206w\u2206v = 0. Then for 0 \u2264 \u03c6 \u2208 C4(B\u0304) \u2229H20 (B) one has\u222b B \u2206u\u0303\u2206\u03c6 = \u222b B \u2206(u\u2212 U)\u2206\u03c6 \u2264 \u03bb \u222b B (f(u)\u2212 f(U))\u03c6. In particular, we have that\u222b B \u2206u\u0303\u2206w \u2264 \u03bb \u222b B (f(u)\u2212 f(U))w. Since by semi-stability of u \u03bb \u222b B f \u2032(u)w2 \u2264 \u222b B (\u2206w)2 = \u222b B \u2206u\u0303\u2206w, 117 5.2. The effect of boundary conditions on the pull-in voltage we get that \u222b B f \u2032(u)w2 \u2264 \u222b B (f(u)\u2212 f(U))w. By Lemma 5.5 we have v \u2264 0 and then w \u2265 u\u0303 \u2265 u\u2212 U a.e. in B. So we see that 0 \u2264 \u222b B (f(u)\u2212 f(U)\u2212 f \u2032(u)(u\u2212 U))w. The strict convexity of f implies as in Lemma 5.16 that U \u2265 u a.e. in B. We shall need the following a-priori estimates along the minimal branch u\u03bb. Lemma 5.18. Let (\u03b1, \u03b2) be an admissible pair. Then one has 2 \u222b B (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 \u2264 \u222b B u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 , where \u03a6 is given in (5.8). In particular, there is a constant C > 0 so that for every \u03bb \u2208 (0, \u03bb\u2217), we have \u222b B (\u2206u\u03bb)2 + \u222b B 1 (1\u2212 u\u03bb)3 \u2264 C. (5.19) Proof. Testing (P )\u03bb,\u03b1,\u03b2 on u\u03bb \u2212 \u03a6 \u2208 C4(B\u0304) \u2229H20 (B), we see that \u03bb \u222b B u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 = \u222b B \u2206u\u03bb\u2206(u\u03bb \u2212 \u03a6) = \u222b B (\u2206(u\u03bb \u2212 \u03a6))2 \u2265 2\u03bb \u222b B (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 in view of \u22062\u03a6 = 0. In particular, for \u03b4 > 0 small we have that\u222b {|u\u03bb\u2212\u03a6|\u2265\u03b4} 1 (1\u2212 u\u03bb)3 \u2264 1 \u03b42 \u222b {|u\u03bb\u2212\u03a6|\u2265\u03b4} (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 \u2264 1 \u03b42 \u222b B 1 (1\u2212 u\u03bb)2 \u2264 \u03b4 \u222b {|u\u03bb\u2212\u03a6|\u2265\u03b4} 1 (1\u2212 u\u03bb)3 + C\u03b4 by means of Young\u2019s inequality. Since for \u03b4 small,\u222b {|u\u03bb\u2212\u03a6|\u2264\u03b4} 1 (1\u2212 u\u03bb)3 \u2264 C \u2032 for some C \u2032 > 0, we can deduce that for every \u03bb \u2208 (0, \u03bb\u2217),\u222b B 1 (1\u2212 u\u03bb)3 \u2264 C for some C > 0. By Young\u2019s and Ho\u0308lder\u2019s inequalities, we now have\u222b B (\u2206u\u03bb)2 = \u222b B \u2206u\u03bb\u2206\u03a6 + \u03bb \u222b B u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 \u2264 \u03b4 \u222b B (\u2206u\u03bb)2 + C\u03b4 + C (\u222b B 1 (1\u2212 u\u03bb)3 ) 2 3 and estimate (5.19) is therefore established. 118 5.3. Regularity of the extremal solution for 1 \u2264 N \u2264 8 We are now ready to establish Theorem 5.15. Proof (of Theorem 5.15): (1) Since \u2016u\u03bb\u2016\u221e < 1, the infimum defining \u00b51(u\u03bb) is achieved at a first eigenfunction for every \u03bb \u2208 (0, \u03bb\u2217). Since \u03bb 7\u2192 u\u03bb(x) is increasing for every x \u2208 B, it is easily seen that \u03bb 7\u2192 \u00b51(u\u03bb) is an increasing, continuous function on (0, \u03bb\u2217). Define \u03bb\u2217\u2217 := sup{0 < \u03bb < \u03bb\u2217 : \u00b51(u\u03bb) > 0}. We have that \u03bb\u2217\u2217 = \u03bb\u2217. Indeed, otherwise we would have that \u00b51(u\u03bb\u2217\u2217) = 0, and for every \u00b5 \u2208 (\u03bb\u2217\u2217, \u03bb\u2217) u\u00b5 would be a classical super-solution of (P )\u03bb\u2217\u2217,\u03b1,\u03b2 . A contradiction arises since Lemma 5.16 implies u\u00b5 = u\u03bb\u2217\u2217 . Finally, Lemma 5.16 guarantees uniqueness in the class of semi-stable H2(B)\u2212weak solutions. (2) By estimate (5.19) it follows that u\u03bb \u2192 u\u2217 in a pointwise sense and weakly in H2(B), and 11\u2212u\u2217 \u2208 L3(B). In particular, u\u2217 is a H2(B)\u2212weak solution of (P )\u03bb\u2217,\u03b1,\u03b2 which is also semi-stable as limiting function of the semi-stable solutions {u\u03bb}. (3) Whenever \u2016u\u2217\u2016\u221e < 1, the function u\u2217 is a classical solution, and by the Implicit Function Theorem we have that \u00b51(u\u2217) = 0 to prevent the continuation of the minimal branch beyond \u03bb\u2217. By Lemma 5.16 u\u2217 is then the unique H2(B)\u2212weak solution of (P )\u03bb\u2217,\u03b1,\u03b2 . An alternative approach \u2013which we do not pursue here\u2013 based on the very definition of the extremal solution u\u2217 is available in [4] when \u03b1 = \u03b2 = 0 (see also [15]) to show that u\u2217 is the unique weak solution of (P )\u03bb\u2217 , regardless of whether u\u2217 is regular or not. (4) If \u03bb < \u03bb\u2217, by uniqueness v = u\u03bb. So v is not singular and a contradiction arises. By Theorem 5.13(3) we have that \u03bb = \u03bb\u2217. Since v is a semi-stable H2(B)\u2212weak solution of (P )\u03bb\u2217,\u03b1,\u03b2 and u\u2217 is a H2(B)\u2212weak super-solution of (P )\u03bb\u2217,\u03b1,\u03b2 , we can apply Lemma 5.16 to get v \u2264 u\u2217 a.e. in B. Since u\u2217 is a semi-stable solution too, we can reverse the roles of v and u\u2217 in Lemma 5.16 to see that v \u2265 u\u2217 a.e. in B. So equality v = u\u2217 holds and the proof is done. 5.3 Regularity of the extremal solution for 1 \u2264 N \u2264 8 We now return to the issue of the regularity of the extremal solution in problem (P )\u03bb. Unless stated otherwise, u\u03bb and u\u2217 refer to the minimal and extremal so- lutions of (P )\u03bb. We shall show that the extremal solution u\u2217 is regular provided 1 \u2264 N \u2264 8. We first begin by showing that it is indeed the case in small dimensions: Theorem 5.20. u\u2217 is regular in dimensions 1 \u2264 N \u2264 4. Proof. As already observed, estimate (5.19) implies that f(u\u2217) = (1 \u2212 u\u2217)\u22122 \u2208 L 3 2 (B). Since u\u2217 is radial and radially decreasing, we need to show that u\u2217(0) < 1 119 5.3. Regularity of the extremal solution for 1 \u2264 N \u2264 8 to get the regularity of u\u2217. The integrability of f(u\u2217) along with elliptic regularity theory shows that u\u2217 \u2208W 4, 32 (B). By the Sobolev imbedding Theorem we get that u\u2217 is a Lipschitz function in B. Now suppose u\u2217(0) = 1 and 1 \u2264 N \u2264 3. Since 1 1\u2212 u \u2265 C |x| in B for some C > 0, one sees that +\u221e = C3 \u222b B 1 |x|3 \u2264 \u222b B 1 (1\u2212 u\u2217)3 < +\u221e. A contradiction arises and hence u\u2217 is regular for 1 \u2264 N \u2264 3. For N = 4 we need to be more careful and observe that u\u2217 \u2208 C1, 13 (B\u0304) by the Sobolev Imbedding Theorem. If u\u2217(0) = 1, then \u2207u\u2217(0) = 0 and 1 1\u2212 u\u2217 \u2265 C |x| 43 in B for some C > 0. We now obtain a contradiction exactly as above. We now tackle the regularity of u\u2217 for 5 \u2264 N \u2264 8. We start with the following crucial result: Theorem 5.21. Let N \u2265 5 and (u\u2217, \u03bb\u2217) be the extremal pair of (P )\u03bb. When u\u2217 is singular, then 1\u2212 u\u2217(x) \u2264 C0|x| 43 in B, where C0 := ( \u03bb\u2217 \u03bb ) 1 3 and \u03bb\u0304 = \u03bb\u0304N := 89 (N \u2212 23 )(N \u2212 83 ). Proof. First note that Theorem 5.6(4) gives the lower bound: \u03bb\u2217 \u2265 \u03bb\u0304 = 8 9 (N \u2212 2 3 )(N \u2212 8 3 ). (5.22) For \u03b4 > 0, we define u\u03b4(x) := 1\u2212 C\u03b4|x| 43 with C\u03b4 := ( \u03bb\u2217 \u03bb\u0304 + \u03b4 ) 1 3 > 1. Since N \u2265 5, we have that u\u03b4 \u2208 H2loc(RN ), 11\u2212u\u03b4 \u2208 L3loc(RN ) and u\u03b4 is a H2\u2212weak solution of \u22062u\u03b4 = \u03bb\u2217 + \u03b4\u03bb\u0304 (1\u2212 u\u03b4)2 in R N . We claim that u\u03b4 \u2264 u\u2217 in B, which will finish the proof by just letting \u03b4 \u2192 0. Assume by contradiction that the set \u0393 := {r \u2208 (0, 1) : u\u03b4(r) > u\u2217(r)} is non-empty, and let r1 = sup \u0393. Since u\u03b4(1) = 1\u2212 C\u03b4 < 0 = u\u2217(1), 120 5.3. Regularity of the extremal solution for 1 \u2264 N \u2264 8 we have that 0 < r1 < 1 and one infers that \u03b1 := u\u2217(r1) = u\u03b4(r1) , \u03b2 := (u\u2217)\u2032(r1) \u2265 u\u2032\u03b4(r1). Setting u\u03b4,r1(r) = r \u2212 43 1 (u\u03b4(r1r)\u2212 1) + 1, we easily see that u\u03b4,r1 is a H2(B)\u2212weak super-solution of (P )\u03bb\u2217+\u03b4\u03bb\u0304N ,\u03b1\u2032,\u03b2\u2032 , where \u03b1\u2032 := r\u2212 4 3 1 (\u03b1\u2212 1) + 1 , \u03b2\u2032 := r\u2212 1 3 1 \u03b2. Similarly, let us define u\u2217r1(r) = r \u2212 43 1 (u \u2217(r1r)\u2212 1) + 1. The dilation map w \u2192 wr1(r) = r\u2212 4 3 1 (w(r1r)\u2212 1) + 1 (5.23) is a correspondence between solutions of (P )\u03bb on B and of (P ) \u03bb,1\u2212r\u2212 4 3 1 ,0 on Br\u221211 which preserves the H2\u2212integrability. In particular, (u\u2217r1 , \u03bb\u2217) is the extremal pair of (P ) \u03bb,1\u2212r\u2212 4 3 1 ,0 on Br\u221211 (defined in the obvious way). Moreover, u \u2217 r1 is a singular semi-stable H2(B)\u2212 weak solution of (P )\u03bb\u2217,\u03b1\u2032,\u03b2\u2032 . Since u\u2217 is radially decreasing, we have that \u03b2\u2032 \u2264 0. Define the function w as w(x) := (\u03b1\u2032 \u2212 \u03b2\u20322 ) + \u03b2 \u2032 2 |x|2 + \u03b3(x), where \u03b3 is a solution of \u22062\u03b3 = \u03bb\u2217 in B with \u03b3 = \u2202\u03bd\u03b3 = 0 on \u2202B. Then w is a classical solution of{ \u22062w = \u03bb\u2217 in B w = \u03b1\u2032 , \u2202\u03bdw = \u03b2\u2032 on \u2202B. Since \u03bb \u2217 (1\u2212u\u2217r1 )2 \u2265 \u03bb\u2217, by Lemma 5.5 we have u\u2217r1 \u2265 w a.e. in B. Since w(0) = \u03b1\u2032 \u2212 \u03b2\u20322 + \u03b3(0) and \u03b3(0) > 0, the bound u\u2217r1 \u2264 1 a.e. in B yields to \u03b1\u2032 \u2212 \u03b2 \u2032 2 < 1. Namely, (\u03b1\u2032, \u03b2\u2032) is an admissible pair and by Theorem 5.15(4) we get that (u\u2217r1 , \u03bb \u2217) coincides with the extremal pair of (P )\u03bb,\u03b1\u2032,\u03b2\u2032 in B. Since (\u03b1\u2032, \u03b2\u2032) is an admissible pair and u\u03b4,r1 is a H 2(B)\u2212weak super-solution of (P )\u03bb\u2217+\u03b4\u03bb\u0304N ,\u03b1\u2032,\u03b2\u2032 , by Proposition 5.2.1 we get the existence of a weak solution of (P )\u03bb\u2217+\u03b4\u03bb\u0304N ,\u03b1\u2032,\u03b2\u2032 . Since \u03bb \u2217+\u03b4\u03bb\u0304N > \u03bb\u2217, we contradict the fact that \u03bb\u2217 is the extremal parameter of (P )\u03bb,\u03b1\u2032,\u03b2\u2032 . Thanks to this lower estimate on u\u2217, we get the following result. Theorem 5.24. If 5 \u2264 N \u2264 8, then the extremal solution u\u2217 of (P )\u03bb is regular. Proof. Assume that u\u2217 is singular. For \u03b5 > 0 set \u03c8(x) := |x| 4\u2212N2 +\u03b5 and note that (\u2206\u03c8)2 = (HN +O(\u03b5))|x|\u2212N+2\u03b5 where HN := N2(N \u2212 4)2 16 . 121 5.4. The extremal solution is singular for N \u2265 9 Given \u03b7 \u2208 C\u221e0 (B), and since N \u2265 5, we can use the test function \u03b7\u03c8 \u2208 H20 (B) into the stability inequality to obtain 2\u03bb \u222b B \u03c82 (1\u2212 u\u2217)3 \u2264 \u222b B (\u2206\u03c8)2 +O(1), where O(1) is a bounded function as \u03b5\u2198 0. By Theorem 5.21 we find that 2\u03bb\u0304N \u222b B \u03c82 |x|4 \u2264 \u222b B (\u2206\u03c8)2 +O(1), and then 2\u03bb\u0304N \u222b B |x|\u2212N+2\u03b5 \u2264 (HN +O(\u03b5)) \u222b B |x|\u2212N+2\u03b5 +O(1). Computing the integrals one arrives at 2\u03bb\u0304N \u2264 HN +O(\u03b5). As \u03b5 \u2192 0 finally we obtain 2\u03bb\u0304N \u2264 HN . Graphing this relation one sees that N \u2265 9. We can now slightly improve the lower bound (5.22). Corollary 5.25. In any dimension N \u2265 1, we have \u03bb\u2217 > \u03bb\u0304N = 8 9 (N \u2212 2 3 )(N \u2212 8 3 ). (5.26) Proof. The function u\u0304 := 1\u2212 |x| 43 is a H2(B)\u2212 weak solution of (P )\u03bb\u0304N ,0,\u2212 43 . If by contradiction \u03bb\u2217 = \u03bb\u0304N , then u\u0304 is a H2(B)\u2212weak super-solution of (P )\u03bb for every \u03bb \u2208 (0, \u03bb\u2217). By Lemma 5.16 we get that u\u03bb \u2264 u\u0304 for all \u03bb < \u03bb\u2217, and then u\u2217 \u2264 u\u0304 a.e. in B. If 1 \u2264 N \u2264 8, u\u2217 is then regular by Theorems 5.20 and 5.24. By Theorem 5.15(3) there holds \u00b51(u\u2217) = 0. Lemma 5.16 then yields that u\u2217 = u\u0304, which is a contradic- tion since then u\u2217 will not satisfy the boundary conditions. If now N \u2265 9 and \u03bb\u0304 = \u03bb\u2217, then C0 = 1 in Theorem 5.21, and we then have u\u2217 \u2265 u\u0304. It means again that u\u2217 = u\u0304, a contradiction that completes the proof. 5.4 The extremal solution is singular for N \u2265 9 We prove in this section that the extremal solution is singular for N \u2265 9. For that we have to distinguish between three different ranges for the dimension. For each range, we will need a suitable Hardy-Rellich type inequality that will be established in the last section, by using the recent results of Ghoussoub-Moradifam [10]. As in the previous section (u\u2217, \u03bb\u2217) denotes the extremal pair of (P )\u03bb. 122 5.4. The extremal solution is singular for N \u2265 9 \u2022 Case N \u2265 17: To establish the singularity of u\u2217 for these dimensions we shall need the following well known improved Hardy-Rellich inequality, which is valid for N \u2265 5. There exists C > 0, such that for all \u03c6 \u2208 H20 (B)\u222b B (\u2206\u03c6)2 dx \u2265 N 2(N \u2212 4)2 16 \u222b B \u03c62 |x|4 dx+ C \u222b B \u03c62 dx. (5.27) \u2022 Case 10 \u2264 N \u2264 16: For this case, we shall need the following inequality valid for all \u03c6 \u2208 H20 (B)\u222b B (\u2206\u03c6)2 \u2265 (N \u2212 2) 2(N \u2212 4)2 16 \u222b B \u03c62 (|x|2 \u2212 |x|N2 +1)(|x|2 \u2212 |x|N2 ) (5.28) + (N \u2212 1)(N \u2212 4)2 4 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) . \u2022 Case N = 9: This case is the trickiest and will require the following inequality for all \u03c6 \u2208 H20 (B) \u222b B (\u2206\u03c6)2 \u2265 \u222b B Q(|x|) ( P (|x|) + N \u2212 1|x|2 ) \u03c62, (5.29) where P (r) = \u2206N\u03d5\u03d5 and Q(r) = \u2206N\u22122\u03c8 \u03c8 , with \u03d5 and \u03c8 being two appropriately chosen polynomials, namely \u03d5(r) := r\u2212 N 2 +1 + r \u2212 1.9 and \u03c8(r) := r\u2212 N 2 +2 + 20r\u22121.69 + 10r\u22121 + 10r + 7r2 \u2212 48. Recall that for a radial function \u03d5, we set \u2206N\u03d5(r) = \u03d5\u2032\u2032(r) + (N\u22121) r \u03d5 \u2032(r). We shall first show the following upper bound on u\u2217. Lemma 5.29. If N \u2265 9, then u\u2217 \u2264 1\u2212 |x| 43 in B. Proof. Recall from Corollary 5.25 that \u03bb\u0304 := 89 (N \u2212 23 )(N \u2212 83 ) < \u03bb\u2217. We now claim that u\u03bb \u2264 u\u0304 for all \u03bb \u2208 (\u03bb\u0304, \u03bb\u2217). Indeed, fix such a \u03bb and assume by contradiction that R1 := inf{0 \u2264 R \u2264 1 : u\u03bb < u\u0304 in (R, 1)} > 0. From the boundary conditions, one has that u\u03bb(r) < u\u0304(r) as r \u2192 1\u2212. Hence, 0 < R1 < 1, \u03b1 := u\u03bb(R1) = u\u0304(R1) and \u03b2 := u\u2032\u03bb(R1) \u2264 u\u0304\u2032(R1). Introduce, as in the proof of Theorem 5.21, the functions u\u03bb,R1 and u\u0304R1 . We have that u\u03bb,R1 is a classical super-solution of (P )\u03bb\u0304N ,\u03b1\u2032,\u03b2\u2032 , where \u03b1\u2032 := R\u2212 4 3 1 (\u03b1\u2212 1) + 1 , \u03b2\u2032 := R\u2212 1 3 1 \u03b2. 123 5.4. The extremal solution is singular for N \u2265 9 Note that u\u0304R1 is aH 2(B)\u2212weak sub-solution of (P )\u03bb\u0304N ,\u03b1\u2032,\u03b2\u2032 which is also semi-stable in view of the Hardy-Rellich inequality (5.27) and the fact that 2\u03bb\u0304N \u2264 HN := N 2(N \u2212 4)2 16 . By Lemma 5.17, we deduce that u\u03bb,R1 \u2265 u\u0304R1 in B. Note that, arguing as in the proof of Theorem 5.21, (\u03b1\u2032, \u03b2\u2032) is an admissible pair. We have therefore shown that u\u03bb \u2265 u\u0304 in BR1 and a contradiction arises in view of the fact that lim x\u21920 u\u0304(x) = 1 and \u2016u\u03bb\u2016\u221e < 1. It follows that u\u03bb \u2264 u\u0304 in B for every \u03bb \u2208 (\u03bb\u0304N , \u03bb\u2217), and in particular u\u2217 \u2264 u\u0304 in B. The following lemma is the key for the proof of the of u\u2217 in higher dimensions. Lemma 5.30. Let N \u2265 9. Suppose there exist \u03bb\u2032 > 0, \u03b2 > 0 and a singular radial function w \u2208 H2(B)with 11\u2212w \u2208 L\u221eloc(B\u0304 \\ {0}) such that{ \u22062w \u2264 \u03bb\u2032(1\u2212w)2 for 0 < r < 1, w(1) = 0, w\u2032(1) = 0, (5.31) and 2\u03b2 \u222b B \u03c62 (1\u2212 w)3 \u2264 \u222b B (\u2206\u03c6)2 for all \u03c6 \u2208 H20 (B), (5.32) 1. If \u03b2 \u2265 \u03bb\u2032, then \u03bb\u2217 \u2264 \u03bb\u2032. 2. If either \u03b2 > \u03bb\u2032 or if \u03b2 = \u03bb\u2032 = HN2 , then the extremal solution u \u2217 is necessarily singular. Proof: 1) First, note that (5.32) and 11\u2212w \u2208 L\u221eloc(B\u0304 \\{0}) yield to 1(1\u2212w)2 \u2208 L1(B). By a density argument, (5.31) implies now that w is a H2(B)\u2212weak sub-solution of (P )\u03bb\u2032 whenever N \u2265 4. If now \u03bb\u2032 < \u03bb\u2217, then by Lemma 5.17 w would necessarily be below the minimal solution u\u03bb\u2032 , which is a contradiction since w is singular while u\u03bb\u2032 is regular. 2) Suppose first that \u03b2 = \u03bb\u2032 = HN2 and that N \u2265 9. Since by part 1) we have \u03bb\u2217 \u2264 HN2 , we get from Lemma 5.29 and the improved Hardy-Rellich inequality (5.27) that there exists C > 0 so that for all \u03c6 \u2208 H20 (B)\u222b B (\u2206\u03c6)2 \u2212 2\u03bb\u2217 \u222b B \u03c62 (1\u2212 u\u2217)3 \u2265 \u222b B (\u2206\u03c6)2 \u2212HN \u222b B \u03c62 |x|4 \u2265 C \u222b B \u03c62. It follows that \u00b51(u\u2217) > 0 and u\u2217 must therefore be singular since otherwise, one could use the Implicit Function Theorem to continue the minimal branch beyond \u03bb\u2217. Suppose now that \u03b2 > \u03bb\u2032, and let \u03bb \u2032 \u03b2 < \u03b3 < 1 in such a way that \u03b1 := ( \u03b3\u03bb\u2217 \u03bb\u2032 )1\/3 < 1. (5.33) 124 5.4. The extremal solution is singular for N \u2265 9 Setting w\u0304 := 1\u2212 \u03b1(1\u2212 w), we claim that u\u2217 \u2264 w\u0304 in B. (5.34) Note that by the choice of \u03b1 we have \u03b13\u03bb\u2032 < \u03bb\u2217, and therefore to prove (5.34) it suffices to show that for \u03b13\u03bb\u2032 \u2264 \u03bb < \u03bb\u2217, we have u\u03bb \u2264 w\u0304 in B. Indeed, fix such \u03bb and note that \u22062w\u0304 = \u03b1\u22062w \u2264 \u03b1\u03bb \u2032 (1\u2212 w)2 = \u03b13\u03bb\u2032 (1\u2212 w\u0304)2 \u2264 \u03bb (1\u2212 w\u0304)2 . Assume that u\u03bb \u2264 w\u0304 does not hold in B, and consider R1 := sup{0 \u2264 R \u2264 1 | u\u03bb(R) > w\u0304(R)} > 0. Since w\u0304(1) = 1 \u2212 \u03b1 > 0 = u\u03bb(1), we then have R1 < 1, u\u03bb(R1) = w\u0304(R1) and (u\u03bb)\u2032(R1) \u2264 (w\u0304)\u2032(R1). Introduce, as in the proof of Theorem 5.21, the functions u\u03bb,R1 and w\u0304R1 . We have that u\u03bb,R1 is a classical solution of (P )\u03bb,\u03b1\u2032,\u03b2\u2032 , where \u03b1\u2032 := R\u2212 4 3 1 (u\u03bb(R1)\u2212 1) + 1 , \u03b2\u2032 := R\u2212 1 3 1 (u\u03bb) \u2032(R1). Since \u03bb < \u03bb\u2217 and then 2\u03bb (1\u2212 w\u0304)3 \u2264 2\u03bb\u2217 \u03b13(1\u2212 w)3 = 2\u03bb\u2032 \u03b3(1\u2212 w)3 < 2\u03b2 (1\u2212 w)3 , by (5.32) w\u0304R1 is a stable H 2(B)\u2212weak sub-solution of (P )\u03bb,\u03b1\u2032,\u03b2\u2032 . By Lemma 5.17, we deduce that u\u03bb \u2265 w\u0304 in BR1 which is impossible, since w\u0304 is singular while u\u03bb is regular. Note that, arguing as in the proof of Theorem 5.21, (\u03b1\u2032, \u03b2\u2032) is an admissible pair. This establishes claim (5.34) which, combined with the above inequality, yields 2\u03bb\u2217 (1\u2212 u\u2217)3 \u2264 2\u03bb\u2217 \u03b13(1\u2212 w)3 < 2\u03b2 (1\u2212 w)3 , and therefore inf \u03c6\u2208H20 (B) \u222b B (\u2206\u03c6)2 \u2212 2\u03bb\u2217\u03c62(1\u2212u\u2217)3\u222b B \u03c62 > 0. It follows that again \u00b51(u\u2217) > 0 and u\u2217 must be singular, since otherwise, one could use the Implicit Function Theorem to continue the minimal branch beyond \u03bb\u2217. Consider for any m > 0 the following function: wm := 1\u2212 3m3m\u2212 4r 4\/3 + 4 3m\u2212 4r m, (5.35) which satisfies the right boundary conditions: wm(1) = w\u2032m(1) = 0. We can now prove that the extremal solution is singular for N \u2265 9. Theorem 5.36. Let N \u2265 9. The following upper bounds on \u03bb\u2217 hold: 125 5.4. The extremal solution is singular for N \u2265 9 1. If N \u2265 31, then Lemma 5.30 holds with w := w2, \u03bb\u2032 = 27\u03bb\u0304N and \u03b2 = HN2 , and therefore \u03bb\u2217(N) \u2264 27\u03bb\u0304N . 2. If 17 \u2264 N \u2264 30, then Lemma 5.30 holds with w := w3, \u03bb\u2032 = \u03b2 = HN2 , and therefore \u03bb\u2217(N) \u2264 HN2 . 3. If 10 \u2264 N \u2264 16, then Lemma 5.30 holds with w := w3, \u03bb\u2032N < \u03b2N given in Table 5.1, and therefore \u03bb\u2217(N) \u2264 \u03bb\u2032N . 4. If N = 9, then Lemma 5.30 holds with w := w2.8, \u03bb\u20329 := 366 < \u03b29 := 368.5, and therefore \u03bb\u2217(9) \u2264 366. The extremal solution is therefore singular for dimension N \u2265 9. Table 5.1: Summary 2 N w \u03bb\u2032N \u03b2N 9 w2.8 366 366.5 10 w3 450 487 11 w3 560 739 12 w3 680 1071 13 w3 802 1495 14 w3 940 2026 15 w3 1100 2678 16 w3 1260 3469 17 \u2264 N \u2264 30 w3 HN\/2 HN\/2 N \u2265 31 w2 27\u03bb\u0304N HN\/2 Proof. 1) Assume first that N \u2265 31, then 27\u03bb\u0304 \u2264 HN2 . We shall show that w2 is a singular H2(B)\u2212weak sub-solution of (P )27\u03bb\u0304 so that (5.32) holds with \u03b2 = HN2 . Indeed, write w2 := 1\u2212 |x| 43 \u2212 2(|x| 43 \u2212 |x|2) = u\u0304\u2212 \u03c60, where \u03c60 := 2(|x| 43 \u2212 |x|2), and note that w2 \u2208 H20 (B), 11\u2212w2 \u2208 L3(B), 0 \u2264 w2 \u2264 1 in B, and \u22062w2 = 3\u03bb\u0304 r 8 3 \u2264 27\u03bb\u0304 (1\u2212 w2)2 in B \\ {0}. So w2 is H2(B)\u2212weak sub-solution of (P )27\u03bb\u0304. Moreover, by \u03c60 \u2265 0 and (5.27) we get that HN \u222b B \u03c62 (1\u2212 w2)3 = HN \u222b B \u03c62 (|x| 43 + \u03c60)3 \u2264 HN \u222b B \u03c62 |x|4 \u2264 \u222b B (\u2206\u03c6)2 126 5.4. The extremal solution is singular for N \u2265 9 for all \u03c6 \u2208 H20 (B). It follows from Lemma 5.30 that u\u2217 is singular and that \u03bb\u2217 \u2264 27\u03bb\u0304 \u2264 HN2 . 2) Assume 17 \u2264 N \u2264 30 and consider the function w3 := 1\u2212 95r 4 3 + 4 5 r3. We show that w3 is a semi-stable singular H2(B)\u2212weak sub-solution of (P )HN 2 . Indeed, we clearly have that 0 \u2264 w3 \u2264 1 in B, w3 \u2208 H20 (B) and 11\u2212w3 \u2208 L3(B). To show the stability condition, we consider \u03c6 \u2208 H20 (B) and write HN \u222b B \u03c62 (1\u2212 w3)3 = 125HN \u222b B \u03c62 (9r 4 3 \u2212 4r3)3 \u2264 125HN sup0 \u03bb\u2032N , we deduce from Lemma 5.30 that the extremal solution is singular for 10 \u2264 N \u2264 16. 4) Suppose now N = 9 and consider w := w2.8. Using Maple on can see that \u22062w \u2264 366 (1\u2212 w)2 in B 127 5.5. Improved Hardy-Rellich Inequalities and 723 (1\u2212 w)3 \u2264 Q(r) ( P (r) + N \u2212 1 r2 ) for all r \u2208 (0, 1), where P and Q are given in (5.29). Since 723 > 2 \u00d7 366, by Lemma 5.30 the extremal solution u\u2217 is singular in dimension N = 9. 5.5 Improved Hardy-Rellich Inequalities We now prove the improved Hardy-Rellich inequalities used in section 4. They rely on the results of Ghoussoub-Moradifam in [10] which provide necessary and suffi- cient conditions for such inequalities to hold. At the heart of this characterization is the following notion of a Bessel pair of functions. Definition 5.37. Assume that B is a ball of radius R in RN , V,W \u2208 C1(0, 1), and \u222b R 0 1 rN\u22121V (r)dr = +\u221e. Say that the couple (V,W ) is a Bessel pair on (0, R) if the ordinary differential equation (BV,W ) y\u2032\u2032(r) + (N\u22121r + Vr(r) V (r) )y \u2032(r) + W (r)V (r) y(r) = 0 has a positive solution on the interval (0, R). The space of radial functions in C\u221e0 (B) will be denoted by C \u221e 0,r(B). The needed inequalities will follow from the following result. Theorem 5.38. (Ghoussoub-Moradifam [10]) Let V and W be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in RN (N \u2265 1) such that \u222b R 0 1 rN\u22121V (r)dr = +\u221e and \u222b R 0 rN\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R). 2. \u222b B V (|x|)|\u2207\u03c6|2dx \u2265 \u222b B W (|x|)\u03c62dx for all \u03c6 \u2208 C\u221e0 (B). 3. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < N \u2212 2, then the above are equivalent to\u222b B V (|x|)(\u2206\u03c6)2dx \u2265 \u222b B W (|x|)|\u2207\u03c6|2dx+ (N \u2212 1) \u222b B ( V (|x|) |x|2 \u2212 Vr(|x|) |x| )|\u2207\u03c6| 2dx for all \u03c6 \u2208 C\u221e0,r(B). 4. If in addition, W (r) \u2212 2V (r)r2 + 2Vr(r)r \u2212 Vrr(r) \u2265 0 on (0, R), then the above are equivalent to\u222b B V (|x|)(\u2206\u03c6)2dx \u2265 \u222b B W (|x|)|\u2207\u03c6|2dx+ (N \u2212 1) \u222b B ( V (|x|) |x|2 \u2212 Vr(|x|) |x| )|\u2207\u03c6| 2dx for all \u03c6 \u2208 C\u221e0 (B). 128 5.5. Improved Hardy-Rellich Inequalities We shall now deduce the following corollary. Corollary 5.39. Let N \u2265 5 and B be the unit ball in RN . Then the following improved Hardy-Rellich inequality holds for all \u03c6 \u2208 C\u221e0 (B):\u222b B (\u2206\u03c6)2 \u2265 (N \u2212 2) 2(N \u2212 4)2 16 \u222b B \u03c62 (|x|2 \u2212 |x|N2 +1)(|x|2 \u2212 |x|N2 ) (5.40) + (N \u2212 1)(N \u2212 4)2 4 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) Proof. Let 0 < \u03b1 < 1 and define y(r) := r\u2212 N 2 +1 \u2212 \u03b1. Since \u2212y \u2032\u2032 + (N\u22121)r y \u2032 y = (N \u2212 2)2 4 1 r2 \u2212 \u03b1rN2 +1 , the couple ( 1, (N\u22122) 2 4 1 r2\u2212\u03b1rN2 +1 ) is a Bessel pair on (0, 1). By Theorem 5.38(4) the following inequality then holds:\u222b B (\u2206\u03c6)2dx \u2265 (N \u2212 2) 2 4 \u222b B |\u2207\u03c6|2 |x|2 \u2212 \u03b1|x|N2 +1 + (N \u2212 1) \u222b B |\u2207\u03c6|2 |x|2 (5.41) for all \u03c6 \u2208 C\u221e0 (B). Set V (r) := 1 r2\u2212\u03b1rN2 +1 and note that Vr V = \u22122 r + \u03b1(N \u2212 2) 2 r N 2 \u22122 1\u2212 \u03b1rN2 \u22121 \u2265 \u2212 2 r . The function y(r) = r\u2212 N 2 +2 \u2212 1 is decreasing and is then a positive super-solution on (0, 1) for the ODE y\u2032\u2032 + ( N \u2212 1 r + Vr V )y\u2032(r) + W1(r) V (r) y = 0, where W1(r) = (N \u2212 4)2 4(r2 \u2212 rN2 )(r2 \u2212 \u03b1rN2 +1) . Hence, by Theorem 5.38(2) we deduce\u222b B |\u2207\u03c6|2 |x|2 \u2212 \u03b1|x|N2 +1 \u2265 ( N \u2212 4 2 )2 \u222b B \u03c62 (|x|2 \u2212 \u03b1|x|N2 +1)(|x|2 \u2212 |x|N2 ) for all \u03c6 \u2208 C\u221e0 (B). Similarly, for V (r) = 1r2 we have that\u222b B |\u2207\u03c6|2 |x|2 \u2265 ( N \u2212 4 2 )2 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) for all \u03c6 \u2208 C\u221e0 (B). Combining the above two inequalities with (5.41) and letting \u03b1\u2192 1 we get inequality (5.40). 129 5.5. Improved Hardy-Rellich Inequalities Corollary 5.42. Let N = 9 and B be the unit ball in RN . Define \u03d5(r) := r\u2212N2 +1+ r\u2212 1.9 and \u03c8(r) := r\u2212N2 +2+20r\u22121.69+10r\u22121+10r+7r2\u2212 48. Then the following improved Hardy-Rellich inequality holds for all \u03c6 \u2208 C\u221e0 (B):\u222b B (\u2206\u03c6)2 \u2265 \u222b B Q(|x|) ( P (|x|) + N \u2212 1|x|2 ) \u03c62, (5.43) where P (r) := \u2212\u03d5 \u2032\u2032(r) + N\u22121r \u03d5 \u2032(r) \u03d5(r) and Q(r) := \u2212\u03c8 \u2032\u2032(r) + N\u22123r \u03c8 \u2032(r) \u03c8(r) . Proof. By definition (1, P (r)) is a Bessel pair on (0, 1). One can easily see that P (r) \u2265 2r2 . Hence, by Theorem 5.38(4) the following inequality holds:\u222b B (\u2206\u03c6)2dx \u2265 \u222b B P (|x|)|\u2207\u03c6|2 + (N \u2212 1) \u222b B |\u2207\u03c6|2 |x|2 (5.44) for all \u03c6 \u2208 C\u221e0 (B). Using Maple it is easy to see that Pr P \u2265 \u22122 r in (0, 1), and therefore \u03c8(r) is a positive super-solution for the ODE y\u2032\u2032 + ( N \u2212 1 r + Pr(r) P (r) )y\u2032(r) + P (r)Q(r) P (r) y = 0, on (0, 1). Hence, by Theorem 5.38(2) we have for all \u03c6 \u2208 C\u221e0 (B)\u222b B P (|x|)|\u2207\u03c6|2 \u2265 \u222b B P (|x|)Q(|x|)\u03c62, and similarly \u222b B |\u2207\u03c6|2 |x|2 \u2265 \u222b B Q(|x|) |x|2 \u03c6 2, since \u03c8(r) is a positive solution for the ODE y\u2032\u2032 + N \u2212 3 r y\u2032(r) +Q(r)y = 0. Combining the above two inequalities with (5.44) we get (5.43). 130 Bibliography [1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. [2] G. Arioli, F. Gazzola, H.-C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (2005), no. 4, 1226-1258. [3] T. Boggio, Sulle funzioni di Green d\u2019ordine m, Rend. Circ. Mat. Palermo (1905), 97-135. [4] D. Cassani, J. do O and N. 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Systems Appl. 3 (1994), no. 4, 465\u2013487. 132 Chapter 6 On the critical dimension of a fourth order elliptic problem with negative exponent 5 6.1 Introduction Consider the fourth order elliptic problem\uf8f1\uf8f2\uf8f3 \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bb(1\u2212u)2 in \u2126, 0 < u \u2264 1 in \u2126, u = \u2206u = 0 on \u2202\u2126, (G\u03bb) where \u03bb > 0 is a parameter, \u03c4 > 0, \u03b2 > 0 are fixed constants, and \u2126 \u2282N (N \u2265 2) is a bounded smooth domain. This problem with \u03b2 = 0 models a simple electrostatic Micro-Electromechanical Systems (MEMS) device which has been recently studied by many authors. For instance, see [3], [5], [7], [8], [9], [10], [11], [14], [15], [16], and the references cited therein. Recently, Lin and Yang [18] derived the equation (G\u03bb) in the study of the charged plates in electrostatic actuators. They showed that there exists 0 < \u03bb\u2217 <\u221e such that for \u03bb \u2208 (0, \u03bb\u2217) (G\u03bb) has a minimal regular solutions u\u03bb (supB u\u03bb < 1) while for \u03bb > \u03bb\u2217, (G\u03bb) does not have any regular solution. Moreover, the branch \u03bb\u2192 u\u03bb(x) is increasing for each x \u2208 B, and therefore the function u\u2217 = lim\u03bb\u2197\u03bb\u2217 u\u03bb can be considered as a generalized solution that corresponds to the pull-in voltage \u03bb\u2217. Now the important question is whether the extremal solution u\u2217 is regular or not. In a recent paper Guo and Wei [17] proved that the extremal solution u\u2217 is regular for dimensions N \u2264 4. In this paper we consider the problem (G\u03bb) on the unit ball in N : \uf8f1\uf8f2\uf8f3 \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bb(1\u2212u)2 in B, 0 < u \u2264 1 in B, u = \u2206u = 0 on \u2202B, (P\u03bb) 5A version of this chapter has been accepted for publication. A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Journal of Differential Equations, 248 (2010), 594-616. 133 6.1. Introduction and show that the critical dimension for (P\u03bb) is N = 9. Indeed we prove that the extremal solution of (P\u03bb) is regular (supB u\u2217 < 1) for N \u2264 8 and \u03b2, \u03c4 > 0 and it is singular (supB u\u2217 = 1) for N \u2265 9, \u03b2 > 0, and \u03c4 > 0 with \u03c4\u03b2 small. Our proof of regularity of the extremal solution in dimensions 5 \u2264 N \u2264 8 is heavily inspired by [4] and [6]. On the other hand we shall use certain improved Hardy- Rellich inequalities to prove that the extremal solution is singular in dimensions N \u2265 9. Our improve Hardy-Rellich inequalities follow from the recent result of Ghoussoub-Moradifam [12] about Hardy and Hardy-Rellich inequalities. We now start by recalling some of the results from [17] concerning (P\u03bb) that will be needed in the sequel. Define \u03bb\u2217(B) := sup{\u03bb > 0 : (P\u03bb) has a classical solution}. We now introduce the following notion of solution. We say that u is a weak solution of (G\u03bb), if 0 \u2264 u \u2264 1 a.e. in \u2126, 1(1\u2212u)2 \u2208 L1(\u2126) and if\u222b \u2126 u(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) dx = \u03bb \u222b \u2126 \u03c6 (1\u2212 u)2 dx, \u2200\u03c6 \u2208W 4,2(\u2126) \u2229H10 (\u2126), Say that u is a weak super-solution (resp. weak sub-solution) of (G\u03bb), if the equality is replaced with \u2265 (resp. \u2264) for \u03c6 \u2265 0. We now introduce the notion of stability. First, we equip the function space H := H2(\u2126) \u2229H10 (\u2126) =W 2,2(\u2126) \u2229H10 (\u2126) with the norm \u2016\u03c8\u2016 = (\u222b \u2126 [\u03c4 |\u2207\u03c8|2 + \u03b2|\u2206\u03c8|2]dx )1\/2 . We say that a weak solution u\u03bb of (G\u03bb) is stable (respectively semi-stable) if the first eigenvalue \u00b51,\u03bb(u\u03bb) of the problem \u2212\u03c4\u2206h+ \u03b2\u22062h\u2212 2\u03bb (1\u2212 u\u03bb)3h = \u00b5h in \u2126, h = \u2206h = 0 on \u2202\u2126 (6.1) is positive (resp., nonnegative). The operator \u03b2\u22062u\u2212\u03c4\u2206u satisfies the following maximum principle which will be frequently used in the sequel. Lemma 6.2. ([17]) Let u \u2208 L1(\u2126). Then u \u2265 0 a.e. in \u2126, provided one of the following conditions hold: 1. u \u2208 C4(\u2126), \u03b2\u22062u\u2212 \u03c4\u2206u \u2265 0 on \u2126, and u = \u2206u = 0 on \u2202\u2126. 2. \u222b \u2126 u(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) dx \u2265 0 for all 0 \u2264 \u03c6 \u2208W 4,2(\u2126) \u2229H10 (\u2126). 3. u \u2208 W 2,2(\u2126), u = 0, \u2206u \u2264 0 on \u2202B, and \u222b \u2126 [ \u03b2\u2206u\u2206\u03c6 + \u03c4\u2207u\u2207\u03c6]dx \u2265 0 for all 0 \u2264 \u03c6 \u2208W 2,2(\u2126) \u2229H10 (\u2126). Moreover, either u \u2261 0 or u > 0 a.e. in \u2126. 134 6.2. The pull-in voltage 6.2 The pull-in voltage As in [6] and [4], we are led here to examine problem (P\u03bb) with non-homogeneous boundary conditions such as\uf8f1\uf8f2\uf8f3 \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bb(1\u2212u)2 in B, \u03b1 < u \u2264 1 in B, u = \u03b1, \u2206u = \u03b3 on \u2202B, (P\u03bb, \u03b1, \u03b3) where \u03b1, \u03b3 are given. Whenever we need to emphasis the parameters \u03b2 and \u03c4 we will refer to problem (P\u03bb,\u03b1,\u03b3) as (P\u03bb,\u03b2,\u03c4,\u03b1,\u03b3). In this section and Section 3 we will obtain several results for the following general form of (P\u03bb, \u03b1, \u03b3)\uf8f1\uf8f2\uf8f3 \u03b2\u22062u\u2212 \u03c4\u2206u = \u03bb(1\u2212u)2 in \u2126, \u03b1 < u \u2264 1 in \u2126, u = \u03b1, \u2206u = \u03b3 on \u2202\u2126, (G\u03bb, \u03b1, \u03b3) which are analogous to the results obtained by Gui and Wei for (G\u03bb) in [17]. Let \u03a6 denote the unique solution of{ \u03b2\u22062\u03a6\u2212 \u03c4\u2206\u03a6 = 0 in \u2126, \u03a6 = \u03b1 , \u2206\u03a6 = \u03b3 on \u2202\u2126. (6.3) We will say that the pair (\u03b1, \u03b3) is admissible if \u03b3 \u2264 0, \u03b1 < 1, and sup\u2126 \u03a6 < 1. We now introduce a notion of weak solution. We say that u is a weak solution of (P\u03bb,\u03b1,\u03b3), if \u03b1 \u2264 u \u2264 1 a.e. in \u2126, 1(1\u2212u)2 \u2208 L1(\u2126) and if\u222b \u2126 (u\u2212 \u03a6)(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) = \u03bb \u222b \u2126 \u03c6 (1\u2212 u)2 \u2200\u03c6 \u2208W 4,2(\u2126) \u2229H10 (\u2126), where \u03a6 is given in (6.3). We say u is a weak super-solution (resp. weak sub- solution) of (P\u03bb,\u03b1,\u03b3), if the equality is replaced with \u2265 (resp. \u2264) for \u03c6 \u2265 0. We say a weak solution u of (P\u03bb,\u03b1,\u03b3) is regular (resp. singular) if \u2016u\u2016\u221e < 1 (resp. \u2016u\u2016\u221e = 1). We now define \u03bb\u2217(\u03b1, \u03b3) := sup { \u03bb > 0 : (P\u03bb,\u03b1,\u03b3) has a classical solution } and \u03bb\u2217(\u03b1, \u03b3) := sup { \u03bb > 0 : (P\u03bb,\u03b1,\u03b3) has a weak solution } . Observe that by the Implicit Function Theorem, we can classically solve (P\u03bb,\u03b1,\u03b3) for small \u03bb\u2019s. Therefore, \u03bb\u2217(\u03b1, \u03b3) and \u03bb\u2217(\u03b1, \u03b3) are well defined for any admissible pair (\u03b1, \u03b3). To cut down on notations we won\u2019t always indicate \u03b1 and \u03b3. For example, \u03bb\u2217 and \u03bb\u2217 will denote the \u201cweak and strong critical voltages\u201d of (P\u03bb,\u03b1,\u03b3). Now let U be a weak super-solution of (P\u03bb,\u03b1,\u03b3) and recall the following existence result. ([17]) For every 0 \u2264 f \u2208 L1(\u2126) there exists a unique 0 \u2264 u \u2208 L1(\u2126) which satisfies \u222b \u2126 u(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) dx = \u222b \u2126 f\u03c6 dx, 135 6.2. The pull-in voltage for all \u03c6 \u2208W 4,2(\u2126) \u2229H10 (\u2126). We can introduce the following \u201cweak\u201d iterative scheme: u0 = U and (induc- tively) let un, n \u2265 1, be the solution of\u222b \u2126 (un \u2212 \u03a6)(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) = \u03bb \u222b \u2126 \u03c6 (1\u2212 un\u22121)2 \u2200 \u03c6 \u2208W 4,2(\u2126\u0304) \u2229H10 (\u2126) given by Theorem 6.2. Since 0 is a sub-solution of (P\u03bb,\u03b1,\u03b3), inductively it is easily shown by Lemma 6.2 that \u03b1 \u2264 un+1 \u2264 un \u2264 U for every n \u2265 0. Since (1\u2212 un)\u22122 \u2264 (1\u2212 U)\u22122 \u2208 L1(\u2126), by Lebesgue Theorem the function u = lim n\u2192+\u221eun is a weak solution of (P\u03bb,\u03b1,\u03b3) so that \u03b1 \u2264 u \u2264 U . We therefore have the following result. Lemma 6.4. Assume the existence of a weak super-solution U of (P\u03bb,\u03b1,\u03b3). Then there exists a weak solution u of (P\u03bb,\u03b1,\u03b3) so that \u03b1 \u2264 u \u2264 U a.e. in \u2126. In particular, for every \u03bb \u2208 (0, \u03bb\u2217), we can find a weak solution of (P\u03bb,\u03b1,\u03b3). In the same range of \u03bb\u2032s, this is still true for regular weak solutions as shown in the following lemma. Lemma 6.5. Let (\u03b1, \u03b3) be an admissible pair and u be a weak solution of (P\u03bb,\u03b1,\u03b3). Then, there exists a regular solution for every 0 < \u00b5 < \u03bb. Proof: Let \u000f \u2208 (0, 1) be given and let u\u0304 = (1\u2212 \u000f)u+ \u000f\u03a6, where \u03a6 is given in (6.3). By Lemma 6.2 sup\u2126 \u03a6 < sup\u2126 u \u2264 1. Hence sup \u2126 u\u0304 \u2264 (1\u2212 \u000f) + \u000f sup \u2126 \u03a6 < 1 , inf \u2126 u\u0304 \u2265 (1\u2212 \u000f)\u03b1+ \u000f inf \u2126 \u03a6 = \u03b1, and for every 0 \u2264 \u03c6 \u2208W 4,2(\u2126\u0304) \u2229H10 (\u2126) there holds:\u222b \u2126 (u\u0304\u2212 \u03a6)(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) = (1\u2212 \u000f) \u222b \u2126 (u\u2212 \u03a6)(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) = (1\u2212 \u000f)\u03bb \u222b \u2126 \u03c6 (1\u2212 u)2 = (1\u2212 \u000f)3\u03bb \u222b \u2126 \u03c6 (1\u2212 u\u0304+ \u000f(\u03a6\u2212 1))2 \u2265 (1\u2212 \u000f)3\u03bb \u222b \u2126 \u03c6 (1\u2212 u\u0304)2 . Note that 0 \u2264 (1\u2212\u000f)(1\u2212u) = 1\u2212 u\u0304+\u000f(\u03a6\u22121) < 1\u2212 u\u0304. So u\u0304 is a weak super-solution of (P(1\u2212\u000f)3\u03bb,\u03b1,\u03b3) so that sup \u2126 u\u0304 < 1. By Lemma 6.4 we get the existence of a weak solution w of (P(1\u2212\u000f)3\u03bb,\u03b1,\u03b3) so that \u03b1 \u2264 w \u2264 u\u0304. In particular, sup \u2126 w < 1 and w is a regular weak solution. Since \u000f \u2208 (0, 1) is arbitrarily chosen, the proof is done. \u0003 136 6.2. The pull-in voltage Lemma 6.5 implies the existence of a regular weak solution U\u03bb for every \u03bb \u2208 (0, \u03bb\u2217). Introduce now a \u201cclassical\u201d iterative scheme: u0 = 0 and (inductively) un = vn +\u03a6, n \u2265 1, where vn \u2208W 4,2(\u2126) \u2229H10 (\u2126) is the solution of \u03b2\u22062vn \u2212 \u03c4\u2206vn = \u03b2\u22062un \u2212 \u03c4\u2206un = \u03bb(1\u2212 un\u22121)2 in \u2126 and \u2206vn = 0 on \u2202\u2126. (6.6) Since vn \u2208W 4,2(\u2126)\u2229H10 (\u2126), un is also a weak solution of (6.6), and by Lemma 6.2 we know that \u03b1 \u2264 un \u2264 un+1 \u2264 U\u03bb for every n \u2265 0. Since sup \u2126 un \u2264 sup \u2126 U\u03bb < 1 for n \u2265 0, we get that (1\u2212un\u22121)\u22122 \u2208 L2(\u2126) and the existence of vn is guaranteed. Since vn is easily seen to be uniformly bounded in H2(\u2126), we have that u\u03bb := lim n\u2192+\u221eun does hold pointwise and weakly in H2(\u2126). By Lebesgue theorem, we have that u\u03bb is a radial weak solution of (P\u03bb) so that sup \u2126 u\u03bb \u2264 sup \u2126 U\u03bb < 1. By elliptic regularity theory [1], u\u03bb \u2208 C\u221e(\u2126\u0304) and u\u03bb = \u2206u\u03bb = 0 on \u2202\u2126. So we can integrate by parts to get \u222b \u2126 \u03b2(\u22062u\u03bb \u2212 \u03c4\u2206u\u03bb)\u03c6dx = \u222b \u2126 u\u03bb(\u03b2\u22062\u03c6\u2212 \u03c4\u2206\u03c6) dx = \u03bb \u222b \u2126 \u03c6 (1\u2212 u\u03bb)2 for every \u03c6 \u2208 W 4,2(\u2126) \u2229H10 (\u2126). Hence, u\u03bb is a classical solution of (P\u03bb) showing that \u03bb\u2217 = \u03bb\u2217. Since the argument above shows that u\u03bb < U for any other classical solution U of (P\u00b5, \u03b1, \u03b3) with \u00b5 \u2265 \u03bb, we have that u\u03bb is exactly the minimal solution and u\u03bb is strictly increasing as \u03bb \u2191 \u03bb\u2217. In particular, we can define u\u2217 in the usual way: u\u2217(x) = lim \u03bb\u2197\u03bb\u2217 u\u03bb(x). Lemma 6.7. \u03bb\u2217(\u2126) < +\u221e. Proof: Let u be a classical solution of (P\u03bb,\u03b1,\u03b3) and let (\u03c8, \u00b51) with \u2206\u03c8 = 0 on \u2202\u2126 denote the first eigenpair of \u03b2\u22062 \u2212 \u03c4\u2206 in H2(\u2126) \u2229H10 (\u2126) with \u03c8 > 0. Now let C be such that \u222b \u2202\u2126 ((\u03c4\u03b1\u2212 \u03b2\u03b3)\u2202\u03bd\u03c8 \u2212 \u03b2\u03b1\u2202\u03bd(\u2206\u03c8)) = C \u222b \u2126 \u03c8. Multiplying (P\u03bb,\u03b1,\u03b3) by \u03c8 and then integrating by parts one arrives at\u222b \u2126 ( \u03bb (1\u2212 u)2 \u2212 \u00b51u\u2212 C ) \u03c8 = 0. Since \u03c8 > 0 there must exist a point x\u0304 \u2208 \u2126 where \u03bb(1\u2212u(x\u0304))2 \u2212\u00b51u(x\u0304)\u2212C \u2264 0. Since \u03b1 < u(x\u0304) < 1, hence one can conclude that \u03bb \u2264 sup0____ \u03bb\u2217 there are no weak solutions of (P\u03bb,\u03b1,\u03b3). 6.3 Stability of the minimal solutions This section is devoted to the proof of the following stability result for minimal solutions. We shall need the following notion of H\u2212weak solutions, which is an intermediate class between classical and weak solutions. We say that u is an H\u2212weak solution of (P\u03bb,\u03b1,\u03b3) if u \u2212 \u03a6 \u2208 H2(\u2126) \u2229 H10 (\u2126), 0 \u2264 u \u2264 1 a.e. in \u2126, 1(1\u2212u)2 \u2208 L1(\u2126) and\u222b \u2126 [ \u03b2\u2206u\u2206\u03c6+ \u03c4\u2207u\u2207\u03c6]dx = \u03bb \u222b \u2126 \u03c6 (1\u2212 u)2 , \u2200\u03c6 \u2208W 2,2(\u2126) \u2229H10 (\u2126). where \u03a6 is given by (6.3). We say that u is an H\u2212weak super-solution (resp. an H\u2212weak sub-solution) of (P\u03bb,\u03b1,\u03b3) if for \u03c6 \u2265 0 the equality is replaced with \u2265 (resp. \u2264) and u \u2265 0 (resp. \u2264), \u2206u \u2264 0 (resp. \u2265) on \u2202\u2126. Suppose that (\u03b1, \u03b3) is an admissible pair. 1. The minimal solution u\u03bb is stable, and is the unique semi-stable H\u2212weak solution of (P\u03bb,\u03b1,\u03b3). 2. The function u\u2217 := lim \u03bb\u2197\u03bb\u2217 u\u03bb is a well-defined semi-stable H\u2212weak solution of (P\u03bb\u2217,\u03b1,\u03b3). 3. u\u2217 is the unique H\u2212weak solution of (P\u03bb\u2217,\u03b1,\u03b3), and when u\u2217 is classical solution, then \u00b51(u\u2217) = 0. 4. If v is a singular, semi-stable H\u2212weak solution of (P\u03bb,\u03b1,\u03b3), then v = u\u2217 and \u03bb = \u03bb\u2217 The main tool is the following comparison lemma which is valid exactly in the class H. Lemma 6.8. Let (\u03b1, \u03b3) be an admissible pair and u be a semi-stable H\u2212weak solution of (P\u03bb,\u03b1,\u03b3). Assume U is a H\u2212weak super-solution of (P\u03bb,\u03b1,\u03b3). Then 1. u \u2264 U a.e. in \u2126; 2. If u is a classical solution and \u00b51(u) = 0 then U = u. 138 6.3. Stability of the minimal solutions Proof: (i) Define w := u \u2212 U . Then by means of the Moreau decomposition for the biharmonic operator (see [19] and [2]), there exist w1 and w2 \u2208 H2(\u2126)\u2229H10 (\u2126), with w = w1 + w2, w1 \u2265 0 a.e., \u03b2\u22062w2 \u2212 \u03c4\u2206w2 \u2264 0 in the H\u2212weak sense and\u222b \u2126 \u03b2\u2206w1\u2206w2 + \u03c4\u2207w1.\u2207w2 = 0. Lemma 6.2 gives that w2 \u2264 0 a.e. in \u2126. Given 0 \u2264 \u03c6 \u2208 C\u221ec (\u2126), we have\u222b \u2126 \u03b2\u2206w\u2206\u03c6+ \u03c4\u2207w.\u2207\u03c6 \u2264 \u03bb \u222b \u2126 (f(u)\u2212 f(U))\u03c6, where f(u) := (1\u2212 u)\u22122. Since u is semi-stable, one has \u03bb \u222b \u2126 f \u2032(u)w21 \u2264 \u222b \u2126 \u03b2(\u2206w1)2 + \u03c4 |\u2207w1|2 = \u222b \u2126 \u03b2\u2206w\u2206w1 + \u03c4\u2207w.\u2207w1 \u2264 \u03bb \u222b \u2126 (f(u)\u2212 f(U))w1. Since w1 \u2265 w one has \u222b \u2126 f \u2032(u)ww1 \u2264 \u222b \u2126 (f(u)\u2212 f(U))w1, which re-arranged gives \u222b \u2126 f\u0303w1 \u2265 0, where f\u0303(u) = f(u) \u2212 f(U) \u2212 f \u2032(u)(u \u2212 U). The strict convexity of f gives f\u0303 \u2264 0 and f\u0303 < 0 whenever u 6= U . Since w1 \u2265 0 a.e. in \u2126, one sees that w \u2264 0 a.e. in \u2126. The inequality u \u2264 U a.e. in \u2126 is then established. (ii) Since u is a classical solution, it is easy to see that the infimum of \u00b51(u) is attained at some \u03c6. The function \u03c6 is then the first eigenfunction of \u03b2\u22062 \u2212 \u03c4\u2206\u2212 2\u03bb (1\u2212u)3 in H 2(\u2126) \u2229 H10 (\u2126). Now we show that \u03c6 is of fixed sign. Using the above decomposition, one has \u03c6 = \u03c61+\u03c62 where \u03c6i \u2208 H2(\u2126)\u2229H10 (\u2126) for i = 1, 2, \u03c61 \u2265 0,\u222b \u2126 \u03b2\u2206\u03c61\u2206\u03c62 + \u03c4\u2207\u03c61.\u2207\u03c62 = 0 and \u03b2\u22062\u03c62 \u2212 \u03c4\u2206\u03c62 \u2264 0 in the H\u2212weak sense. If \u03c6 changes sign, then \u03c61 6\u2261 0 and \u03c62 < 0 in \u2126 (recall that either \u03c62 < 0 or \u03c62 = 0 a.e. in \u2126). We can write now 0 = \u00b51(u) \u2264 \u222b \u2126 \u03b2(\u2206(\u03c61 \u2212 \u03c62))2 + \u03c4 |\u2207(\u03c61 \u2212 \u03c62)|2 \u2212 \u03bbf \u2032(u)(\u03c61 \u2212 \u03c62)2\u222b \u2126 (\u03c61 \u2212 \u03c62)2 < \u222b \u2126 \u03b2(\u2206\u03c6)2 + \u03c4 |\u2207\u03c6|2 \u2212 \u03bbf \u2032(u)\u03c62\u222b \u2126 \u03c62 = \u00b51(u), in view of \u03c61\u03c62 < \u2212\u03c61\u03c62 in a set of positive measure, leading to a contradiction. So we can assume \u03c6 \u2265 0, and by Lemma 6.2 we have \u03c6 > 0 in \u2126. For 0 \u2264 t \u2264 1, define g(t) = \u222b \u2126 \u03b2\u2206 [tU + (1\u2212 t)u]\u2206\u03c6+ \u03c4\u2207 [tU + (1\u2212 t)u] .\u2207\u03c6\u2212 \u03bb \u222b \u2126 f(tU + (1\u2212 t)u)\u03c6, 139 6.3. Stability of the minimal solutions where \u03c6 is the above first eigenfunction. Since f is convex one sees that g(t) \u2265 \u03bb \u222b \u2126 [tf(U) + (1\u2212 t)f(u)\u2212 f(tU + (1\u2212 t)u)]\u03c6 \u2265 0 for every t \u2265 0. Since g(0) = 0 and g\u2032(0) = \u222b \u2126 \u03b2\u2206(U \u2212 u)\u2206\u03c6+ \u03c4\u2207(U \u2212 u).\u2207\u03c6\u2212 \u03bbf \u2032(u)(U \u2212 u)\u03c6 = 0, we get that g\u2032\u2032(0) = \u2212\u03bb \u222b \u2126 f \u2032\u2032(u)(U \u2212 u)2\u03c6 \u2265 0. Since f \u2032\u2032(u)\u03c6 > 0 in \u2126, we finally get that U = u a.e. in \u2126. \u0003 A more general version of Lemma 6.8 is available in the following. Lemma 6.9. Let (\u03b1, \u03b3) be an admissible pair and \u03b3\u2032 \u2264 0. Let u be a semi-stable H\u2212weak sub-solution of (P\u03bb,\u03b1,\u03b3) with u = \u03b1\u2032 \u2264 \u03b1, \u2206u = \u03b2\u2032 \u2265 \u03b2 on \u2202\u2126. Assume that U is a H\u2212weak super-solution of (P\u03bb,\u03b1,\u03b3) with U = \u03b1, \u2206U = \u03b2 on \u2202\u2126. Then U \u2265 u a.e. in \u2126. Proof: Let u\u0303 \u2208 H2(\u2126)\u2229H10 (\u2126) denote a weak solution of \u03b2\u22062u\u0303\u2212 \u03c4\u2206u\u0303 = \u03b2\u22062(u\u2212 U)\u2212\u03c4\u2206(u\u2212U) in \u2126 and u\u0303 = \u2206u\u0303 = 0 on \u2202\u2126. Since u\u0303\u2212u+U \u2265 0 and \u2206(u\u0303\u2212u+U) \u2264 0 on \u2202\u2126, by Lemma 6.2 one has that u\u0303 \u2265 u\u2212 U a.e. in \u2126. By means of the Moreau decomposition (see [19] and [2]) we write u\u0303 as u\u0303 = w+v, where w, v \u2208 H20 (\u2126), w \u2265 0 a.e. in \u2126, \u03b2\u22062v \u2212 \u03c4\u2206v \u2264 0 in a H\u2212weak sense and \u222b \u2126 \u03b2\u2206w\u2206v + \u03c4\u2207w.\u2207v = 0. Then for 0 \u2264 \u03c6 \u2208W 4,2(\u2126\u0304) \u2229H10 (\u2126), one has\u222b \u2126 \u03b2\u2206u\u0303\u2206\u03c6+ \u03c4\u2207u\u0303.\u2207\u03c6 \u2264 \u03bb \u222b \u2126 (f(u)\u2212 f(U))\u03c6. In particular, we have\u222b \u2126 \u03b2\u2206u\u0303\u2206w + \u03c4\u2207u\u0303.\u2207w \u2264 \u03bb \u222b \u2126 (f(u)\u2212 f(U))w. Since the semi-stability of u gives that \u03bb \u222b \u2126 f \u2032(u)w2 \u2264 \u222b \u2126 \u03b2(\u2206w)2 + \u03c4 |\u2207w|2 = \u222b \u2126 \u03b2\u2206u\u0303\u2206w + \u03c4\u2207u\u0303.\u2207w, we get that \u222b \u2126 f \u2032(u)w2 \u2264 \u222b \u2126 (f(u)\u2212 f(U))w. By Lemma 6.2 we have v \u2264 0 and then w \u2265 u\u0303 \u2265 u\u2212U a.e. in \u2126. So we obtain that 0 \u2264 \u222b \u2126 (f(u)\u2212 f(U)\u2212 f \u2032(u)(u\u2212 U))w. The strict convexity of f implies that U \u2265 u a.e. in \u2126. \u0003 140 6.3. Stability of the minimal solutions We need also some a-priori estimates along the minimal branch u\u03bb. Lemma 6.10. Let (\u03b1, \u03b3) be an admissible pair. Then for every \u03bb \u2208 (0, \u03bb\u2217), we have 2 \u222b \u2126 (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 \u2264 \u222b \u2126 u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 , where \u03a6 is given by (6.3). In particular, there is a constant C > 0 independent of \u03bb so that \u222b \u2126 (\u03c4 |\u2207u\u03bb|2 + \u03b2|\u2206u\u03bb|2)dx+ \u222b \u2126 1 (1\u2212 u\u03bb)3 \u2264 C, (6.11) for every \u03bb \u2208 (0, \u03bb\u2217). Proof: Testing (P\u03bb,\u03b1,\u03b3) on u\u03bb \u2212 \u03a6 \u2208W 4,2(\u2126) \u2229H10 (\u2126), we see that \u03bb \u222b \u2126 u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 = \u222b \u2126 (\u03c4 |\u2207(u\u03bb \u2212 \u03a6)|2 + \u03b2(\u2206(u\u03bb \u2212 \u03a6))2)dx \u2265 2\u03bb \u222b \u2126 (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 . In the view of \u03b2\u22062\u03a6\u2212 \u03c4\u2206\u03a6 = 0. In particular, for \u03b4 > 0 small we have that\u222b {|u\u03bb|\u2265\u03b4} 1 (1\u2212 u\u03bb)3 \u2264 1 \u03b42 \u222b {|u\u03bb\u2212\u03a6|\u2265\u03b4} (u\u03bb \u2212 \u03a6)2 (1\u2212 u\u03bb)3 \u2264 1 \u03b42 \u222b \u2126 1 (1\u2212 u\u03bb)2 \u2264 \u03b4 \u222b {|u\u03bb\u2212\u03a6|\u2265\u03b4} 1 (1\u2212 u\u03bb)3 + C\u03b4 by means of Young\u2019s inequality. Since for \u03b4 small\u222b {|u\u03bb\u2212\u03a6|\u2264\u03b4} 1 (1\u2212 u\u03bb)3 \u2264 C, for some C > 0, we get that \u222b \u2126 1 (1\u2212 u\u03bb)3 \u2264 C, for some C > 0 and for every \u03bb \u2208 (0, \u03bb\u2217). Since\u222b \u2126 (\u03c4 |\u2207u\u03bb|2 + \u03b2|\u2206u\u03bb|2)dx = \u222b \u2126 (\u03b2\u2206u\u03bb\u2206\u03a6+ \u03c4\u2207u\u03bb.\u2207\u03a6) + \u03bb \u222b \u2126 u\u03bb \u2212 \u03a6 (1\u2212 u\u03bb)2 \u2264 \u03b4 \u222b \u2126 (\u03c4 |\u2207u\u03bb|2 + \u03b2|\u2206u\u03bb|2)dx + C\u03b4 + C (\u222b \u2126 1 (1\u2212 u\u03bb)3 ) 2 3 in view of Young\u2019s and Ho\u0308lder\u2019s inequalities, estimate (6.11) is finally established. \u0003 Proof of Theorem 6.3: (1) Since \u2016u\u03bb\u2016\u221e < 1, the infimum defining \u00b51(u\u03bb) is achieved at a first eigenfunction for every \u03bb \u2208 (0, \u03bb\u2217). Since \u03bb 7\u2192 u\u03bb(x) is increasing 141 6.4. Regularity of the extremal solutions in dimensions N \u2264 8 for every x \u2208 \u2126, it is easily seen that \u03bb 7\u2192 \u00b51(u\u03bb) is a decreasing and continuous function on (0, \u03bb\u2217). Define \u03bb\u2217\u2217 := sup{0 < \u03bb < \u03bb\u2217 : \u00b51(u\u03bb) > 0}. We have that \u03bb\u2217\u2217 = \u03bb\u2217. Indeed, otherwise we would have \u00b51(u\u03bb\u2217\u2217) = 0, and for every \u00b5 \u2208 (\u03bb\u2217\u2217, \u03bb\u2217), u\u00b5 would be a classical super-solution of (P\u03bb\u2217\u2217,\u03b1,\u03b3). A contra- diction arises since Lemma 6.8 implies u\u00b5 = u\u03bb\u2217\u2217 . Finally, Lemma 6.8 guarantees the uniqueness in the class of semi-stable H\u2212weak solutions. (2) It follows from (6.11) that u\u03bb \u2192 u\u2217 in a pointwise sense and weakly in H2(\u2126), and 11\u2212u\u2217 \u2208 L3(\u2126). In particular, u\u2217 is a H2\u2212weak solution of (P\u03bb\u2217,\u03b1,\u03b3) which is also semi-stable as the limiting function of the semi-stable solutions {u\u03bb}. (3) Whenever \u2016u\u2217\u2016\u221e < 1, the function u\u2217 is a classical solution, and by the Implicit Function Theorem we have that \u00b51(u\u2217) = 0 to prevent the continuation of the minimal branch beyond \u03bb\u2217. By Lemma 6.8, u\u2217 is then the unique H\u2212weak solution of (P\u03bb\u2217,\u03b1,\u03b3). (4) If \u03bb < \u03bb\u2217, we get by uniqueness that v = u\u03bb. So v is not singular and a contradiction arises. Now, by Theorem 6.2(3) we have that \u03bb = \u03bb\u2217. Since v is a semi-stable H\u2212 weak solution of (P\u03bb\u2217,\u03b1,\u03b3) and u\u2217 is a H\u2212 weak super-solution of (P\u03bb\u2217,\u03b1,\u03b3), we can apply Lemma 6.8 to get v \u2264 u\u2217 a.e. in \u2126. Since u\u2217 is also a semi-stable solution, we can reverse the roles of v and u\u2217 in Lemma 6.8 to see that v \u2265 u\u2217 a.e. in \u2126. So equality v = u\u2217 holds and the proof is done. \u0003 6.4 Regularity of the extremal solutions in dimensions N \u2264 8 In this section we shall show that the extremal solution is regular in small dimen- sions. Let us begin with the following lemma. Lemma 6.12. Let N \u2265 5 and (u\u2217, \u03bb\u2217) be the extremal pair of (P\u03bb). If u\u2217 is singular, and he set \u0393 := {r \u2208 (0, 1) : u\u03b4(r) > u\u2217(r)} (6.13) is non-empty, where u\u03b4(x) := 1 \u2212 C\u03b4|x| 43 and C\u03b4 > 1 is a constant. Then there exists r1 \u2208 (0, 1) such that u\u03b4(r1) \u2265 u\u2217(r1) and \u2206u\u03b4(r1) \u2264 \u2206u\u2217(r1). Proof. Assume by contradiction that for every r with u\u03b4(r1) \u2265 u\u2217(r1) one has \u2206u\u03b4(r1) > \u2206u\u2217(r1). Since \u0393 is non-empty and u\u03b4(1) = 1\u2212 C\u03b4 < 0 = u\u2217(1), there exists s1 \u2208 (0, 1) such that u\u03b4(s1) = u\u2217(s1). We claim that u\u03b4(s) > u\u2217(s), 142 6.4. Regularity of the extremal solutions in dimensions N \u2264 8 for 0 < s < s1. Assume that there exist s3 < s2 \u2264 s1 such that u\u2217(s2) = u\u03b4(s2), u\u2217(s3) = u\u03b4(s3) and u\u03b4(s) \u2265 u\u2217(s) for s \u2208 (s3, s2). By our assumption \u2206us > \u2206u\u2217(s) for s \u2208 (s3, s2) which contradicts the maximum principle and justifies the claim. Therefore u\u03b4(s) > u\u2217(s) for 0 < s < s1. Now set w := u\u03b4 \u2212 u\u2217. Then w \u2265 0 on Bs1 and \u2206w \u2264 0 in Bs1 . Since w(0) = 0, by strong maximum principle we get w \u2261 0 on Bs1 . This is a contradiction and completes the proof. \u0003 Let N \u2265 5 and (u\u2217, \u03bb\u2217) be the extremal pair of (P\u03bb). When u\u2217 is singular, then 1\u2212 u\u2217 \u2264 C|x| 43 in B, where C := ( \u03bb \u2217 \u03b2\u03bb\u0304 ) 1 3 and \u03bb\u0304 := 8(N\u2212 2 3 )(N\u2212 83 ) 9 . Proof. For \u03b4 > 0, define u\u03b4(x) := 1\u2212C\u03b4|x| 43 with C\u03b4 := ( \u03bb\u2217\u03b2\u03bb\u0304 + \u03b4) 1 3 > 1. Since N \u2265 5, we have that u\u03b4 \u2208 H2loc(N ) and u\u03b4 is a H\u2212weak solution of \u03b2\u22062u\u03b4 \u2212 \u03c4\u2206u\u03b4 = \u03bb \u2217 + \u03b2\u03b4\u03bb\u0304 (1\u2212 u\u03b4)2 + 4 3 \u03c4C\u03b4(N \u2212 23)|x| \u2212 23 in N . We claim that u\u03b4 \u2264 u\u2217 in B, which will finish the proof by just letting \u03b4 \u2192 0. Assume by contradiction that the set \u0393 := {r \u2208 (0, 1) : u\u03b4(r) > u\u2217(r)} is non-empty. By Lemma 6.12 the set \u039b := {r \u2208 (0, 1) : u\u03b4(r) \u2265 u\u2217(r) and \u2206u\u03b4(r) \u2264 \u2206u\u2217(r)} is non-empty. Let r1 \u2208 \u039b. Since u\u03b4(1) = 1\u2212 C\u03b4 < 0 = u\u2217(1), we have that 0 < r1 < 1. Define \u03b1 := u\u2217(r1) \u2264 u\u03b4(r1), \u03b3 := \u2206u\u2217(r1) \u2265 \u2206u\u03b4(r1). Setting u\u03b4,r1 = r \u2212 43 1 (u\u03b4(r1r)\u2212 1) + 1, we see that u\u03b4,r1 is a H\u2212weak super-solution of (P\u03bb\u2217+\u03b4\u03bb,\u03b2,r\u221221 \u03c4,\u03b1\u2032,\u03b3\u2032), where \u03b1\u2032 := r\u2212 4 3 1 (\u03b1\u2212 1) + 1, \u03b3\u2032 = r 2 3 1 \u03b3. Similarly, define u\u2217r1(r) = r \u2212 43 1 (u \u2217(r1r) \u2212 1) + 1. Note that \u22062u\u2217 \u2212 \u03b1\u2206u\u2217 \u2265 0 in B and \u2206u\u2217 = 0 on \u2202B. Hence by maximum principle we have \u2206u\u2217 \u2264 0 in B and therefore \u03b3\u2032 \u2264 0. Also obviously \u03b1\u2032 < 1. So, (\u03b1\u2032, \u03b3\u2032) is an admissible pair and by Theorem 6.3(4) we get that (u\u2217r1 , \u03bb \u2217) coincides with the extremal pair of (P\u03bb,\u03b2,r\u221221 \u03c4,\u03b1\u2032,\u03b3\u2032) in B. Also by Lemma 6.4 we get the existence of a week solution of (P\u03bb\u2217+\u03b4\u03bb,\u03b2,r\u221221 \u03c4,\u03b1\u2032,\u03b3\u2032). Since \u03bb \u2217 + \u03b4\u03bb > \u03bb\u2217, we contradict the fact that \u03bb\u2217 is the extremal parameter of (P\u03bb,\u03b2,r\u221221 \u03c4,\u03b1\u2032,\u03b3\u2032). \u0003 Now we are ready to prove the following result. If 5 \u2264 N \u2264 8, then the extremal solution u\u2217 of (P )\u03bb is regular. 143 6.5. The extremal solution is singular in dimensions N \u2265 9 Proof. Assume that u\u2217 is singular. For \u000f > 0 define \u03d5(x) := |x| 4\u2212N2 +\u000f and note that (\u2206\u03d5)2 = (HN +O(\u000f))|x|\u2212N+2\u000f, where HN := N 2(N \u2212 4)2 16 . Given \u03b7 \u2208 C\u221e0 (B), and since N \u2265 5, we can use the test function \u03b7\u03d5 \u2208 H20 (B) into the stability inequality to obtain 2\u03bb\u2217 \u222b B \u03d52 (1\u2212 u\u2217)3 \u2264 \u03b2 \u222b B (\u2206\u03d5)2 + \u03c4 \u222b B |\u2207\u03d5|2 +O(1), where O(1) is a bounded function as \u000f\u2192 0. By Theorem 6.4 we find 2\u03bb\u0304 \u222b B \u03d52 |x|4 \u2264 \u222b B (\u2206\u03d5)2 +O(1), and then 2\u03bb\u0304 \u222b B |x|\u2212N+2\u000f \u2264 (HN +O(\u000f)) \u222b B |x|\u2212N+2\u000f +O(1). Computing the integrals on obtains 2\u03bb\u0304 \u2264 HN +O(\u000f). Letting \u000f\u2192 0 we get 2\u03bb\u0304 \u2264 HN . Graphing this relation we see that N \u2265 9. \u0003 6.5 The extremal solution is singular in dimensions N \u2265 9 In this section we will show that the extremal solution u\u2217 of (P\u03bb,\u03b2,\u03c4,0,0) in dimen- sions N \u2265 9 is singular for \u03c4 > 0 sufficiently small. To do this, first we shall show that the extremal solution of (P\u03bb,1,0,0,0) is singular in dimensions N \u2265 9. Again to cut down the notation we won\u2019t always indicate that \u03b2 = 1 and \u03c4 = 0. We have to distinguish between three different ranges for the dimension. For each range, we will need a suitable Hardy-Rellich type inequality that will be es- tablished in the last section, by using the recent results of Ghoussoub-Moradifam [12]. \u2022 Case N \u2265 16: To establish the singularity of u\u2217 for these dimensions we shall need the classical Hardy-Rellich inequality, which is valid for all \u03c6 \u2208 H2(B)\u2229H10 (B):\u222b B (\u2206\u03c6)2 dx \u2265 N 2(N \u2212 4)2 16 \u222b B \u03c62 |x|4 dx. (6.14) 144 6.5. The extremal solution is singular in dimensions N \u2265 9 \u2022 Case 10 \u2264 N \u2264 16: For this case, we shall need the following inequality valid for all \u03c6 \u2208 H2(B) \u2229H10 (B)\u222b B (\u2206\u03c6)2 \u2265 (N \u2212 2) 2(N \u2212 4)2 16 \u222b B \u03c62 (|x|2 \u2212 N2(N\u22121) |x| N 2 +1)(|x|2 \u2212 |x|N2 ) + (N \u2212 1)(N \u2212 4)2 4 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) . (6.15) \u2022 Case N = 9: This case is the trickiest and will require the following inequality for all \u03c6 \u2208 H2(B) \u2229H10 (B), which is valid for N \u2265 7\u222b B |\u2206u|2 \u2265 \u222b B W (|x|)u2. (6.16) where where W (r) = K(r)( (N \u2212 2)2 4(r2 \u2212 N2(N\u22121)r N 2 +1) + (N \u2212 1) r2 ), K(r) = \u2212\u03d5 \u2032\u2032(r) + (n\u22123)r \u03d5 \u2032(r) \u03d5(r) , and \u03d5(r) = r\u2212 N 2 +2 + 9r\u22122 + 10r \u2212 20. The next lemma will be our main tool to guarantee that u\u2217 is singular for N \u2265 9. The proof is based on an upper estimate by a singular stable sub-solution. Lemma 6.17. Suppose there exist \u03bb\u2032 > 0 and a radial function u \u2208 H2(B) \u2229 W 4,\u221eloc (B \\ {0}) such that \u22062u \u2264 \u03bb \u2032 (1\u2212 u)2 for 0 < r < 1, (6.18) u(1) = 0, \u2206u|r=1 = 0, (6.19) u is singular, (6.20) and 2\u03b2 \u222b B \u03d52 (1\u2212 u)3 \u2264 \u222b B (\u2206\u03d5)2 for all \u03d5 \u2208 H2(B) \u2229H10 (B), (6.21) for some \u03b2 > \u03bb\u2032. Then u\u2217 is singular and \u03bb\u2217 \u2264 \u03bb\u2032 (6.22) Proof. By Lemma 6.9 we have (6.22). Let \u03bb \u2032 \u03b2 < \u03b3 < 1 and \u03b1 := ( \u03b3\u03bb\u2217 \u03bb\u2032 )1\/3, (6.23) 145 6.5. The extremal solution is singular in dimensions N \u2265 9 and define u\u0304 := 1\u2212 \u03b1(1\u2212 u). We claim that u\u2217 \u2264 u\u0304 in B. (6.24) To prove this, we shall show that for \u03bb < \u03bb\u2217 u\u03bb \u2264 u\u0304 in B. (6.25) Indeed, we have \u22062(u\u0304) = \u03b1\u22062(u\u0304) \u2264 \u03b1\u03bb \u2032 (1\u2212 u)2 = \u03b13\u03bb\u2032 (1\u2212 u\u0304)2 . By (6.22) and the choice of \u03b1 \u03b13\u03bb\u2032 < \u03bb\u2217. To prove (6.24) it suffices to prove it for \u03b13\u03bb\u2032 < \u03bb < \u03bb\u2217. Fix such \u03bb and assume that (6.24) is not true. Then \u039b = {0 \u2264 R \u2264 1 | u\u03bb(R) > u\u0304(R)}, in non-empty. There exists 0 < R1 < 1, such that u\u03bb(R1) \u2265 u\u2217(R1) and \u2206u\u03bb(R1) \u2264 \u2206u\u2217(R1), since otherwise we can find 0 < s1 < s2 < 1 so that u\u03bb(s1) = u\u0304(s1), u\u03bb(s2) = u\u0304(s2), u\u03bb(R) > u\u0304(R), and \u2206u\u03bb(R1) > \u2206u\u2217(R1) which contradict the maximum principle. Now consider the following problem \u22062u = \u03bb (1\u2212 u)2 in B u = u\u03bb(R1) on \u2202B \u2206u = \u2206u\u03bb on \u2202B. Then u\u03bb is a solution to the above problem while u\u0304 is a sub-solution to the same problem. Moreover u\u0304 is stable since, \u03bb < \u03bb\u2217 and hence 2\u03bb (1\u2212 u\u0304)3 \u2264 2\u03bb\u2217 \u03b13(1\u2212 u)3 = 2\u03bb\u2032 \u03b3(1\u2212 u)3 < 2\u03b2 (1\u2212 u)3 . We deduce u\u0304 \u2264 u\u03bb in BR1 which is impossible, since u\u0304 is singular while u\u03bb is smooth. This establishes (6.24). From (6.24) and the above two inequalities we have 2\u03bb\u2217 (1\u2212 u\u2217)3 \u2264 2\u03bb\u2032 \u03b3(1\u2212 u)3 < \u03b2 (1\u2212 u)3 . Thus inf \u03d5\u2208C\u221e0 (B) \u222b B (\u2206\u03d5)2 \u2212 2\u03bb\u2217\u03d52(1\u2212u\u2217)3\u222b B \u03d52 > 0. 146 6.5. The extremal solution is singular in dimensions N \u2265 9 This is not possible if u\u2217 is a smooth solution. \u0003 For any m > 43 define wm := 1\u2212 aN,mr 43 + bN,mrm, where aN,m := m(N +m\u2212 2) m(N +m\u2212 2)\u2212 43 (N \u2212 2\/3) , and bN,m := 4 3 (N \u2212 2\/3) m(N +m\u2212 2)\u2212 43 (N \u2212 2\/3) . Now we are ready to prove the main result of this section. The following upper bounds on \u03bb\u2217 hold in large dimensions. 1. If N \u2265 31, then Lemma 6.17 holds with u := w2, \u03bb\u2032N = 27\u03bb\u0304 and \u03b2 = HN2 > 27\u03bb\u0304. 2. If 16 \u2264 N \u2264 30, then Lemma 6.17 holds with u := w3, \u03bb\u2032N = HN2 \u2212 1, \u03b2N = HN 2 . 3. If 10 \u2264 N \u2264 15, then Lemma 6.17 holds with u := w3, \u03bb\u2032N < \u03b2N given in Table 6.1. 4. If N = 9, then Lemma 6.17 holds with u := w2.8, \u03bb\u20329 := 249 < \u03b29 := 251. The extremal solution is therefore singular for dimensions N \u2265 9. Proof. 1) Assume first that N \u2265 31, then it is easy to see that aN,2 < 3 and a3N,2\u03bb\u0304 \u2264 27\u03bb\u0304 < HN2 . We shall show that w2 is a singular H\u2212weak sub-solution of (P )a3N,2\u03bb\u0304 which is stable. Note that w2 \u2208 H2(B), 1 1\u2212w2 \u2208 L3(B), 0 \u2264 w2 \u2264 1 in B, and \u22062w2 \u2264 a3N,2\u03bb\u0304 (1\u2212 w2)2 in B \\ {0}. So w2 is a H\u2212weak sub-solution of (P )27\u03bb\u0304. Moreover, w2 = 1\u2212 |x| 43 + (aN,2 \u2212 1)(|x| 43 \u2212 |x|2) \u2264 1\u2212 |x| 43 . Since 27\u03bb\u0304 \u2264 HN2 , we get that 54\u03bb\u0304 \u222b B \u03d52 (1\u2212 w2)3 \u2264 HN \u222b B \u03d52 (1\u2212 w2)3 \u2264 HN \u222b B \u03d52 |x|4 \u2264 \u222b B (\u2206\u03d5)2 for all \u03d5 \u2208 C\u221e0 (B). Hence, w2 is stable. Thus it follows from Lemma 6.17 that u\u2217 is singular and \u03bb\u2217 \u2264 27\u03bb\u0304. 2) Assume 16 \u2264 N \u2264 30 and consider w3 := 1\u2212 aN,3r 43 + bN,3r3. 147 6.5. The extremal solution is singular in dimensions N \u2265 9 We show that it is a singular H\u2212weak sub-solution of (PHN 2 \u22121 ) which is stable. Indeed, we clearly have 0 \u2264 w3 \u2264 1 a.e. in B, w3 \u2208 H2(B) and 11\u2212w3 \u2208 L3(B). Note that HN \u222b B \u03d52 (1\u2212 w3)3 = HN \u222b B \u03d52 (aN,mr 4 3 \u2212 bN,mrm)3 \u2264 sup 0 \u03bb\u2032N such that (N \u2212 2)2(N \u2212 4)2 16 1 (|x|2 \u2212 N2(N\u22121) |x| N 2 +1)(|x|2 \u2212 |x|N2 ) + (N \u2212 1)(N \u2212 4)2 4 1 |x|2(|x|2 \u2212 |x|N2 ) \u2265 2\u03b2N (1\u2212 w3)3 . The above inequality and improved Hardy-Rellich inequality (6.40) guarantee that the stability condition (6.29) holds for \u03b2N > \u03bb\u2032. Hence by Lemma 6.17 the extremal solution is singular for 10 \u2264 N \u2264 15. The values of \u03bbN and \u03b2N are shown in Table 6.1. 4) Let u:=w2.8. Using Maple on can see that \u22062u \u2264 249 (1\u2212 u)2 in B and 502 (1\u2212 u(r))3 \u2264W (r) for all r \u2208 (0, 1), where W is given by (6.42). Since, 502 > 2 \u00d7 249, by Lemma 6.17 the extremal solution u\u2217 is singular in dimension N = 9. \u0003 Remark 6.5.1. It follows from the proof of Theorem 6.5 that for N \u2265 9 and \u03c4\u03b2 sufficiently small, there exists u \u2208 H2(B) \u2229W 4,\u221eloc (B \\ {0}) such that \u22062u\u2212 \u03c4 \u03b2 \u2206u \u2264 \u03bb \u2032\u2032 N (1\u2212 u)2 for 0 < r < 1, (6.26) 148 6.6. Improved Hardy-Rellich Inequalities Table 6.1: Summary 1 N \u03bb\u2032N \u03b2N 9 249 251 10 320 367 11 405 574 12 502 851 13 610 1211 14 730 1668 15 860 2235 16 \u2264 N \u2264 30 HN2 \u2212 1 HN2 N \u2265 31 27\u03bb\u0304 HN2 u(1) = 0, \u2206u|r=1 = 0, (6.27) u is singular, (6.28) and 2\u03b2\u2032N \u222b B \u03d52 (1\u2212 u)3 \u2264 \u222b B (\u2206\u03d5)2 + \u03c4 \u03b2 |\u2207\u03d5|2 for all \u03d5 \u2208 H2(B) \u2229H10 (B), (6.29) where \u03b2\u2032N > \u03bb \u2032\u2032 N > 0 are constants. Indeed, for each dimension N \u2265 9, it is enough to take u to be the sub-solution we constructed in the proof of Theorem 6.5, \u03b2\u2032N := \u03b2N , \u03bb \u2032 < \u03bb\u2032\u2032 < \u03b2. If \u03c4\u03b2 is sufficiently small so that \u2212 \u03c4\u03b2\u2206u < \u03bb \u2032\u2032\u2212\u03bb\u2032 (1\u2212u)2 on (0, 1), then with an argument similar to that of Lemma 6.17 we deduce that the extremal solution u\u2217 of (P\u03bb,\u03b2,\u03c4,0,0) is singular. We believe that the extremal solution of (P\u03bb,\u03b2,\u03c4,0,0) is singular for all \u03b2, \u03c4 > 0 in dimensions N \u2265 9. 6.6 Improved Hardy-Rellich Inequalities We now prove the improved Hardy-Rellich inequalities used in section 4. They rely on the results of Ghoussoub-Moradifam in [12] which provide necessary and suffi- cient conditions for such inequalities to hold. At the heart of this characterization is the following notion of a Bessel pair of functions. Assume that B is a ball of radius R in N , V,W \u2208 C1(0, 1), and \u222b R 0 1 rN\u22121V (r)dr = +\u221e. Say that the couple (V,W ) is a Bessel pair on (0, R) if the ordinary differential equation (BV,W ) y\u2032\u2032(r) + (N\u22121r + Vr(r) V (r) )y \u2032(r) + W (r)V (r) y(r) = 0 has a positive solution on the interval (0, R). The needed inequalities will follow from the following two results. (Ghoussoub-Moradifam [12]) Let V and W 149 6.6. Improved Hardy-Rellich Inequalities be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in N (N \u2265 1) such that \u222b R 0 1 rN\u22121V (r)dr = +\u221e and \u222b R 0 rN\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R). 2. \u222b B V (|x|)|\u2207\u03c6|2dx \u2265 \u222b B W (|x|)\u03c62dx for all \u03c6 \u2208 C\u221e0 (B). Let B be the unit ball in N (N \u2265 5). Then the inequality\u222b B |\u2206u|2dx \u2265 \u222b B |\u2207u|2 |x|2\u2212 N2(N\u22121) |x| N 2 +1 dx+ (N \u2212 1) \u222b B |\u2207u|2 |x|2 dx, (6.30) holds for all u \u2208 C\u221e0 (B\u0304). We shall need the following result to prove (6.30). Lemma 6.31. For every u \u2208 C1([0, 1]) the following inequality holds\u222b 1 0 |u\u2032(r)|2rN\u22121dr \u2265 \u222b 1 0 u2 r2 \u2212 N2(N\u22121)r N 2 +1 rN\u22121dr \u2212 (N \u2212 1)(u(1))2. (6.32) Proof. Let \u03d5 := r\u2212 N 2 +1 \u2212 N2(N\u22121) and k(r) := rN\u22121. Define \u03c8(r) = u(r)\/\u03d5(r), r \u2208 [0, 1]. Then\u222b 1 0 |u\u2032(r)|2k(r)dr = \u222b 1 0 |\u03c8(r)|2|\u03d5\u2032(r)|2k(r)dr + \u222b 1 0 2\u03d5(r)\u03d5\u2032(r)\u03c8(r)\u03c8\u2032(r)k(r)dr + \u222b 1 0 |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr = \u222b 1 0 |\u03c8(r)|2(|\u03d5\u2032(r)|2k(r)\u2212 (k\u03d5\u03d5\u2032)\u2032(r))dr + \u222b 1 0 |\u03d5(r)|2|\u03c8\u2032(r)|2k(r)dr + \u03c82(1)\u03d5\u2032(1)\u03d5(1) \u2265 \u222b 1 0 |\u03c8(r)|2(|\u03d5\u2032(r)|2k(r)\u2212 (k\u03d5\u03d5\u2032)\u2032(r))dr + \u03c82(1)\u03d5\u2032(1)\u03d5(1) Note that \u03c82(1)\u03d5\u2032(1)\u03d5(1) = u2(1)\u03d5 \u2032(1) \u03d5(1) = \u2212(N \u2212 1)u2(1). Hence, we have\u222b 1 0 |u\u2032(r)|2k(r)dr \u2265 \u222b 1 0 \u2212u2(r)k \u2032(r)\u03d5\u2032(r) + k(r)\u03d5\u2032\u2032(r) \u03d5 )dr \u2212 (N \u2212 1)u2(1) Simplifying the above inequality we get (6.32). \u0003 150 6.6. Improved Hardy-Rellich Inequalities The decomposition of a function into its spherical harmonics will be one of our tools to prove Theorem 6.6. Let u \u2208 C\u221e0 (B\u0304). By decomposing u into spherical harmonics we get u = \u03a3\u221ek=0uk where uk = fk(|x|)\u03d5k(x) and (\u03d5k(x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(N + k \u2212 2), k \u2265 0. The functions fk belong to u \u2208 C\u221e([0, 1]), fk(1) = 0, and satisfy fk(r) = O(rk) and f \u2032(r) = O(rk\u22121) as r \u2192 0. In particular, \u03d50 = 1 and f0 = 1N\u03c9NrN\u22121 \u222b \u2202Br uds = 1N\u03c9N \u222b |x|=1 u(rx)ds. (6.33) We also have for any k \u2265 0, and any continuous real valued W on (0, 1),\u222b B |\u2206uk|2dx = \u222b B ( \u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2 dx, (6.34) and \u222b B W (|x|)|\u2207uk|2dx = \u222b B W (|x|)|\u2207fk|2dx+ ck \u222b B W (|x|)|x|\u22122f2kdx. (6.35) Now we are ready to prove Theorem 6.6. We shall use the inequality \u222b 1 0 |x\u2032(r)|2rN\u22121dr \u2265 (N\u22122)24 \u222b 1 0 x2(r) r2\u2212 N2(N\u22121) r N 2 +1 rN\u22121dr for all x \u2208 C1([0, 1]), with x(1) = 0. Proof of Theorem 6.6: For all N \u2265 5 and k \u2265 0 we have 1 NwN \u222b B |\u2206uk|2dx = 1 NwN \u222b B ( \u2206fk(|x|)\u2212 ck fk(|x|)|x|2 )2 dx = \u222b 1 0 ( f \u2032\u2032k (r) + N \u2212 1 r f \u2032k(r)\u2212 ck fk(r) r2 )2 rN\u22121dr = \u222b 1 0 (f \u2032\u2032k (r)) 2rN\u22121dr + (N \u2212 1)2 \u222b 1 0 (f \u2032k(r)) 2rN\u22123dr +c2k \u222b 1 0 f2k (r)r N\u22125 + 2(N \u2212 1) \u222b 1 0 f \u2032\u2032k (r)f \u2032 k(r)r N\u22122 \u22122ck \u222b 1 0 f \u2032\u2032k (r)fk(r)r N\u22123dr \u22122ck(N \u2212 1) \u222b 1 0 f \u2032k(r)fk(r)r N\u22124dr. 151 6.6. Improved Hardy-Rellich Inequalities Integrate by parts and use (6.33) for k = 0 to get 1 N\u03c9N \u222b B |\u2206uk|2dx \u2265 \u222b 1 0 (f \u2032\u2032k (r)) 2rN\u22121dr + (N \u2212 1 + 2ck) \u222b 1 0 (f \u2032k(r)) 2rN\u22123dr + (2ck(n\u2212 4) + c2k) \u222b 1 0 rn\u22125f2k (r)dr + (N \u2212 1)(f \u2032k(1))2 (6.36) Now define gk(r) = fk(r) r and note that gk(r) = O(r k\u22121) for all k \u2265 1. We have\u222b 1 0 (f \u2032k(r)) 2rN\u22123 = \u222b 1 0 (g\u2032k(r)) 2rN\u22121dr + \u222b 1 0 2gk(r)g\u2032k(r)r N\u22122dr + \u222b 1 0 g2k(r)r N\u22123dr = \u222b 1 0 (g\u2032k(r)) 2rN\u22121dr \u2212 (N \u2212 3) \u222b 1 0 g2k(r)r N\u22123dr Thus,\u222b 1 0 (f \u2032k(r)) 2rN\u22123 \u2265 (N \u2212 2) 2 4 \u222b 1 0 f2k (r) r2 (6.37) \u2212 N 2(N \u2212 1)r N 2 +1rN\u22123dr \u2212 (N \u2212 3) \u222b 1 0 f2k (r)r N\u22125dr (6.39) 152 6.6. Improved Hardy-Rellich Inequalities Substituting 2ck \u222b 1 0 (f \u2032k(r)) 2rN\u22123 in (6.36) by its lower estimate in the last inequality (6.37), and using Lemma 6.31 we get 1 N\u03c9N \u222b B |\u2206uk|2dx \u2265 (N \u2212 2) 2 4 \u222b 1 0 (f \u2032k(r)) 2 r2 \u2212 N2(N\u22121)r N 2 +1 rN\u22121dr + 2ck (N \u2212 2)2 4 \u222b 1 0 f2k (r) r2 \u2212 N2(N\u22121)r N 2 +1 rn\u22123dr + (N \u2212 1) \u222b 1 0 (f \u2032k(r)) 2rN\u22123dr + ck(N \u2212 1) \u222b 1 0 (fk(r))2rN\u22125dr + ck(ck \u2212 (N \u2212 1)) \u222b 1 0 rN\u22125f2k (r)dr + ck \u222b 1 0 (N \u2212 2)2 4(r2 \u2212 N2(N\u22121)r N 2 +1) \u2212 2 r2 )dr. \u2265 (N \u2212 2) 2 4 \u222b 1 0 (f \u2032k(r)) 2 r2 \u2212 N2(N\u22121)r N 2 +1 rN\u22121dr + ck (N \u2212 2)2 4 \u222b 1 0 f2k (r) r2 \u2212 N2(N\u22121)r N 2 +1 rn\u22123dr + (N \u2212 1) \u222b 1 0 (f \u2032k(r)) 2rN\u22123dr + ck(N \u2212 1) \u222b 1 0 (fk(r))2rN\u22125dr The proof is complete in the view of (6.35). \u0003 We shall now deduce the following corollary. Let N \u2265 5 and B be the unit ball in N . Then the following improved Hardy-Rellich inequality holds for all \u03c6 \u2208 H2(B) \u2229H10 (B):\u222b B (\u2206\u03c6)2 \u2265 (N \u2212 2) 2(N \u2212 4)2 16 \u222b B \u03c62 (|x|2 \u2212 N2(N\u22121) |x| N 2 +1)(|x|2 \u2212 |x|N2 ) + (N \u2212 1)(N \u2212 4)2 4 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) (6.40) Proof. Let \u03b1 := N2(N\u22121) and V (r) := 1 r2\u2212\u03b1rN2 +1 and note that Vr V = \u22122 r + \u03b1(N \u2212 2) 2 r N 2 \u22122 1\u2212 \u03b1rN2 \u22121 \u2265 \u2212 2 r . 153 6.6. Improved Hardy-Rellich Inequalities The function y(r) = r\u2212 N 2 +2 \u2212 1 is decreasing and is then a positive super-solution on (0, 1) for the ODE y\u2032\u2032 + ( N \u2212 1 r + Vr V )y\u2032(r) + W1(r) V (r) y = 0, where W1(r) = (N \u2212 4)2 4(r2 \u2212 rN2 )(r2 \u2212 \u03b1rN2 +1) . Hence, by Theorem 6.6 we deduce\u222b B |\u2207\u03c6|2 |x|2 \u2212 \u03b1|x|N2 +1 \u2265 ( N \u2212 4 2 )2 \u222b B \u03c62 (|x|2 \u2212 \u03b1|x|N2 +1)(|x|2 \u2212 |x|N2 ) for all \u03c6 \u2208 H2(B) \u2229H10 (B). Similarly, for V (r) = 1r2 we have that\u222b B |\u2207\u03c6|2 |x|2 \u2265 ( N \u2212 4 2 )2 \u222b B \u03c62 |x|2(|x|2 \u2212 |x|N2 ) for all \u03c6 \u2208 H2(B) \u2229H10 (B). Combining the above two inequalities with (6.32) we get (6.40). \u0003 Let N \u2265 7 and B be the unit ball in N . Then the following improved Hardy- Rellich inequality holds for all \u03c6 \u2208 H2(B) \u2229H10 (B):\u222b B |\u2206u|2 \u2265 \u222b B W (|x|)u2. (6.41) where W (r) = K(r)( (N \u2212 2)2 4(r2 \u2212 N2(N\u22121)r N 2 +1) + (N \u2212 1) r2 ), (6.42) K(r) = \u2212\u03d5 \u2032\u2032(r) + (n\u22123)r \u03d5 \u2032(r) \u03d5(r) , and \u03d5(r) = r\u2212 N 2 +2 + 9r\u22122 + 10r \u2212 20. Proof. Let \u03b1 := N2(N\u22121) and V (r) := 1 r2\u2212\u03b1rN2 +1 . Then \u03d5 is a sub-solution for the ODE y\u2032\u2032 + ( N \u2212 1 r + Vr V )y\u2032(r) + W2(r) V (r) y = 0, where W2(r) = K(r) r2 \u2212 \u03b1rN2 +1 , Hence by Theorem 6.6 we have\u222b B |\u2207u|2 |x|2 \u2212 \u03b1|x|N2 +1 \u2265 \u222b B W2(|x|)u2. (6.43) 154 6.6. Improved Hardy-Rellich Inequalities Similarly \u222b B |\u2207u|2 |x|2 \u2265 \u222b B W3(|x|)u2. (6.44) where W3(r) = K(r) r2 . Combining the above two inequalities with (6.32) we get improved Hardy-Rellich inequality (6.41). \u0003 155 Bibliography [1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. [2] T. Brau, A decomposition method with respect to dual cones and its applica- tion to higher order Sobolev spaces, preprint. [3] C. Cowan and N. 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Paris 255 (1962), 238-240. 157 Chapter 7 The singular extremal solutions of the bilaplacian with exponential nonlinearity 6 7.1 Introduction Consider the fourth order elliptic problem{ \u22062u = \u03bbeu in B u = \u2202u\u2202n = 0 on \u2202B, (7.1) where B is the unit ball in N , N \u2265 1, n is the exterior unit normal vector and \u03bb \u2265 0 is a parameter. This problem is the fourth order analogue of the classical Gelfand problem (see [2], [4], and [9]). Recently, many authors are intrested in fourth order equations and interesting results can be found in [1], [2], [3], [5], [8], [10], [11] and the references cited therein. In [1], Arioli et al. studied the problem (7.1) and showed that for each dimension N \u2265 1 there exists a \u03bb\u2217 > 0 such that for every 0 < \u03bb < \u03bb\u2217, there exists a smooth minimal (smallest) solution of (7.1), while for \u03bb > \u03bb\u2217 there is no solution even in a weak sense. Moreover, the branch \u03bb 7\u2192 u\u03bb(x) is increasing for each x \u2208 B, and therefore the function u\u2217(x) := lim\u03bb\u2197\u03bb\u2217 u\u03bb(x) can be considered as a generalized solution that corresponds to \u03bb\u2217. Now the important question is whether u\u2217 is regular (u\u2217 \u2208 L\u221e(B)) or singular (u\u2217 \/\u2208 L\u221e(B)). Even though there are similarities between (7.1) and the Gelfand problem, several tools which have been developed for the Gelfand problem, are no longer available for (7.1). In [5] the authors developed a new method to prove the regularity of the extremal solutions in low dimensions and showed that for N \u2264 12, u\u2217 is regular. But unlike the Gelfand problem the natural candidate u = \u22124 ln(|x|), for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close 6A version of this chapter has been accepted for publication. A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math Soc., 138 (2010), 1287-1293. 158 7.2. An improved Hardy-Rellich inequality to the critical dimension is challenging. Da\u0301vila et al. [5] used a computer assisted proof to show that the extremal solution is singular in dimensions 13 \u2264 N \u2264 31 while they gave a mathematical proof in dimensions N \u2265 32. In this paper we introduce a unified mathematical approach to deal with this problem and show that for N \u2265 13, the extremal solution is singular. One of our main tools is an improved Hardy-Rellich inequality that follows from the recent result of Ghoussoub- Moradifam about improved Hardy and Hardy-Rellich inequalities developed in [7] and [6]. 7.2 An improved Hardy-Rellich inequality In this section we shall prove an improvement of classical Hardy-Rellich inequality which will be used to prove our main result in Section 3. It relies on the results of Ghoussoub-Moradifam in [6] which provide necessary and sufficient conditions for such inequalities to hold. At the heart of this characterization is the following notion of a Bessel pair of functions. Assume that B is a ball of radius R in N , V,W \u2208 C1(0, 1), and \u222b R 0 1 rN\u22121V (r)dr = +\u221e. Say that the couple (V,W ) is a Bessel pair on (0, R) if the ordinary differential equation (BV,W ) y\u2032\u2032(r) + (N\u22121r + Vr(r) V (r) )y \u2032(r) + W (r)V (r) y(r) = 0 has a positive solution on the interval (0, R). (Ghoussoub-Moradifam [6]) Let V and W be positive radial C1-functions on B\\{0}, where B is a ball centered at zero with radius R in N (N \u2265 1) such that\u222b R 0 1 rN\u22121V (r)dr = +\u221e and \u222b R 0 rN\u22121V (r)dr < +\u221e. The following statements are then equivalent: 1. (V,W ) is a Bessel pair on (0, R) and \u03b2(V,W ;R) \u2265 1. 2. \u222b B V (x)|\u2207u|2dx \u2265 \u222b B W (x)u2dx for all u \u2208 C\u221e0 (B). 3. If limr\u21920 r\u03b1V (r) = 0 for some \u03b1 < N\u22122 andW (r)\u2212 2V (r)r2 + 2Vr(r)r \u2212Vrr(r) \u2265 0 on (0, R), then the above are equivalent to\u222b B V (x)|\u2206u|2dx \u2265 \u222b B W (x)|\u2207u|2dx+ (N \u2212 1) \u222b B (V (x)|x|2 \u2212 Vr(|x|)|x| )|\u2207u|2dx, for all u \u2208 C\u221e0 (B). As an application we have the following improvement of the classical Hardy- Rellich inequality. Let N \u2265 5 and B be the unit ball in N . Then the following improved Hardy- Rellich inequality holds for all u \u2208 C\u221e0 (B).\u222b B |\u2206u|2 \u2265 (N \u2212 2) 2(N \u2212 4)2 16 \u222b B u2 (|x|2 \u2212 910 |x| N 2 +1)(|x|2 \u2212 |x|N2 ) + (N \u2212 1)(N \u2212 4)2 4 \u222b B u2 |x|2(|x|2 \u2212 |x|N2 ) . (7.2) 159 7.2. An improved Hardy-Rellich inequality As a consequence the following improvement of classical Hardy-Rellich inequality holds: \u222b B |\u2206u|2 \u2265 N 2(N \u2212 4)2 16 \u222b B u2 |x|2(|x|2 \u2212 |x|N2 ) . (7.3) Proof. Let \u03d5 := r\u2212 N 2 +1 \u2212 910 . Since \u2212\u03d5 \u2032\u2032 + (N\u22121)r \u03d5 \u2032 \u03d5 = (N \u2212 2)2 4 . 1 r2 \u2212 910r N 2 +1 , (1, (N\u22122) 2 4 1 r2\u2212 910 r N 2 +1 ) is a bessel pair on (0, 1). By Theorem 7.2 the following inequality holds for all u \u2208 C\u221e0 (B).\u222b B |\u2206u|2dx \u2265 (N \u2212 2) 2 4 \u222b B |\u2207u|2 |x|2 \u2212 910 |x| N 2 +1 + (N \u2212 1) \u222b B |\u2207u|2 |x|2 . (7.4) Let V (r) := 1 r2\u2212 910 r N 2 +1 . Then Vr V = \u22122 r + 9 10 ( N \u2212 2 2 ) r N 2 \u22122 1\u2212 910r N 2 \u22121 \u2265 \u22122 r , (7.5) and \u03c8(r) = r\u2212 N 2 +2 \u2212 1 is a positive super-solution for the ODE y\u2032\u2032 + ( N \u2212 1 r + Vr V )y\u2032(r) + W1(r) V (r) y = 0, (7.6) where W1(r) = (N \u2212 4)2 4(r2 \u2212 rN2 )(r2 \u2212 910r N 2 +1) . Hence the ODE (7.6) has actually a positive solution and by Theorem 7.2 we have \u222b B |\u2207u|2 |x|2 \u2212 910 |x| N 2 +1 \u2265 (N \u2212 4 2 )2 \u222b B u2 (|x|2 \u2212 910 |x| N 2 +1)(|x|2 \u2212 |x|N2 ) . (7.7) Similarly \u222b B |\u2207u|2 |x|2 \u2265 ( N \u2212 4 2 )2 \u222b B u2 |x|2(|x|2 \u2212 |x|N2 ) . (7.8) Combining the above two inequalities with (7.4) we get (7.2). \u0003 160 7.3. Main results 7.3 Main results In this section we shall prove that the extremal solution u\u2217 of the problem (7.1) is singular in dimensions N \u2265 13. The next lemma will be our main tool to guarantee that u\u2217 is singular for N \u2265 13. The proof is based on an upper estimate by a singular stable sub-solution. Lemma 7.9. Suppose there exist \u03bb\u2032 > 0 and a radial function u \u2208 H2(B) \u2229 W 4,\u221eloc (B \\ {0}) such that \u22062u \u2264 \u03bb\u2032eu for all 0 < r < 1, (7.10) u(1) = 0, \u2202u \u2202n (1) = 0, (7.11) u \/\u2208 L\u221e(B), (7.12) and \u03b2 \u222b B eu\u03d52 \u2264 \u222b B (\u2206\u03d5)2 for all \u03d5 \u2208 C\u221e0 (B), (7.13) for some \u03b2 > \u03bb\u2032. Then u\u2217 is singular and \u03bb\u2217 \u2264 \u03bb\u2032 (7.14) Proof. By Lemma 2.6 in [5] we have (7.14). Define \u03b1 := ln( \u03bb\u2032 \u03b3\u03bb\u2217 ), (7.15) where \u03bb \u2032 \u03b2 < \u03b3 < 1 and let u\u0304 := u+ \u03b1. We claim that u\u2217 \u2264 u\u0304 in B. (7.16) To prove this, we shall show that for \u03bb < \u03bb\u2217 u\u03bb \u2264 u\u0304 in B. (7.17) Indeed, we have \u22062(u\u0304) = \u22062(u) \u2264 \u03bb\u2032eu = \u03bb\u2032e\u2212\u03b1eu\u0304 = \u03b3\u03bb\u2217eu\u0304. To prove (7.16) it suffices to prove it for \u03b3\u03bb\u2217 < \u03bb < \u03bb\u2217. Fix such \u03bb and assume that (7.16) is not true. Let R1 := sup{0 \u2264 R \u2264 1 | u\u03bb(R) = u\u0304(R)}. Since u\u0304(1) = \u03b1 > 0 = u\u03bb(1), we have 0 < R1 < 1, u\u03bb(R1) = u\u0304(R1), and u\u2032\u03bb(R1) \u2264 u\u0304\u2032(R1). Now consider the following problem\uf8f1\uf8f2\uf8f3 \u22062u = \u03bbeu in \u2126 u = u\u03bb(R1) on \u2202\u2126 \u2202u \u2202n = u \u2032 \u03bb(R1) on \u2202\u2126. 161 7.3. Main results Obviously u\u03bb is a solution to the above problem while u\u0304 is a sub-solution to the same problem. Moreover u\u0304 is stable since, \u03bb < \u03bb\u2217 and hence \u03bbeu\u0304 \u2264 \u03bb\u2217e\u03b1eu = \u03bb \u2032 \u03b3 eu < \u03b2eu. We deduce u\u0304 \u2264 u\u03bb in BR1 which is impossible, since u\u0304 is singular while u\u03bb is smooth. This establishes (7.16). From (7.16) and the above two inequalities we have \u03bb\u2217eu \u2217 \u2264 \u03bb\u2217eaeu = \u03bb \u2032 \u03b3 eu. Since \u03bb \u2032 \u03b3 < \u03b2, inf \u03d5\u2208C\u221e0 (B) \u222b B (\u2206\u03d5)2 \u2212 \u03bb\u2217eu\u2217\u222b B \u03d52 > 0. This is not possible if u\u2217 is a smooth solution. \u0003 In the following, for each dimension N \u2265 13, we shall construct u satisfying all the assumptions of Lemma 7.9. Define wm := \u22124 ln(r)\u2212 4 m + 4 m rm, m > 0, and let HN := N2(N\u22124)2 16 . Now we are ready to prove our main result. The following upper bounds on \u03bb\u2217 hold in large dimensions. 1. If N \u2265 32, then Lemma 7.9 holds with u := w2, \u03bb\u2032N = 8(N \u2212 2)(N \u2212 4)e2 and \u03b2 = HN > \u03bb\u2032N . 2. If 13 \u2264 N \u2264 31, then Lemma 7.9 holds with u := w3.5 and \u03bb\u2032N < \u03b2N given in Table 1. The extremal solution is therefore singular for dimensions N \u2265 13. Proof. 1) Assume first that N \u2265 32, then 8(N \u2212 2)(N \u2212 4)e2 < N 2(N \u2212 4)2 16 , and \u22062w2 = 8(N \u2212 2)(N \u2212 4) 1 r4 \u2264 8(N \u2212 2)(N \u2212 4)e2ew2 . Moreover, 8(N \u2212 2)(N \u2212 4)e2 \u222b B ew2\u03d52 \u2264 Hn \u222b B e\u22124 ln(|x|)\u03d52 = Hn \u222b B \u03d52 |x|2 \u2264 \u222b B |\u2206\u03d5|2. 162 7.3. Main results Thus it follows from Lemma 7.9 that u\u2217 is singular and \u03bb\u2217 \u2264 8(N \u2212 2)(N \u2212 4)e2. 2) Assume 13 \u2264 N \u2264 31. We shall show that u = w3.5 satisfies the assumptions of Lemma 7.9 for each dimension 13 \u2264 N \u2264 31. Using Maple, for each dimension 13 \u2264 N \u2264 31, one can verify that inequality (7.10) holds for \u03bb\u2032N given by Table 7.1. Then, by using Maple again, we show that there exists \u03b2N > \u03bb\u2032N such that (N \u2212 2)2(N \u2212 4)2 16 1 (|x|2 \u2212 0.9|x|N2 +1)(|x|2 \u2212 |x|N2 ) + (N \u2212 1)(N \u2212 4)2 4 1 |x|2(|x|2 \u2212 |x|N2 ) \u2265 \u03b2New3.5 . The above inequality and improved Hardy-Rellich inequality (7.2) guarantee that the stability condition (7.13) holds for \u03b2N > \u03bb\u2032. Hence by Lemma 7.9 the extremal solution is singular for 13 \u2264 N \u2264 31. The values of \u03bbN and \u03b2N are shown in Table 7.1. Table 7.1: Summary 3 N \u03bb\u2032N \u03b2N N \u2265 32 8(N \u2212 2)(N \u2212 4)e2 Hn 31 20000 86900 30 18500 76500 29 17000 67100 28 16000 58500 27 14500 50800 26 13500 43870 25 12200 37630 24 11100 32050 23 10100 27100 22 9050 22730 21 8150 18890 20 7250 15540 19 6400 12645 18 5650 10155 17 4900 8035 16 4230 6250 15 3610 4765 14 3050 3545 13 2525 2560 163 7.3. Main results Remark 7.3.1. The values of \u03bb\u2032N and \u03b2N in Table 7.1 are not optimal. Remark 7.3.2. The improved Hardy-Rellich inequality (7.2) is crucial to prove that u\u2217 is singular in dimensions N \u2265 13. Indeed by the classical Hardy-Rellich inequality and u := w3.5, Lemma 7.9 only implies that u\u2217 is singular in dimensions N \u2265 22. 164 Bibliography [1] G. Arioli, F. Gazzola, H.-C. Grunau, E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity. SIAM J. Math. Anal. 36 (2005), No. 4, 1226-1258. [2] H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Compl. Madrid 10 (1997), 443-468. [3] C. Cowan, P. Esposito, N. Ghoussoub, A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., to appear. [4] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational meth- ods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ra- tion. Mech. Anal., 58 (1975), pp. 207-218. [5] J. Da\u0301vila, L. Dupaigne, I. Guerra, and M. Montenegro, Stable Solutions for the Bilaplacian with Exponential Nonlinearity, SIAM J. Math. Anal. 39 (2007) 565-592. [6] N. Ghoussoub, A. Moradifam, Bessel pairs and optimal Hardy and Hardy- Rellich inequalities, submitted. [7] N. Ghoussoub, A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA 105 (2008) no 37, 13746-13751. [8] Z. Gui, J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2009), 2034-2054. [9] F. Mignot, J.P. Puel, Solution radiale singuliere de \u2212\u2206u = \u03bbeu. C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), no. 8, 379-382. [10] A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, Journal of Differential Equations, 248 (2010), 594-616. [11] J. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Differential Equations 21 (1996), 1451-1467. 165 Part III Preconditioning of Nonsymmetric Linear Systems 166 Chapter 8 Simultaneous preconditioning and symmetrization of non-symmetric linear systems 7 8.1 Introduction and main results Many problems in scientific computing lead to systems of linear equations of the form, Ax = b, (8.1) where A \u2208 Rn\u00d7n is a nonsingular but sparse matrix, and b is a given vector in Rn and various iterative methods have been developed for a fast and efficient resolution of such systems. The Conjugate Gradient Method (CG) which is the oldest and best known of the nonstationary iterative methods, is highly effective in solving symmetric positive definite systems. For indefinite matrices, the minimization fea- ture of CG is no longer an option, but the Minimum Residual (MINRES) and the Symmetric LQ (SYMMLQ) methods are often computational alternatives for CG, since they are applicable to systems whose coefficient matrices are symmetric but possibly indefinite. The case of non-symmetric linear systems is more challenging, and again meth- ods such as CGNE, CGNR, GMRES, BiCG, QMR, CGS, and Bi-CGSTAB have been developed to deal with these situations (see the survey books [9] and [11]). One approach to deal with the non-symmetric case, consists of reducing the prob- lem to a symmetric one to which one can apply the above mentioned schemes. The one that is normally used consists of simply applying CG to the normal equations ATAx = AT b or AAT y = b, x = AT y. (8.2) 7A version of this chapter has been accepted for publication. N. Ghoussoub and A. Moradifam, Simultaneous preconditioning and symmetrization of non-symmetric linear systems, Numer. Linear Algebra Appl., to appear (2010). 167 8.1. Introduction and main results It is easy to understand and code this approach, and the CGNE and CGNRmethods are based on this idea. However, the convergence analysis of these methods depends closely on the condition number of the matrix under study. For a general matrix A, the condition number is defined as \u03ba(A) = \u2016A\u2016 \u00b7 \u2016A\u22121\u2016, (8.3) and in the case where A is positive definite and symmetric, the condition number is then equal to \u03ba(A) = \u03bbmax(A) \u03bbmin(A) , (8.4) where \u03bbmin(A) (resp., \u03bbmax(A)) is the smallest (resp., largest) eigenvalue of A). The two expressions can be very different for non-symmetric matrices, and these are precisely the systems that seem to be the most pathological from numerical point of view. Going back to the crudely symmetrized system (8.2), we echo Greenbaum\u2019s statement [9] that numerical analysts cringe at the thought of solving these normal equations because the condition number (see below) of the new matrix ATA is the square of the condition number of the original matrix A. In this paper, we shall follow a similar approach that consists of symmetriz- ing the problem so as to be able to apply CG, MINRES, or SYMMLQ. However, we argue that for a large class of non-symmetric, ill-conditionned matrices, it is sometimes beneficial to replace problem (8.1) by one of the form ATMAx = ATMb, (8.5) whereM is a symmetric and positive definite matrix that can be chosen properly so as to obtain good convergence behavior for CG when it is applied to the resulting symmetric ATMA. This reformulation should not only be seen as a symmetriza- tion, but also as preconditioning procedure. While it is difficult to obtain general conditions on M that ensure higher efficiency by minimizing the condition num- ber \u03ba(ATMA), we shall show theoretically and numerically that by choosing M to be either the inverse of the symmetric part of A, or its resolvent, one can get surprisingly good numerical schemes to solve (8.1). The basis of our approach originates from the selfdual variational principle developed in [6, 7] to provide a variational formulation and resolution for non self- adjoint partial differential equations that do not normally fit in the standard Euler- Lagrangian theory. Applied to the linear system (8.1), the new principle yields the following procedure. Split the matrix A into its symmetric Aa (resp., anti- symmetric part Aa) A = As +Aa, (8.6) where As := 1 2 (A+AT ) and Aa := 1 2 (A\u2212AT ). (8.7) Proposition 8.8. (Selfdual symmetrization) Assume the matrix A is positive def- inite, i.e., for some \u03b4 > 0, \u3008Ax, x\u3009 \u2265 \u03b4|x|2 for all x \u2208 Rn. (8.9) 168 8.1. Introduction and main results The convex continuous functional I(x) = 1 2 \u3008Ax, x\u3009+ 1 2 \u3008A\u22121s (b\u2212Aax), b\u2212Aax\u3009 \u2212 \u3008b, x\u3009 (8.10) then attains its minimum at some x\u0304 in Rn, in such a way that I(x\u0304) = inf x\u2208Rn I(x) = 0 (8.11) Ax\u0304 = b. (8.12) Here \u3008x, y\u3009 = xT y and |x|2 = \u3008x, x\u3009. Symmetrization and preconditioning via selfduality: Note that the func- tional I can be written as I(x) = 1 2 \u3008A\u0303x, x\u3009+ \u3008AaA\u22121s b\u2212 b, x\u3009+ 1 2 \u3008A\u22121s b, b\u3009, (8.13) where A\u0303 := As \u2212AaA\u22121s Aa = ATA\u22121s A. (8.14) By writing that DI(x\u0304) = 0 (DI is the subdifferential of the functional I), one gets the following equivalent way of solving (8.1). If both A \u2208 Rn\u00d7n and its symmetric part As are nonsingular, then x is a solution of the equation (8.1) if and only if it is a solution of the linear symmetric equation ATA\u22121s Ax = (As \u2212AaA\u22121s Aa)x = b\u2212AaA\u22121s b = ATA\u22121s b. (8.15) One can therefore apply to (8.15) all known iterative methods for symmetric sys- tems to solve the non-symmetric linear system (8.1). As mentioned before, the new equation (8.15) can be seen as a new symmetrization of problem (8.1) which also preserves positivity, i.e., ATA\u22121s A is positive definite if A is. This will then allow for the use of the Conjugate Gradient Method (CG) for the functional I. More important and less obvious than the symmetrization effect of A\u0303, is our observation that for a large class of matrices, the convergence behavior of the system (8.15) is more favorable than the original one. The Conjugate Gradient method \u2013which can now be applied to the symmetrized matrix A\u0303\u2013 has the potential of providing an efficient algorithm for resolving non-symmetric linear systems. We shall call this scheme the Self-Dual Conjugate Gradient for Non-symmetric matrices and we will refer to it as SD-CGN. As mentioned above, the convergence analysis of this method depends closely on the condition number \u03ba(A\u0303) of A\u0303 = ATA\u22121s A. We observe in section 2.3 that for a large class of ill-conditioned matrices, \u03ba(A\u0303) may be very small and hence SD-CGN can be very efficient. In other words, the inverse C of ATA\u22121s can be an efficient preconditioning matrix, in spite of the additional cost involved in finding the inverse of As. Moreover, the efficiency of C seems to surprisingly improve in many cases 169 8.1. Introduction and main results as the norm of the anti-symmetric part gets larger (Proposition 2.2). A typical example is when the anti-symmetric matrix Aa is a multiple of the symplectic matrix J (i.e. JJ\u2217 = \u2212J2 = I). Consider then a matrix A\u000f = As + 1\u000fJ which has an arbitrarily large anti-symmetric part. One can show that \u03ba(A\u0303\u000f) \u2264 \u03ba(As) + \u000f2\u03bbmax(As)2, (8.16) which means that the larger the anti-symmetric part, the smaller our upper bound for \u03ba(A\u0303\u000f) and consequently the more efficient is our proposed selfdual precondi- tioning. Needless to say that this method is of practical interest only when the equation Asx = d can be solved with less computational effort than the original system, which is not always the case. Now the relevance of this approach stems from the fact that non-symmetric Krylov subspace solvers are costly since they require the storage of previously cal- culated vectors. It is however worth noting that Concus and Golub [3] and Widlund [15] have also proposed another way to combine CG with a preconditioning using the symmetric part As, which does not need this extended storage. Their method has essentially the same cost per iteration as the preconditioning with the inverse of ATA\u22121s that we propose for SD-CGN and both schemes converge to the solution in at most N iterations. Iterated preconditioning: Another way to see the relevance of As as a pre- conditioner, is by noting that the convergence of \u201csimple iteration\u201d Asxk = \u2212Aaxk\u22121 + b (8.17) applied to the decomposition of A into its symmetric and anti-symmetric parts, requires that the spectral radius \u03c1(I \u2212 A\u22121s A) = \u03c1(A\u22121s Aa) < 1. By multiplying (8.17) by A\u22121s , we see that this is equivalent to the process of applying simple iteration to the original system (8.1) conditioned by A\u22121s , i.e., to the system A\u22121s Ax = A \u22121 s b. (8.18) On the other hand, \u201csimple iteration\u201d applied to the decomposition of A\u0303 into As and AaA\u22121s Aa is given by Asxk = AaA\u22121s Aaxk\u22121 + b\u2212AaA\u22121s b. (8.19) Its convergence is controlled by \u03c1(I \u2212 A\u22121s A\u0303) = \u03c1((A\u22121s Aa)2) = \u03c1(A\u22121s Aa)2 which is strictly less than \u03c1(A\u22121s Aa), i.e., an improvement when the latter is strictly less than one, which the mode in which we have convergence. In other words, the linear system (8.15) can still be preconditioned one more time as follows: If both A \u2208 Rn\u00d7n and its symmetric part As are nonsingular, then x is a solution of the equation (8.1) if and only if it is a solution of the linear symmetric equation A\u0304x := A\u22121s A TA\u22121s Ax = [I \u2212 (A\u22121s Aa)2]x = (I \u2212A\u22121s Aa)A\u22121s b = A\u22121s ATA\u22121s b. (8.20) 170 8.1. Introduction and main results Note however that with this last formulation, one has to deal with the potential loss of positivity for the matrix A\u0303. Anti-symmetry in transport problems: Numerical experiments on standard linear ODEs (Example 3.1) and PDEs (Example 3.2), show the efficiency of SD- CGN for non-selfadjoint equations. Roughly speeking, discretization of differential equations normally leads to a symmetric component coming from the Laplace oper- ator, while the discretization of the non-self-adjoint part leads to the anti-symmetric part of the coefficient matrix. As such, the symmetric part of the matrix is of order O( 1h2 ), while the anti-symmetric part is of order O( 1 h ), where h is the step size. The coefficient matrix A in the original system (8.1) is therefore an O(h) perturbation of its symmetric part. However, for the new system (8.15) we have roughly A\u0303 = As \u2212AaA\u22121s Aa = O( 1 h2 )\u2212O( 1 h )O(h2)O( 1 h ) = O( 1 h2 )\u2212O(1), (8.21) making the matrix A\u0303 an O(1) perturbation of As, and therefore a matrix of the form As + \u03b1I becomes a natural candidate to precondition the new system (8.15). Resolvents of As as preconditioners: One may therefore consider precondi- tioned equations of the form ATMAx = ATMb, where M is of the form M\u03b1 = ( \u03b1As + (1\u2212 \u03b1)I )\u22121 or N\u03b2 = \u03b2A\u22121s + (1\u2212 \u03b2)I, (8.22) for some 0 \u2264 \u03b1, \u03b2 \u2208 R, and where I is the unit matrix. Note that we obviously recover (8.2) when \u03b1 = 0, and (8.15) when \u03b1 = 1. As \u03b1\u2192 0 the matrix \u03b1As + (1\u2212 \u03b1)I becomes easier to invert, but the matrix A1,\u03b1 = AT (\u03b1As + (1\u2212 \u03b1)I)\u22121A (8.23) may become more ill conditioned, eventually leading (for \u03b1 = 0) to ATAx = AT b. There is therefore a trade-off between the efficiency of CG for the system (8.5) and the condition number of the inner matrix \u03b1As+(1\u2212\u03b1)I, and so by an appropriate choice of the parameter \u03b1 we may minimize the cost of finding a solution for the system (8.1). In the case where As is positive definite, one can choose \u2013and it is sometimes preferable as shown in example (3.4)\u2013 \u03b1 > 1, as long as \u03b1 < 11\u2212\u03bbsmin , where \u03bbsmin is the smallest eigenvalue of As. Moreover, in the case where the matrix A is not positive definite or if its symmetric part is not invertible, one may take \u03b1 small enough, so that the matrix M\u03b1 (and hence A1,\u03b1) becomes positive definite, and therefore making CG applicable (See example 3.4). Similarly, the matrix N\u03b2 = \u03b2A\u22121s + (1 \u2212 \u03b2)I provides another choice for the matrix M in (8.5), for \u03b2 < \u03bb s max \u03bbsmax\u22121 where \u03bb s max is the largest eigenvalue of As. Again we may choose \u03b1 close to zero to make the matrix N\u03b2 positive definite. As we will see in the last section, appropriate choices of \u03b2, can lead to better convergence of CG for equation (8.5). One can also combine both effects by considering matrices of the form L\u03b1,\u03b2 = ( \u03b1As + (1\u2212 \u03b1)I )\u22121 + \u03b2I, (8.24) 171 8.2. Selfdual methods for non-symmetric systems as is done in example (3.4). We also note that the matricesM \u2032\u03b1 := (\u03b1A \u2032 s+(1\u2212\u03b1)I)\u22121 and N \u2032\u03b2 := \u03b2(A\u2032s)\u22121+ (1\u2212\u03b2)I can be other options for the matrixM , where A\u2032s is a suitable approximation of As, chosen is such a way that M \u2032\u03b1q and N \u2032 \u03b2q can be relatively easier to compute for any given vector q. Finally, we observe that the above reasoning applies to any decomposition A = B + C of the non-singular matrix A \u2208 Rn\u00d7n, where B and (B \u2212 C) are both invertible. In this case, B(B\u2212C)\u22121 can be a preconditioner for the equation (8.1). Indeed, since B \u2212 CB\u22121C = (B \u2212 C)B\u22121A, x is a solution of (8.1) if and only of it is a solution of the system (B \u2212 C)B\u22121Ax = (B \u2212 CB\u22121C)x = b\u2212 CB\u22121b. (8.25) In the next section, we shall describe a general framework based on the ideas ex- plained above for the use of iterative methods for solving non-symmetric linear systems. In section 3 we present various numerical experiments to test the effec- tiveness of the proposed methods. 8.2 Selfdual methods for non-symmetric systems By selfdual methods we mean the ones that consist of first associating to problem (8.1) the equivalent system (8.5) with appropriate choices of M , then exploiting the symmetry of the new system by using the various existing iterative methods for symmetric systems such as CG, MINRES, and SYMMLQ, leading eventually to the solution of the original problem (8.1). In the case where the matrix M is positive definite and symmetric, one can then use CG on the equivalent system (8.5). This scheme (SD-CGN) is illustrated in Table (1) below, in the case where the matrixM is chosen to be the inverse of the symmetric part of A. IfM is not positive definite, then one can use MINRES (or SYMMLQ) to solve the system (8.15). We will then refer to them as SD-MINRESN (i.e., Self-Dual MINRES for Nonsymmetric linear equations). 8.2.1 Exact methods In each iteration of CG, MINRES, or SYMMLQ, one needs to compute Mq for certain vectors q. Since selfdual methods call for a preconditioner matrix M that involves inverting another one, the computation of Mq can therefore be costly, and therefore not necessarily efficient for all linear equations. But as we will see in section 3, M can sometimes be chosen so that computing Mq is much easier than solving the original equation itself. This is the case for example when the symmetric part is either diagonal or tri-diagonal, or when we are dealing with several linear systems all having the same symmetric part, but with different anti-symmetric components. Moreover, one need not find the whole matrixM , in order to compute Mq. The following scheme illustrates the exact SD-CGN method applied in the case 172 8.2. Selfdual methods for non-symmetric systems Given an initial guess x0, Solve Asy = b Compute b = b\u2212Aay. Solve Asy0 = Aax0 Compute r0 = b\u2212Asx0 +Aay0 and set p0 = r0. For k=1,2, . . . , Solve Asz = Aapk\u22121 Compute w = Aspk\u22121 \u2212Aaz . Set xk = xk\u22121 + \u03b1k\u22121pk\u22121, where \u03b1k\u22121 = . Cpmpute rk = rk\u22121 \u2212 \u03b1k\u22121w. Set pk = rk + bk\u22121pk\u22121, where bk\u22121 = . Check convergence; continue if necessary. Table 8.1: GCGN where the coefficient matrix A in (8.1) is positive definite, and when ATA\u22121s Aq can be computed exactly for any given vector q. In the case where A is not positive definite, or when it is preferable to choose a non-positive definite conditioning matrix M , then one can apply MINRES or SYMMLQ to the equivalent system (8.5). These schemes will be then called SD- MINRESN and SD-SYMMLQN respectively. 8.2.2 Inexact methods The SD-CGN, SD-MINRESN and SD-SYMMLQN are of practical interest when for example, the equation Asx = q (8.26) can be solved with less computational effort than the original equation (8.1). Ac- tually, one can use CG, MINRES, or SYMMLQ to solve (8.26) in every iteration of SD-CGN, SD-MINRESN, or SD-SYMMLQN. But since each sub-iteration may lead to an error in the computation of (8.26), one needs to control such errors, in order for the method to lead to a solution of the system (8.1) with the desired tolerance. This leads to the Inexact SD-CGN, SD-MINRESN and SD-SYMMLQN methods (denoted below by ISD-CGN, ISD-MINRESN and ISD-SYMMLQN respectively). The following proposition \u2013which is a direct consequence of Theorem 4.4.3 in [9]\u2013 shows that if we solve the inner equations (8.26) \u201caccurately enough\u201d then ISD-CGN and ISD-MINRESN can be used to solve (8.1) with a pre-determined accuracy. Indeed, given \u000f > 0, we assume that in each iteration of ISD-CGN or ISD-MINRESN, we can solve the inner equation \u2013corresponding to As\u2013 accurately 173 8.2. Selfdual methods for non-symmetric systems enough in such a way that \u2016(As \u2212AaA\u22121s Aa)p\u2212 (Asp\u2212Aay)\u2016 = \u2016AaA\u22121s Aap\u2212Aay\u2016 < \u000f, (8.27) where y is the (inexact) solution of the equation Asy = Aap. (8.28) In other words, we assume CG and MINRES are implemented on (8.28) in a finite precision arithmetic with machine precision \u000f. Set \u000f0 := 2(n+ 4)\u000f, \u000f1 := 2(7 + n \u2016 |As \u2212AaA\u22121s Aa| \u2016| \u2016As \u2212AaA\u22121s Aa\u2016 )\u000f, (8.29) where |D| denotes the matrix whose terms are the absolute values of the correspond- ing terms in the matrix D. Let \u03bb1 \u2264 ... \u2264 \u03bbn be the eigenvalues of (As\u2212AaA\u22121s Aa) and let Tk+1,k be the (k+ 1)\u00d7 k tridiagonal matrix generated by a finite precision Lanczos computation. Suppose that there exists a symmetric tridiagonal matrix T , with Tk+1,k as its upper left (k + 1) \u00d7 k block, whose eigenvalues all lie in the intervals S = \u222aki=1[\u03bbi \u2212 \u03b4, \u03bbi + \u03b4], (8.30) where none of the intervals contain the origin. Let d denote the distance from the origin to the set S, and let pk denote a polynomial of degree k. Proposition 8.31. The ISD-MINRESN residual rIMk then satisfies ||rIMk || ||r0|| \u2264 \u221a (1 + 2\u000f0)(k + 1) min pk max z=S |pk(z)|+ 2 \u221a k( \u03bbn d )\u000f1. (8.32) If A is positive definite, then the ISD-CGN residual rIC satisfies ||rICk || ||r0|| \u2264 \u221a (1 + 2\u000f0)(\u03bbn + \u03b4)\/d min pk max z=S |pk(z)|+ \u221a k( \u03bbn d )\u000f1. (8.33) It is shown by Greenbaum [6] that Tk+1,k can be extended to a larger symmetric tridiagonal matrix T whose eigenvalues all lie in tiny intervals about the eigenvalues of (As \u2212 AaA\u22121s Aa). Hence the above proposition guarantees that if we solve the inner equations accurate enough, then ISD-CGN and ISD-MINRESN converges to the solution of the system (8.1) with the desired relative residual (see the last section for numerical experiments). 8.2.3 Preconditioning As mentioned in the introduction, the convergence of iterative methods depends heavily on the spectral properties of the coefficient matrix. Preconditioning tech- niques attempt to transform the linear system (8.1) into an equivalent one of the 174 8.2. Selfdual methods for non-symmetric systems form C\u22121Ax = C\u22121b, in such a way that it has the same solution, but hopefully with more favorable spectral properties. As such the reformulation of (1) as ATA\u22121s Ax = A TA\u22121s b, (8.34) can be seen as a preconditioning procedure with C being the inverse of ATA\u22121s . The spectral radius, and more importantly the condition number of the coefficient matrix in linear systems, are crucial parameters for the convergence of iterative methods. The following simple proposition gives upper bounds on the condition number of A\u0303 = ATA\u22121s A. Proposition 8.35. Assume A is an invertible positive definite matrix, then \u03ba(A\u0303) \u2264 min{\u03ba1, \u03ba2}, (8.36) where \u03ba1 := \u03ba(As) + \u2016Aa\u20162 \u03bbmin(As)2 and \u03ba2 := \u03ba(As)\u03ba(\u2212A2a) + \u03bbmax(As) 2 \u03bbmin(\u2212A2a) . (8.37) Proof: We have \u03bbmin(A\u0303) = \u03bbmin(As \u2212AaA\u22121s Aa) \u2265 \u03bbmin(As). We also have \u03bbmax(A\u0303) = sup x6=0 xT A\u0303x xTx = sup x6=0 xt(As \u2212AaA\u22121s Aa)x xTx \u2264 \u03bbmax(As) + ||Aa|| 2 \u03bbmin(As) . Since \u03ba(A\u0303) = \u03bbmax(A\u0303) \u03bbmin(A\u0303) , it follows that \u03ba(A\u0303) \u2264 \u03ba1. To obtain the second estimate, observe that \u03bbmin(A\u0303) = \u03bbmin(As \u2212AaA\u22121s Aa) > \u03bbmin(\u2212AaA\u22121s Aa) = inf x6=0 \u2212xTAaA\u22121s Aax xTx = inf x6=0 { (Aax) TA\u22121s (Aax) (Aax)T (Aax) \u00d7 (Aax) T (Aax) xTx } \u2265 inf x6=0 (Aax)TA\u22121s (Aax) (Aax)T (Aax) \u00d7 inf x6=0 xT (Aa)T (Aa)x xTx = 1 \u03bbmax(As) \u00d7 \u03bbmin((Aa)TAa) = 1 \u03bbmax(As) \u00d7 \u03bbmin(\u2212A2a) With the same estimate for \u03bbmax(A\u0303) we get \u03ba(A\u0303) \u2264 \u03ba2. 175 8.2. Selfdual methods for non-symmetric systems Remark 8.2.1. Inequality (8.36) shows that SD-CGN and SD-MINRES can be very efficient schemes for a large class of ill conditioned non-symmetric matrices, even those that are almost singular and with arbitrary large condition numbers. It suffices that either \u03ba1 or \u03ba2 be small. Indeed, \u2022 The inequality \u03ba(A\u0303) \u2264 \u03ba1 shows that the condition number \u03ba(A\u0303) is reasonable as long as the anti-symmetric part Aa is not too large. On the other hand, even if \u2016Aa\u2016 is of the order of \u03bbmax(As), and \u03ba(A\u0303) is then as large as \u03ba(As)2, it may still be an improved situation, since this can happen for cases when \u03ba(A) is exceedingly large. This can be seen in example 2.2 below. \u2022 The inequality \u03ba(A\u0303) \u2264 \u03ba2 is even more interesting especially in situations when \u03bbmin(\u2212A2a) is arbitrarily large while remaining of the same order as ||Aa||2. This means that \u03ba(A\u0303) can remain of the same order as \u03ba(As) regardless how large is Aa. A typical example is when the anti-symmetric matrix Aa is a multiple of the sym- plectic matrix J (i.e. JJ\u2217 = \u2212J2 = I). Consider then a matrix A\u000f = As + 1\u000fJ which has an arbitrarily large anti-symmetric part. By using that \u03ba(A\u0303) \u2264 \u03ba2, one gets \u03ba(A\u0303\u000f) \u2264 \u03ba(As) + \u000f2\u03bbmax(As)2. (8.38) Here are other examples where the larger the condition number of A is, the more efficient is the proposed selfdual preconditioning. Consider the matrix A\u000f = [ 1 \u22121 1 \u22121 + \u000f ] (8.39) which is a typical example of an ill-conditioned non-symmetric matrix. One can actually show that \u03ba(A\u000f) = O( 1\u000f ) \u2192 \u221e as \u000f \u2192 0 with respect to any norm. However, the condition number of the associated selfdual coefficient matrix A\u0303\u000f = As \u2212Aa(As)\u22121Aa = [ \u000f \u000f\u22121 0 0 \u000f ] is \u03ba(A\u0303\u000f) = 11\u2212\u03b5 , and therefore goes to 1 as \u03b5 \u2192 0. Note also that the condition number of the symmetric part of A\u000f goes to one as \u000f\u2192 0. In other words, the more ill-conditioned problem (8.1) is, the more efficient the selfdual conditioned system (8.15) is. We also observe that \u03ba(A\u22121s A) goes to \u221e as \u000f goes to zero, which means that besides making the problem symmetric, our proposed conditioned matrix ATA\u22121s A has a much smaller condition number than the matrix A\u22121s A, which uses As as a preconditioner. Similarly, consider the non-symmetric linear system with coefficient matrix A\u000f = [ 1 \u22121 + \u000f 1 \u22121 ] . (8.40) As \u000f \u2192 0, the matrix becomes again more and more ill-conditioned, while the condition number of its symmetric part converges to one. Observe now that the 176 8.3. Numerical experiments condition number of A\u0303\u000f also converges to 1 as \u000f goes to zero. This example shows that self-dual preconditioning can also be very efficient for non-positive definite problems. 8.3 Numerical experiments In this section we present some numerical examples to illustrate the proposed schemes and to compare them to other known iterative methods for non-symmetric linear systems. Our experiments have been carried out on Matlab (7.0.1.24704 (R14) Service Pack 1). In all cases the iteration was started with x0 = 0. Consider the ordinary differential equation \u2212\u000fy\u2032\u2032 + y\u2032 = f(x), on [0, 1], y(0) = y(1) = 0. (8.41) By discretizing this equation with stepsize 1\/65 and by using backward difference for the first order term, one obtains a nonsymmetric system of linear equations with 64 unknowns. We present in Table 2 below, the number of iterations needed for various decreasing values of the residual \u000f. We use ESD-CGN and ISD-CGN (with relative residual 10\u22127 for the solutions of the inner equations). We then compare them to the known methods CGNE, BiCG, QMR, CGS, and BiCGSTAB for solving non-symmetric linear systems. We also test preconditioned version of these methods by using the symmetric part of the corresponding matrix as a preconditioner. Table 8.2: Number of iterations for (8.41) with the solution y = x sin(pix). N=64 \u000f = 10\u22122 \u000f = 10\u22123 \u000f = 10\u22124 \u000f = 10\u22126 \u000f = 10\u221210 \u000f = 10\u221216 ESD-CGN 22 8 5 4 3 2 ISD-CGN 24 9 6 4 3 2 GCNE 88 64 64 64 64 64 QMR 114 > 1000 > 1000 > 1000 > 1000 > 1000 PQMR 34 51 50 52 52 52 BiCGSTAB 63.5 78.5 92.5 98.5 100.5 103.5 PBiCGSTAB 26.5 46.5 50.5 50 51.5 51.5 BiCG 125 > 1000 > 1000 > 1000 > 1000 > 1000 PBiCG 31 44 50 50 52 52 CGS > 1000 > 1000 > 1000 > 1000 > 1000 > 1000 PCGS 27 51 46 46 46 48 As we see in Tables 2 and and 3, a phenomenon similar to Example 8.2.3 is occuring. As the problem gets harder (\u000f smaller), SD-CGN becomes more efficient. These results can be compared with the number of iterations that the HSS iteration method needs to solve equation (8.41) (Tables 3,4, and 5 in [2]). Consider the partial differential equation \u2212\u2206u+ a\u2202u \u2202x = f(x, y), 0 \u2264 x \u2264 1, 0 \u2264 y \u2264 1, (8.42) 177 8.3. Numerical experiments with Dirichlet boundary condition. The number of iterations that ESD-CGN and ISD-CGN needed to find a solution with relative residual 10\u22126, are presented in Table 4 below for different coefficients a. Table 8.3: Number of iterations for equation (8.41) with the solution y = x(1\u2212x) cos(x) . N=128 \u000f = 10\u22122 \u000f = 10\u22123 \u000f = 10\u22124 \u000f = 10\u22126 \u000f = 10\u221210 \u000f = 10\u221216 ESD-CGN 37 11 6 4 3 2 ISD-CGN(10\u22127) 38 12 7 4 3 2 GCNE 266 140 128 128 128 128 QMR > 1000 > 1000 > 1000 > 1000 > 1000 > 1000 PQMR 40 77 87 92 90 85 BiCGSTAB 136.5 167.5 241 226.5 233.5 237.5 PBiCGSTAB 35.5 87.5 106.5 109 110.5 110.5 BiCG > 1000 > 1000 > 1000 > 1000 > 1000 > 1000 PBiCG 37 76 84 89 85 91 CGS > 1000 > 1000 > 1000 > 1000 > 1000 > 1000 PCGS 34 80 96 91 94 90 Table 4 and 5 can be compared with Table 1 in [15], where Widlund had tested his Lanczos method for non-symmetric linear systems. Comparing Table 5 with Table 1 in [15] we see that for small a (1 and 10) Widlund\u2019s method is more efficient than SD-CGN, but for large values of a, SD-CGN turns out to be more efficient than Widlund\u2019s Lanczos method. Remark 8.3.1. As we see in Tables 2,3, and 4, the number of iterations for ESD- CGN and ISD-CGN (with relative residual 10\u22127 for the solutions of the inner equa- tions) are almost the same One might choose dynamic relative residuals for the so- lutions of inner equations to decrease the average cost per iterations of ISD-CGN. It is interesting to figure out whether there is a procedure to determine the accuracy of solutions for the inner equations to minimize the total cost of finding a solution. Consider the partial differential equation \u2212\u2206u+ 10\u2202(exp(3.5(x 2 + y2)u) \u2202x + 10 exp(3.5(x2 + y2)) \u2202u \u2202x = f(x), (8.43) on [0, 1]\u00d7 [0, 1] with Dirichlet boundary condition, and choose f so that sin(pix) sin(piy) exp((x\/2 + y)3) is the solution of the equation. We take the stepsize h = 1\/31 which leads to a linear system Ax = b with 900 unknowns. Table 5 includes the number of iterations 178 8.3. Numerical experiments Table 8.4: Number of iterations for the backward scheme method (Example 3.2) a N I (ESD-CGN) I (ISD-CGN) Solution 100 49 18 18 random 100 225 40 37 random 100 961 44 46 random 100 961 52 51 sinpix sinpiy. exp((x\/2 + y)3) 1000 49 10 10 random 1000 225 31 31 random 1000 961 36 37 random 1000 961 31 39 sinpix sinpiy. exp((x\/2 + y)3) 106 49 4 4 random 106 225 6 6 random 106 961 6 6 random 106 961 6 6 sinpix sinpiy. exp((x\/2 + y)3) 1016 961 2 2 sinpix sinpiy. exp((x\/2 + y)3) which CG needs to converge to a solution with relative residual 10\u22126 when applied to the preconditioned matrix AT (\u03b1A\u22121s + (1\u2212 \u03b1)I)A. (8.44) Table 5 can be compared with Table 1 in [15], where Widlund has presented the number of iterations needed to solve equation (8.43). Remark 8.3.2. As we see in Table 5, for \u03bbsmax( 1\u2212\u03b1 \u03b1 ) = \u2212.99 we have the minimum number of iterations. Actually, this is the case in some other experiments, but for many other system the minimum number of iterations accrues for some other \u03b1 with \u22121 < \u03bbsmax( 1\u2212\u03b1\u03b1 ) \u2264 0. Our experiments show that for a well chosen \u03b1 > 1, one may considerably decrease the number of iterations. Obtaining theoretical results on how to choose parameter \u03b1 in 8.44 seems to be an interesting problem. Note that the coefficient matrix of the linear system corresponding to (8.43) is positive definite. Hence we may also apply CG with the preconditioned symmetric system of equations AT (As \u2212 \u03b1\u03bbsminI)\u22121A = AT (As \u2212 \u03b1\u03bbsminI)\u22121b, (8.45) where \u03bbsmin is the smallest eigenvalue of As and \u03b1 < 1. The number of iterations function of \u03b1, that CG needs to converges to a solution with relative residual 10\u22126 are presented in Table 7. 179 8.3. Numerical experiments Table 8.5: Number of iterations for the centered difference scheme method (Example 3.2) a N I (ESD-CGN) Solution Relative Residoual 1 49 21 random 6.71\u00d7 10\u22126 1 225 73 random 9.95\u00d7 10\u22126 1 961 91 random 8.09\u00d7 10\u22126 1 961 72 sinpix sinpiy. exp((x\/2 + y)3) 9.70\u00d7 10\u22126 10 49 18 random 9.97\u00d7 10\u22126 10 225 65 random 5.90\u00d7 10\u22126 10 961 78 random 8.95\u00d7 10\u22126 10 961 65 sinpix sinpiy. exp((x\/2 + y)3) 7.78\u00d7 10\u22126 100 49 31 random 6.07\u00d7 10\u22126 100 225 42 random 5.20\u00d7 10\u22126 100 961 43 random 5.03\u00d7 10\u22126 100 961 38 sinpix sinpiy. exp((x\/2 + y)3) 4.69\u00d7 10\u22126 1000 49 65 random 4.54\u00d7 10\u22126 1000 225 130 random 8.66\u00d7 10\u22126 1000 961 140 random 2.12\u00d7 10\u22126 100 961 150 sinpix sinpiy. exp((x\/2 + y)3) 5.98\u00d7 10\u22126 Remark 8.3.3. As we see in the above table, for \u03b1 = 0.99 in (8.45) we have the minimum number of iterations. Obtaining theoretical results on how to choose the parameter \u03b1 seems to be an interesting problem to study. We also repeat the experiment by applying CG to the system of equations AT ( As \u2212 0.99\u03bbsminI)\u22121 \u2212 0.99 \u03bbsmax I ) A = AT ( (As \u2212 o.99\u03bbsminI)\u22121 \u2212 0.99 \u03bbsmax I ) b. (8.46) Then CG needs 131 iterations to converge to a solution with relative residual 10\u22126. As another experiment we apply CG to the preconditioned linear system A\u22121s A TA\u22121s Ax = A \u22121 s A TA\u22121s b, to solve the non-symmetric linear system obtained from discritization of the Equa- tion (8.43). The CG converges in 31 iterations to a solution with relative residual less than 10\u22126. Since, we need to solve two equations with the coefficient matrix As, the cost of each iteration in this case is twice as much as SD-CGN. So, by the 180 8.3. Numerical experiments above preconditioning we decrease cost of finding a solution to less that 62\/131 of that of SD-CGN (System (8.46)). Consider now the following equation \u2212\u2206u+10\u2202(exp(3.5(x 2 + y2)u) \u2202x +10 exp(3.5(x2+y2)) \u2202u \u2202x \u2212200u = f(x), on [0, 1]\u00d7[0, 1], (8.47) If we discretize this equation with stepsize 1\/31 and use backward differences for the first order term, we get a linear system of equations Ax = b with A being a non-symmetric and non-positive definite coefficient matrix. We then apply CG to the following preconditioned, symmetrized and positive definite matrix AT ((As \u2212 \u03b1\u03bbsminI)\u22121 + \u03b2I)A = AT ((As \u2212 \u03b1\u03bbsminI)\u22121 + \u03b2I)b, (8.48) with \u03b1 < 1. For different values of \u03b1 the number of iterations which CG needs to converge to a solution with the relative residual 10\u22126 are presented in Table 8. We Table 8.6: Number of iterations for SD-CGN with different values of \u03b1. \u03bbsmax( 1\u2212\u03b1 \u03b1 ) I \u03bbsmax( 1\u2212\u03b1 \u03b1 ) I \u221e(\u03b1 = 0) > 5000 0.1 232 0(\u03b1 = 1) 229 0.2 237 -0.1 221 0.4 249 -0.25 216 0.8 263 -0.5 201 1 272 -0.7 191 5 384 -0.8 186 10 474 -0.9 180 20 642 -0.95 179 50 890 -0.99 177 100 1170 -0.999 180 1000 2790 -0.9999 234 10000 4807 repeat our experiment with stepsize 1\/61 and get a system with 3600 unknowns. With \u03b1 = \u22121.00000001 and \u03b2 = 0, CG converges in one single iteration to a solution with relative residual less than 10\u22126. We also apply QMR, BiCGSTAB, BiCG, and CGS (also preconditioned with the symmetric part as well) to solve the corresponding system of linear equations with stepsize 1\/31. The number of iterations needed to converge to a solution with relative residual 10\u22126 are presented in Table 9. Acknowledgments: This paper wouldn\u2019t have seen the light without the gentle prodding and constant encouragement of Anthony Peirce, and the expert guidance and generous support of Chen Greif. They have our deep and sincere gratitude. 181 8.3. Numerical experiments Table 8.7: Number of iterations for (8.43) with different values of \u03b1. \u03b1 I 0 229 0.5 204 0.9 177 0.99 166 0.999 168 0.9999 181 0.99999 194 0.999999 222 0.9999999 248 0.99999999 257 182 Bibliography [1] O. Axelsson, Z.-Z. Bai, and S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms, to appear. [2] Z.-Z. Bai, G. H. Golub, and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603-626. [3] P. Concus and G. H. 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Van Der Vorst, The convergence behaviour of preconditioned CG and CG-S in the presence of rounding errors, Lecture Notes in Math., 1457 (1990), pp. 126-136. [15] O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801-812. 184 Chapter 9 Conclusion In chapters 2, 3, and 4 we presented necessary and sufficient conditions under which one can improve Hardy and Hardy-Rellich type inequalities. Indeed we made a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equations that allowed us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities such as those in [1], [3], [4], [7], [11], and [15]. In chapter 4, we developed an approach to prove various classes of optimal weighted Hardy-Rellich inequalities on H2\u2229H10 which are crucial in the study of fourth order nonlinear elliptic equations and systems of elliptic partial differential equations. The approach developed in [8], [9], and[14] basically finishes the problem of improving Hardy and Hardy-Rellich inequalities in Rn. In chapters 5 and 6, we studied the critical dimension of the fourth order elliptic equation with negative exponent under Drichlet and Navier boundary conditions. In [5], and [8] we showed that under both boundary conditions the critical dimen- sion is N = 9. For a general domain our problem suffers from the lack of energy estimates. Also the blow-up analysis that we use to prove the regularity of the extremal solution in dimensions 5 \u2264 N \u2264 8 does not work. So determining the crit- ical dimension on general domains remains a very interesting and important open problem which probably needs new ideas and techniques. However we conjecture that the critical dimension is N = 9. Improved Hardy-Rellich inequalities obtain in chapters 3 and 4 play an impor- tant role to prove the singular nature of the extremal solutions in large dimensions close to the critical dimension in chapter 5, 6, and 7. In chapter 7 these inequalities allow us to provide a unified mathematical proof for the singularity of the extremal solutions of the bi-laplacian with exponential nonlinearity in dimension N \u2265 13. This result was first proved in [6] by a computer assisted proof. There are many open problems about the singularity of the extremal solutions of nonlinear eigen- value problems. I believe that the above approach can be modified to prove the singularity of the extremal solutions in these problems. In chapter 8, motivated by the theory of self-duality, we proposed new templates for solving large non-symmetric linear systems. Our approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems. Our scheme in [10] is surprisingly efficient when dealing with certain ill-conditioned systems. However, obtaining theoretical result seem to be hard. It is interesting to obtain theoretical results about the SD-CGN scheme that we developed in chapter 8. 185 Bibliography [1] Adimurthi, N. Chaudhuri, and N. Ramaswamy, An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc. 130 (2002), 489-505. [2] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vasquez, Hardy- Poincare\u0301 inequalities and applications to nonlinear diffusions, C. R. Acad. Sci. Paris, Ser. I 344 (2007), 431-436. [3] H. Brezis and J. L. Va\u0301zquez, Blowup solutions of some nonlinear elliptic prob- lems, Revista Mat. Univ. Complutense Madrid 10 (1997), 443-469. [4] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compositio Mathematica 53 (1984), 259-275. [5] C. Cowan, P. Esposito, N. Ghoussoub, A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., to appear. [6] J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal. 39 (2007), 565-592. [7] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002), no. 1, 186-233. [8] N. Ghoussoub, A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Nat. Acad. Sci., vol. 105, no. 37 (2008) p. 13746-13751. [9] N. Ghoussoub, A. Moradifam,Bessel pairs and optimal Hardy-Rellich Inequal- ities, Math. Annalen, Published Online (2010) 57 pp. [10] N. Ghoussoub, A. Moradifam,N. Ghoussoub and A. Moradifam, On simulta- neous preconditioning and symmetrization of non-symmetric linear systems, Numer. Linear Algebra Appl., to appear (2010). [11] Liskevich V., Lyakhova S.and Moroz V. Positive solutions to nonlinear p- Laplace equations with Hardy potential in exterior domains, Journal of Differ- ential Equations 232 (2007), 212-252. [12] A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, Journal of Differential Equations, 248 (2010), 594-616. 186 Chapter 9. Bibliography [13] A. Moradifam, The singular extremal solutions of the bilaplacian with expo- nential nonlinearity, Proc. Amer. Math Soc., 138 (2010), 1287-1293. [14] A. Moradifam, Optimal weighted Hardy-Rellich inequalities on H2 \u2229H10 , sub- mitted. [15] A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequal- ities and related improvements, Advances in Mathematics, 209 (2007) 407-459. 187","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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