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Hardy-Rellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems Moradifam, Amir 2010

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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 Amir Moradifam 2010  Abstract This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigenvalue 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 ≥ 1, so that the following inequalities hold for all u ∈ C0∞ (B): 2 2 B V (x)|∇u| dx ≥ B W (x)u dx,  B  V (x)|∆u|2 dx ≥  B  W (x)|∇u|2 dx + (n − 1)  V (x) B ( |x|2  −  Vr (|x|) 2 |x| )|∇u| dx.  This 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 various classes of Hardy-Rellich inequalities on H 2 ∩ H01 . The second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6 we study the λ extremal solution uλ∗ of the semilinear biharmonic equation ∆2 u = (1−u) 2, which models a simple Micro-Electromechanical System (MEMS) device on a ball B ⊂ RN , under Dirichlet or Navier boundary conditions. We show that u∗ is regular provided N ≤ 8 while uλ∗ is singular for N ≥ 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 ≥ 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 Optimal improved Hardy and Hardy-Rellich inequalities . . . 1.2 Fourth order nonlinear eigenvalue problems . . . . . . . . . . 1.3 Preconditioning and symmetrization of non-symmetric linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 1 8 10  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11  I  13  Hardy and Hardy-Rellich Inequalities  2 On the best possible remaining term in the Hardy inequality 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Two dimensional inequalities . . . . . . . . . . . . . . . . . . 2.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .  14 14 18 20 25  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  iii  Table of Contents 3 Bessel pairs and optimal Hardy and Hardy-Rellich inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 General Hardy Inequalities . . . . . . . . . . . . . . . . . . . 3.2.1 Integral criteria for Bessel pairs . . . . . . . . . . . . 3.2.2 New weighted Hardy inequalities . . . . . . . . . . . . 3.2.3 Improved Hardy and Caffarelli-Kohn-Nirenberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 General Hardy-Rellich inequalities . . . . . . . . . . . . . . . 3.3.1 The non-radial case . . . . . . . . . . . . . . . . . . . 3.3.2 The case of power potentials |x|m . . . . . . . . . . . 3.4 Higher order Rellich inequalities . . . . . . . . . . . . . . . . 3.5 The class of Bessel potentials . . . . . . . . . . . . . . . . . . 3.6 The evaluation of an,m . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Optimal weighted Hardy-Rellich inequalities 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 General Hardy Inequalities . . . . . . . . . . 4.3 General Hardy-Rellich inequalities . . . . . . 4.3.1 The non-radial case . . . . . . . . . . 4.3.2 The case of power potentials |x|m . .  on . . . . . . . . . .  H 2 ∩ H01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . .  34 34 39 44 47 51 55 56 62 70 72 76 81 84 84 88 91 93 96  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105  II  Fourth Order Nonlinear Eigenvalue Problems  5 The critical dimension for a fourth order elliptic problem 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The effect of boundary conditions on the pull-in voltage . . . 5.2.1 Stability of the minimal branch of solutions . . . . . . 5.3 Regularity of the extremal solution for 1 ≤ N ≤ 8 . . . . . . 5.4 The extremal solution is singular for N ≥ 9 . . . . . . . . . . 5.5 Improved Hardy-Rellich Inequalities . . . . . . . . . . . . . .  107 108 108 112 115 119 122 128  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131  iv  Table of Contents 6 On the critical dimension for a fourth order elliptic problem 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The pull-in voltage . . . . . . . . . . . . . . . . . . . . . . . 6.3 Stability of the minimal solutions . . . . . . . . . . . . . . . 6.4 Regularity of the extremal solutions in dimensions N ≤ 8 . . 6.5 The extremal solution is singular in dimensions N ≥ 9 . . . . 6.6 Improved Hardy-Rellich Inequalities . . . . . . . . . . . . . .  133 133 135 138 142 144 149  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7 The 7.1 7.2 7.3  singular extremal solutions of the Introduction . . . . . . . . . . . . . . An improved Hardy-Rellich inequality Main results . . . . . . . . . . . . . .  bilaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . .  . . . .  158 158 159 161  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165  III  Preconditioning of Nonsymmetric Linear Systems  8 Simultaneous preconditioning and symmetrization 8.1 Introduction and main results . . . . . . . . . . . . 8.2 Selfdual methods for non-symmetric systems . . . . 8.2.1 Exact methods . . . . . . . . . . . . . . . . . 8.2.2 Inexact methods . . . . . . . . . . . . . . . . 8.2.3 Preconditioning . . . . . . . . . . . . . . . . 8.3 Numerical experiments . . . . . . . . . . . . . . . .  . . . . . .  . . . . . . .  . . . . . . .  166 . . . . . . .  . . . . . . .  167 167 172 172 173 174 177  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 8.2 8.3  GCGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Number of iterations for (8.41) with the solution y = x sin(πx).177 Number of iterations for equation (8.41) with the solution y = x(1−x) cos(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Number of iterations for the backward scheme method (Example 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Number of iterations for the centered difference scheme method (Example 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Number of iterations for SD-CGN with different values of α. . 181 Number of iterations for (8.43) with different values of α. . . 182  8.4 8.5 8.6 8.7  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 discussions 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 • Chapters 2, 3, and 8 were jointly authored by Nassif Ghoussoub and Amir Moradifam. • 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 inequalities, 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 nonlinear 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 dimension. 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 Ω be smooth bounded domain in Rn and 0 ∈ Ω. Hardy and HardyRellich inequalities assert that 2 Ω |∇u| dx  2 ≥ ( n−2 2 )  |u|2 Ω |x|2 dx,  (1.1)  n2 (n−4)2 16  u2 Ω |x|4 dx,  (1.2)  for all u ∈ H01 (Ω) and n ≥ 3, and 2 Ω |∆u| dx  ≥  for u ∈ H02 (Ω) and n ≥ 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 Schr¨odinger 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 Br´ezisVazquez [4] showed that Hardy’s inequality can be improved once restricted to a smooth bounded domain Ω in Rn , there was a flurry of activity about possible improvements of the following type: If n ≥ 3  then  2 Ω |∇u| dx  |u|2 Ω |x|2 dx  2 − ( n−2 2 )  ≥  ΩV  (x)|u|2 dx,  (1.3)  2 dx  (1.4)  for all u ∈ H01 (Ω), as well as its fourth order counterpart If n ≥ 5  then  2 Ω |∆u| dx  −  n2 (n−4)2 16  u2 Ω |x|4 dx  ≥  Ω W (x)u  for u ∈ H02 (Ω), where V, W are certain explicit radially symmetric potentials of order lower than r12 (for V ) and r14 (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: B  V (x)|∇u|2 dx ≥ c  B  W (x)u2 dx for all u ∈ C0∞ (B).  (1.5)  R  Assuming that the ball B has radius R and that 0 rn−11V (r) dr = +∞, the condition is simply that the ordinary differential equation (BV,cW )  y (r) + ( n−1 r +  Vr (r) V (r) )y  (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 β(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 example, by using recent results on ordinary differential equations, we can then infer that an integral condition on V, W of the form R  lim sup r2(n−1) V (r)W (r) r→0  r  dτ τ n−1 V  2  (τ )  <  1 4  (1.7) 2  1.1. Optimal improved Hardy and Hardy-Rellich inequalities is sufficient (and “almost necessary”) for (V, W ) to be a Bessel pair on a ball of sufficiently small radius ρ. (r) Applied in particular, to a pair (V, r12 V ) where the function rV V (r) is assumed to decrease to −λ on (0, R), we obtain the following extension of Hardy’s inequality: If λ ≤ n − 2, then B  V (x)|∇u|2 dx ≥ ( n−λ−2 )2 2  2  B  u V (x) |x| 2 dx  for all u ∈ C0∞ (B)  (1.8)  and ( n−λ−2 )2 is the best constant. The case where V (x) ≡ 1 is obviously 2 the classical Hardy inequality and when V (x) = |x|−2a for −∞ < a < n−2 2 , 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 • If αβ > 0 and m ≤  n−2 2 ,  then for all u ∈ C0∞ (Rn )  (a + b|x|α )β 2 u dx, |x|2m+2 Rn Rn (1.9) 2 is the best constant in the inequality. and ( n−2m−2 ) 2 (a + b|x|α )β n − 2m − 2 2 |∇u|2 dx ≥ ( ) 2m |x| 2  • If αβ < 0 and 2m − αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) (a + b|x|α )β n − 2m + αβ − 2 2 |∇u|2 dx ≥ ( ) |x|2m 2  (a + b|x|α )β 2 u dx, |x|2m+2 Rn Rn (1.10) n−2m+αβ−2 2 and ( ) is the best constant in the inequality. 2 We can also extend some of the recent results of Blanchet-Bonforte-DolbeaultGrillo-Vasquez [3]. • If αβ < 0 and −αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ b α ( Rn  n − αβ − 2 2 ) 2  2  (a + b|x|α )β− α u2 dx, Rn  (1.11) 2  and b α ( n−αβ−2 )2 is the best constant in the inequality. 2 • If αβ > 0, and n ≥ 2, then there exists a constant C > 0 such that for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ C Rn  (a + b|x|α )β− α u2 dx.  (1.12)  Rn 2  2  n+αβ−2 2 2 α Moreover, b α ( n−2 ) . 2 ) ≤C ≤b ( 2  3  1.1. Optimal improved Hardy and Hardy-Rellich inequalities On the other hand, by considering the pair V (x) = |x|−2a  Wa,c (x) = ( n−2a−2 )2 |x|−2a−2 + c|x|−2a W (x) 2  and  we get the following improvement of the Caffarelli-Kohn-Nirenberg inequalities: |x|−2a |∇u|2 dx − ( B  n − 2a − 2 2 ) 2  |x|−2a−2 u2 dx ≥ c B  |x|−2a W (x)u2 dx, B  (1.13)  for all u ∈  C0∞ (B),  if and only if the following ODE y + 1r y + cW (r)y = 0  (BcW )  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 optimal Hardy-Rellich inequalities of arbitrary high order, therefore extending and completing a series of new results by Adimurthi et al. [2], TertikasZographopoulos [19] and others. They are mostly based on the following theorem. Let V and W be positive radial C 1 -functions on B\{0}, where B is a ball R centered at zero with radius R in Rn (n ≥ 1) such that 0 rn−11V (r) dr = +∞ and  R n−1 V 0 r  (r)dr < +∞. The following statements are then equivalent:  1. (V, W ) is a Bessel pair on (0, R) and β(V, W ; R) ≥ 1. 2.  B  V (x)|∇u|2 dx ≥  B  W (x)u2 dx for all u ∈ C0∞ (B).  3. If limr→0 rα V (r) = 0 for some α < n−2, then the above are equivalent to B  V (x)|∆u|2 dx ≥  B  W (x)|∇u|2 dx + (n − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| dx,  ∞ (B). for all radial u ∈ C0,r  4. If in addition, W (r) − 2Vr2(r) + above are equivalent to B  V (x)|∆u|2 dx ≥  B  2Vr (r) r  − Vrr (r) ≥ 0 on (0, R), then the  W (x)|∇u|2 dx + (n − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| dx,  for all u ∈ C0∞ (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 – 2 −2 + cW (x) is a Bessel pair– is equivalently those W such that 1, ( n−2 2 ) |x| quite rich and contains several important potentials. Here are some of the most relevant properties of the class of C 1 Bessel potentials W on (0, R), that we shall denote by B(0, R). First, the class is a closed convex solid subset of C 1 (0, R), that is if W ∈ B(0, R) and 0 ≤ V ≤ W , then V ∈ B(0, R). The ”weight” of each W ∈ B(R), that is β(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 corresponding functional inequalities. Here are some basic examples of Bessel potentials and their corresponding weights. • W ≡ 0 is a Bessel potential on (0, R) for any R > 0. • W ≡ 1 is a Bessel potential on (0, R) for any R > 0, and β(1; R) = where z0 = 2.4048... is the first zero of the Bessel function J0 .  z02 R2  • If a < 2, then there exists Ra > 0 such that W (r) = r−a is a Bessel potential on (0, Ra ). .e((k−1)−times) e.  • For k ≥ 1, R > 0 and ρ = R(ee Wk,ρ (r) =  1 Σkj=1 2 r  ), let  j  log (i) i=1  ρ r  −2  ,  where the functions log (i) are defined iteratively as follows: log (1) (.) = log(.) and for k ≥ 2, log (k) (.) = log(log (k−1) (.)). Wk,ρ is then a Bessel potential on (0, R) with β(Wk,ρ ; R) = 14 . • For k ≥ 1, R > 0 and ρ ≥ R, define ˜ k;ρ (r) = Σk 1 X 2 ( r )X 2 ( r ) . . . X 2 ( r )X 2 ( r ), W j=1 2 1 j−1 r ρ 2 ρ ρ j ρ where the functions Xi are defined iteratively as follows: X1 (t) = ˜ k,ρ (1 − log(t))−1 and for k ≥ 2, Xk (t) = X1 (Xk−1 (t)). Then again W 1 ˜ k,ρ ; R) = . is a Bessel potential on (0, R) with β(W 4 5  1.1. Optimal improved Hardy and Hardy-Rellich inequalities • More generally, if W is any positive function on R such that r  sW (s)ds > −∞,  lim inf ln(r) r→0  0  then for every R > 0, there exists α := α(R) > 0 such that Wα (x) := α2 W (αx) 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 ≥ 3) of the following type:  B  |∆u|2 dx − C(n)  |∇u|2 B |x|2 dx  ≥ c(W, R)  B  W (x)|∇u|2 dx ,  (1.15)  2  n for all u ∈ H02 (B), where C(3) = 25 36 , C(4) = 3 and C(n) = 4 for n ≥ 5. ˜ k,ρ and n ≥ 5 was recently established by TertikasThe case when W ≡ W 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 β(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 ≥ 4. For example, if W is a r (r) Bessel potential on (0, R) such that the function r W W (r) decreases to −λ, and if λ ≤ n − 2, then we have for all u ∈ C0∞ (BR )  |∆u|2 dx − B  ≥  n2 (n − 4)2 u2 dx 4 16 B |x| n2 (n − λ − 2)2 + β(W ; R) 4 4  (1.16)  B  W (x) 2 u dx. |x|2  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 ˜ k defined above. For example (1 + n(n−4) ) for all the potentials Wk and W 8 we have for n ≥ 4, |∆u(x)|2 dx ≥ B  n2 (n − 4)2 16  B  n(n − 4) ) + (1 + 8  u2 dx |x|4 k j=1  B  (1.17) u2 |x|4  j  log (i) i=1  ρ |x|  −2  dx.  6  1.1. Optimal improved Hardy and Hardy-Rellich inequalities More generally, we show that for any m < n−2 2 , and any W Bessel n potential on a ball BR ⊂ R of radius R, the following inequality holds for all u ∈ C0∞ (BR )  BR  |∆u|2 ≥ an,m |x|2m  BR  |∇u|2 dx + β(W ; R) |x|2m+2  W (x) BR  |∇u|2 dx, |x|2m  (1.18)  where am,n and β(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 (r) n−4 again that rW W (r) decreases to −λ on (0, R), and provided m ≤ 2 and λ ≤ n − 2m − 2, we then have for all u ∈ C0∞ (BR ),  BR  |∆u|2 dx ≥ βn,m |x|2m  + β(W ; R)(  BR  u2 dx |x|2m+4  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4  (1.19)  BR  W (x) 2 u dx, |x|2m+2  where again the best constants βn,m are computed in chapter 3. This completes the results in Theorem 1.6 of [19], where the inequality is established ˜ for n ≥ 5, 0 ≤ m < n−4 2 , and the particular potential Wk,ρ . Another inequality that relates the Hessian integral to the Dirichlet energy is the following: Assuming −1 < m ≤ n−4 2 and W is a Bessel potential on a ball B of radius R in Rn , then for all u ∈ C0∞ (B), |∆u|2 (n + 2m)2 (n − 2m − 4)2 u2 dx − dx ≥ (1.20) 2m 2m+4 16 B |x| B |x| (n + 2m)2 W (x) 2 β(W ; R) u dx + β(|x|2m ; R)||u||H01 . 2m+2 4 |x| B This improves considerably Theorem A.2. in [2] where it is established – for m = 0 and without best constants – with the potential W1,ρ in dimension n ≥ 5, and the potential W2,ρ when n = 4. Finally, we establish several higher order Rellich inequalities for integrals m u|2 of the form BR |∆|x|2k dx, improving in many ways several recent results in [19]. Hardy-Rellich inequalities on H 2 ∩ H01 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 H 2 ∩ H01 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 problem of improving Hardy and Hardy-Rellich inequalities in Rn .  1.2  Fourth order nonlinear eigenvalue problems  Consider the fourth order elliptic problem β∆2 u − τ ∆u = λf (u)  in Ω, .  (Gλ )  with either Dirichlet boundary condition u = ∂ν u = 0 or Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ > 0 is a parameter, τ > 0, β > 0 are fixed constants, and Ω ⊂ RN (N ≥ 2) is a bounded smooth domain. The case β = 0, τ = 1, and f (u) = eu is the well known Gelfand problem. Under some technical assumptions on f one can show that there exists λ∗ > 0 such that for every 0 < λ < λ∗ , there exists a smooth minimal (smallest) solution of (S)λ,f , while for λ > λ∗ there is no solution even in a weak sense. Moreover, the branch λ → uλ (x) is increasing for each x ∈ Ω, and therefore the function u∗ (x) := limλ λ∗ uλ (x) can be considered as a generalized solution that corresponds to the pull-in voltage λ∗ . Now the issue of the regularity of this extremal solution 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∗ , λ∗ ). 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 nonlinearities f with both Dirichlet and Navier boundary conditions. Consider the fourth order elliptic problem  λ 2 in Ω,  β∆ u − τ ∆u = (1−u)2 (Gλ ) 0<u≤1 in Ω,  u = ∆u = 0 on ∂Ω, where λ > 0 is a parameter, τ > 0, β > 0 are fixed constants, and Ω ⊂ RN (N ≥ 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.  2 λ  ∆ u = (1−u)2 0<u<1  u = ∂ν u = 0  in B in B on ∂B,  (P )λ  where B is the unit ball in RN . This problem models a simple electrostatic Micro-Electromechanical Systems (MEMS) device. We proved that the extremal solution uλ∗ is regular (supB uλ∗ < 1) provided N ≤ 8 while uλ∗ is singular (supB uλ∗ = 1) for N ≥ 9, in which case 1 − C0 |x|4/3 ≤ uλ∗ (x) ≤ 1  ∗ 3 ¯ := 8 (N − 2 )(N − 8 ). 1 − |x|4/3 on the unit ball, where C0 := λλ and λ 9 3 3 This completes the results of F.H. Lin and Y.S. Yang [15]. In chapter 5, we study the problem (Gλ ) on the unit ball in RN and showed that the critical dimension for (Pλ ) is N = 9. Indeed I proved that the extremal solution of (Pλ ) is regular (supB u∗ < 1) for N ≤ 8 and β, τ > 0 and it is singular (supB u∗ = 1) for N ≥ 9, β > 0, and τ > 0 with βτ 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 D´avila et al. [7]. To show the singularity of the exterimal solutions in dimensions N ≥ 9 we use various improved Hardy-Rellich inequalities that are consequences of our main results in the first part of the thesis. Recently D´avila et al. [7] studied the fourth order counter part of the Gelfand problem  ∆2 u = λeu in B ∂u u = ∂n = 0 on ∂B,  (1.21)  They developed a new method to prove the regularity of the extremal solutions in low dimensions and showed that for N ≤ 12, u∗ is regular. They used a computer assisted proof to show that the extremal solution is singular in dimensions 13 ≤ N ≤ 31 while they gave a mathematical proof in dimensions N ≥ 32. In [17], I introduced a unified mathematical approach to deal with this problem and showed that for N ≥ 13, the extremal solution is singular. The following lemma plays an essential role in our proof. Lemma 1.22. Suppose there exist λ > 0 and a radial function u ∈ H 2 (B)∩ 4,∞ Wloc (B \ {0}) such that ∆2 u ≤ λ eu for all 0 < r < 1,  (1.23) 9  1.3. Preconditioning and symmetrization of non-symmetric linear systems  ∂u (1) = 0, ∂n u∈ / L∞ (B),  (1.25)  (∆ϕ)2 for all ϕ ∈ C0∞ (B),  (1.26)  u(1) = 0,  (1.24)  and eu ϕ2 ≤  β B  B  for some β > λ . Then u∗ is singular and λ∗ ≤ λ  (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 ∈ Rn×n 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 selfduality ([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 AT M Ax = AT M b,  (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 AT M A. 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 inequalities, 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, HardyPoincar´e inequalities and applications to nonlinear diffusions, C. R. Acad. Sci. Paris, Ser. I, 344 (2007), 431-436. [4] H. Brezis and J. L. V´azquez, 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 inequalities 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 Solutions 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. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 61916203. [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 symmetrization of non-symmetric linear systems, submitted to Numer. Linear Algebra Appl., 14 pages. [14] Z. Gui, J. Wei, On a fourth order nonlinear elliptic equation with negative 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 problem 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 H 2 ∩ H01 , 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 remainder 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 Ω be a bounded domain in Rn , n ≥ 3, with 0 ∈ Ω. The classical Hardy inequality asserts that 2 Ω |∇u| dx  2 ≥ ( n−2 2 )  |u|2 Ω |x|2 dx  for all u ∈ H01 (Ω).  (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 Schr¨odinger operators [10, 12]. 2 Now it is well known that ( n−2 2 ) is the best constant for inequality (2.1), and that this constant is however not attained in H01 (Ω). 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 V´azquez [6]. 2 Ω |∇u| dx  2 ≥ ( n−2 2 )  |u|2 Ω |x|2 dx  + λΩ  2 Ω |u| dx  for every u ∈ H01 (Ω). (2.2)  The constant λΩ in (2.2) is given by 2  λΩ = z02 ωn2/n |Ω|− n ,  (2.3)  1  A version of this chapter has been accepted for publication. N. Ghoussoub, A. Moradifam, 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 ωn and |Ω| denote the volume of the unit ball and Ω respectively, and z0 is the first zero of the bessel function J0 (z). Moreover, λΩ is optimal when Ω is a ball, but is –again– not achieved in H01 (Ω). 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 Ω such that 2 Ω |∇u| dx  2 ≥ ( n−2 2 )  |u|2 Ω |x|2 dx  +c  |u|2 k j=1 Ω |x|2  j (i) ρ −2 dx, i=1 log |x|  (2.4)  e(k−times) ..  e for u ∈ H01 (Ω), where ρ = (supx∈Ω |x|)(ee notations log (1) (.) = log(.) and log (k) (.) =  ). Here we have used the log(log (k−1) (.)) for k ≥ 2. Also motivated by the question of Brezis and V´azquez, 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’s inequality. More precisely, by defining iteratively the following functions, X1 (t) = (1 − log(t))−1 ,  Xk (t) = X1 (Xk−1 (t))  k = 2, 3, ...,  they prove that for any D ≥ supx∈Ω |x|, the following inequality holds for any u ∈ H01 (Ω): |∇u|2 dx ≥ ( Ω  +  n−2 2 ) 2  1 4  ∞ i=1  Ω  Ω  |u|2 dx |x|2  (2.5)  |x| |x| |x| 1 X12 ( )X22 ( )...Xi2 ( )|u|2 dx. 2 |x| D D D  Moreover, they proved that the constant 41 is the best constant for the corresponding k−improved Hardy inequality which is again not attained in H01 (Ω). In this paper, we show that all the above results –and more– 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 Ω 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 (r) +  y (r) r  + v(r)y(r) = 0  has a positive solution on the interval (0, R). 2. The following improved Hardy inequality holds (HV )  2 Ω |∇u| dx  2 − ( n−2 2 )  |u|2 Ω |x|2 dx  ≥  ΩV  (|x|)|u|2 dx,  for u ∈ H01 (Ω). Moreover, the best constant c(V ) := sup c; (HcV ) holds is the largest c so that y (r) + y 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 2 1 domain Ω in n containing 0, provided v(r) + ( n−2 2 ) r2 is non-increasing on (0, supx∈Ω |x|) and R is the radius of the ball which has the same volume as 1 n Ω (i.e. R = ( |Ω| ωn ) ). 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 + yr + v(r)y(r) = 0. We shall see that the results of Brezis-V´azquez, Adimurthi et al, and FilippasTertikas 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 constants in the above mentioned improvements. Indeed, the case when V ≡ 1 will follow immediately from Theorem 2.1. A slightly more involved reasoning – but also based of the above characterization – 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 (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 Ω in Rn . r  1. If lim inf r→0 ln(r) 0 sV (s)ds > −∞, then there exists α := α(Ω) > 0 such that an improved Hardy inequality (HVα ) holds for the scaled potential Vα (x) := α2 V (αx). 16  2.1. Introduction r  2. If limr→0 ln(r) 0 sV (s)ds = −∞, then there are no β, c > 0, for which (HVβ,c ) holds with Vβ,c = cV (βx). The following is a consequence of the two results above. For any α < 2, inequality (HcV ) holds on a bounded domain Ω for Vα (x) = |x|1α and some c > 0. Moreover, the best constant c( |x|1α ) is equal to the largest c such that the equation 1 1 y (r) + y (r) + c α = 0, r |x| has a positive solution on (0, R), where R is the radius of the ball wich has the same volume as Ω. Moreover, if α ≥ 2 inequality (HV ) does not hold for Vα,c (x) = c |x|1α for any c > 0. Note that the above corollary gives another 2 proof of the fact that ( n−2 2 ) is the best constant for the classical Hardy inequality. Define now the class AΩ = {v : R → R+ ; v is non-increasing on(0, sup |x|), x∈Ω  (Dv )  has a positive solution on(0, (  |Ω| 1 ) n )}. ωn  An immediate application of Theorem 2.1 coupled with H¨older’s inequality gives the following duality statement, which should be compared to inequalities dual to those of Sobolev, recently obtained via the theory of mass transport [2, 9]. Suppose that Ω is a smooth bounded domain in Rn containing 0. Then for any 0 < p ≤ 2, we have |∇u|2 dx − (  inf ≥ sup  Ω   1  ||V −1 (|x|)||  |u|2 dx; u ∈ H01 (Ω), ||u||p = 1 2 |x| Ω   ; V ∈ AΩ . (2.6)  .  n−2 2 ) 2  p  L p−2 (Ω)  Finally, consider the following classes of radial potentials: r  X = {V : Ω →+ ; V ∈ L∞ loc (Ω \ {0}), lim inf ln(r) r→0  sV (s)ds > −∞}, 0  (2.7) 17  2.2. Two dimensional inequalities and r  Y = {V : Ω →+ ; V ∈ L∞ loc (Ω \ {0}), lim ln(r) r→0  For any 0 < µ < µn := eigenvalue problem,  (n−2) 2 2 )  we consider the following weighted  −∆u −  (EV,µ )  sV (s)ds = −∞}. (2.8) 0  µ u |x|2  = λV u in Ω, u = 0 on Ω.  (2.9) Our results above combine with standard arguments to yield the following. For any 0 < µ < µn , and V : Ω →+ with V ∈ L∞ loc (Ω \ {0}) and 2 lim|x|→0 |x| V (x) = 0, the problem (EV,µ ) admits a positive weak solution uµ ∈ H01 (Ω) corresponding to the first eigenvalue λ = λ1µ (V ). Moreover, by letting λ1 (V ) = limµ↑µn λ1µ (V ), we have • If V ∈ X, then there exists c > o such that λ1 (Vc ) > 0. • If V ∈ Y , then λ1 (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 “twodimensional” Poincar´e and Poincar´e-Wirtinger inequalities. Let a < b, k > 0 is a differentiable function on (a, b), and ϕ be a strictly positive real valued differentiable function on (a, b). Then, every h ∈ C 1 ([a, b]) with −∞ < lim k(r)|h(r)|2 r→a  ϕ (r) ϕ (r) = lim k(r)|h(r)|2 < ∞, r→b ϕ(r) ϕ(r)  (2.10)  satisfies the following inequality: b  b  |h (r)|2 k(r)dr ≥ a  −|h(r)|2 ( a  k (r)ϕ (r) + k(r)ϕ (r) )dr. ϕ(r)  (2.11)  18  2.2. Two dimensional inequalities Moreover, assuming (2.10), the equality holds if and only if h(r) = ϕ(r) for all r ∈ (a, b). Proof. Define ψ(r) = h(r)/ϕ(r), r ∈ [a, b]. Then b  b  |h (r)|2 k(r)dr = a  b  |ψ(r)|2 |ϕ (r)|2 k(r)dr + a  2ϕ(r)ϕ (r)ψ(r)ψ (r)k(r)dr a  b  |ϕ(r)|2 |ψ (r)|2 k(r)dr  + a  b  b  |ψ(r)|2 (kϕϕ ) (r)dr  |ψ(r)|2 |ϕ (r)|2 k(r)dr −  =  a  a b  |ϕ(r)|2 |ψ (r)|2 k(r)dr  + a b  |ψ(r)|2 (|ϕ (r)|2 k(r) − (kϕϕ ) (r)dr  = a  b  |ϕ(r)|2 |ψ (r)|2 k(r)dr.  + a  Hence, we have b  b  |h (r)|2 k(r)dr =  −|h(r)|2 ( a  a  k (r)ϕ (r) + k(r)ϕ (r) )dr ϕ  b  |ϕ(r)|2 |ψ (r)|2 k(r)dr  + a b  −|h(r)|2 (  ≥ a  k (r)ϕ (r) + k(r)ϕ (r) )dr. ϕ(r)  Hence (2.11) holds. Note that the last inequality is obviously an idendity if and only if h(r) = ϕ(r) for all r ∈ (a, b). The proof is complete. By applying Theorem 2.2 to the weight k(r) = r, we obtain the following generalization of the 2-dimensional Poincar´e inequality. (Generalized 2-dimensional Poincar´e inequality) Let 0 ≤ a < b and ϕ be a strictly positive real valued differentiable function on (a, b). Then every h ∈ C 1 ([a, b]) with ϕ (r) ϕ (r) −∞ < lim r|h(r)|2 = lim r|h(r)|2 < ∞, (2.12) r→a r→b ϕ(r) ϕ(r) satisfies the following inequality: b  b  |h (r)|2 rdr ≥ a  −|h(r)|2 ( a  ϕ (r) + rϕ (r) )dr. ϕ(r)  (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) = ϕ(r) for all r ∈ (a, b). By applying Theorem 2.2 to the weight k(r) = 1, we obtain the following generalization of the 2-dimensional Poincar´eWirtinger inequality. (Generalized Poincar´e-Wirtinger inequality) Let a < b and ϕ be a strictly positive real valued differentiable function on (a, b). Then, every h ∈ C 1 ([a, b]) with ϕ (r) ϕ (r) −∞ < lim |h(r)|2 = lim |h(r)|2 < ∞, (2.14) r→a r→b ϕ(r) ϕ(r) satisfies the following inequality: b  b  |h (r)|2 dr ≥ a  −|h(r)|2 a  ϕ (r) dr. ϕ(r)  (2.15)  Moreover, under assumption (2.14), the equality holds if and only if h(r) = ϕ(r) for all r ∈ (a, b). Remark 2.2.1. Note that all of inequalities presented in the above theorems hold when we replace the condtions (2.10), (2.12), and (2.14) with the following weaker conditions ϕ (r) ϕ(r) ϕ (r) lim inf r|h(r)|2 r→b ϕ(r) ϕ (r) lim inf |h(r)|2 r→b ϕ(r)  lim inf k(r)|h(r)|2 r→b  ϕ (r) , ϕ(r) r→a ϕ (r) ≥ lim sup r|h(r)|2 , ϕ(r) r→a ϕ (r) ≥ lim sup |h(r)|2 , ϕ(r) r→a ≥ lim sup k(r)|h(r)|2  respectively, provided both sides in the above inequalities are not equal to −∞ or +∞.  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 Ω be a bounded smooth domain in Rn with 0 ∈ Ω, and set R = (|Ω|/ωn )1/n . Suppose V is a radially symmetric function on Ω and ϕ is a C 2 -function on (0, R) such that (|x|) 0 ≤ V (|x|) ≤ − ϕ (|x|)+rϕ for all x ∈ Ω, 0 < |x| < R, |x|ϕ(|x|)  (2.17) 20  2.3. Proof of the main theorem (r) lim inf r ϕϕ(r) ≥0  (r) lim sup ϕϕ(r) < ∞,  and  r→0  (2.18)  r→R  2 1 ( n−2 2 ) |x|2 + V (|x|) is a decreasing function of |x|.  (2.19)  Then for any u ∈ H01 (Ω), we have |∇u|2 dx ≥ ( Ω  n−2 2 ) 2  Ω  |u|2 dx + |x|2  V (|x|)|u|2 dx.  (2.20)  Ω  Moreover, if limr→0 rϕ(r)ϕ (r) = limr→R ϕ(r)ϕ (r) = 0, then equality holds if and only if u is a radial function on Ω such that u(x) = ϕ(|x|) for all x ∈ Ω. Proof: We first prove the inequality for smooth radial positive functions on the ball Ω = BR . For such u ∈ C02 (BR ), we define v(r) = u(r)r(n−2)/2 , r = |x|. In view of Corollary 2.2, we can write |∇u(x)|2 dx − ( Ω  n−2 2 ) 2  Ω  u2 (x) dx |x|2  R  n − 2 −n/2 r v(r) − r1−n/2 v (r)|2 rn−1 dr 2 0 R 2 n−2 2 v (r) − ( ) ωn dr 2 r 0 2v (r)r 2 n−2 2 R 2 dr v ((1 − = ωn ( ) ) − 1) 2 (n − 2)v(r) r 0 |  = ωn  R  (v (r))2 r − ωn (  = ωn 0  n−2 ) 2  R  v(r)v (r)dr 0  R  (v (r))2 r  = ωn 0 R  −v 2 (r)(  ≥ ωn 0  ϕ (r) + rϕ (r) )dr ϕ(r)  R  ϕ (r) + rϕ (r) n−2 )r dr ϕ(r) 0 ϕ (|x|) + |x|ϕ (|x|) = − u2 (x)( )dx. |x|ϕ(|x|) Ω  = ωn  −u2 (r)(  Hence, the inequality (2.20) holds for radial smooth positive functions. By density arguments, inequality (2.20) is valid for any u ∈ H01 , u ≥ 0. For 21  2.3. Proof of the main theorem u ∈ H01 which is not positve and general domain Ω, we use symmetrization arguments. Let BR be a ball having the same volume as Ω with R = (|Ω|/ωn )1/n and let |u|∗ be the symmetric decreasing rearrangement of the function |u|. Now note that for any u ∈ H01 (Ω), |u|∗ ∈ H01 (BR ) and |u|∗ > 0. It is well known that the symmetrization does not change the Lp -norm, and that it decreases the Dirichlet energy, while increasing the n−2 2 1 2 1 2 integrals Ω (( n−2 2 ) |x|2 + V (|x|)|u| dx, since the weight ( 2 ) |x|2 + V (|x|) is a decreasing function of |x|. Hence, (2.20) holds for any u ∈ H01 (Ω). We shall need the following lemmas. Lemma 2.21. Let x(r) be a function in C 1 (0, R] that is a solution of rx (r) + x2 (r) ≤ −F (r),  0 < r ≤ R,  (2.22)  where F is a nonnegative continuous function. Then lim x(r) = 0.  (2.23)  r↓0  Proof: Divide equation (2.22) by r and integrate once. Then we have R  x(r) ≥ r  R  |x(s)|2 ds + x(1) + s  r  F (s) ds. s  (2.24)  It follows that limr↓0 x(r) exists. In order to prove that this limit is zero, we claim that R r  x2 (s) ds < ∞. s  Indeed, otherwise we have G(r) := we have  R x2 (s) s ds r  (2.25) → ∞ as r → 0. From (2.22) R  1  (−rG (r)) 2 ≥ G(r) + x(1) + r  F (s) ds. s  Note that F ≥ 0, and G goes to infinity as r goes to zero. Thus, for r 1 sufficiently small we have −rG (r) ≥ 21 G2 (r) hence, ( G(r) ) ≥ 21 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. Lemma 2.26. If the equation φ + φr + v(r)φ = 0 has a positive solution on some interval (0, R), then we have necessarily, (r) lim inf r ϕϕ(r) ≥0 r→0  and  (r) lim sup ϕϕ(r) < ∞.  (2.27)  r→R  22  2.3. Proof of the main theorem Proof: Since ϕ(δ) ≥ 0 and ϕ(r) > 0 for 0 < r < δ, it is obvious that ϕ (r) satisfies the second condition. To obtain the first condition, set x(r) = r ϕϕ(r) . one may verify that x(r) satisfies the ODE: rx (r) + x2 (r) = −F (r),  for 0 < r ≤ δ,  (r) where F (r) = r2 v(r) ≥ 0. By Lemma 2.21 we conclude that limr↓0 r ϕϕ(r) = limr↓0 x(t) = 0.  Lemma 2.28. Let V be positive radial potential on the ball Ω of radius R in Rn (n ≥ 3). Assume that 2  Ω  2 |u| 2 dx ≥ 0 for all u ∈ H 1 (Ω). |∇u|2 − ( n−2 0 2 ) |x|2 − V (|x|)|u|  Then there exists a C 2 -supersolution to the equation n−2 2  −∆u −  2  u − V (|x|)u = 0, |x|2 u > 0 u = 0  in Ω,  (2.29)  in Ω \ {0},  (2.30)  in ∂Ω.  (2.31)  Proof: Define λ1 (V ) := inf{  2 Ω |∇ψ|  2 2 2 − ( n−2 2 ) |ψ| − V |ψ| ; ψ ∈ C0∞ (Ω \ {0})}. 2 |ψ| Ω  By our assumption λ(V ) ≥ 0. Let (φn , λn1 ) be the first eigenpair for the problem (L − λ1 (V ) − λr1 )φr = 0 on Ω \ B R n  φ(r) = 0 on ∂(Ω \ B R ), n  2 1 where L = −∆ − ( n−2 is a ball of radius R 2 ) |x|2 − V , and B R n, n ≥ 2 . n The eigenfunctions can be chosen in such a way that φn > 0 on Ω \ B R and n  ϕn (b) = 1, for some b ∈ Ω with R2 < |b| < R. Note that λn1 ↓ 0 as n → ∞. Harnak’s inequality yields that for any K φn compact subset K, max maxK φn ≤ C(K) with the later constant being independant of φn . Also standard elliptic estimates also yields that the family (φn ) have also uniformly bounded derivatives on compact sets Ω − B R . n Therefore, there exists a subsequence (ϕnl2 )l2 of (ϕn )n such that (ϕnl2 )l2 23  2.3. Proof of the main theorem converges to some ϕ2 ∈ C 2 (Ω \ B( R2 )). Now consider (ϕnl2 )l2 on Ω \ B( R3 ). Again there exists a subsequence (ϕnl3 )l3 of (ϕnl2 )l2 which converges to ϕ3 ∈ C 2 (Ω \ B( R3 )), and ϕ3 (x) = ϕ2 (x) for all x ∈ Ω \ B( R2 ). By repeating this argument we get a supersolution ϕ ∈ C 2 (Ω \ {0}) i.e. Lϕ ≥ 0, such that ϕ > 0 on Ω \ {0}. Lemma 2.32. Let a be a locally integrable function on , then the following statements are equivalent. 1. z (s) + a(s)z(s) = 0, has a strictly positive solution on (b, ∞). 2. There exists a function ψ ∈ C 1 (b, ∞) such that ψ (r) + ψ 2 (r) + a(t) ≤ 0, f or r > b. Consequently, the equation y + 1r y + v(r)y = 0 has a positive supersolution on (0, δ) if and only if it has a positive solution on (0, δ). 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−s ) maps the equation y + 1r y + v(r)y = 0 into z + e−2s v(e−s )z(s) = 0. On the −t  −t  other hand, the change of variables ψ(t) = −e y(ey−t(e) ) maps y + 1r y + v(r)y into ψ (t) + ψ 2 (t) + e−2t v(e−t ). This proves the lemma. Proof of Theorem 2.1: The implication 1) implies 2) follows immediately 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 Ω of radius R, and then apply Lemma (2.28) to obtain a C 2 -supersolution for the equation (2.29). Now take the surface average of u, that is w(r) =  1 nωw rn−1  u(x)dS = ∂Br  1 nωn  u(rω)dω > 0,  (2.33)  |ω|=1  where ωn denotes the volume of the unit ball in Rn . We may assume that the unit ball is contained in Ω (otherwise we just use a smaller ball). By a standard calculation we get w (r) +  1 n−1 w (r) ≤ r nωn rn−1  ∆u(x)dS.  (2.34)  ∂Br  24  2.4. Applications Since u(x) is a supersolution of (2.29), w satisfies the inequality: w (r)+  n−1 n − 2 2 w(r) w (r)+( ) ≤ −v(r)w(r), r 2 r2  Now define ϕ(r) = r  n−2 2  w(r),  f or 0 < r < R. (2.35)  in 0 < r < R.  (2.36)  Using (2.35), a straightforward calculation shows that ϕ satisfies the following inequality ϕ (r) +  ϕ (r) ≤ −ϕ(r)v(r), r  f or 0 < r < R.  (2.37)  By Lemma 2.32 we may conclude that the equation y (r) + 1r y + v(r)y = 0 has actually a positive solution φ on (0, R). It is clear that by the sufficient condition c(V ) ≥ c whenever y (r) + 1 y + cv(r)y = 0 has a positive solution on (0, R). On the other hand, the r necessary condition yields that y (r) + 1r y + c(V )v(r)y = 0 has a positive solution on (0, R). The proof is now complete.  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’s inequality. For that we need to investigate whether the ordinary differential equation y y + + v(r)y(r) = 0, (2.38) r corresponding to a potential v has a positive solution φ on (0, δ) for some 1 δ2 δ δ > 0. In this case, ψ(r) = φ( δr R ) is a solution for y (r)+ r y + R2 v( R r)y = 0 2 on (0, R), which means that the scaled potential Vδ (x) = Rδ 2 V ( Rδ x) yields an improved Hardy formula (HVδ ) on a ball of radius R, with constant larger than one. Here is an immediate application of this criterium. 1) The Brezis-V´ azquez 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 +  y + cy(r) = 0, r  (2.39)  25  2.4. Applications 1  2/n  has a positive solution on (0, R), with R = (|Ω|/ωn ) n is z02 ωn |Ω|−2/n . Indeed, if z0 is the first root of the solution of the Bessel equation y + y r + y(r) = 0, then the solution of (2.39) in this case is the Bessel function ϕ(r) = J0 ( rzR0 ). This readily gives the result of Brezis-V´azquez mentioned in the introduction. 2) The Adimurthi et al. improvement [1]: In this case, one easily 1 sees that the functions ϕj (r) = ( ji=1 log (i) ρr ) 2 is a solution of the equation −  ϕj (r) + rϕj (r) 1 = 2( rϕj (r) 4r  n i=1  ρ log(i) )−2 , r  on (0, R), which means that the inequality (HV ) holds for the potential n (i) ρ −2 1 which yields the result of Adimurthi et al. In V (x) = 4|x| 2( i=1 log |x| ) the following, we use our characterization to show that the constant appearing in the above improvement is indeed the best constant in the following sense: 1 = 4 inf 1  Ω  2 |∇u|2 dx − ( n−2 2 )  u∈H0 (Ω)\{0}  2 m−1 |u|2 dx − 14 j=1 Ω |u| |x|2 Ω |x|2 m |u|2 (i) R −2 i=1 log |x| Ω |x|2  j i=1  ρ log (i) |x|  −2  j i=1  ρ log (i) |x|  −2  ,  for all 1 ≤ m ≤ k. We proceed by contradiction, and assume that 1 +λ= 4 inf  Ω  2 |∇u|2 dx − ( n−2 2 )  u∈H01 (Ω)\{0}  2 m−1 |u|2 dx − 14 j=1 Ω |u| |x|2 Ω |x|2 m |u|2 (i) ρ −2 i=1 log |x| Ω |x|2  ,  and λ > 0. From Theorem 2.1 we deduce that there exists a positive function ϕ such that −  ϕ (r) + rϕ (r) 1 = ϕ(r) 4  Now define f (r) =  ϕ(r) ϕm (r)  m−1  j=1  1 r  j  log (i) i=1  ρ r  −2  1 1 + ( + λ) 4 r  m  log (i) i=1  ρ r  −2  .  > 0, and calculate,  ϕ (r) + rϕ (r) ϕ (r) + rϕm (r) f (r) + rf (r) f (r) = m + − ϕ(r) ϕm (r) f (r) f (r)  m  i=1  1 i j=1  logj ( ρr )  .  26  2.4. Applications Thus, f (r) + rf (r) f (r) − f (r) f (r)  m  i=1  1 i j=1  logj ( ρr )  = −λ  m  1 r  log (i) i=1  ρ r  −2  .  (2.40)  If now f (αn ) = 0 for some sequence {αn }∞ n=1 that converges to zero, then there exists a sequence {βn }∞ that also converges to zero, such that f (βn ) = 0, and n=1 f (βn ) > 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 (r) > 0 for r > 0 sufficiently small. Then we will have (rf (r)) ≤ rf (r)  m  1  i=1  r  r  m j=1  Integrating once we get f (r) ≥  i j=1  logj ( ρr )  .  c , logj ( ρr )  for some c > 0. Hence, limr→0 f (r) = −∞ which is a contradiction. Case II: Assume f (r) < 0 for r > 0 sufficiently small. Then (rf (r)) ≥ rf (r)  m  1  i=1  i j=1  r  Thus, f (r) ≥ −  r  logj ( ρr )  .  c , logj ( ρr )  (2.41)  m j=1  for some c > 0 and r > 0 sufficiently small. On the other hand j  m  f (r) + rf (r) 1 ≤ −λ f (r) r j=1  log (i) i=1  R r  −2  ≤ −λ(  m j=1  1 ). logj ( ρr )  Since f (r) < 0, there exists l such that f (r) > l > 0 for r > 0 sufficiently small. From the above inequality we then have bf (b) − af (a) < −λl(  m j=1  1 − logj ( ρb )  m j=1  1 ). logj ( aρ )  From (2.41) we have lima→0 af (a) = 0. Hence, bf (b) < −  m j=1  λl , logj ( ρb )  27  2.4. Applications for every b > 0, and f (r) < −  r  λl , logj ( ρr )  m j=1  for r > 0 sufficiently small. Therefore, lim f (r) = +∞,  r→0  and by choosing l large enouph (e.g., l > proof is now complete.  c λ)  we get to contradict (2.41) and the  3) The Filippas and Tertikas improvement [11]: Let D ≥ supx∈Ω |x|, and define 1 r r r r i = 1, 2, . . . . ϕk (r) = (X1 ( )X2 ( ) . . . Xi−1 ( )Xi ( ))− 2 , D D D D Using the fact that Xk (r) = 1r X1 (r)X2 (r) . . . Xk−1 (r)Xk2 (r) for k = 1, 2, . . ., we get −  ϕk (r) + rϕk (r) 1 r r r r 2 = X12 ( )X22 ( ) . . . Xk−1 ( )Xk2 ( ). ϕk (r) 4r D D D D  This means that the inequality (HV ) holds for the potential V (x) =  1 |x| |x| |x| |x| 2 X12 ( )X22 ( ) . . . Xk−1 ( )Xk2 ( ), 2 4|x| D D D D  which yields the result of Filippas and Tertikas [11]. We now identify the best constant by showing that: 1 = inf 4 u∈H01 (Ω)\{0} Ω  2 |∇u|2 dx − ( n−2 2 )  2 m−1 |u|2 |x| 2 |x| 2 |x| 2 2 |x| dx − 14 j=1 Ω |u| |x|2 X1 ( D )X2 ( D ) . . . Xj−1 ( D )Xj ( D ) Ω |x|2 , |u|2 |x| 2 2 |x| X 2 ( |x| )X22 ( |x| D ) . . . Xm−1 ( D )Xm ( D ) Ω |x|2 1 D  for all 1 ≤ m ≤ k. We proceed again by contradiction and in a way very similar to the above case. Indeed, assuming that 1 +λ= inf 4 u∈H01 (Ω)\{0} Ω  2 |∇u|2 dx − ( n−2 2 )  2 m−1 |u|2 |x| 2 |x| 2 |x| 2 2 |x| dx − 14 j=1 Ω |u| |x|2 X1 ( D )X2 ( D ) . . . Xj−1 ( D )Xj ( D ) Ω |x|2 , |u|2 |x| 2 ( |x| )X 2 ( |x| ) . . . X 2 2 ( |x| ) ( X )X 2 m 1 2 m−1 |x| D D D D Ω  and λ > 0, we use again Theorem 2.1 to find a positive function ϕ such that −  ϕ (r) + rϕ (r) ϕ(r)  m−1  =  1 4  +  1 1 r r r 2 2 r ( + λ) X12 ( )X22 ( ) . . . Xm−1 ( )Xm ( ). 4 r D D D D  j=1  1 2 r r r r 2 X1 ( )X22 ( ) . . . Xj−1 ( )Xj2 ( ) r D D D D  28  2.4. Applications Setting f (r) =  ϕ(r) ϕm (r)  > 0, we have  ϕ (r) + rϕ (r) ϕ (r) + rϕm (r) f (r) + rf (r) f (r) = m + − ϕ(r) ϕm (r) f (r) f (r)  m  i  Xj ( i=1 j=1  r ). D  Thus, f (r) + rf (r) f (r) − f (r) f (r)  m  m  i  Xj ( i=1 j=1  1 r r ) = −λ Xj2 ( ). D r j=1 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 (r) > 0 for r > 0 sufficiently small, then we will have (rf (r)) ≤ rf (r)  m  i  i=1  1 r Xj ( ). r j=1 D  Integrating once we get c f (r) ≥ r  m  Xj ( j=1  r ), D  for some c > 0, and therefore limr→0 f (r) = −∞ which is a contradiction. Case II: Assume f (r) < 0 for r > 0 sufficiently small. Then (rf (r)) ≥ rf (r)  m  i  1 r Xj ( ) r j=1 D  i=1  Thus, c r  f (r) ≥ −  m  Xj ( j=1  r ), D  (2.43)  for some c > 0 and r > 0 sufficiently small. On the other hand j  m  m  f (r) + rf (r) 1 r ≤ −λ Xj2 ≤ −λ( Xj ( )) . f (r) r D j=1 i=1 j=1 Since f (r) < 0, we may assume f (r) > l > 0 for r > 0 sufficiently small, and from the above inequality we have m  m  bf (b) − af (a) < −λl( j=1  Xj (  b a )− Xj ( )). D D j=1  29  2.4. Applications From (2.43) we have lima→0 af (a) = 0. Hence, f (r)) < −  λl r  m  Xj ( j=1  r ), D  for r > 0 sufficiently small. Therefore, lim f (r) = +∞,  r→0  and by choosing l large enouph (i.e. l > complete.  c λ)  we contradict (2.43) and the proof is  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 (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): ∞ t  1. If lim supt→∞ t 2. If lim inf t→∞ t  ∞ t  a(s)ds < 41 , then Eq. (2.44) is non-oscillatory.  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 H¨older’s inequality that 2  Ω  where s ≥ 1 and  1 s  +  u s (x)dx  1  V (|x|)u2 (x)dx) s ≥  ( 1 r  (  Ω  Ω r 1 V − s (|x|)dx) r  ,  = 1. Letting p = 2s , we get 2  V (|x|)u2 (x)dx ≥ ( Ω  up (x)dx) p Ω  1 ||V −1 (|x|)||  . p  L 2−p (Ω)  Inequality (2.1) now follows from Theorem 2.1. Proof of Corollary 2.1: Define the functional |∇u(x)|2 dx − µ  Fµ (u) = Ω  Ω  u2 (x) dx, |x|2  (2.45)  which is continuous, Gateaux differentiable and coercive on H01 (Ω). Let uµ > 0 be a minimizer of Fµ over the manifold M = {u ∈ H01 (Ω)| Ω u2 (x)V (x) = 1} and  30  2.4. Applications assume λ1µ is the infimum. It is clear that λ1µ > 0. By standard arguments we can conclude that uµ is a weak solution of (EV,µ ). The rest of the proof follows from Corollary 2.1 and the fact that 2  λ1 (V ) = lim  µ→µn  λ1µ  =  inf  u∈H01 (Ω)\{0}  Ω  (|∇u|2 − µn u|x|(x) 2 )dx Ω  |u(x)|2 V (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. 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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 Br´ezis-Vazquez [10] showed that Hardy’s inequality can be improved once restricted to a smooth bounded domain Ω in Rn , there was a flurry of activity about possible improvements of the following type: If n ≥ 3  then  Ω  2 |∇u|2 dx − ( n−2 2 )  |u|2 dx Ω |x|2  ≥  Ω  V (x)|u|2 dx,  (3.1)  W (x)u2 dx  (3.2)  for all u ∈ H01 (Ω), as well as its fourth order counterpart If n ≥ 5  then  Ω  |∆u|2 dx −  n2 (n−4)2 16  u2 dx Ω |x|4  ≥  Ω  for u ∈ H02 (Ω), where V, W are certain explicit radially symmetric potentials of order lower than r12 (for V ) and r14 (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 functions V and W on a ball B in Rn , so that the following inequality holds for some c > 0: V (x)|∇u|2 dx ≥ c B W (x)u2 dx for all u ∈ C0∞ (B). (3.3) B R  Assuming that the ball B has radius R and that 0 is simply that the ordinary differential equation (BV,cW )  y (r) + ( n−1 r +  Vr (r) V (r) )y  (r) +  1 r n−1 V (r) dr  cW (r) V (r) y(r)  = +∞, the condition  =0  2  A version of this chapter has been accepted for publication. N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Annalen, 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 β(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 inequalities 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 R  lim sup r2(n−1) V (r)W (r) r→0  r  dτ τ n−1 V  2  (τ )  <  1 4  (3.5)  is sufficient (and “almost necessary”) for (V, W ) to be a Bessel pair on a ball of sufficiently small radius ρ. (r) Applied in particular, to a pair (V, r12 V ) where the function rV V (r) is assumed to decrease to −λ on (0, R), we obtain the following extension of Hardy’s inequality: If λ ≤ n − 2, then B  )2 V (x)|∇u|2 dx ≥ ( n−λ−2 2  2  B  u V (x) |x| 2 dx  for all u ∈ C0∞ (B)  (3.6)  and ( n−λ−2 )2 is the best constant. The case where V (x) ≡ 1 is obviously the 2 classical Hardy inequality and when V (x) = |x|−2a for −∞ < a < n−2 2 , 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 • If αβ > 0 and m ≤  Rn  n−2 2 ,  then for all u ∈ C0∞ (Rn )  n − 2m − 2 2 (a + b|x|α )β |∇u|2 dx ≥ ( ) |x|2m 2  Rn  (a + b|x|α )β 2 u dx, |x|2m+2  (3.7)  and ( n−2m−2 )2 is the best constant in the inequality. 2 • If αβ < 0 and 2m − αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) (a + b|x|α )β 2 u dx, |x|2m+2 Rn Rn (3.8) 2 and ( n−2m+αβ−2 ) is the best constant in the inequality. 2 n − 2m + αβ − 2 2 (a + b|x|α )β |∇u|2 dx ≥ ( ) 2m |x| 2  We can also extend some of the recent results of Blanchet-Bonforte-DolbeaultGrillo-Vasquez [4]. • If αβ < 0 and −αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ b α ( Rn  n − αβ − 2 2 ) 2  2  (a + b|x|α )β− α u2 dx, (3.9) Rn  2  and b α ( n−αβ−2 )2 is the best constant in the inequality. 2  35  3.1. Introduction • If αβ > 0, and n ≥ 2, then there exists a constant C > 0 such that for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ C Rn  (a + b|x|α )β− α u2 dx.  (3.10)  Rn  2  2  n+αβ−2 2 2 α Moreover, b α ( n−2 ) . 2 ) ≤C ≤b ( 2  On the other hand, by considering the pair V (x) = |x|−2a  )2 |x|−2a−2 + c|x|−2a W (x) Wa,c (x) = ( n−2a−2 2  and  we get the following improvement of the Caffarelli-Kohn-Nirenberg inequalities: |x|−2a |∇u|2 dx − ( B  n − 2a − 2 2 ) 2  |x|−2a−2 u2 dx ≥ c B  |x|−2a W (x)u2 dx, B  (3.11) for all u ∈ C0∞ (B), if and only if the following ODE y + 1r y + cW (r)y = 0  (BcW )  has a positive solution on (0, R). Such a function W 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’s inequalities (on bounded domains) established by Brezis-V´azquez [10], Adimurthi et al. [1], and Filippas-Tertikas [17]. Our results here include in addition those 1 proved by Wang-Willem [34] in the case where a < n−2 2 and W (r) = r 2 (ln R )2 , but r  also cover the previously unknown limiting case corresponding to a = n−2 2 as well as the critical dimension n = 2. More importantly, we establish here that Bessel pairs lead to a myriad of optimal 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 C 1 -functions on B\{0}, where B is a ball R centered at zero with radius R in Rn (n ≥ 1) such that 0 rn−11V (r) dr = +∞ and R n−1 r V 0  (r)dr < +∞. The following statements are then equivalent:  1. (V, W ) is a Bessel pair on (0, R) and β(V, W ; R) ≥ 1. 2.  B  V (x)|∇u|2 dx ≥  B  W (x)u2 dx for all u ∈ C0∞ (B).  3. If limr→0 rα V (r) = 0 for some α < n − 2, then the above are equivalent to B  V (x)|∆u|2 dx ≥  B  W (x)|∇u|2 dx + (n − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| dx,  ∞ for all radial u ∈ C0,r (B).  36  3.1. Introduction 4. If in addition, W (r) − are equivalent to B  V (x)|∆u|2 dx ≥  2V (r) r2  B  +  2Vr (r) r  − Vrr (r) ≥ 0 on (0, R), then the above  W (x)|∇u|2 dx + (n − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| dx,  for all u ∈ C0∞ (B). 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 –equivalently those W such that 2 −2 + cW (x) is a Bessel pair– is quite rich and contains several impor1, ( n−2 2 ) |x| tant potentials. Here are some of the most relevant properties of the class of C 1 Bessel potentials W on (0, R), that we shall denote by B(0, R). First, the class is a closed convex solid subset of C 1 (0, R), that is if W ∈ B(0, R) and 0 ≤ V ≤ W , then V ∈ B(0, R). The ”weight” of each W ∈ B(R), that is β(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. • W ≡ 0 is a Bessel potential on (0, R) for any R > 0. • W ≡ 1 is a Bessel potential on (0, R) for any R > 0, and β(1; R) = z0 = 2.4048... is the first zero of the Bessel function J0 .  z02 R2  where  • If a < 2, then there exists Ra > 0 such that W (r) = r−a is a Bessel potential on (0, Ra ). • For k ≥ 1, R > 0 and ρ = R(ee  . e.  e((k−1)−times)  Wk,ρ (r) = Σkj=1  ), let 1 r2  j  log (i) i=1  ρ r  −2  ,  where the functions log (i) are defined iteratively as follows: log (1) (.) = log(.) and for k ≥ 2, log (k) (.) = log(log (k−1) (.)). Wk,ρ is then a Bessel potential on (0, R) with β(Wk,ρ ; R) = 14 . • For k ≥ 1, R > 0 and ρ ≥ R, define r r 2 ˜ k;ρ (r) = Σkj=1 1 X12 ( r )X22 ( r ) . . . Xj−1 W ( )Xj2 ( ), r2 ρ ρ ρ ρ where the functions Xi are defined iteratively as follows: X1 (t) = (1 − ˜ k,ρ is a Bessel log(t))−1 and for k ≥ 2, Xk (t) = X1 (Xk−1 (t)). Then again W 1 ˜ k,ρ ; R) = . potential on (0, R) with β(W 4  37  3.1. Introduction • More generally, if W is any positive function on R such that r  sW (s)ds > −∞,  lim inf ln(r) r→0  0  then for every R > 0, there exists α := α(R) > 0 such that Wα (x) := α2 W (αx) 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 ≥ 3) of the following type: B  |∆u|2 dx − C(n)  B  |∇u|2 |x|2 dx  ≥ c(W, R)  B  W (x)|∇u|2 dx ,  (3.13)  2  25 for all u ∈ H02 (B), where C(3) = 36 , C(4) = 3 and C(n) = n4 for n ≥ 5. The case ˜ when W ≡ Wk,ρ and n ≥ 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 β(W ; R) are the best constants in the above inequality. Appropriate combinations of (3.3) and (3.13) then lead to a myriad of HardyRellich inequalities in dimension n ≥ 4. The For example, if W is a Bessel potential r (r) on (0, R) such that the function r W W (r) decreases to −λ, and if λ ≤ n − 2, then we have for all u ∈ C0∞ (BR )  n2 (n − 4)2 u2 dx 4 16 B |x| n2 (n − λ − 2)2 + β(W ; R) 4 4  |∆u|2 dx − B  ≥  (3.14)  B  W (x) 2 u dx. |x|2  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−4) ) for all the potentials 8 ˜ k defined above. For example we have for n ≥ 4, Wk and W |∆u(x)|2 dx ≥ B  n2 (n − 4)2 16  B  u2 dx |x|4 k  +  (1 +  n(n − 4) ) 8 j=1  B  (3.15) u2 |x|4  j  log (i) i=1  ρ |x|  −2  dx.  More generally, we show that for any m < n−2 2 , and any W Bessel potential on a ball BR ⊂ Rn of radius R, the following inequality holds for all u ∈ C0∞ (BR )  BR  |∆u|2 ≥ an,m |x|2m  BR  |∇u|2 dx + β(W ; R) |x|2m+2  W (x) BR  |∇u|2 dx, |x|2m  (3.16)  38  3.2. General Hardy Inequalities where am,n and β(W ; R) are best constants that we compute in the appendices for all m and n and for many Bessel potentials W . Worth noting is Corollary 3.3 where we show that inequality (3.16) restricted to radial functions in C0∞ (BR ) holds with 2 a best constant equal to ( n+2m 2 ) , but that an,m can however be strictly smaller n+2m 2 than ( 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 (r) n−4 that rW W (r) decreases to −λ on (0, R), and provided m ≤ 2 and λ ≤ n − 2m − 2, ∞ we then have for all u ∈ C0 (BR ), BR  |∆u|2 dx ≥ βn,m |x|2m  + β(W ; R)(  u2 BR  |x|2m+4  dx  (3.17)  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4  BR  W (x) 2 u dx, |x|2m+2  where again the best constants βn,m are computed in section 3. This completes the results in Theorem 1.6 of [31], where the inequality is established for n ≥ 5, ˜ 0 ≤ m < n−4 2 , and the particular potential Wk,ρ . Another inequality that relates the Hessian integral to the Dirichlet energy is the following: Assuming −1 < m ≤ n−4 and W is a Bessel potential on a ball B 2 of radius R in Rn , then for all u ∈ C0∞ (B), |∆u|2 u2 (n + 2m)2 (n − 2m − 4)2 dx − dx ≥ 2m 2m+4 16 B |x| B |x| W (x) 2 (n + 2m)2 u dx + β(|x|2m ; R)||u||H01 . β(W ; R) 2m+2 4 |x| B  (3.18)  This improves considerably Theorem A.2. in [2] where it is established – for m = 0 and without best constants – with the potential W1,ρ in dimension n ≥ 5, and the potential W2,ρ when n = 4. Finally, we establish several higher order Rellich inequalities for integrals of the m u|2 form BR |∆|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 and W be positive radial C 1 -functions on BR \{0}, where BR is a ball centered at zero with radius R (0 < T ≤ +∞) in a a Rn (n ≥ 1). Assume that 0 rn−11V (r) dr = +∞ and 0 rn−1 V (r)dr < ∞ for some 0 < a < R. Then the following two statements are equivalent: 1. The ordinary differential equation (BV,W )  y (r) + ( n−1 r +  Vr (r) V (r) )y  (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 ϕ(R) = 0). 2. For all u ∈ C0∞ (BR ) (HV,W )  BR  V (x)|∇u(x)|2 dx ≥  BR  W (x)u2 dx.  Before proceeding with the proofs, we note the following immediate but useful corollary. Let V and W be positive radial C 1 -functions on B\{0}, where B is a ball with R radius R in Rn (n ≥ 1) and centered at zero, such that 0 rn−11V (r) dr = +∞ and R n−1 r V (r)dr < ∞. 0 u ∈ C0∞ (BR ), we have  Then (V, W ) is a Bessel pair on (0, R) if and only if for all  V (x)|∇u|2 dx ≥ β(V, W ; R) BR  W (x)u2 dx, BR  with β(V, W ; R) being the best constant. For the proof of Theorem 3.2, we shall need the following lemmas. Lemma 3.19. Let Ω be a smooth bounded domain in Rn with n ≥ 1 and let ϕ ∈ C 1 (0, R := supx∈∂Ω |x|) be a positive solution of the ordinary differential equation y +(  n − 1 Vr (r) W (r) + )y + y = 0, r V (r) V (r)  on (0, R) for some V (r), W (r) ≥ 0 where ∞. Setting ψ(x) =  u(x) ϕ(|x|)  for any u ∈  1.  R n−1 r V 0  2.  Ω  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx < ∞.  3.  Ω  V (|x|)ϕ2 (|x|)|∇ψ|2 (x)dx < ∞.  4. |  Ω  R 1 dr 0 r n−1 V (r)  C0∞ (Ω),  (3.20)  = +∞ and  R n−1 r V 0  (r)dr <  we then have the following properties:  (r) 2 (r) (r)( ϕϕ(r) ) dr < ∞ and limr→0 rn−1 V (r) ϕϕ(r) = 0.  x V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) |x| .∇ψ(x)dx| < ∞.  5. limr→0 | ∂Br V (|x|)ϕ (|x|)ϕ(|x|)ψ 2 (x)ds| = 0, where Br ⊂ Ω is a ball of radius r centered at 0. (r) Proof: 1) Setting x(r) = rn−1 V (r) ϕϕ(r) , we have  rn−1 V (r)x (r) + x2 (r)  =  r2(n−1) V 2 (r) n − 1 Vr (r) (ϕ (r) + ( + )ϕ (r)) ϕ r V (r)  = −  r2(n−1) V (r)W (r) ≤ 0, ϕ(r)  0 < r < R.  40  3.2. General Hardy Inequalities Dividing by rn−1 V (r) and integrating once, we obtain R  x(r) ≥ r  |x(s)|2 ds + x(R). sn−1 V (s)  To prove that limr→0 G(r) < ∞, where G(r) := contrary and use (3.21) to write that  (3.21)  R x2 (s) ds, r sn−1 V (s)  we assume the  1  (−rn−1 V (r))G (r)) 2 ≥ G(r) + x(R). 1 Thus, for r sufficiently small we have −rn−1 V (r)G (r) ≥ 21 G2 (r) and hence, ( G(r) ) ≥ 1 , which contradicts the fact that G(r) goes to infinity as r tends to zero. 2r n−1 V (r) Also in view of (3.21), we have that x0 := limr→0 x(r) exists, and since limr→0 G(r) < ∞, we necessarily have x0 = 0 and 1) is proved. For assertion 2), we use 1) to see that  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx ≤ ||u||2∞ Ω  V (|x|) Ω  (ϕ (|x|))2 dx < ∞. ϕ2 (|x|)  3) Note that |∇ψ(x)| ≤  |∇u(x)| ϕ(|x|)  (|x|)| + |u(x)| |ϕ ϕ2 (|x|) ≤  C1 ϕ(|x|)  (|x|)| + C2 |ϕ ϕ2 (|x|) ,  for all x ∈ Ω,  where C1 = maxx∈Ω |∇u| and C2 = maxx∈Ω |u|. Hence we have (C1 ϕ(|x|) + C2 ϕ (|x|))2 dx ϕ2 (|x|) Ω ω ϕ (|x|) 2 |ϕ (|x|)| = C12 V (|x|)dx + 2C1 C2 V (|x|)dx + C22 ( ) V (|x|)dx ϕ(|x|) ϕ(|x|) Ω Ω Ω 1 1 ϕ (|x|) 2 ≤ L1 + 2C1 C2 V (|x|)( ) dx 2 V (|x|)dx 2 + L2 ϕ(|x|) Ω Ω < ∞, V (|x|)ϕ2 (|x|)|∇ψ|2 (x)dx ≤  V (|x|)  which proves 3). 4) now follows from 2) and 3) since V (|x|)|∇u|2  = V (|x|)(ϕ (|x|))2 ψ 2 (x) +  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x)  x .∇ψ(x) + V (|x|)ϕ2 (|x|)|∇ψ|2 . |x|  Finally, 5) follows from 1) since V (|x|)ϕ (|x|)ϕ(|x|)ψ 2 (x)ds| < ||u||2∞ |  | ∂Br  V (|x|) ∂Br  = ||u||2∞ V (r)  |ϕ (r)| ϕ(r)  ϕ (|x|) ds ϕ(|x|) 1ds ∂Br  = nωn ||u||2∞ rn−1 V (r)  |ϕ (r)| . ϕ(r)  41  3.2. General Hardy Inequalities Lemma 3.22. Let V and W be positive radial C 1 -functions on a ball B\{0}, where B is a ball with radius R in Rn (n ≥ 1) and centered at zero. Assuming B  V (x)|∇u|2 − W (x)|u|2 dx ≥ 0 for all u ∈ C0∞ (B),  then there exists a C 2 -supersolution to the following linear elliptic equation −div(V (x)∇u) − W (x)u = 0, u > 0 u = 0  in B, in B \ {0}, in ∂B.  (3.23) (3.24) (3.25) (3.26)  Proof: Define λ1 (V ) := inf{  B  V (x)|∇ψ|2 − W (x)|ψ|2 ; ψ ∈ C0∞ (B \ {0})}. 2 |ψ| B  By our assumption λ1 (V ) ≥ 0. Let (φn , λn1 ) be the first eigenpair for the problem (L − λ1 (V ) − λn1 )φn  =  0 on B \ B R  φn  =  0 on ∂(B \ B R ),  n  n  where Lu = −div(V (x)∇u) − W (x)u, and B R is a ball of radius R n , n ≥ 2 . The n eigenfunctions can be chosen in such a way that φn > 0 on B \ B R and ϕn (b) = 1, n for some b ∈ B with R2 < |b| < R. Note that λn1 ↓ 0 as n → ∞. Harnak’s inequality yields that for any compact K φn subset K, max minK φn ≤ C(K) with the later constant being independant of φn . Also standard elliptic estimates also yields that the family (φn ) have also uniformly bounded derivatives on the compact sets B − B R . n Therefore, there exists a subsequence (ϕnl2 )l2 of (ϕn )n such that (ϕnl2 )l2 converges to some ϕ2 ∈ C 2 (B \ B( R2 )). Now consider (ϕnl2 )l2 on B \ B( R3 ). Again there exists a subsequence (ϕnl3 )l3 of (ϕnl2 )l2 which converges to ϕ3 ∈ C 2 (B \ B( R3 )), and ϕ3 (x) = ϕ2 (x) for all x ∈ B \ B( R2 ). By repeating this argument we get a supersolution ϕ ∈ C 2 (B \ {0}) i.e. Lϕ ≥ 0, such that ϕ > 0 on B \ {0}. Proof of Theorem 3.2: First we prove that 1) implies 2). Let φ ∈ C 1 (0, R] be u(x) = ψ(x). a solution of (BV,W ) such that φ(x) > 0 for all x ∈ (0, R). Define ϕ(|x|) Then |∇u|2 = (ϕ (|x|))2 ψ 2 (x) + 2ϕ (|x|)ϕ(|x|)ψ(x)  x .∇ψ + ϕ2 (|x|)|∇ψ|2 . |x|  Hence, V (|x|)|∇u|2 ≥ V (|x|)(ϕ (|x|))2 ψ 2 (x) + 2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x)  x .∇ψ(x). |x|  42  3.2. General Hardy Inequalities Thus, we have V (|x|)|∇u|2 dx ≥  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx  B  B  +  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B  Let B be a ball of radius  x .∇ψdx. |x|  centered at the origin. Integrate by parts to get  V (|x|)|∇u|2 dx B  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx +  ≥  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x)  B  B  +  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B\B  x .∇ψdx |x|  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx +  = B  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B  − B\B  x .∇ψdx |x|  x .∇ψdx |x|  (n − 1)V (|x|) + Vr (|x|))ϕ (|x|)ϕ(|x|) ψ 2 (x)dx (V (|x|)ϕ (|x|)ϕ(|x|) + ( r V (|x|)ϕ (|x|)ϕ(|x|)ψ 2 (x)ds  + ∂(B\B )  Let  → 0 and use Lemma 3.19 and the fact that φ is a solution of (Dv,w ) to get V (|x|)|∇u|2 dx ≥  −  [V (|x|)ϕ (|x|)  B  B  +  (  (n − 1)V (|x|) u2 (x) + Vr (|x|))ϕ (|x|)] dx r ϕ(|x|) W (|x|)u2 (x)dx.  = B  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 C 2 -supersolution for the equation (3.23). Now take the surface average of u, that is y(r) =  1 nωw rn−1  u(x)dS = ∂Br  1 nωn  u(rω)dω > 0,  (3.27)  |ω|=1  where ωn 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 (r) +  n−1 1 y (r) = r nωn rn−1  ∆u(x)dS.  (3.28)  ∂Br  Since u(x) is a supersolution of (3.23), we have div(V (|x|)∇u)ds − ∂Br  W (|x|)udx ≥ 0, ∂B  43  3.2. General Hardy Inequalities and therefore, ∆udS − Vr (r)  V (r) ∂Br  ∇u.xds − W (r) ∂Br  u(x)ds ≥ 0. ∂Br  It follows that ∆udS − Vr (r)y (r) − W (r)y(r) ≥ 0,  V (r)  (3.29)  ∂Br  and in view of (3.27), we see that y satisfies the inequality V (r)y (r) + (  (n − 1)V (r) + Vr (r))y (r) ≤ −W (r)y(r), r  f or 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−1 V (r)y ) + rn−1 W (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, δ) if and only if x is a positive solution for the equation (s−(n−3) V ( 1s )x (s)) + s−(n+1) W ( 1s )x(s) = 0  on  ( 1δ , ∞).  (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 }∞ n=1 such that an → +∞ 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 C 1 -functions on BR \{0}, where BR is a ball R centered at 0 with radius R in Rn (n ≥ 1). Assume 0 τ n−11V (τ ) dτ = +∞ and R n−1 r v(r)dr 0  < ∞.  • Assume  R  lim sup r2(n−1) V (r)W (r) r→0  r  1 dτ τ n−1 V (τ )  2  <  1 4  (3.32)  then (V, W ) is a Bessel pair on (0, ρ) for some ρ > 0 and consequently, inequality (HV,W ) holds for all u ∈ C0∞ (Bρ ), where Bρ is a ball of radius ρ.  44  3.2. General Hardy Inequalities • On the other hand, if R  lim inf r2(n−1) V (r)W (r) r→0  r  1 dτ τ n−1 V (τ )  2  >  1 4  (3.33)  then there is no interval (0, ρ) on which (V, W ) is a Bessel pair and consequently, there is no smooth domain Ω on which inequality (HV,W ) holds. A typical Bessel pair is (|x|−λ , |x|−λ−2 ) for λ ≤ n − 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|−λ , |x|−λ (|x|−2 + 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 C 1 -function on (0, R) such that for some λ ∈ R rVr (r) V (r)  rVr (r) r→0 V (r)  + λ ≥ 0 on (0, R) and lim  + λ = 0.  (3.35)  If λ ≤ n − 2, then for any Bessel potential W on (0, R), and any c ≤ β(W ; R), the couple (V, Wλ,c ) is a Bessel pair, where Wλ,c (r) = V (r)((  n − λ − 2 2 −2 ) r + cW (r)). 2  (3.36)  Moreover, β V, Wλ,c ; R = 1 for all c ≤ β(W ; R). We need the following easy lemma. Lemma 3.37. Assume the equation a y + y + V (r)y = 0, r has a positive solution on (0, R), where a ≥ 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, δ), for δ sufficiently small. Assume y is increasing. Under this assumption if y (a) = 0 for some a > 0, then y (a) = 0 which contradicts the fact that y is a positive solution of the above ODE. So we have yy ≤ − ar , thus, y ≥  c . ra  Therefore, x(r) → −∞ as r → 0 which is a contradiction. Since, y can not have a local minimum it should be strictly decreasing on (0, R). (r) Proof of Theorem 3.2.1: Write VVr(r) = − λr + f (r) where f (r) ≥ 0 on (0, R)  45  3.2. General Hardy Inequalities and lim rf (r) = 0. In order to prove that V (r), V (r)(( n−λ−2 )2 r−2 + cW (r)) is 2 r→0  a Bessel pair, we need to show that the equation n−λ−1 n − λ − 2 2 −2 + f (r))y + (( ) r + cW (r))y(r) = 0, r 2 has a positive solution on (0, R). But first we note that the equation y +(  (3.38)  n−λ−1 n − λ − 2 2 −2 )x + (( ) r + cW (r))x(r) = 0, r 2 has a positive solution on (0, R), whenever c ≤ β(W ; R). Since now f (r) ≥ 0 and since, by the proceeding lemma, x (r) ≤ 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 β(V, Wλ,c ; R) ≥ 1. For the reverse inequality, we shall use the criterium in Theorem 3.2.1. Indeed to get apply criteria (3.32) to V (r) and W1 (r) = C Vr(r) 2 x +(  R  lim r2(n−1) V (r)W1 (r)  r→0  r R  = C lim r2(n−2) V 2 (r) r→0  r R  = C lim r(n−2) V (r) r→0  = C lim  r→0  = C lim  r→0  =  r  1 dτ τ n−1 V (τ ) 1  τ n−1 V  (τ )  dτ  1 dτ n−1 τ V (τ )  2  2  2  1 2 r n−1 V (r) (r)+r n−2 Vr (r) r 2(n−2) V 2 (r)  (n−2)r n−3 V  1 (n − 2) +  2 (r) r VVr(r)  C . (n − λ − 2)2  1 C For V, CV (r−2 + cW ) to be a Bessel pair, it is necessary that (n−λ−2) 2 ≤ 4 , and the proof for the best constant is complete. With a similar argument one can also prove the following. Let V and W be positive radial C 1 -functions on BR \{0}, where BR is a ball centered at zero with radius R in Rn (n ≥ 1). Assume that (r) = −λ and λ ≤ n − 2. lim r VVr(r)  r→0  (3.39)  (r) n−λ−2 2 • If lim sup r2 W ) , then (V, W ) is a Bessel pair on some interval V (r) < ( 2 r→0  (0, ρ), and consequently there exists a ball Bρ ⊂ Rn such that inequality (HV,W ) holds for all u ∈ C0∞ (Bρ ). (r) n−λ−2 2 ) , then there is no smooth • On the other hand, if lim inf r2 W V (r) > ( 2 r→0  domain Ω ⊂ Rn such that inequality (HV,W ) holds on Ω.  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 Ω containing 0 such that R = supx∈Ω |x|. Assume that for some λ ∈ R rVr (r) V (r)  rVr (r) r→0 V (r)  + λ ≥ 0 on (0, R) and lim  + λ = 0.  (3.40)  1. If λ ≤ n − 2, then the following inequality holds for any Bessel potential W on (0, R): V (x)|∇u|2 dx ≥ ( Ω  n−λ−2 2 ) 2  Ω  V (x) 2 u dx |x|2  V (x)W (x)u2 dx,  + β(W ; R)  (3.41)  Ω  for all u ∈ C0∞ (Ω) and both ( n−λ−2 )2 and β(W ; R) are the best constants. 2 2. In particular, β(V, r−2 V ; R) = ( n−λ−2 )2 is the best constant in the following 2 inequality Ω  )2 V (x)|∇u|2 dx ≥ ( n−λ−2 2  V (x) 2 u dx Ω |x|2  for all u ∈ C0∞ (Ω).  (3.42)  ˜ k,ρ (r) where Applied to V1 (r) = r−m Wk,ρ (r) and V2 (r) = r−m W Wk,ρ (r) =  1 Σkj=1 2 r  j  log (i) i=1  ρ r  −2  ˜ k;ρ (r) = Σk 12 X 2 ( r )X 2 ( r ) . . . X 2 ( r )X 2 ( r ) are the iterated logs introand W 1 ρ 2 ρ j=1 r j−1 ρ j ρ duced in the introduction, and noting that in both cases the corresponding λ is equal to 2m + 2, we get the following new Hardy inequalities. Let Ω be a smooth bounded domain in Rn (n ≥ 1) and m ≤ n−4 2 . Then the following inequalities hold. n − 2m − 4 2 Wk,ρ (x) |∇u|2 dx ≥ ( ) 2m |x| 2 Ω ˜ k,ρ (x) n − 2m − 4 2 W ) |∇u|2 dx ≥ ( 2m 2 Ω |x|  Wk,ρ (x) 2 u dx 2m+2 Ω |x| ˜ k,ρ (x) W u2 dx. 2m+2 Ω |x|  (3.43) (3.44)  Moreover, the constant ( n−2m−4 )2 is the best constant in both inequalities. 2 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 ∈ C0∞ (Rn ) if and only if the ODE (BV,W ) has a positive solution on (0, ∞). 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 ) + b(r)y(r) = 0,  (3.45)  where a(r) and b(r) are positive real valued functions. Assuming that ∞ for some d > 0, and that the following limit ∞  L := lim a(r)b(r) r→∞  r  ∞ 1 dτ d a(τ )  <  2  1 dr a(r)  ,  exists. Then for the equation (3.45) equation to be non-oscillatory, it is necessary that L ≤ 41 . Let a, b > 0, and α, β, m be real numbers. • If αβ > 0, and m ≤  Rn  n−2 2 ,  then for all u ∈ C0∞ (Rn )  (a + b|x|α )β n − 2m − 2 2 |∇u|2 dx ≥ ( ) 2m |x| 2  Rn  (a + b|x|α )β 2 u dx, |x|2m+2  (3.46)  and ( n−2m−2 )2 is the best constant in the inequality. 2 • If αβ < 0, and 2m − αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) (a + b|x|α )β n − 2m + αβ − 2 2 |∇u|2 dx ≥ ( ) |x|2m 2  (a + b|x|α )β 2 u dx, |x|2m+2 Rn Rn (3.47) 2 ) is the best constant in the inequality. and ( n−2m+αβ−2 2 Proof: Letting V (r) = r  (a+br α )β , r 2m  then  V (r) bαβrα aαβ = −2m + = −2m + αβ − . α V (r) a + br a + brα  Hence, in the case α, β > 0 and 2m ≤ n − 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 α < 0, β > 0 and 2m − αβ ≤ n − 2. For the remaining two other cases, we will use Theorem 3.2. Indeed, in this case the equation (BV,W ) becomes y +(  c n − 2m − 1 bαβrα−1 + )y + 2 y = 0, α r a + br r  (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, +∞). Note that by Lemma 3.37, we  48  3.2. General Hardy Inequalities have that y < 0 on (0, +∞). Hence, if α < 0 and β < 0, then the positive solution of the equation )2 ( n−2m−2 n − 2m − 1 2 y=0 y + y + r r2 is a positive super-solution for (3.48) and therefore the latter ODE has a positive solution on (0, +∞), from which we conclude that (3.46) holds. To prove now that )2 is the best constant in (3.46), we use the fact that if the equation (3.48) ( n−2m−2 2 has a positive solution on (0, +∞), then the equation is necessarily non-oscillatory. By rewriting (3.48) as rn−2m−1 (a + brα )β y and by noting that  + crn−2m−3 (a + brα )β y = 0,  ∞ d  1 rn−2m−1 (a  + brα )β  (3.49)  < ∞,  and ∞  lim cr2(n−2m−2) (a + brα )2β  r→∞  r  1 dr rn−2m−1 (a + brα )β  2  =  c , (n − 2m − 2)2  we can use Theorem 2.1 in [29] to conclude that for equation (3.49) to be nonoscillatory it is necessary that 1 c ≤ . 2 (n − 2m − 2) 4 2  Thus, (n−2m−2) is the best constant in the inequality (3.46). 4 A very similar argument applies in the case where α > 0, β < 0, and 2m < n−2, to obtain that inequality (3.47) holds for all u ∈ C0∞ (Rn ) and that ( n−2m+αβ−2 )2 2 is indeed the best constant. 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-GrilloVasquez [4] and address some of their questions regarding best constants. Let a, b > 0, and α, β be real numbers. • If αβ < 0 and −αβ ≤ n − 2, then for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ b α ( Rn  n − αβ − 2 2 ) 2  2  (a + b|x|α )β− α u2 dx, (3.50) Rn  2  and b α ( n−αβ−2 )2 is the best constant in the inequality. 2 • If αβ > 0 and n ≥ 2, then there exists a constant C > 0 such that for all u ∈ C0∞ (Rn ) 2  (a + b|x|α )β |∇u|2 dx ≥ C Rn  (a + b|x|α )β− α u2 dx.  (3.51)  Rn  49  3.2. General Hardy Inequalities 2  2  n+αβ−2 2 2 α Moreover, b α ( n−2 ) . 2 ) ≤C ≤b ( 2  Proof: Letting V (r) = (a + brα )β , then we have r  aαβ V (r) bαβrα = αβ − . = V (r) a + brα a + brα  Inequality (3.50) and its best constant in the case when α < 0 and β > 0, then follow immediately from Theorem 3.2.2 with λ = −αβ. 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 αβ < 0, then with an argument similar to that of Theorem 3.2.2 above, one can show that the positive solution of the equation y + 2 )y + (n+αβ−2) y = 0 on (0, +∞) is a positive supersolution for the equa( n+αβ−1 r 4r 2 tion 2 n − 1 V (r) b α (n + αβ − 2)2 y +( + )y + y = 0. 2 r V (r) 4(a + brα ) α Theorem 3.2 then yields that the inequality (3.50) holds for all u ∈ C0∞ (Rn ). To 2 )2 is the best constant in (3.50) it is enough to show prove now that b α ( n+αβ−2 2 that if the following equation 2  rn−1 (a + brα )β y  + crn−1 (a + brα )β− α y = 0  (3.52)  2  )2 . If now α > 0 and β < 0, has a positive solution on (0, +∞), then c ≤ b α ( n+αβ−2 2 then we have ∞  2  lim cr2(n−1) (a + brα )2β− α  r→∞  r  1 dr n−1 r (a + brα )β  2  =  c 2 α  b (n + αβ − 2)2  .  Hence, by Theorem 2.1 in [29] again, the non-oscillatory aspect of the equation 2 α  2  holds for c ≤ b (n+αβ−2) which completes the proof of the first part. 4 A similar argument applies in the case where αβ > 0 to prove that (3.51) holds 2 n+αβ−2 2 2 α for all u ∈ C0∞ (Rn ) and b α ( n−2 )2 . The best constants are 2 ) ≤ C ≤ b ( 2 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 α = 2) under the additional condition: (1 + |x|2 )β−1 u(x)dx = 0, f or β < Rn  n−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 β ≤ 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: Rn  |x|−bp |u|p dx  2 p  ≤ Ca,b  Rn  |x|−2a |∇u|2 dx for all u ∈ C0∞ (Rn ),  (3.54)  where for n ≥ 3, −∞ < a <  n−2 2 ,  a ≤ b ≤ a + 1, and p =  2n n−2+2(b−a) .  (3.55)  For the cases n = 2 and n = 1 the conditions are slightly different. For n = 2 −∞ < a < 0, a < b ≤ a + 1, and p =  2 b−a ,  (3.56)  2 −1+2(b−a) .  (3.57)  and for n = 1 −∞ < a < − 12 , a +  1 2  < b ≤ a + 1, and p =  Let Da1,2 be the completion of C0∞ (Rn ) for the inner product |x|−2a ∇u.∇vdx  (u, v) = Rn  and let S(a, b) =  inf 1,2  u∈Da \{0}  Rn  (  Rn  |x|−2a |∇u|2 dx |x|−bp |u|p dx  (3.58)  2/p  denote the best embedding constant. We are concerned here with the “Hardy critical” case of the above inequalities, that is when b = a + 1. In this direction, Catrina and Wang [14] showed that for n ≥ 3 we have S(a, a + 1) = ( n−2a−2 )2 and 2 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, 2 S(a, a + 1) = ( 1+2a 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 ≥ 1. Our results cover also the critical case when a = n−2 2 which is not allowed by the methods of [11]. Let W be a positive radial function on the ball B in Rn (n ≥ 1) with radius R and centered at zero. Assume a ≤ n−2 2 . 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 ∈ C0∞ (B) |x|−2a |∇u(x)|2 dx ≥ (  (Ha,cW ) B  n − 2a − 2 2 ) 2 −2a  |x|  + c  |x|−2a−2 u2 dx B  W (x)u2 dx,  B  51  3.2. General Hardy Inequalities Moreover, ( n−2a−2 )2 is the best constant and β(W ; R) = sup{c; (Ha,cW )holds}, 2 where β(W ; R) is the weight of W on (0, R). On the other hand, there is no strictly positive W ∈ C 1 (0, ∞), such that the following inequality holds for all u ∈ C0∞ (Rn ), |x|−2a |∇u(x)|2 dx ≥ ( Rn  n − 2a − 2 2 ) 2  |x|−2a−2 u2 dx Rn  W (|x|)u2 dx.  + c  (3.59)  Rn  (3.60) Proof: It suffices to use Theorems 3.2 and 3.2.2 with V (r) = r−2a to get that W is a Bessel function if and only if the pair r−2a , Wa,c (r) is a Bessel pair on (0, R) for some c > 0, where Wa,c (r) = (  n − 2a − 2 2 −2−2a ) r + cr−2a W (r). 2  For the last part, assume that (3.59) holds for some W . Then it follows from Theorem 3.2.3 that for V = cr2a W (r) the equation y (r) + 1r y + v(r)y = 0 has a positive solution on (0, ∞). From Lemma 3.37 we know that y is strictly decreasing (r) on (0, +∞). Hence, yy (r) ≥ − 1r which yields y (r) ≤ rb , for some b > 0. Thus y(r) → −∞ as r → +∞. This is a contradiction and the proof is complete. Remark 3.2.3. Theorem 3.2.3 characterizes the best constant only when Ω is a ball, while for general domain Ω, 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 ) ≤ CΩ (W ) ≤ CBρ (W ), where BR is the smallest ball containing Ω and Bρ is the largest ball contained in it. If now W is a Bessel potential such that β(W, R) is independent of R, then clearly β(W, R) is also the best constant in inequality (Ha,cW ) for any smooth ˜ k,ρ where bounded domain. This is clearly the case for the potentials Wk,ρ and W 1 β(W, R) = 4 for all R, while for W ≡ 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 Ω be a bounded smooth domain in Rn with n ≥ 1, and let W be a nonnegative function in C 1 (0, R =: supx∈∂Ω |x|] and a ≤ n−2 2 . 1. If lim inf ln(r) r→0  r 0  sW (s)ds > −∞, then there exists α := α(Ω) > 0 such  that an improved Hardy inequality (Ha,Wα ) holds for the scaled potential Wα (x) := α2 W (α|x|). 2. If lim ln(r) r→0  r 0  sW (s)ds = −∞, then there are no α, c > 0, for which (Ha,Wα,c )  holds with Wα,c = cW (α|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 established by Brezis and V´ azquez [10] in the case where a = 0, and b = 0. Let Ω be a bounded smooth domain in Rn with n ≥ 1 and a ≤ n−2 2 . Then, for any b < 2a + 2 there exists c > 0 such that for all u ∈ C0∞ (Ω) Ω  |x|−2a |∇u|2 dx ≥ ( n−2a−2 )2 2  Ω  |x|−2a−2 u2 dx + c  Ω  |x|−b u2 dx.  (3.61)  Moreover, when Ω is a ball B of radius R the best constant c for which (3.61) holds is equal to the weight β(r2a−b ; R) of the Bessel potential W (r) = r2a−b on (0, R]. In particular, B  |x|−2a |∇u|2 dx ≥ ( n−2a−2 )2 2  |x|−2a−2 u2 dx + λB  B  B  |x|−2a u2 dx,  (3.62)  2/n  where the best constant λB is equal to z0 ωn |Ω|−2/n , where ωn and |Ω| denote the volume of the unit ball and Ω 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 > −2a − 2 since then r  s2a+1 W (s)ds > −∞  lim inf ln(r) r→0  0 z2  . In the case where b = −2a and therefore W ≡ 1, we use the fact that β(1; R) = R02 (established in section 3.5) to deduce that the best constant is then equal to 2/n z0 ωn |Ω|−2/n . 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 ≥ 1 and a ≤ n−2 2 . Then for every integer k, and ρ = (supx∈Ω |x|)(ee u ∈ H01 (Ω), |x|−2a |∇u|2 dx ≥ ( Ω  +  n − 2a − 2 2 ) 2  1 4  k  j=1  2  Ω  . e.  Ω  |u| |x|2a+2  e((k−1)−times)  ), we have for any u2 dx |x|2a+2 j  log (i) i=1  ρ |x|  Moreover, 41 is the best constant which is not attained in H01 (Ω). k j (i) ρ Proof: As seen in section 3.5, Wk,ρ (r) = j=1 r12 i=1 log |x| potential on (0, R) where R = supx∈Ω |x|, and β(Wk,ρ ; R) = 14 .  −2  dx.  −2  (3.63)  dx is a Bessel  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 Ω be a bounded . Then for every integer k, and any smooth domain in Rn with n ≥ 1 and a ≤ n−2 2 D ≥ supx∈Ω |x|, we have for u ∈ H01 (Ω),  Ω  n − 2a − 2 2 |∇u|2 dx ≥ ( ) 2a |x| 2 +  1 4  ∞  i=1  Ω  Ω  u2 dx |x|2a+2  1 |x| |x| |x| X 2 ( )X22 ( )...Xi2 ( )|u|2 dx, |x|2a+2 1 D D D  (3.64)  1 4  is the best constant which is not attained in H01 (Ω). The classical Hardy inequality is valid for dimensions n ≥ 3. We now present optimal Hardy type inequalities for dimension two in bounded domains, as well as the corresponding best constants. Let Ω be a smooth domain in R2 and 0 ∈ Ω. Then we have the following inequalities. and  • Let D ≥ supx∈Ω |x|, then for all u ∈ H01 (Ω), Ω  and  1 4  |∇u|2 dx ≥  1 4  Ω 1 4  (3.65)  is the best constant.  • Let ρ = (supx∈Ω |x|)(ee  and  ∞ 1 2 |x| 2 |x| 2 |x| 2 i=1 Ω |x|2 X1 ( D )X2 ( D )...Xi ( D )|u| dx  e.  .e((k−1)−times)  ), then for all u ∈ H01 (Ω)  |∇u|2 dx ≥  1 4  k |u|2 j=1 Ω |x|2  j i=1  ρ log (i) |x|  −2  dx,  (3.66)  is the best constant for all k ≥ 1.  • If α < 2, then there exists c > 0 such that for all u ∈ H01 (Ω), Ω  |∇u|2 dx ≥ c  u2 Ω |x|α  dx,  (3.67)  and the best constant is larger or equal to β(rα ; sup |x|). x∈Ω  An immediate application of Theorem 3.2 coupled with H¨older’s inequality gives the following duality statement, which should be compared to inequalities dual to those of Sobolev’s, recently obtained via the theory of mass transport [3, 13]. Suppose that Ω is a smooth bounded domain containing 0 in Rn (n ≥ 1) with R := supx∈Ω |x|. Then, for any a ≤ n−2 2 and 0 < p ≤ 2, we have the following dual inequalities: |x|−2a |∇u|2 dx − (  inf Ω  n − 2a − 2 2 ) 2  |x|−2a−2 |u|2 dx; u ∈ C0∞ (Ω), ||u||p = 1 Ω  ≥ sup  ( Ω  p |x|−2a p−2 ) dx W (x)  2−p p  ; W ∈ B(0, R) .  54  3.3. General Hardy-Rellich inequalities  3.3  General Hardy-Rellich inequalities  Let 0 ∈ Ω ⊂ Rn be a smooth domain, and denote k C0,r (Ω) = {v ∈ C0k (Ω) : v is radial and supp v ⊂ Ω}, m H0,r (Ω) = {u ∈ H0m (Ω) : u is radial}.  We start by considering a general inequality for radial functions. Let V and W be positive radial C 1 -functions on a ball B\{0}, where B is a R ball with radius R in Rn (n ≥ 1) and centered at zero. Assume 0 rn−11V (r) dr = ∞ and limr→0 rα V (r) = 0 for some α < n − 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 ∈ C0,r (B) V (x)|∆u|2 dx ≥ c  (HRV,cW ) B  W (x)|∇u|2 dx B  +  (n − 1)  ( B  V (x) Vr (|x|) − )|∇u|2 dx. |x|2 |x|  Moreover, the best constant is given by β(V, W ; R) = sup c; (HRV,cW ) holds for radial functions .  (3.68)  ∞ Proof: Assume u ∈ C0,r (B) and observe that R  V (x)|∆u|2 dx = nωn { B  V (r)u2rr rn−1 dr 0 R  +  (n − 1)2  V (r) 0  u2r n−1 r dr + 2(n − 1) r2  R  V (r)uur rn−2 dr}. 0  Setting ν = ur , we then have V (x)|∆u|2 dx = B  V (x)|∇ν|2 dx + (n − 1) B  ( B  V (|x|) Vr (|x|) − )|ν|2 dx. |x|2 |x|  Thus, (HRV,W ) for radial functions is equivalent to V (x)|∇ν|2 dx ≥ B  W (x)ν 2 dx. B  Letting x(r) = ν(x) where |x| = r, we then have R  R  V (r)(x (r))2 rn−1 dr ≥ 0  W (r)x2 (r)rn−1 dr.  (3.69)  0  55  3.3. General Hardy-Rellich inequalities It therefore follows from Theorem 3.2 that 1) and 2) are equivalent. By applying the above theorem to the Bessel pair V (x) = |x|−2m  Wm (x) = V (x) ( n−2m−2 )2 |x|−2 + W (x) 2  and  where W is a Bessel potential, and by using Theorem 3.2.2, we get the following result in the case of radial functions. n Suppose n ≥ 1 and m < n−2 2 . Let BR ⊂ R 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 ∈ C0,r (BR )  BR  n + 2m 2 |∆u|2 ≥( ) |x|2m 2  BR  |∇u|2 dx + β(W ; R) |x|2m+2  W (x) BR  |∇u|2 dx. |x|2m  (3.70)  2 Moreover, ( n+2m 2 ) and β(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 ∈ C0∞ (Ω) could be extended by zero outside Ω, and could therefore be considered as a function in C0∞ (Rn ). By decomposing u into spherical harmonics we get u = Σ∞ k=0 uk where uk = fk (|x|)ϕk (x) and (ϕk (x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(n + k − 2), k ≥ 0. The functions fk belong to C0∞ (Ω) and satisfy fk (r) = O(rk ) and f (r) = O(rk−1 ) as r → 0. In particular, ϕ0 = 1 and f0 =  1 nωn r n−1  ∂Br  uds =  1 nωn  |x|=1  u(rx)ds.  (3.71)  We also have for any k ≥ 0, and any continuous real valued functions v and w on (0, ∞), V (|x|)|∆uk |2 dx = Rn  V (|x|) ∆fk (|x|) − ck Rn  fk (|x|) |x|2  2  dx,  (3.72)  and W (|x|)|∇uk |2 dx = Rn  W (|x|)|∇fk |2 dx + ck Rn  Rn  W (|x|)|x|−2 fk2 dx.  (3.73)  Let V and W be positive radial C 1 -functions on a ball B\{0}, where B is a R ball with radius R in Rn (n ≥ 1) and centered at zero. Assume 0 rn−11V (r) dr = ∞ and limr→0 rα V (r) = 0 for some α < (n − 2). If 2V (r) 2Vr (r) + − Vrr (r) ≥ 0 f or 0 ≤ r ≤ R, r2 r then the following statements are equivalent. W (r) −  (3.74)  56  3.3. General Hardy-Rellich inequalities 1. (V, W ) is a Bessel pair with β(V, W ; R) ≥ 1. 2. The following inequality holds for all u ∈ C0∞ (B), V (x)|∆u|2 dx ≥  (HRV,W )  W (x)|∇u|2 dx  B  B  (n − 1)  +  ( B  V (x) Vr (|x|) − )|∇u|2 dx. |x|2 |x|  Moreover, if β(V, W ; R) ≥ 1, then the best constant is given by β(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 ∈ C0∞ (B) by frequently using that R 0  V (r)|x (r)|2 rn−1 dr ≥  R 0  W (r)x2 (r)rn−1 dr for all x ∈ C 1 (0, R].  (3.76)  Indeed, for all n ≥ 1 and k ≥ 0 we have 1 nwn  V (x)|∆uk |2 dx = Rn  R  =  V (r) fk (r) + 0  1 nwn  V (x) ∆fk (|x|) − ck Rn  n−1 fk (r) fk (r) − ck 2 r r  R  r  2  dx  dr  R  V (r)(fk (r))2 rn−1 dr + (n − 1)2  =  2 n−1  fk (|x|) |x|2  V (r)(fk (r))2 rn−3 dr  0  0 R  + c2k  R  V (r)fk2 (r)rn−5 + 2(n − 1) 0  V (r)fk (r)fk (r)rn−2 0  R  R  V (r)fk (r)fk (r)rn−3 dr − 2ck (n − 1)  − 2ck 0  V (r)fk (r)fk (r)rn−4 dr. 0  Integrate by parts and use (3.71) for k = 0 to get 1 nωn  V (x)|∆uk |2 dx  (3.77)  Rn  R  R  V (r)(fk (r))2 rn−1 dr + (n − 1 + 2ck )  = 0  V (r)(fk (r))2 rn−3 dr 0  R  +  (2ck (n − 4) + c2k )  R  V (r)rn−5 fk2 (r)dr − (n − 1) 0  R  R  Vr (r)fk2 (r)rn−4 dr − ck  − ck (n − 5) 0  Vr (r)rn−2 (fk )2 (r)dr 0  Vrr (r)fk2 (r)rn−3 dr. 0  57  3.3. General Hardy-Rellich inequalities fk (r) r  Now define gk (r) =  and note that gk (r) = O(rk−1 ) for all k ≥ 1. We have  R  V (r)(fk (r))2 rn−3 0 R  R  V (r)(gk (r))2 rn−1 dr +  =  R  2V (r)gk (r)gk (r)rn−2 dr +  0  0 R  R  V (r)(gk (r))2 rn−1 dr − (n − 3)  =  V (r)gk2 (r)rn−3 dr 0  0  R  V (r)gk2 (r)rn−3 dr − 0  Vr (r)gk2 (r)rn−2 dr 0  Thus, R  R  V (r)(fk (r))2 rn−3 ≥ 0  W (r)fk2 (r)rn−3 dr 0  R  R  V (r)fk2 (r)rn−5 dr −  − (n − 3) 0  Vr (r)fk2 (r)rn−4 dr.  (3.78)  0  R  Substituting 2ck 0 V (r)(fk (r))2 rn−3 in (3.78) by its lower estimate in the last inequality (3.78), we get 1 nωn  V (x)|∆uk |2 dx Rn  R  R  W (r)(fk (r))2 rn−1 dr +  ≥ 0  W (r)(fk (r))2 rn−3 dr 0  R  +  R 2 n−3  (n − 1)  V (r)(fk (r)) r  V (r)(fk (r))2 rn−5 dr  dr + ck (n − 1)  0  0 R  R  Vr (r)rn−2 (fk )2 (r)dr − ck (n − 1)  − (n − 1) 0  Vr (r)rn−4 (fk )2 (r)dr 0  R  V (r)rn−5 fk2 (r)dr  + ck (ck − (n − 1)) 0 R  (W (r) −  + ck 0  2V (r) 2Vr (r) + − Vrr (r))fk2 (r)rn−3 dr. r2 r  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 β(V, W ; R) ≥ 1, since if W satisfies (3.74) then cW satisfies it for any c ≥ 1. Remark 3.3.1. In order to apply the above theorem to the Bessel pair V (x) = |x|−2m  and  Wm,c (x) = V (x) ( n−2m−2 )2 |x|−2 + cW (x) 2  where W is a Bessel potential, we see that even in the simplest case V ≡ 1 and 2 −2 2 −2 Wm,c (x) = ( n−2 + W (x), condition (3.74) reduces to ( n−2 + W (x) ≥ 2 ) |x| 2 ) |x| −2 2|x| , which is then guaranteed only if n ≥ 5.  58  3.3. General Hardy-Rellich inequalities More generally, if V (x) = |x|−2m , then in order to satisfy (3.74) we need to have √ √ −(n + 4) − 2 n2 − n + 1 −(n + 4) + 2 n2 − n + 1 ≤m≤ , (3.79) 6 6 and in this case, we have for m < all u ∈ C0∞ (BR )  BR  n + 2m 2 |∆u|2 ≥( ) 2m |x| 2  n−2 2  and any Bessel potential W on BR , that for  |∇u|2 dx + β(W ; R) |x|2m+2  BR  W (x) BR  |∇u|2 dx. |x|2m  (3.80)  2 Moreover, ( n+2m 2 ) and β(W ; R) are the best constant. Therefore, inequality (3.80) in the case where m = 0 and n ≥ 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 √ −(n + 4) + 2 n2 − n + 1 0≤m≤ (3.81) 6  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 dif2 ferent from ( n+2m 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 25 36 in dimension 3. We shall now give a few immediate applications of the above in the case where m = 0 and n ≥ 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 ⊂ Rn with n ≥ 3, then for all u ∈ ∞ C0 (BR ) we have |∆u|2 dx ≥ C(n) BR  BR  |∇u|2 dx + β(W ; R) |x|2  W (x)|∇u|2 dx,  (3.82)  BR  2  25 where C(3) = 36 , C(4) = 3 and C(n) = n4 for all n ≥ 5. Moreover, C(n) and β(W ; R) are the best constants. In particular, the following holds for any smooth bounded domain Ω in Rn with R = supx∈Ω |x|, and any u ∈ H02 (Ω).  • For any α < 2, |∆u|2 dx ≥ C(n) Ω  Ω  |∇u|2 dx + β(|x|α ; R) |x|2  Ω  |∇u|2 dx, |x|α  (3.83)  and for α = 0, Ω  |∆u|2 dx ≥ C(n)  |∇u|2 dx Ω |x|2  +  z02 R2  Ω  |∇u|2 dx,  (3.84)  the constants being optimal when Ω is a ball.  59  3.3. General Hardy-Rellich inequalities ..  e(k−times)  ee  • For any k ≥ 1, and ρ = R(e  ), we have  Ω  |∇u|2 dx |x|2  Ω  |∇u|2 |x|2  |∆u(x)|2 dx ≥ C(n) Ω  +  1 4  k  j=1  j  log (i) i=1  ρ |x|  −2  dx,  (3.85)  • For D ≥ R, and Xi is defined as (3.113) we have  Ω  |∇u|2 dx |x|2  Ω  |∇u| 2 |x| 2 |x| |x| X ( )X2 ( )...Xi2 ( )dx, |x|2 1 D D D  |∆u(x)|2 dx ≥ C(n) Ω  +  1 4  ∞  i=1  (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 λ r (r) with n ≥ 4, and such that W W (r) = r + f (r), where f (r) ≥ 0 and limr→0 rf (r) = 0. If λ < n − 2, then the following Hardy-Rellich inequality holds: |∆u|2 dx ≥ B  +  n2 (n − 4)2 u2 dx 4 16 B |x| (n − λ − 2)2 n2 + )β(W ; R) ( 4 4  B  W (x) 2 u dx, |x|2  (3.87)  Proof: Use first Theorem 3.3.1 with the Bessel potential W , then Theorem 3.2.3 2 with the Bessel pair (|x|−2 , |x|−2 ( (n−4) |x|−2 + W ), then Theorem 3.2.2 with the 4 2  Bessel pair (W, (n−λ−2) )|x|−2 W ) to obtain 4 |∇u|2 dx + β(W, R) W (x)|∇u|2 dx 2 B |x| B u2 (n − 4)2 dx C(n) 4 4 B |x| W (x) 2 C(n)β(W, R) u + β(W, R) W (x)|∇u|2 dx 2 B |x| (n − 4)2 u2 C(n) dx + (C(n) 4 4 B |x| (n − λ − 2)2 W (x) 2 )β(W, R) u dx. 2 4 B |x|  |∆u|2 dx ≥ C(n) B  ≥ + ≥ + 2  Recall that C(n) = n4 for n ≥ 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 λ = 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 Ω be a smooth bounded domain in Rn , n ≥ 4 and R = supx∈Ω |x|. Then the following holds for all u ∈ H02 (Ω) 1. If ρ = R(ee  . e.  e(k−times)  ) and log (i) (.) is defined as (3.112), then |∆u(x)|2 dx ≥ Ω  n2 (n − 4)2 16  k  +  (1 +  n(n − 4) ) 8 j=1  j  2  Ω  Ω  u2 dx |x|4  u |x|4  log (i) i=1  ρ |x|  −2  dx.  (3.88)  2. If D ≥ R and Xi is defined as (3.113), then |∆u(x)|2 dx ≥ Ω  n2 (n − 4)2 16  ∞  +  (1 +  n(n − 4) ) 8 i=1  Ω  u2 dx |x|4  2  Ω  u |x| |x| |x| X 2 ( )X22 ( )...Xi2 ( )dx. |x|4 1 D D D  (3.89)  Let W1 (x) and W2 (x) be two radial Bessel potentials on a ball B of radius R in Rn with n ≥ 4. If a < 1, then there exists c(a, R) > 0 such that for all u ∈ H02 (B) |∆u|2 dx ≥ B  u2 n2 (n − 4)2 n2 u2 dx + β(W ; R) W (x) dx 1 1 4 16 4 |x|2 B |x| B u2 n − 2a − 2 2 u2 +c( ) dx + cβ(W ; R) W (x) dx, 2 2 2a+2 2 |x|2a B |x| B  Proof: Here again we shall give a proof when n ≥ 5. The case n = 4 will be handled in the next section. We again first use Theorem 3.3.1 (for n ≥ 5) with the Bessel potential |x|−2a where a < 1, then Theorem 3.2.3 with the Bessel 2 pair (|x|−2 , |x|−2 ( (n−4) |x|−2 + W )), then again Theorem 3.2.3 with the Bessel pair 4  61  3.3. General Hardy-Rellich inequalities (|x|−2a , |x|−2a (( n−2a−2 )2 |x|−2 + W ) to obtain 2 |∆u|2 dx ≥ B  ≥ + ≥ + +  |∇u|2 |∇u|2 n2 −2a dx + β(|x| ; R) dx −2a 4 B |x|2 B |x| n2 (n − 4)2 u2 dx 4 16 B |x| n2 |∇u|2 u2 dx β(W1 ; R) W1 (x) 2 dx + β(|x|−2a ; R) −2a 4 |x| B |x| B n2 (n − 4)2 n2 u2 u2 dx + β(W1 ; R) W1 (x) 2 dx 4 16 4 |x| B |x| B 2 n − 2a − 2 2 u β(|x|−2a ; R)( ) dx 2a+2 2 B |x| u2 β(|x|−2a ; R)β(W2 ; R) W2 (x) 2a dx. |x| B  The following theorem will be established in full generality (i.e with V (r) = r−m ) in the next section. Let W (x) = W (|x|) be a radial Bessel potential on a smooth bounded domain Ω in Rn , n ≥ 4. Then, Ω  |∆u(x)|2 dx −  n2 (n−4)2 16  u2 dx Ω |x|4  −  n2 4  Ω  W (x)u2 dx ≥  z02 2 R2 ||u||W01,2 (Ω) ,  u ∈ H02 (Ω).  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 restricted interval of powers m. We shall now prove that the same holds for all m < n−2 2 . The following theorem improves considerably Theorem 1.7, Theorem 1.8, and Theorem 6.4 in [31]. Suppose n ≥ 1 and m < n−2 2 , and let W be a Bessel potential on a ball BR ⊂ Rn of radius R. Then for all u ∈ C0∞ (BR )  BR  |∆u|2 ≥ an,m |x|2m  |∇u|2 dx + β(W ; R) |x|2m+2  BR  where an,m = inf      |∆u|2 dx BR |x|2m ; |∇u|2 dx 2m+2 BR |x|  W (x) BR  |∇u|2 dx, |x|2m  (3.90)    u ∈ C0∞ (BR ) \ {0} .   Moreover, β(W ; R) and am,n are the best constants to be computed in section 3.6. Proof: Assuming the inequality  BR  |∆u|2 ≥ an,m |x|2m  BR  |∇u|2 dx, |x|2m+2  62  3.3. General Hardy-Rellich inequalities holds for all u ∈ C0∞ (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. R  rα (f (r))2 dr 0  ≥ (  R  α−1 2 ) 2  rα−2 f 2 (r)dr 0 R  rα W (r)f 2 (r)dr, α ≥ 1,  + β(W ; R)  (3.91)  0 2 for all f ∈ C ∞ (0, R), where both ( α−1 2 ) and β(W ; R) are best constants. ∞ Decompose u ∈ C0 (BR ) into its spherical harmonics Σ∞ k=0 uk , where uk = 2 k| fk (|x|)ϕk (x). We evaluate Ik = nw1 n Rn |∆u dx in the following way |x|2m R  Ik  R  rn−2m−1 (fk (r))2 dr + [(n − 1)(2m + 1) + 2ck ]  = 0  rn−2m−3 (fk )2 dr 0  R  rn−2m−5 (fk (r))2 dr  +ck [ck + (n − 2m − 4)(2m + 2)] 0 R  rn−2m−1 W (x)(fk )2 dr + [(  ≥ β(W ) 0  n + 2m 2 ) + 2ck ] 2  R  rn−2m−3 (fk )2 dr 0  R  rn−2m−5 (fk (r))2 dr  +ck [ck + (n − 2m − 4)(2m + 2)] 0 R  R  rn−2m−1 W (x)(fk )2 dr + an,m  ≥ β(W ) 0  rn−2m−3 (fk )2 dr 0  R n + 2m 2 ) + 2ck − an,m ] rn−2m−3 W (x)(fk )2 dr +β(W )[( 2 0 n − 2m − 4 2 n + 2m 2 + ( ) [( ) + 2ck − an,m ] 2 2 R  rn−2m−5 (fk (r))2 dr.  + ck [ck + (n − 2m − 4)(2m + 2)] 0  Now by (3.122) we have (  n − 2m − 4 2 n + 2m 2 ) [( ) + 2ck − an,m ] + ck [ck + (n − 2m − 4)(2m + 2)] ≥ ck an,m , 2 2  63  3.3. General Hardy-Rellich inequalities for all k ≥ 0. Hence, we have R  Ik  R  rn−2m−3 (fk )2 dr + an,m ck  ≥ an,m 0  rn−2m−5 (fk (r))2 dr 0  R  rn−2m−1 W (x)(fk )2 dr  + β(W ) 0  n + 2m 2 ) + 2ck − an,m ] 2  + β(W )[(  R  rn−2m−3 W (x)(fk )2 dr 0  R  R  rn−2m−3 (fk )2 dr + an,m ck  ≥ an,m 0  rn−2m−5 (fk (r))2 dr 0  R  R  rn−2m−1 W (x)(fk )2 dr + β(W )ck  +β(W ) 0  = an,m BR  rn−2m−3 W (x)(fk )2 dr 0  |∇u|2 dx + β(W ) |x|2m+2  |∇u|2 W (x) 2m dx. |x| BR  Moreover, it is easy to see from Theorem 3.2 and the above calculation that β(W ; R) is the best constant. Let Ω be a smooth domain in Rn with n ≥ 1 and let V ∈ C 2 (0, R =: supx∈Ω |x|) be a non-negative function that satisfies the following conditions: Vr (r) ≤ 0  R 1 dr 0 r n−3 V (r)  and  =−  R 1 dr 0 r n−4 Vr (r)  = +∞.  (3.92)  There exists λ1 , λ2 ∈ R such that rVr (r) V (r)  + λ1 ≥ 0 on (0, R) and lim  rVr (r) r→0 V (r)  rVrr (r) Vr (r)  + λ2 ≥ 0 on (0, R) and lim  + λ1 = 0,  rVrr (r) r→0 Vr (r)  (3.93)  + λ2 = 0,  (3.94)  and 1 2 (n  − λ1 − 2)2 + 3(n − 3) V (r) − (n − 5)rVr (r) − r2 Vrr (r) ≥ 0,  (3.95)  for all r ∈ (0, R). Then the following inequality holds: (n − λ1 − 2)2 (n − λ1 − 4)2 + (n − 1)) 4 4 Vr (|x|) 2 (n − 1)(n − λ2 − 2)2 − u dx. 3 4 Ω |x|  V (|x|)|∆u|2 dx ≥ ( Ω  Ω  V (|x|) 2 u dx |x|4 (3.96)  64  3.3. General Hardy-Rellich inequalities Proof: We have by Theorem 3.2.2 and condition (3.95), R  1 nωn  V (x)|∆uk |2 dx = Rn  V (r)(fk (r))2 rn−1 dr 0  R  V (r)(fk (r))2 rn−3 dr  (n − 1 + 2ck )  +  0 R  (2ck (n − 4) + c2k )  +  R  V (r)rn−5 fk2 (r)dr − (n − 1) 0  0  R  R  Vr (r)fk2 (r)rn−4 dr  − ck (n − 5)  Vrr (r)fk2 (r)rn−3 dr  − ck  0  0 R  R  V (r)(fk (r))2 rn−1 dr + (n − 1)  ≥  Vr (r)rn−2 (fk )2 (r)dr  0  V (r)(fk (r))2 rn−3 dr 0  R  Vr (r)rn−2 (fk )2 (r)dr  − (n − 1) 0 R  1 (n − λ1 − 2)2 + 3(n − 3) V (r) − (n − 5)rVr (r) − r2 Vrr (r) fk2 (r)rn−5 2  + ck 0  The rest of the proof follows from the above inequality combined with Theorem 3.2.2. Remark 3.3.3. Let V (r) = r−2m with m ≤ n−4 2 . Then in order to satisfy condition √ 1+(n−1)2 ≤ m ≤ n−4 (3.95) we must have −1 − 2 2 . Under this assumption the inequality (3.96) gives the following weighted second order Rellich inequality: |∆u|2 (n + 2m)(n − 4 − 2m) 2 ) dx ≥ ( 2m |x| 4  B  u2 B  |x|2m+4  dx.  In the following theorem we will show that the constant appearing in the above √ 1+(n−1)2 inequality is optimal. Moreover, we will see that if m < −1 − , then the 2 (n+2m)(n−4−2m) 2 best constant is strictly less than ( ) . This shows that inequality 4 (3.96) is actually sharp. Let m ≤  n−4 2  and define βn,m =  inf  u∈C0∞ (B)\{0}  |∆u|2 dx B |x|2m . u2 dx B |x|2m+4  (3.97)  Then βn,m  = +  (  (n + 2m)(n − 4 − 2m) 2 ) 4 min  k=0,1,2,...  {k(n + k − 2)[k(n + k − 2) +  (n + 2m)(n − 2m − 4) ]}. 2  Consequently the values of βn,m are as follows.  65  3.3. General Hardy-Rellich inequalities √ 1. If −1 −  1+(n−1)2 2  ≤m≤  n−4 2 ,  βn,m = ( √ 2. If  n 2  − 3 ≤ m ≤ −1 −  βn,m = ( 3. If k :=  4. If k <  (n + 2m)(n − 4 − 2m) 2 ) . 4  1+(n−1)2 , 2  then  (n + 2m)(n − 4 − 2m) 2 (n + 2m)(n − 2m − 4) ) + (n − 1)[(n − 1) + ]. 4 2  n−2m−4 2  βn,m = (  then  ∈ N , then  (n + 2m)(n − 4 − 2m) 2 (n + 2m)(n − 2m − 4) ) +k(n+k−2)[k(n+k−2)+ ]. 4 2  n−2m−4 2  < k + 1 for some k ∈ N , then  βn,m =  (n + 2m)2 (n − 2m − 4)2 + a(m, n, k) 16  where a(m, n, k)  =  (n + 2m)(n − 2m − 4) ], 2 (n + 2m)(n − 2m − 4) ]}. (k + 1)(n + k − 1)[(k + 1)(n + k − 1) + 2 min{k(n + k − 2)[k(n + k − 2) +  Proof: Decompose u ∈ C0∞ (BR ) into spherical harmonics Σ∞ k=0 uk , where uk = fk (|x|)ϕk (x). we have 1 nωn  Rn  |∆uk |2 dx = |x|2m  R  rn−2m−1 (fk (r))2 dr 0 R  +  rn−2m−3 (fk )2 dr  [(n − 1)(2m + 1) + 2ck ] 0  R  rn−2m−5 (fk (r))2 dr  + ck [ck + (n − 2m − 4)(2m + 2)] 0  (n + 2m)(n − 4 − 2m) 2 ) 4 (n + 2m)(n − 2m − 4) + ck [ck + ] 2 ≥  (  R  rn−2m−5 (fk (r))2 dr, 0  by Hardy inequality. Hence, βn,m ≥ B(n, m, k)  (n + 2m)(n − 4 − 2m) 2 ) + min {k(n + k − 2)[k(n + k − 2) k=0,1,2,... 4 (n + 2m)(n − 2m − 4) ]}. 2  := ( +  66  3.3. General Hardy-Rellich inequalities To prove that βn,m is the best constant, let k be such that βn,m  =  (n + 2m)(n − 4 − 2m) 2 ) 4  + k(n + k − 2)[k(n + k − 2) + Set u = |x|−  n−4 2 +m+  (n + 2m)(n − 2m − 4) ]. 2  (3.98)  ϕk (x)ϕ(|x|),  where ϕk (x) is an eigenfunction corresponding to the eigenvalue ck and ϕ(r) is a smooth cutoff function, such that 0 ≤ ϕ ≤ 1, with ϕ ≡ 1 in [0, 21 ]. We have |∆u|2 dx BR |x|2m 2 u dx BR |x|2m+4  Let now  = (−  (n + 2m)(n − 4 − 2m) − ck + (2 + 2m + ))2 + O(1). 4  → 0 to obtain the result. Thus the inequality  BR  |∆u|2 ≥ βn,m |x|2m  u2 BR  |x|2m+4  dx,  holds for all u ∈ C0∞ (BR ). To calculate explicit values of βn,m we need to find the minimum point of the function (n + 2m)(n − 2m − 4) f (x) = x(x + ), x ≥ 0. 2 Observe that (n + 2m)(n − 2m − 4) ) = 0. f (− 4 To find minimizer k ∈ N we should solve the equation k 2 + (n − 2)k +  (n + 2m)(n − 2m − 4) = 0. 4 n+2m 2  and x2 = n−2m−4 . 1) follows from 2 √ 1+(n−1)2 Theorem 3.3.2. It is easy to see that if m ≤ −1 − , then x1 < 0. Hence, 2 √ 1+(n−1)2 for m ≤ −1 − the minimum of the function f is attained in x2 . Note √2 1+(n−1)2 that if m ≤ −1 − , then B(n, m1) ≤ B(n, m, 0). Therefore claims 2), 3), 2 and 4) follow. The roots of the above equation are x1 =  The following theorem extends Theorem 1.6 of [31] in many ways. First, we do not assume that n ≥ 5 or m ≥ 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 n Let m ≤ n−4 2 and let W (x) be a Bessel potential on a ball B of radius R in R W (r) = − λr + f (r), where f (r) ≥ 0 and limr→0 rf (r) = 0. with radius R. Assume W r (r) Then the following inequality holds for all u ∈ C0∞ (B)  B  |∆u|2 dx ≥ βn,m |x|2m  u2 B  |x|2m+4  + β(W ; R)(  dx  (3.99)  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4  B  W (x) 2 u dx. |x|2m+2  Proof: Again we will frequently use inequality (3.91) in the proof. Decomposing u ∈ C0∞ (BR ) into spherical harmonics Σ∞ k=0 uk , where uk = fk (|x|)ϕk (x), we can write 1 nωn  Rn  |∆uk |2 dx |x|2m  R  rn−2m−1 (fk (r))2 dr + [(n − 1)(2m + 1)  = 0  R  rn−2m−3 (fk )2 dr  +2ck ] 0  R  rn−2m−5 (fk (r))2 dr  +ck [ck + (n − 2m − 4)(2m + 2)] 0  n + 2m 2 ) ≥ ( 2 +ck [ck + 2(  R  R  rn−2m−3 (fk )2 dr + β(W ; R) 0  rn−2m−1 W (x)(fk )2 dr 0  n−λ−4 2 ) + (n − 2m − 4)(2m + 2)] 2  R  rn−2m−5 (fk (r))2 dr, 0  68  3.3. General Hardy-Rellich inequalities where we have used the fact that ck ≥ 0 to get the above inequality. We have 1 nωn  Rn R  |∆uk |2 dx |x|2m rn−2m−5 (fk )2 dr  ≥ βn,m 0  +β(W ; R)  (n + 2m)2 4  R  rn−2m−3 W (x)(fk )2 dr 0  R  rn−2m−1 W (x)(fk )2 dr  +β(W ; R) 0 R  rn−2m−5 (fk )2 dr  ≥ βn,m 0  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4 βn,m u2k dx ≥ nωn B |x|2m+4 β(W ; R) (n + 2m)2 (n − 2m − λ − 2)2 + + ) ( nωn 4 4  R  rn−2m−3 W (x)(fk )2 dr  +β(W ; R)(  0  B  W (x) 2 u dx, |x|2m+2 k  by Theorem 3.2.2. Hence, (3.99) holds and the proof is complete. Assume −1 < m ≤ n−4 2 and let W (x) be a Bessel potential on a ball B of radius R and centered at zero in Rn (n ≥ 1). Then the following holds for all u ∈ C0∞ (B):  B  |∆u|2 dx ≥ |x|2m  (n + 2m)2 (n − 2m − 4)2 u2 dx (3.100) 2m+4 16 B |x| (n + 2m)2 W (x) 2 + β(W ; R) u dx + β(|x|2m ; R)||u||H01 . 2m+2 4 |x| B  Proof: Decomposing again u ∈ C0∞ (BR ) into its spherical harmonics Σ∞ k=0 uk  69  3.4. Higher order Rellich inequalities where uk = fk (|x|)ϕk (x), we calculate 1 nωn  Rn  |∆uk |2 dx |x|2m  R  R  rn−2m−1 (fk (r))2 dr + [(n − 1)(2m + 1) + 2ck ]  = 0  rn−2m−3 (fk )2 dr 0  R  rn−2m−5 (fk (r))2 dr  + ck [ck + (n − 2m − 4)(2m + 2)] 0  ≥ (  n + 2m 2 ) 2  R  R  rn−1 (fk )2 dr  rn−2m−3 (fk )2 dr + β(|x|2m ; R) 0  0  R  rn−2m−3 (fk )2 dr  + ck 0  ≥  (n + 2m)2 (n − 2m − 4)2 16 (n + 2m)2 +β(W ; R) 4  R  rn−2m−5 (fk )2 dr 0  R  W (r)rn−2m−3 (fk )2 dr 0  R  + β(|x|2m ; R)  R  rn−1 (fk )2 dr + ck β(|x|2m ; R) 0  = +  rn−3 (fk )2 dr 0  (n + 2m)2 (n − 2m − 4)2 u2k dx 2m+4 16nωn Rn |x| β(W ; R) (n + 2m)2 W (x) 2 ) u dx + β(|x|2m ; R)||uk ||W 1,2 . ( 2m+2 k 0 nωn 4 Rn |x|  Hence (3.100) holds. We note that even for m = 0 and n ≥ 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, βn,m be defined as in Theorem 3.3.2 and σn,m = β(W ; R)(  (n + 2m)2 (n − 2m − λ − 2)2 + ). 4 4 0  For the sake of convenience we make the following convention:  ai = 1. Let BR be i=1 W (r) that W = − λr + f (r), r (r)  a ball of radius R and W be a Bessel potential on BR such where f (r) ≥ 0 and limr→0 rf (r) = 0. Assume m ∈ N , 1 ≤ l ≤ m, and 2k+4m ≤ n.  70  3.4. Higher order Rellich inequalities Then the following inequality holds for all u ∈ C0∞ (BR )  BR  l−1  |∆m u|2 dx ≥ |x|2k  |∆m−l u|2 dx |x|2k+4l  βn,k+2i BR  i=0 l−1  +  l−1  σn,k+2i i=0  βn,k+2j−2 BR  j=1  (3.101) W (x)|∆m−i−1 u|2 dx |x|2k+4i+2  Proof: Follows directly from theorem 3.3.2. Let BR be a ball of radius R and W be a Bessel potential on BR such that W (r) λ Wr (r) = − r + f (r), where f (r) ≥ 0 and limr→0 rf (r) = 0. Assume m ∈ N , 1 ≤ l ≤ m, and 2k + 4m + 2 ≤ n. Then the following inequality holds for all u ∈ C0∞ (BR ) l−1  BR  n − 2k − 2 2 |∇∆m u|2 dx ≥ ( ) βn,k+2i+1 |x|2k 2 i=0 l−1  +  (  l−1  n − 2k − 2 2 ) σn,k+2i+1 βn,k+2j−1 2 i=0 j=1  + β(W ; R)  W (x) BR  BR  BR  |∆m−l u|2 dx |x|2k+4l+2  W (x)|∆m−i−1 u|2 dx |x|2k+4i+4  |∆m u|2 dx |x|2k  (3.102)  Proof: Follows directly from Theorem 3.2.3 and the previous theorem. 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 = − λr + f (r), where f (r) ≥ 0 and limr→0 rf (r) = 0. Assume m ∈ N , 1 ≤ l ≤ m − 1, and 2k + 4m ≤ n. Then the following inequality holds for all u ∈ C0∞ (BR ) W (r) Wr (r)  l−1  BR  |∆m u|2 n − 2k − 4 2 dx ≥ an,k ( ) βn,k+2i+2 |x|2k 2 i=0 l−1  + an,k (  l−1  n − 2k − 4 2 ) σn,k+2i+2 βn,k+2j 2 i=0 j=1 m−1  + β(W ; R)an,k  W (x) BR  + β(W ; R)  W (x) BR  BR  BR  |∆m−l−1 u|2 dx |x|2k+4l+4  W (x)|∆m−i−2 u|2 dx |x|2k+4i+6  2  |∆ u| dx |x|2k+2  |∇∆m−1 u|2 dx, |x|2k  where an,m is defined in Theorem 3.3.2. 3.3.2 and the previous theorem.  (3.103)  Proof: Follows directly from Theorem  71  3.5. The class of Bessel potentials The following improves Theorem 1.10 in [31] in many ways, since it is assumed √ −n+8+2 n2 −n+1 there that l ≤ and 4m < n. Even for k = 0, Theorem 3.4 below 12 shows that we can drop the first condition and replace the second one by 4m ≤ n. Let BR be a ball of radius R and W be a Bessel potential on BR such that . Assume m ∈ N , 1 ≤ l ≤ m, and 2k + 4m ≤ n. Then the following inequality holds for all u ∈ C0∞ (BR )  BR  |∆m u|2 dx |x|2k  (3.104)  l  ≥  an,k+2i−2 (n − 2k − 4i)2 4 i=1 l  + β(W ; R)  BR  |∆m−l u|2 dx |x|2k+4l  l−1  an,k+2j−2 (n − 2k − 4j)2 4 i=1 j=1 an,k+2i−2 i=1  W (x) BR  |∇∆m−i u|2 dx |x|2k+4i−4  l−1  l  + β(W ; R)  (3.105)  an,k+2j−2 (n − 2k − 4j)2 4 j=1  W (x) BR  |∆m−i u|2 dx, |x|2k+4i−2  where an,m are the best constants in inequality (3.90). Proof: Follows directly from Theorem 3.3.2.  3.5  The class of Bessel potentials  The Bessel equation associated to a potential W (BW )  y + 1r y + 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 C 1 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−s ) maps the equation y + 1r y + W (r)y = 0 into (BW )  z + e−2s W (e−s )z(s) = 0.  On the other hand, the change of variables ψ(t) = nonlinear equation (BW )  −e−t y (e−t ) y(e−t )  (3.106) maps it into the  ψ (t) + ψ 2 (t) + e−2t W (e−t ) = 0.  (3.107) This will allow us to relate the existence of positive solutions of (BW ) to the nonoscillatory behaviour of equations (BW ) and (BW ).  72  3.5. The class of Bessel potentials The theory of sub/supersolutions –applied to (BW ) (See Wintner [35, 36, 20])– already yields, that if (BW ) has a positive solution on an interval (0, R) for some non-negative potential W ≥ 0, then for any W such that 0 ≤ V ≤ W , the equation (BV ) has also a positive solution on (0, R). This leads to the definition of the weight of a potential W ∈ B(0, R) as: β(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 C 1 (0, R). 2) For every W ∈ B(0, R), the equation y + 1r y + β(W ; R)W (r)y = 0  (BW )  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 ln(r) r→0  r 0  sW (s)ds > −∞, then for every R > 0, there exists α := α(R) > 0  such that the scaled function Wα (x) := α2 W (αx) is a Bessel potential on (0, R). 2. If lim ln(r) r→0  r 0  sW (s)ds = −∞, then there are no α, c > 0, for which Wα,c =  cW (α|x|) is a Bessel potential on (0, R). Proof: This relies on well known results concerning the existence of nonoscillatory solutions (i.e., those z(s) such that z(s) > 0 for s > 0 sufficiently large) for the second order linear differential equations z (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→∞ t  ∞ t  a(s)ds < 41 , then Eq. (3.111) is non-oscillatory.  ii) If lim inf t→∞ t  ∞ t  a(s)ds > 14 , then Eq. (3.111) is oscillatory.  It follows that if lim inf ln(r) r→0  r 0  sW (s)ds > −∞ holds, then there exists δ > 0 such  that (BW ) has a positive solution on (0, δ). An easy scaling argument then shows that there exists α > 0 such that Wα (x) := α2 W (αx) is a Bessel potential on (0, R). The rest of the proof is similar.  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−1) (.))  for k ≥ 2.  (3.112)  and X1 (t) = (1 − log(t))−1 ,  Xk (t) = X1 (Xk−1 (t))  k = 2, 3, ...,  (3.113)  Explicit Bessel potentials 1. W ≡ 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 ≡ 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, β(1; R) =  z02 . R2  (3.114)  3. If a < 2, then there exists Ra > 0 such that W (r) = r−a is a Bessel potential on (0, Ra ). 4. For each k ≥ 1 and ρ > R(ee ing to the potential  e.  .e(k−times)  ), the equation (B 41 Wk,ρ ) correspond-  Wk,ρ (r) = Σkj=1 Uj where Uj (r) =  1 r2  j i=1  log (i) ρr  −2 k  1  has a positive solution on (0, R) that is explicitly given by ϕk,ρ (r) = ( i=1 log (i) ρr ) 2 . On the other hand, the equation (B 1 W +λU ) corresponding to the potential 1 4 Wk,ρ  k,ρ  4  k  + λUk has no positive solution for any λ > 0. In other words, Wk,ρ is a Bessel potential on (0, R) with 1 4  β(Wk;ρ , R) =  for any k ≥ 1.  (3.115)  5. For each k ≥ 1 and R > 0, the equation (B 1 W ˜ k,R ) corresponding to the 4 potential ˜ k,R (r) = Σk U ˜ ˜ W j=1 j where Uj (r) =  1 r 2 r 2 r 2 2 r r 2 X1 ( R )X2 ( R ) . . . Xj−1 ( R )Xj ( R )  has a positive solution on (0, R) that is explicitly given by 1 r r r r ϕk (r) = (X1 ( )X2 ( ) . . . Xk−1 ( )Xk ( ))− 2 . R R R R On the other hand, the equation (B 1 W ˜ k,R +λU ˜k ) corresponding to the poten4 1 ˜ ˜ tial 4 Wk,R + λUk has no positive solution for any λ > 0. In other words, ˜ k,R is a Bessel potential on (0, R) with W ˜ k,R ; R) = β(W  1 4  for any k ≥ 1.  (3.116)  74  3.5. The class of Bessel potentials Proof: 1) It is clear that φ(r) = −log( Re r) is a positive solution of (B0 ) on (0, R) for any R > 0. 2)The best constant for which the equation y + 1r y + cy = 0 has a positive z2  solution on (0, R) is R02 , where z0 = 2.4048... is the first zero of Bessel function J0 (z). Indeed if α is the first root of the an arbitrary solution of the Bessel equation y + yr +y(r) = 0, then we have α ≤ 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 ≥ 0. Also b ≤ 0, because Y0 (t) → −∞ as t → 0. Finally note that Y0 (z0 ) > 0, so if b < 0, then x(z0 + ) < 0 for sufficiently small. Therefore, b = 0 which is a contradiction. 3) follows directly from the integral criteria. 4) That φk is an explicit solution of the equation (B 41 Wk ) is straightforward. Assume now that there exists a positive function ϕ such that −  1 ϕ (r) + rϕ (r) = ϕ(r) 4  Define f (r) =  ϕ(r) ϕk (r)  k−1  j=1  1 r  j  log (i) i=1  ρ r  −2  1 1 + ( + λ) 4 r  k  log (i) i=1  ρ r  −2  .  > 0, and calculate,  ϕ (r) + rϕk (r) f (r) + rf (r) f (r) ϕ (r) + rϕ (r) = k + − ϕ(r) ϕk (r) f (r) f (r)  k  1  i=1  i j=1  logj ( ρr )  .  Thus, f (r) + rf (r) f (r) − f (r) f (r)  k  i=1  1 i j=1  logj ( ρr )  = −λ  1 r  k  log (i) i=1  ρ r  −2  .  (3.117)  If now f (αn ) = 0 for some sequence {αn }∞ n=1 that converges to zero, then there exists a sequence {βn }∞ that also converges to zero, such that f (βn ) = 0, n=1 and f (βn ) > 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 (r) > 0 for r > 0 sufficiently small. Then we will have (rf (r)) ≤ rf (r)  k  i=1  r  r  k j=1  Integrating once we get f (r) ≥  1 i j=1  logj ( ρr )  c logj ( ρr )  .  ,  75  3.6. The evaluation of an,m for some c > 0. Hence, limr→0 f (r) = −∞ which is a contradiction. Case II: Assume f (r) < 0 for r > 0 sufficiently small. Then (rf (r)) ≥ rf (r)  k  1  i=1  i j=1  r  Thus, f (r) ≥ −  logj ( ρr )  c r  k j=1  logj ( ρr )  .  ,  (3.118)  for some c > 0 and r > 0 sufficiently small. On the other hand j  k  f (r) + rf (r) 1 ≤ −λ f (r) r j=1  R r  log (i) i=1  −2  1  ≤ −λ(  k j=1  logj ( ρr )  ).  Since f (r) < 0, there exists l such that f (r) > l > 0 for r > 0 sufficiently small. From the above inequality we then have bf (b) − af (a) < −λl(  1 k j=1  logj ( ρb )  1  −  k j=1  logj ( aρ )  ).  From (3.118) we have lima→0 af (a) = 0. Hence, bf (b) < −  λl k j=1  logj ( ρb )  ,  for every b > 0, and f (r) < −  λl r  k j=1  logj ( ρr )  ,  for r > 0 sufficiently small. Therefore, lim f (r) = +∞,  r→0  and by choosing l large enouph (e.g., l > λc ) we get to contradict (3.118). The proof of 5) is similar and is left to the interested reader.  3.6  The evaluation of an,m  Here we evaluate the best constants an,m which appear in Theorem 3.3.2. Suppose n ≥ 1 and m ≤ n−2 2 . Then for any R > 0, the constants   |∆u|2   dx BR |x|2m an,m = inf ; u ∈ C0∞ (BR ) \ {0} 2 |∇u|   dx BR |x|2m+2 are given by the following expressions.  76  3.6. The evaluation of an,m 1. For n = 1 • if m ∈ (−∞, − 32 ) ∪ [− 67 , − 12 ], then a1,m = (  1 + 2m 2 ) 2  • if − 32 < m < − 76 , then a1,m = min{( 2. If m =  n−4 2 ,  n + 2m 2 ( ) , 2  (n−4−2m)(n+2m) + 4 n−4−2m 2 ) +2 ( 2  2)2  }.  then am,n = min{(n − 2)2 , n − 1}.  3. If n ≥ 2 and m ≤ 4. If 2 ≤ n ≤ 3 and n−2 2 , then  √ −(n+4)+2 n2 −n+1 , 6 √ −(n+4)+2 n2 −n+1 6  an,m = 5. For n ≥ 4 and  2 then an,m = ( n+2m 2 ) .  <m≤  n−2 2 ,  or n ≥ 4 and  n−4 2  <m≤  ( (n−4−2m)(n+2m) + n − 1)2 4 . n−4−2m 2 ( ) +n−1 2  √ −(n+4)+2 n2 −n+1 6  <m<  n−4 2 ,  define k ∗ = [(  √  3 3  − 21 )(n − 2)].  • If k ∗ ≤ 1, then an,m =  + n − 1)2 ( (n−4−2m)(n+2m) 4 . ( n−4−2m )2 + n − 1 2 √  2  • For k ∗ > 1 the interval (m10 := −(n+4)+26 n −n+1 , m20 := divided in 2k ∗ − 1 subintervals. For 1 ≤ k ≤ k ∗ define m1k := m2k :=  2(n − 5) −  (n − 2)2 − 12k(k + n − 2) , 6  2(n − 5) +  (n − 2)2 − 12k(k + n − 2) . 6  n−4 2 )  can be  If m ∈ (m10 , m11 ] ∪ [m21 , m20 )], then an,m =  ( (n−4−2m)(n+2m) + n − 1)2 4 . ( n−4−2m )2 + n − 1 2  77  3.6. The evaluation of an,m • For k ≥ 1 and m ∈ (m1k , m1k+1 ] ∪ [m2k+1 , m2k ), then an,m  =  min{  ( (n−4−2m)(n+2m) + k(n + k − 2))2 4 , ( n−4−2m )2 + k(n + k − 2) 2  ( (n−4−2m)(n+2m) + (k + 1)(n + k − 1))2 4 }. )2 + (k + 1)(n + k − 1) ( n−4−2m 2 For m ∈ (m1k∗ , m2k∗ ), then an,m  =  min{  ( (n−4−2m)(n+2m) + k ∗ (n + k ∗ − 2))2 4 , n−4−2m 2 ( ) + k ∗ (n + k ∗ − 2) 2  + (k ∗ + 1)(n + k ∗ − 1))2 ( (n−4−2m)(n+2m) 4 }. n−4−2m 2 ( ) + (k ∗ + 1)(n + k ∗ − 1) 2 Proof: Letting V (r) = r−2m then, W (r) −  2V (r) 2Vr (r) n − 2m − 2 2 + − Vrr (r) = (( ) − 2 − 4m − 2m(2m + 1))r−2m−2 . r2 r 2  In order to satisfy condition (3.74) we should have √ √ −(n + 4) + 2 n2 − n + 1 −(n + 4) + 2 n2 − n + 1 ≤m≤ . 6 6  (3.119)  2 So, by Theorem 3.3.1 under the above condition we have an,m = ( n+2m 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 ≥ 5 and for certain intervals of m. Decomposing again u ∈ C0∞ (BR ) into spherical harmonics; u = Σ∞ k=0 uk , where uk = fk (|x|)ϕk (x), one has  Rn  |∆uk |2 dx = |x|2m +  Rn  |x|−2m (fk (|x|))2 dx  (3.120)  ((n − 1)(2m + 1) + 2ck ) Rn  |x|−2m−2 (fk )2 dx |x|−2m−4 (fk )2 dx,  + ck (ck + (n − 4 − 2m)(2m + 2)) Rn  and  Rn  |∇uk |2 dx = |x|2m+2  Rn  |x|−2m−2 (fk )2 dx + ck  |x|−2m−4 (fk )2 dx.  (3.121)  Rn  One can then prove as in [31] that an,m = min {A(k, m, n); k ∈}  (3.122)  78  3.6. The evaluation of an,m where A(k, m, n) =  (  (n−4−2m)(n+2m) +ck )2 4 n−4−2m 2 ( ) +ck 2  if m =  n−4 2  (3.123)  and A(k, m, n) := ck if m =  n−4 2  and n + k > 2.  (3.124)  n−4 2  and n + k > 2, then ck = 0. Actually, this also holds for Note that when m = n + k ≤ 2, in which case one deduces that if m = n−4 2 , then an,m = min{(n − 2)2 = (  n + 2m 2 ) , (n − 1) = c1 } 2  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) =  + x)2 ( (n−4−2m)(n+2m) 4 . ( n−4−2m )2 + x 2  It is easy to check that f (x) = 0 at x1 and x2 , where (n − 4 − 2m)(n + 2m) 4 (n − 4 − 2m)(−n + 6m + 8) . 4  x1  = −  (3.125)  x2  =  (3.126)  n−4 Observe that for for n ≥ 2, n−8 6 ≤ 2 . Hence, for m ≤ n+2m 2 negative and hence an,m = ( 2 ) . Also note that  n−8 6  both x1 and x2 are  √ −(n + 4) − 2 n2 − n + 1 n−8 ≤ f or all n ≥ 1. 6 6 2 Hence, under the condition in 3) we have an,m = ( n+2m 2 ) . 3 Also for n = 1 if m ≤ − 2 both critical points are negative and we have a1,m ≤ 2 ( 1+2m 2 ) . Comparing A(0, m, n) and A(1, m, n) we see that A(1, m, n) ≥ A(0, m, n) if and only if (3.119) holds. For n = 1 and − 32 < m < − 76 both x1 and x2 are positive. Consider the equations (2m + 3)(2m + 1) x(x − 1) = x1 = , 4 and (2m + 3)(6m + 7) x(x − 1) = x2 = − . 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 < − 67 , we have a1,m ≤ min{A(1, m, 1), A(2, m, 1)} and 1) follows.  79  3.6. The evaluation of an,m For n ≥ 2 and equation  n−4 2  < m <  n−2 2  we have x1 > 0 and x2 < 0. Consider the  x(x + n − 2) = x1 = −  (n − 4 − 2m)(n + 2m) . 4  and − (2m+n) are solutions of the above equation and both are less Then 2m+4−n 2 2 than one. Since, for n ≥ 4 √ n−2 −(n + 4) + 2 n2 − n + 1 > , 2 6 √  2  and A(1, m, n) ≤ A(0, m, n) for m ≥ −(n+4)+26 n −n+1 , the best constant is equal to what 4) claims. 5) follows from an argument similar to that of Theorem 6.4 in [31].  80  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 inequalities, maximum principles and related eigenvalue problems, J. Funct. Anal. 240 (2006), 36-83. [3] M. Agueh, N. Ghoussoub, X. S. 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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 H 2 ∩ H01 3 4.1  Introduction  Let Ω be a smooth bounded domain in n and 0 ∈ Ω. Let us recall that the classical Hardy-Rellich inequality assets that |∆u|2 dx ≥ Ω  n2 (n − 4)2 16  Ω  u2 dx, for u ∈ H02 (Ω), |x|4  (4.1)  where the constant appearing in the above inequality is the best constant and it is never achieved in H02 . Recently there has been a flurry of activity about possible improvements of the following type If n ≥ 5  n2 (n−4)2 16  Ω  |∆u|2 dx −  Ω  |∆u|2 dx − C(n)  then  u2 dx Ω |x|4  ≥  W (x)u2 dx,  (4.2)  V (x)|∇u|2 dx,  (4.3)  Ω  for u ∈ H02 (Ω) as well as If n ≥ 3  then  |∇u|2 dx Ω |x|2  ≥  Ω  for all u ∈ H02 (Ω), where V, W are certain explicit radially symmetric potentials of order lower than r12 (for V ) and r14 (for W ) (see [2], [3], [8], [10], [11], [15], and [18]. The inequality (4.1) was first proved by Rellich [17] for u ∈ H02 (Ω) and then it was extended to functions in H 2 (Ω) ∩ H01 (Ω) 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 ∈ H02 (Ω) (see [8], [15], and [18]). The goal of this paper is to provide a general approach to prove optimal weighted Hardy-Rellich inequalities on H 2 (Ω) ∩ H01 (Ω) and inequalities of type (4.3) on H 2 (Ω) which are important in the study of fourth order elliptic equations with Navier boudary condition and systems of second order elliptic equations (see [16]). 3  A version of this chapter has been submitted for publication; A. Moradifam, Optimal weighted Hardy-Rellich inequalities on H 2 ∩ H01 (2009).  84  4.1. Introduction We start – in section 2 – 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 and b < 0: B  V (x)|∇u|2 dx ≥ c  B  W (x)u2 dx + b  ∂B R  Assuming that the ball B has radius R and that 0 is simply that the ordinary differential equation y (r) + ( n−1 r +  (BV,cW )  Vr (r) V (r) )y  (r) +  u2 for all u ∈ H 1 (B).  1 r n−1 V (r) dr  cW (r) V (r) y(r)  (4.4)  = +∞, the condition  =0  (R) has a positive solution φ on the interval (0, R) with V (R) φφ(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  β(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 behavior 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 recently 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 C 1 -functions on B\{0}, where B is a ball R centered at zero with radius R in n (n ≥ 1) such that 0 rn−11V (r) dr = +∞ and R n−1 r V 0  (r)dr < +∞. The following statements are then equivalent:  (R) , where φ is the corre1. (V, W ) is a Bessel pair on (0, R) with θ := V (R) φφ(R) sponding solution of (B(V,W ) ).  2.  B  V (x)|∇u|2 dx ≥  B  W (x)u2 dx + θ  ∂B  ¯ u2 ds for all u ∈ C ∞ (B).  3. If limr→0 rα V (r) = 0 for some α < n − 2, then the above are equivalent to V (x)|∆u|2 dx ≥ B  W (x)|∇u|2 dx + (n − 1) B  −  ( B  V (x) |x|2  Vr (|x|) )|∇u|2 dx + (θ + (n − 1)V (R)) |x|  |∇u|2 , ∂B  ¯ for all radial u ∈ C ∞ (B).  85  4.1. Introduction 2V (r) r2  4. If in addition, W (r) − are equivalent to V (x)|∆u|2 dx ≥  +  2Vr (r) r  − Vrr (r) ≥ 0 on (0, R), then the above  W (x)|∇u|2 dx + (n − 1)  B  B  −  ( B  V (x) |x|2  Vr (|x|) )|∇u|2 dx + (θ + (n − 1)V (R)) |x|  |∇u|2 , ∂B  ¯ for all u ∈ C ∞ (B). Appropriate combinations of 4) and 2) in the above theorem and lead to a myriad of Hardy-Rellich type inequalities on H 2 (Ω) ∩ H01 (Ω). Remark 4.1.1. The condition W (r) − 2Vr2(r) + 2Vrr(r) − Vrr (r) ≥ 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’s main tools to prove singularity of the extremal solutions in dimensions n ≥ 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 ¯ such as those in [15] and [18]. C0∞ (Ω) to corresponding inequalities on C ∞ (Ω) We shall show that for − n2 ≤ m ≤ Hn,m = and for − n2 ≤ m ≤  inf  u∈H 2 (B)\{0}  n−2 2  |∆u|2 B |x|2 m |∇u|2 B |x|2m+2  =  inf  u∈H02 (B)\{0}  |∆u|2 B |x|2 m |∇u|2 B |x|2m+2  ,  (4.6)  n−4 2  an,m =  inf  u∈H 2 (B)∩H01 (B)\{0}  |∆u|2 B |x|2m u2 B |x|2m+4  =  |∆u|2 B |x|2m u2 B |x|2m+4  ,  (4.7)  where the constants Hn,m and an,m have been computed in [18] and then more 2 generally in [15]. For example an,0 = n4 for n ≥ 5, a4,0 = 3, and a3,0 = 25 36 . The above general theorem also allows us to obtain improved Hardy-Rellich inequalities on H 2 (B) ∩ H01 (B). For instance, assume W is a Bessel potential on (R) r (r) (0, R) and φ is the corresponding solution of (B(1,W ) ) with R φφ(R) ≥ − n2 . If r W W (r) decreases to −λ and λ ≤ n − 2, then we have for all H 2 (B) ∩ H01 (B) |∆u|2 dx − B  n2 (n − 4)2 16  B  u2 n2 (n − λ − 2)2 dx ≥ + β(W ; R) |x|4 4 4  B  W (x) 2 u dx. |x|2 (4.8)  86  4.1. Introduction By applying (4.8) to the various examples of Bessel functions, we can various improved Hardy-Rellich inequalities on H 2 (B) ∩ H01 (B). Here are some basic examples of Bessel potentials, their corresponding solution φ of (B(1,W ) ). • W ≡ 0 is a Bessel potential on (0, R) for any R > 0 and φ = 1. • W ≡ 1 is a Bessel potential on (0, R) for any R > 0, φ(r) = J0 ( µr R ), where J0 is the Bessel function and z0 = 2.4048... is the first zero of the Bessel function (R) J0 . Moreover R φφ(R) = − n2 . −2  j  (i) ρ • For k ≥ 1, R > 0, let Wk,ρ (r) = Σkj=1 r12 where the functions i=1 log r (i) (1) log are defined iteratively as follows: log (.) = log(.) and for k ≥ 2, log (k) (.) = log(log (k−1) (.)). Wk,ρ is then a Bessel potential on (0, R) with the corresponding solution j  log (i)  φk = i=1  It is easy to see that for ρ ≥ R(ee φ (R) R φkk (R) ≥ − n2 .  . e.  ρ r  − 12  .  e((k−1)−times)  ) large enough we have  • For k ≥ 1, and R > 0, define r r 2 ˜ k;ρ (r) = Σkj=1 1 X12 ( r )X22 ( r ) . . . Xj−1 W ( )Xj2 ( ) r2 R R R R where the functions Xi are defined iteratively as follows: X1 (t) = (1 − ˜ k,ρ is a Bessel log(t))−1 and for k ≥ 2, Xk (t) = X1 (Xk−1 (t)). Then again W 1 r r r potential on (0, R) with φk = (X1 ( R )X2 ( R ) . . . Xj−1 ( R )Xk ( Rr )) 2 . Moreφ (R)  over, R φkk (R) = − k2 . As an example, let k ≥ 1 and choose ρ ≥ R(ee φ (R) R φ(R) ≥ − n2 , where j  log (i)  φ= i=1  ρ |x|  e.  1 2  .e(k−times)  ) large enough so that  .  (4.9)  Then we have |∆u(x)|2 dx ≥ B  n2 (n − 4)2 16  B  u2 dx |x|4 k  +  (1 +  n(n − 4) ) 8 j=1  B  (4.10) u2 |x|4  j  log (i) i=1  ρ |x|  −2  dx,  for all H 2 (B) ∩ H01 (B) which corresponds to the result od Adimurthi et al. [2].  87  4.2. General Hardy Inequalities More generally, we show that for any − n2 ≤ m < n−2 2 , and any W Bessel potential on a ball BR ⊂ Rn of radius R, if for the corresponding solution φ of (R) ≥ − n2 − m, then the following inequality holds for all (B(1,W ) ) we have R φφ(R) ∞ u ∈ C0 (BR )  BR  |∆u|2 ≥ an,m |x|2m  BR  |∇u|2 dx + β(W ; R) |x|2m+2  W (x) BR  |∇u|2 dx. |x|2m  (4.11)  We also establish a more general version of equation (4.8). Assuming again (r) n−4 that rW and n2 + m ≥ λ ≥ W (r) decreases to −λ on (0, R), and provided m ≤ 2 ∞ n − 2m − 4, we then have for all u ∈ C0 (BR ),  BR  (n + 2m)2 (n − 2m − 4)2 16  |∆u|2 dx ≥ |x|2m  + β(W ; R)(  4.2  u2 BR  |x|2m+4  dx  (4.12)  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4  BR  W (x) 2 u dx. |x|2m+2  General Hardy Inequalities  Here is the main result of this section. Let V and W be positive radial C 1 -functions on BR \{0}, where BR is a ball centered at zero with radius R (0 < R ≤ +∞) in a a n (n ≥ 1). Assume that 0 rn−11V (r) dr = +∞ and 0 rn−1 V (r)dr < ∞ for some 0 < a < R. Then the following two statements are equivalent: 1. The ordinary differential equation y (r) + ( n−1 r +  (BV,W )  Vr (r) V (r) )y  (r) +  W (r) V (r) y(r)  =0  (R) . has a positive solution on the interval (0, R] with θ := V (R) φφ(R)  2. For all u ∈ H 1 (BR ) (HV,W )  BR  V (x)|∇u(x)|2 dx ≥  BR  W (x)u2 dx+θ  ∂B  u2 ds.  The above theorem allows to generalize all Hardy type inequalities on H01 (Ω) to a corresponding inequality on H 1 (Ω). 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 ≤ n−2 2 , then n − 2a − 2 2 ) |x|−2a−2 u2 dx 2 B (n − 2a − 2)R−2a−1 u2 dx, 2 ∂B  |x|−2a |∇u(x)|2 dx ≥ ( B  −  (4.13)  for all u ∈ H 1 (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 C 1 -functions on a ball B\{0}, where B is a ball with radius R in n (n ≥ 1) and centered at zero. Assume B  V (x)|∇u|2 − W (x)|u|2 dx − θ  ∂B  u2 ds ≥ 0 for all u ∈ H 1 (B),  for some θ < 0. Then there exists a C 2 -supersolution to the following linear elliptic equation −div(V (x)∇u) − W (x)u = 0, u > 0 V ∇u.ν = θu  in B, in B \ {0}, in ∂B.  (4.15) (4.16) (4.17)  Proof: Define λ1 (V ) := inf{  B  V (x)|∇ψ|2 − W (x)|ψ|2 − θ |ψ|2 B  ∂B  u2  ; ψ ∈ C0∞ (B \ {0})}.  By our assumption λ1 (V ) ≥ 0. Let (φn , λn1 ) be the first eigenpair for the problem (L − λ1 (V ) − λn1 )φn  =  0 on B \ B R  φn  =  0 on ∂B R  V ∇φn .ν  n  n  = θφn on ∂B,  where Lu = −div(V (x)∇u) − W (x)u, and B R is a ball of radius R n , n ≥ 2 . The n eigenfunctions can be chosen in such a way that φn > 0 on B \ B R and ϕn (b) = 1, n for some b ∈ B with R2 < |b| < R. Note that λn1 ↓ 0 as n → ∞. Harnak’s inequality yields that for any comK φn pact subset K, max minK φn ≤ C(K) with the later constant being independent of φn . Also standard elliptic estimates also yields that the family (φn ) have also uniformly bounded derivatives on the compact sets B − B R . n Therefore, there exists a subsequence (ϕnl2 )l2 of (ϕn )n such that (ϕnl2 )l2 converges to some ϕ2 ∈ C 2 (B \ B( R2 )). Now consider (ϕnl2 )l2 on B \ B( R3 ). Again there exists a subsequence (ϕnl3 )l3 of (ϕnl2 )l2 which converges to ϕ3 ∈ C 2 (B \ B( R3 )), and ϕ3 (x) = ϕ2 (x) for all x ∈ B \ B( R2 ). By repeating this argument we get a supersolution ϕ ∈ C 2 (B \ {0}) i.e. Lϕ ≥ 0, such that ϕ > 0 on B \ {0} and V ∇φ.ν = θφ on ∂B. Proof of Theorem 4.2: First we prove that 1) implies 2). Let φ ∈ C 1 (0, R] be u(x) a solution of (BV,W ) such that φ(x) > 0 for all x ∈ (0, R). Define ϕ(|x|) = ψ(x). Then x |∇u|2 = (ϕ (|x|))2 ψ 2 (x) + 2ϕ (|x|)ϕ(|x|)ψ(x) .∇ψ + ϕ2 (|x|)|∇ψ|2 . |x| Hence, V (|x|)|∇u|2 ≥ V (|x|)(ϕ (|x|))2 ψ 2 (x) + 2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x)  x .∇ψ(x). |x|  89  4.2. General Hardy Inequalities Thus, we have V (|x|)|∇u|2 dx ≥ B  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx B  +  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B  Let B be a ball of radius  centered at the origin. Integrate by parts to get  V (|x|)|∇u|2 dx ≥ B  +  x .∇ψdx. |x|  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx B  x .∇ψdx |x| x 2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) .∇ψdx |x|  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B  + B\B  V (|x|)(ϕ (|x|))2 ψ 2 (x)dx +  =  2V (|x|)ϕ (|x|)ϕ(|x|)ψ(x) B  B  −  V (|x|)ϕ (|x|)ϕ(|x|) + ( B\B  x .∇ψdx |x|  (n − 1)V (|x|) + Vr (|x|))ϕ (|x|)ϕ(|x|) ψ 2 (x) dx r  V (|x|)ϕ (|x|)ϕ(|x|)ψ 2 (x)ds  + ∂(B\B )  Let get  → 0 and use Lemma 2.3 in [15] and the fact that φ is a solution of (Dv,w ) to  V (|x|)|∇u|2 dx ≥  −  B  [V (|x|)ϕ (|x|) + ( B  + Vr (|x|))ϕ (|x|)]  (n − 1)V (|x|) r  u2 (x) dx ϕ(|x|)  W (|x|)u2 (x)dx − θ  = B  u2 ds. ∂B  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 C 2 -supersolution for the equation (4.15). Now take the surface average of u, that is y(r) =  1 nωw rn−1  u(x)dS = ∂Br  1 nωn  u(rω)dω > 0,  (4.18)  |ω|=1  where ωn 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 (R) V (R) yy(R) = θ. We clearly have y (r) +  n−1 1 y (r) = r nωn rn−1  ∆u(x)dS.  (4.19)  ∂Br  90  4.3. General Hardy-Rellich inequalities Since u(x) is a supersolution of (4.15), we have div(V (|x|)∇u)ds − ∂Br  W (|x|)udx ≥ 0, ∂B  and therefore, ∆udS − Vr (r)  V (r) ∂Br  ∇u.xds − W (r) ∂Br  u(x)ds ≥ 0. ∂Br  It follows that ∆udS − Vr (r)y (r) − W (r)y(r) ≥ 0,  V (r)  (4.20)  ∂Br  and in view of (4.18), we see that y satisfies the inequality V (r)y (r) + (  (n − 1)V (r) + Vr (r))y (r) ≤ −W (r)y(r), r  f or 0 < r < R, (4.21)  (R) that is it is a positive supersolution y for (BV,W ) with V (R) yy(R) = θ. 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 4.2 is now complete.  An immediate application of Theorem 2.6 in [15] and Theorem 4.2 is the following very general Hardy inequality. Let V (x) = V (|x|) be a strictly positive radial function on a smooth domain Ω containing 0 such that R = supx∈Ω |x|. Assume that for some λ ∈ rVr (r) V (r)  rVr (r) r→0 V (r)  + λ ≥ 0 on (0, R) and lim  + λ = 0.  (4.22)  If λ ≤ n − 2, then the following inequality holds for any Bessel potential W on (0, R): V (x) 2 n−λ−2 2 ) u dx + β(W ; R) V (x)W (x)u2 dx 2 2 Ω |x| Ω φ (R) n − λ − 2 + V (R)( − ) u2 f or u ∈ H 1 (Ω), φ(R) 2R ∂B  V (x)|∇u|2 dx ≥ ( Ω  where φ is the corresponding solution of (B1,W ). n−λ−2 Proof: Under our assumptions, it is easy to see that y = r 2 φ(r) is a positive super-solution of B(V,V ( n−λ−2 )2 r−2 +W ) . Now apply Theorem 2.6 in [15] 2 and Theorem 4.2 to complete the proof.  4.3  General Hardy-Rellich inequalities  Let 0 ∈ Ω ⊂ Rn be a smooth domain, and denote ¯ = {v ∈ C k (Ω) ¯ : v is radial }. Crk (Ω)  91  4.3. General Hardy-Rellich inequalities We start by considering a general inequality for radial functions. Let V and W be positive radial C 1 -functions on a ball B\{0}, where B is a R ball with radius R in n (n ≥ 1) and centered at zero. Assume 0 rn−11V (r) dr = ∞ and limr→0 rα V (r) = 0 for some α < n − 2. Then the following statements are equivalent: (R) 1. (V, W ) is a Bessel pair on (0, R) with θ := V (R) φφ(R) , where φ is the corresponding solution of (B(V,W ) ).  2. If limr→0 rα V (r) = 0 for some α < n − 2, then the above are equivalent to W (x)|∇u|2 dx  V (x)|∆u|2 dx ≥ B  B  (n − 1)  +  ( B  V (x) Vr (|x|) − )|∇u|2 dx |x|2 |x| |∇u|2 ,  (θ + (n − 1)V (R))  +  ∂B  ¯ for all radial u ∈ C ∞ (B). ¯ and observe that Proof: Assume u ∈ Cr∞ (B) R  V (x)|∆u|2 dx = nωn { B  R  V (r)u2rr rn−1 dr + (n − 1)2 0  V (r) 0  u2r n−1 r dr r2  R  +  V (r)uur rn−2 dr}.  2(n − 1) 0  Setting ν = ur , we then have V (x)|∆u|2 dx = B  V (x)|∇ν|2 dx B  V (|x|) |x|2  +  (n − 1)  −  Vr (|x|) )|ν|2 dx + (n − 1)V (R) |x|  ( B  |ν|2 ds. ∂B  Thus, (HRV,W ) for radial functions is equivalent to V (x)|∇ν|2 dx ≥ B  W (x)ν 2 dx. B  It therefore follows from Theorem 4.2 that 1) and 2) are equivalent.  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 ∈ C ∞ (B). By decomposing u into spherical harmonics we get u = Σ∞ k=0 uk where uk = fk (|x|)ϕk (x) and (ϕk (x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(N + k − 2), k ≥ 0. The functions fk belong to u ∈ C ∞ ([0, R]), fk (R) = 0, and satisfy fk (r) = O(rk ) and f (r) = O(rk−1 ) as r → 0. In particular, ϕ0 = 1 and f0 =  1 nωn r n−1  ∂Br  uds =  1 nωn  |x|=1  u(rx)ds.  (4.23)  We also have for any k ≥ 0, and any continuous real valued functions v and w on (0, ∞), V (|x|)|∆uk |2 dx =  fk (|x|) |x|2  V (|x|) ∆fk (|x|) − ck  Rn  Rn  2  dx,  (4.24)  and W (|x|)|∇uk |2 dx = Rn  W (|x|)|∇fk |2 dx + ck Rn  Rn  W (|x|)|x|−2 fk2 dx.  (4.25)  Let V and W be positive radial C 1 -functions on a ball B\{0}, where B is a R ball with radius R in n (n ≥ 1) and centered at zero. Assume 0 rn−11V (r) dr = ∞ and limr→0 rα V (r) = 0 for some α < (n − 2). If W (r) −  2V (r) 2Vr (r) + − Vrr (r) ≥ 0 f or 0 ≤ r ≤ R, r2 r  (4.26)  and the ordinary differential equation (BV,W ) has a positive solution φ on the interval (0, R] such that (n − 1 + R  φ (R) )V (R) ≥ 0, φ(R)  (4.27)  then the following inequality holds for all u ∈ H 2 (B). (HRV,W )  B  V (x)|∆u|2 dx ≥  B  W (x)|∇u|2 dx + (n − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| .  Moreover, if β(V, W ; R) ≥ 1, then the best constant is given by β(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 ∈ C0∞ (B) by frequently using that R  R  V (r)|x (r)|2 rn−1 dr  W (r)x2 (r)rn−1 dr  ≥  0  0  + V (R)  φ (R) n−1 R (x(R))2 , φ(R)  (4.29)  for all x ∈ C 1 (0, R]. Indeed, for all n ≥ 1 and k ≥ 0 we have 1 nwn  V (x)|∆uk |2 dx = Rn  R  =  V (r) fk (r) + 0  1 nwn  V (x) ∆fk (|x|) − ck Rn  n−1 fk (r) fk (r) − ck 2 r r  R  r  2  dx  dr  R  V (r)(fk (r))2 rn−1 dr + (n − 1)2  =  2 n−1  fk (|x|) |x|2  0 R  +c2k  V (r)(fk (r))2 rn−3 dr 0 R  V (r)fk2 (r)rn−5 + 2(n − 1) 0  V (r)fk (r)fk (r)rn−2 0  R  R  V (r)fk (r)fk (r)rn−3 dr − 2ck (n − 1)  −2ck 0  V (r)fk (r)fk (r)rn−4 dr. 0  Integrate by parts and use (4.23) for k = 0 to get 1 nωn  R  V (x)|∆uk |2 dx = Rn  V (r)(fk (r))2 rn−1 dr 0  R  +  V (r)(fk (r))2 rn−3 dr  (n − 1 + 2ck ) 0  R  +  (2ck (n − 4) + c2k )  R  V (r)rn−5 fk2 (r)dr − (n − 1) 0  0  R  R  Vr (r)fk2 (r)rn−4 dr − ck  − ck (n − 5) 0  +  Vr (r)rn−2 (fk )2 (r)dr  Vrr (r)fk2 (r)rn−3 dr. 0  (n − 1)V (R)(fk (R))2 Rn−2  94  4.3. General Hardy-Rellich inequalities fk (r) r  Now define gk (r) =  and note that gk (r) = O(rk−1 ) for all k ≥ 1. We have R  R  R  V (r)(fk (r))2 rn−3  2V (r)gk (r)gk (r)rn−2 dr  V (r)(gk (r))2 rn−1 dr +  =  0  0  0  R  V (r)gk2 (r)rn−3 dr  + 0 R  R  V (r)(gk (r))2 rn−1 dr − (n − 3)  = 0  V (r)gk2 (r)rn−3 dr 0  R  Vr (r)gk2 (r)rn−2 dr  − 0  Thus, R  R  V (r)(fk (r))2 rn−3  W (r)fk2 (r)rn−3 dr  ≥  0  (4.30)  0 R  R  V (r)fk2 (r)rn−5 dr −  − (n − 3) 0 R 0  2 n−3  Substituting 2ck V (r)(fk (r)) r inequality (4.30), we get  Vr (r)fk2 (r)rn−4 dr. 0  in (4.30) by its lower estimate in the last R  1 nωn  V (x)|∆uk |2 dx ≥ Rn  W (r)(fk (r))2 rn−1 dr 0  R  W (r)(fk (r))2 rn−3 dr  + 0  R  +  R  V (r)(fk (r))2 rn−3 dr + ck (n − 1)  (n − 1) 0  V (r)(fk (r))2 rn−5 dr 0  R  R  Vr (r)rn−2 (fk )2 (r)dr − ck (n − 1)  − (n − 1) 0  Vr (r)rn−4 (fk )2 (r)dr 0  R  V (r)rn−5 fk2 (r)dr  + ck (ck − (n − 1)) 0 R  2V (r) 2Vr (r) + − Vrr (r))fk2 (r)rn−3 dr 2 r r 0 φ (R) n−1 (n − 1)V (R)(fk (R))2 Rn−2 + V (R) R (fk (R))2 φ(R)  + ck +  (W (r) −  The proof is now complete since the last two terms are non-negative by our assumptions. Remark 4.3.1. In order to apply the above theorem to V (x) = |x|−2m  95  4.3. General Hardy-Rellich inequalities 2 −2 we see that even in the simplest case V ≡ 1 condition (4.26) reduces to ( n−2 ≥ 2 ) |x| −2 −2m 2|x| , which is then guaranteed only if n ≥ 5. More generally, if V (x) = |x| , then in order to satisfy (4.26) we need to have √ √ −(n + 4) + 2 n2 − n + 1 −(n + 4) − 2 n2 − n + 1 ≤m≤ . (4.31) 6 6  Also to satisfy the condition (4.27) we need to have m > − n2 . Thus for m satisfying (4.31) the inequality  BR  n + 2m 2 |∆u|2 ≥( ) 2m |x| 2  BR  |∇u|2 dx. |x|2m+2  (4.32)  2 for all u ∈ H 2 (BR ). Moreover, ( n+2m 2 ) is the best constant. We shall see however that this inequality remains true without condition (4.31), but with a constant 2 that is sometimes different from ( n+2m 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 25 36 in dimension 3.  The case of power potentials |x|m  4.3.2  The general Theorem 4.3.1 allowed us to deduce inequality (4.36) below for a restricted interval of powers m. We shall now prove that the same holds for all − n2 ≤ m < n−2 2 . We start with the following result. n Assume − n2 ≤ m < n−2 2 and Ω be a smooth domain in , n ≥ 1. Then   |∆u|2   dx 2m BR |x| an,m = inf ; H 2 (Ω) \ {0} = 2 |∇u|   dx 2m+2 |x| BR   |∆u|2   dx 2m BR |x| 2 inf ; u ∈ H (Ω) \ {0} 0 |∇u|2   dx BR |x|2m+2 ¯R ) into spherical harmonics; u = Σ∞ uk , Proof. Decomposing again u ∈ C ∞ (B k=0 where uk = fk (|x|)ϕk (x), one has  n  |∆uk |2 dx = |x|2m +  n  |x|−2m (fk (|x|))2 dx  ((n − 1)(2m + 1) + 2ck ) n  |x|−2m−2 (fk )2 dx |x|−2m−4 (fk )2 dx  + ck (ck + (n − 4 − 2m)(2m + 2)) n  +  (n − 1)Rn−2m−2 (fk (R))2 ,  (4.34)  96  4.3. General Hardy-Rellich inequalities  and  n  |∇uk |2 dx = |x|2m+2  n  |x|−2m−2 (fk )2 dx + ck  |x|−2m−4 (fk )2 dx.  (4.35)  n  The rest of the proof follows from the inequality (4.13) and an argument similar to that of Theorem 6.1 in [15]. Remark 4.3.2. The constant an,m has been computed explicitly in [15] (Theorem 6.1). n Suppose n ≥ 1 and − n2 ≤ m < n−2 2 , and W is a Bessel potential on BR ⊂ R with n ≥ 3 and φ is the corresponding solution for the (B1,W ). If  R  n φ (R) ≥ − − m, φ(R) 2  then for all u ∈ H 2 (BR ) we have  BR  |∆u|2 ≥ an,m |x|2m  BR  where an,m = inf  |∇u|2 dx + β(W ; R) |x|2m+2      |∆u|2 dx BR |x|2m ; |∇u|2 dx BR |x|2m+2  W (x) BR  |∇u|2 dx, |x|2m  (4.36)    u ∈ H 2 (BR ) \ {0} .   Moreover β(W ; R) and am,n are the best constants. Proof: Assuming the inequality |∇u|2 |∆u|2 ≥ a dx, n,m 2m 2m+2 BR |x| BR |x| ¯R ), we shall prove that it can be improved by any Bessel holds for all u ∈ C ∞ (B potential W . We will use the following inequality in the proof which follows directly from the inequality (4.13) with n=1. R  rα (f (r))2 dr 0  ≥ (  α−1 2 ) 2  R  rα−2 f 2 (r)dr 0 R  rα W (r)f 2 (r)dr + (  + β(W ; R) 0  (4.37) φ (R) α − 1 α − )R , φ(R) 2R  2 for α ≥ 1 and for all f ∈ C ∞ (0, R], where both ( α−1 2 ) and β(W ; R) are best ∞ ¯ constants. Decompose u ∈ C (BR ) into its spherical harmonics Σ∞ k=0 uk , where  97  4.3. General Hardy-Rellich inequalities uk = fk (|x|)ϕk (x). We evaluate Ik =  |∆uk |2 dx Rn |x|2m  1 nwn  in the following way  R  Ik  R  rn−2m−1 (fk (r))2 dr + [(n − 1)(2m + 1) + 2ck ]  = 0  rn−2m−3 (fk )2 dr 0  R  rn−2m−5 (fk (r))2 dr  +ck [ck + (n − 2m − 4)(2m + 2)] 0  +  (n − 1)Rn−2m−2 (fk (R))2 R  rn−2m−1 W (x)(fk )2 dr + [(  ≥ β(W ) 0  n + 2m 2 ) + 2ck ] 2  R  rn−2m−3 (fk )2 dr 0  R  rn−2m−5 (fk (r))2 dr  +ck [ck + (n − 2m − 4)(2m + 2)] 0 R  R  rn−2m−1 W (x)(fk )2 dr + an,m  ≥ β(W ) 0  rn−2m−3 (fk )2 dr 0  R n + 2m 2 ) + 2ck − an,m ] rn−2m−3 W (x)(fk )2 dr 2 0 n − 2m − 4 2 n + 2m 2 ) [( ) + 2ck − an,m ] + ( 2 2  +β(W )[(  R  rn−2m−5 (fk (r))2 dr.  + ck [ck + (n − 2m − 4)(2m + 2)] 0  Now by (115) in [15] we have n − 2m − 4 2 n + 2m 2 ) [( ) + 2ck − an,m ] + ck [ck + (n − 2m − 4)(2m + 2)] ≥ ck an,m , 2 2 for all k ≥ 0. Hence, we have (  R  Ik  R  rn−2m−3 (fk )2 dr + an,m ck  ≥ an,m 0  rn−2m−5 (fk (r))2 dr 0  R  rn−2m−1 W (x)(fk )2 dr  +β(W ) 0  +β(W )[(  n + 2m 2 ) + 2ck − an,m ] 2  R  rn−2m−3 W (x)(fk )2 dr 0  R  R  rn−2m−3 (fk )2 dr + an,m ck  ≥ an,m 0  rn−2m−5 (fk (r))2 dr 0  R  R  rn−2m−1 W (x)(fk )2 dr + β(W )ck  +β(W ) 0  = an,m BR  |∇u|2 dx + β(W ) |x|2m+2  rn−2m−3 W (x)(fk )2 dr 0  W (x) BR  |∇u|2 dx. |x|2m  In the following theorem we prove a very general class of weighted Hardy-Rellich inequalities on H 2 (Ω) ∩ H01 .  98  4.3. General Hardy-Rellich inequalities Let Ω be a smooth domain in Rn with n ≥ 1 and let V ∈ C 2 (0, R =: supx∈Ω |x|) be a non-negative function that satisfies the following conditions: Vr (r) ≤ 0  R 1 dr 0 r n−3 V (r)  and  R 1 dr 0 r n−4 Vr (r)  =−  = +∞.  (4.38)  There exists λ1 , λ2 ∈ R such that rVr (r) V (r)  + λ1 ≥ 0 on (0, R) and lim  rVr (r) r→0 V (r)  rVrr (r) Vr (r)  + λ2 ≥ 0 on (0, R) and lim  + λ1 = 0,  rVrr (r) r→0 Vr (r)  (4.39)  + λ2 = 0,  (4.40)  and 1 2 (n  − λ1 − 2)2 + 3(n − 3) V (r) − (n − 5)rVr (r) − r2 Vrr (r) ≥ 0 for all r ∈ (0, R). (4.41) If λ1 ≤ n, then the following inequality holds: (n − λ1 − 2)2 (n − λ1 − 4)2 + (n − 1)) 4 4 2 (n − 1)(n − λ2 − 2) Vr (|x|) 2 − u dx. 3 4 Ω |x|  V (|x|)|∆u|2 dx ≥ ( Ω  Ω  V (|x|) 2 u dx |x|4 (4.42)  Proof: We have by Theorem 4.2 and condition (4.41), R  1 nωn  V (x)|∆uk |2 dx = Rn  V (r)(fk (r))2 rn−1 dr 0  R  +  V (r)(fk (r))2 rn−3 dr  (n − 1 + 2ck ) 0  R  +  (2ck (n − 4) + c2k )  R  V (r)rn−5 fk2 (r)dr − (n − 1) 0  R  R  Vr (r)fk2 (r)rn−4 dr − ck  − ck (n − 5) 0  +  Vr (r)rn−2 (fk )2 (r)dr 0  Vrr (r)fk2 (r)rn−3 dr 0  (n − 1)V (R)(fk (R))2 Rn−2 R  R  V (r)(fk (r))2 rn−1 dr + (n − 1)  ≥ 0  V (r)(fk (r))2 rn−3 dr 0  R  Vr (r)rn−2 (fk )2 (r)dr  − (n − 1) 0 R  + ck 0  +  1 (n − λ1 − 2)2 + 3(n − 3) V (r) − (n − 5)rVr (r) − r2 Vrr (r) fk2 (r)rn−5 dr 2  (n − 1)V (R)(fk (R))2 Rn−2  The rest of the proof follows from the above inequality combined with Theorem 4.2.  99  4.3. General Hardy-Rellich inequalities Remark 4.3.3. Let V (r) = r−2m with − n2 ≤ m ≤ n−4 to satisfy 2 . Then in order √ √ 2 1+(n−1)2 1+(n−1) ≤ m ≤ n−4 ≤ condition (4.41) we must have −1− 2 2 . Since −1− 2 the inequality (4.42) gives the following weighted second − n2 , if − n2 ≤ m ≤ n−4 2 order Rellich inequality:  B  |∆u|2 dx ≥ Hn,m |x|2m  u2 B  |x|2m+4  dx u ∈ H 2 (Ω) ∩ H01 (Ω),  where  (n + 2m)(n − 4 − 2m) 2 ) . (4.43) 4 The following theorem includes a large class of improved Hardy-Rellich inequalities as special cases. Let − n2 ≤ m ≤ n−4 2 and let W (x) be a Bessel potential on a ball B of radius R in W (r) n R with radius R. Assume W = − λr + f (r), where f (r) ≥ 0 and limr→0 rf (r) = r (r) 0. If λ ≤ n2 + m, then the following inequality holds for all u ∈ H 2 ∩ H01 (B) Hn,m := (  B  |∆u|2 dx ≥ Hn,m |x|2m  u2 B  |x|2m+4  + β(W ; R)(  dx  (4.44)  (n − 2m − λ − 2)2 (n + 2m)2 + ) 4 4  B  W (x) 2 u dx. |x|2m+2  Moreover, both constants are the best constants. Proof: Again we will frequently use inequality (4.37) in the proof. Decompos¯R ) into spherical harmonics Σ∞ uk , where uk = fk (|x|)ϕk (x), we ing u ∈ C ∞ (B k=0 can write 1 nωn  Rn  R  |∆uk |2 dx = |x|2m  rn−2m−1 (fk (r))2 dr 0 R  +  rn−2m−3 (fk )2 dr  [(n − 1)(2m + 1) + 2ck ] 0  R  rn−2m−5 (fk (r))2 dr  + ck [ck + (n − 2m − 4)(2m + 2)] 0  +  (n − 1)(fk (R))2 Rn−2m−2  ≥ (  n + 2m 2 ) 2  R  rn−2m−3 (fk )2 dr 0 R  rn−2m−1 W (x)(fk )2 dr  + β(W ; R) 0  n−λ−4 2 + ck [ck + 2( ) 2 R  + +  rn−2m−5 (fk (r))2 dr  (n − 2m − 4)(2m + 2)] (n − 1)(fk (R))2 R  0 n−2m−2  ,  100  4.3. General Hardy-Rellich inequalities where we have used the fact that ck ≥ 0 to get the above inequality. We have 1 nωn  Rn  |∆uk |2 dx ≥ βn,m |x|2m  (n + 2m)2 +β(W ; R) 4  R  rn−2m−5 (fk )2 dr 0  R  rn−2m−3 W (x)(fk )2 dr 0  R  rn−2m−1 W (x)(fk )2 dr  +β(W ; R) 0 R  rn−2m−5 (fk )2 dr  ≥ βn,m 0  (n + 2m)2 (n − 2m − λ − 2)2 + ) 4 4 βn,m u2k ≥ dx nωn B |x|2m+4 β(W ; R) (n + 2m)2 (n − 2m − λ − 2)2 + + ) ( nωn 4 4  R  rn−2m−3 W (x)(fk )2 dr  +β(W ; R)(  0  B  W (x) 2 u dx, |x|2m+2 k  by Theorem 4.2. Hence, (4.44) holds and the proof is complete. We shall now give a few immediate applications of the above in the case where m = 0 and n ≥ 3. Assume W is a Bessel potential on BR ⊂ Rn with n ≥ 3 and φ is the corresponding solution for the (B1,W ). If R  n φ (R) ≥− , φ(R) 2  then for all u ∈ H 2 (BR ) we have |∆u|2 dx ≥ C(n) BR  BR  |∇u|2 dx + β(W ; R) |x|2  25 where C(3) = 36 , C(4) = 3 and C(n) = β(W ; R) are best constants.  n2 4  W (x)|∇u|2 dx,  (4.45)  BR  for all n ≥ 5. Moreover, C(n) and  The following holds for any smooth bounded domain Ω in Rn with R = supx∈Ω |x|, and any u ∈ H 2 (Ω). 1. Let z0 be the first zero of the Bessel function J0 (z) and choose 0 < µ < z0 so that µ  J0 (µ) n =− . J0 (µ) 2  Then Ω  |∆u|2 dx ≥ C(n)  |∇u|2 dx Ω |x|2  +  (4.46) µ2 R2  Ω  |∇u|2 dx  (4.47)  101  4.3. General Hardy-Rellich inequalities ee  ..  e(k−times)  (R) ≥ ) large enough so that R φφ(R)  2. For any k ≥ 1, choose ρ ≥ R(e − n2 , where  j  log (i)  φ=  ρ |x|  1 2  .  (4.48)  (4.49)  Ω  |∇u|2 dx |x|2  Ω  |∇u|2 |x|2  i=1  Then we have |∆u(x)|2 dx ≥ C(n) Ω  1 4  +  k  j=1  j  log (i) i=1  ρ |x|  −2  dx,  3. We have  Ω  |∇u|2 dx |x|2  Ω  |x| |∇u| 2 |x| 2 |x| X ( )X2 ( )...Xi2 ( )dx. |x|2 1 R R R  |∆u(x)|2 dx ≥ C(n) Ω  1 4  +  n  i=1  (4.50)  The following is immediate from Theorem 4.3.2 and from the fact that λ = 2 for the Bessel potential under consideration. Let Ω be a smooth bounded domain in n , n ≥ 4 and R = supx∈Ω |x|. Then the following holds for all u ∈ H 2 (Ω) ∩ H01 (Ω) ..  e(k−times)  ee  (R) ) so that R φφ(R) ≥ − n2 . Then  1. Choose ρ ≥ R(e  |∆u(x)|2 dx ≥ Ω  n2 (n − 4)2 16  Ω  u2 dx |x|4 k  +  (1 +  n(n − 4) ) 8 j=1  Ω  (4.51) u2 |x|4  j  log (i) i=1  ρ |x|  −2  dx.  2. Let Xi is defined as in the introduction, then |∆u(x)|2 dx ≥ Ω  n2 (n − 4)2 16  Ω  u2 dx |x|4 n  +  (1 +  n(n − 4) ) 8 i=1  Ω  (4.52) u2 2 |x| 2 |x| |x| X ( )X2 ( )...Xi2 ( )dx. |x|4 1 R R R  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 ≥ 4. Then for all u ∈ H 2 (B) ∩ H01 (B) |∆u|2 dx ≥ B  n2 (n − 4)2 16 n−2 2 +µ( ) 2  n2 u2 dx + β(W1 ; R) 4 4 B |x| u2 dx + µβ(W2 ; R) 2 B |x|  W1 (x) B  u2 dx |x|2  W2 (x)u2 dx, B  where µ is defined by (4.46). Proof: Here again we shall give a proof when n ≥ 5. The case n = 4 will be handled in the next section. We again first use Theorem 4.3.2 (for n ≥ 5), then Theorem 2.15 in [15] with the Bessel pair 2 2 −2 (|x|−2 , |x|−2 ( (n−4) |x|−2 +W )), then again Theorem 4.2 with the Bessel pair (1, ( n−2 + 4 2 ) |x| W ) to obtain |∆u|2 dx ≥ B  ≥  |∇u|2 n2 dx + µ |∇u|2 dx 4 B |x|2 B n2 (n − 4)2 u2 n2 β(W1 ; R) dx + 4 16 4 B |x|  W1 (x)  u2 dx |x|2  W1 (x)  u2 dx |x|2  B  |∇u|2 dxdx  +µ B  ≥  n2 (n − 4)2 16 n−2 2 ) +µ( 2  u2 n2 dx + β(W1 ; R) 4 4 B |x| u2 dx + µβ(W2 ; R) 2 B |x|  B  W2 (x)u2 dx. B  Assume n ≥ 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 ∈ H 2 (B) ∩ H01 (B): |∆u|2 dx ≥ B  where  µ2 R2  n2 (n − 4)2 u2 dx 4 16 B |x| n2 W (x) 2 µ2 +β(W ; R) u dx + ||u||H01 , 4 B |x|2 R2  is defined by (4.46).  (4.53) (4.54)  ¯R ) into its Proof: Decomposing again u ∈ C ∞ (B  103  4.3. General Hardy-Rellich inequalities spherical harmonics Σ∞ k=0 uk where uk = fk (|x|)ϕk (x), we calculate 1 nωn  R  |∆uk |2 dx = Rn  R  rn−1 (fk (r))2 dr + [n − 1 + 2ck ] 0  rn−3 (fk )2 dr 0  R  + ck [ck + n − 4]  rn−5 (fk (r))2 dr 0 2 n−2m−2  +  (n − 1)(fk (R)) R  ≥  n2 4  R  rn−3 (fk )2 dr + 0 R  µ2 R2  R  rn−1 (fk )2 dr 0  rn−3 (fk )2 dr  + ck 0  ≥  n2 (n − 4)2 16  R  rn−5 (fk )2 dr 0  n2 +β(W ; R) 4 + = +  R  W (r)rn−3 (fk )2 dr 0  µ2 R n−1 µ2 R n−3 2 r (f ) dr + c r (fk )2 dr k k R2 0 R2 0 n2 (n − 4)2 u2k dx 2m+4 16nωn Rn |x| β(W ; R) n2 W (x) 2 µ2 u dx + ||uk ||W 1,2 . ( ) k 2 0 nωn 4 nωn R2 Rn |x|  Hence (4.53) holds.  104  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 inequalities, maximum principles and related eigenvalue problems, J. Funct. Anal. 240 (2006), 36-83. [3] G. 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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:  λf (x) 2 2 ✑ in Ω   α∆ u = β Ω |∇u| dx + γ ∆u + (1−u)2 ✏1+χ ❘ dx Ω (1−u)2 (5.1) 0<u<1 in Ω   u = α∂ν u = 0 on ∂Ω, where α, β, γ, χ ≥ 0, f ∈ C(Ω, [0, 1]) are fixed, Ω is a bounded domain in RN and λ ≥ 0 is a varying parameter (see for example Bernstein and Pelesko [19]). The function u(x) denotes the height above a point x ∈ Ω ⊂ RN of a dielectric membrane clamped on ∂Ω, once it deflects torwards a ground plate fixed at height z = 1, whenever a positive voltage – proportional to λ – is applied. In studying this problem, one typically makes various simplifying assumptions on the parameters α, β, γ, χ, and the first approximation of (5.1) that has been studied extensively so far is the equation  f (x)  −∆u = λ (1−u)2 in Ω in Ω (S)λ,f  0<u<1 u=0 on ∂Ω, where we have set α = β = χ = 0 and γ = 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 “permittivity profile” f is constant (f = 1) on a general domain was studied in [16], following the pioneering work of 4  A 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 λ∗ (Ω, f ), defined as λ∗ (Ω, f ) := sup λ > 0 : there exists a classical solution of (S)λ,f . It is then shown that for every 0 < λ < λ∗ , there exists a smooth minimal (smallest) solution of (S)λ,f , while for λ > λ∗ there is no solution even in a weak sense. Moreover, the branch λ → uλ (x) is increasing for each x ∈ Ω, and therefore the function u∗ (x) := limλ λ∗ uλ (x) can be considered as a generalized solution that corresponds to the pull-in voltage λ∗ . Now the issue of the regularity of this extremal solution – which, by elliptic regularity theory, is equivalent to whether supΩ u∗ < 1 – 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∗ , λ∗ ). This issue turned out to depend closely on the dimension and on the permittivity profile f . Indeed, it was shown in [9] that u∗ is regular in dimensions 1 ≤ N ≤ 7, while it is not necessarily the case for N ≥ 8. In other words, the dimension N = 7 is critical for equation (S)λ (when f = 1, we simplify the notation (S)λ,1 into (S)λ ). On the other hand, it is shown in [8] that the regularity of u∗ can be restored in any dimension, provided we allow for a power law profile |x|η with η large enough. The case where β = γ = χ = 0 (and α = 1) in the above model, that is when we are dealing with the following fourth order analog of (S)λ  2 λ  ∆ u = (1−u)2 in Ω 0<u<1 in Ω (P )λ  u = ∂ν u = 0 on ∂Ω, was also considered by [4] and [14] but with limited success. One of the reasons is the lack of a “maximum principle” 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 Ω (at least for the clamped boundary conditions u = ∂ν u = 0 on ∂Ω) unless one restricts attention to the unit ball Ω = B in RN , where one can exploit a positivity preserving property of ∆2 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 )λ is developed along the same lines as for (S)λ . 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)λ could not carry to the corresponding problem for (P )λ . 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∗ for (P )λ∗ in B is regular in dimension 1 ≤ N ≤ 8, while it is singular (i.e, supB u∗ = 1) for N ≥ 9. In other words, the critical dimension for (P )λ in B is N = 8, as opposed to being equal to 7 in (S)λ . 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 ∆2 u = λeu in B u = ∂ν u = 0 on ∂B, while the critical dimension for its second order counterpart (i.e., the Gelfand problem) 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 – just above the critical one – 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 )λ on the unit ball B. We start by recalling some of the results from [4] concerning (P )λ , that will be needed in the sequel. We define λ∗ := sup λ > 0 : there exists a classical solution of (P )λ , 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 )λ if 0 ≤ u ≤ 1 a.e. in B, 1 1 (1−u)2 ∈ L (B) and u∆2 φ = λ B  B  φ , (1 − u)2  ¯ ∩ H02 (B). ∀φ ∈ C 4 (B)  We say that u is a weak super-solution (resp. weak sub-solution) of (P )λ if the ¯ ∩ H 2 (B) with equality is replaced with the inequality ≥ (resp. ≤) for all φ ∈ C 4 (B) 0 φ ≥ 0. We also introduce notions of regularity and stability. Definition 5.4. Say that a weak solution u of (P )λ is regular (resp. singular) if u ∞ < 1 (resp. =) and stable (resp. semi-stable) if (∆φ)2 − 2λ  µ1 (u) = inf B  B  φ2 : φ ∈ H02 (B), φ (1 − u)3  L2  =1  is positive (resp. non-negative).  110  5.1. Introduction The following extension of Boggio’s principle will be frequently used in the sequel (see [2, Lemma 16] and [5, Lemma 2.4]): Lemma 5.5 (Boggio’s Principle). Let u ∈ L1 (B). Then u ≥ 0 a.e. in B, provided one of the following conditions hold: 1. u ∈ C 4 (B), ∆2 u ≥ 0 on B, and u = 2.  2  B  ∂u ∂n  = 0 on ∂B.  4  u∆ φ dx ≥ 0 for all 0 ≤ φ ∈ C (B) ∩ H02 (B).  3. u ∈ H 2 (B), u = 0 and  ∂u ∂n  ≤ 0 on ∂B, and  B  ∆u∆φ ≥ 0 for all 0 ≤ φ ∈ H02 (B).  Moreover, either u ≡ 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 < λ < λ∗ there exists a classical minimal solution uλ of (P )λ . Moreover uλ is radial and radially decreasing. 2. For λ > λ∗ , there are no weak solutions of (P )λ . 3. For each x ∈ B the map λ → uλ (x) is strictly increasing on (0, λ∗ ). 4. The pull-in voltage λ∗ satisfies the following bounds: max  2 8 32(10N − N 2 − 12) 8 , (N − )(N − ) 27 9 3 3  ≤ λ∗ ≤  4ν1 27  where ν1 denotes the first eigenvalue of ∆2 in H02 (B). 5. For each 0 < λ < λ∗ , uλ is a stable solution (i.e., µ1 (uλ ) > 0). Using the stability of uλ , it can be shown that uλ is uniformly bounded in H02 (B) 1 and that 1−u is uniformly bounded in L3 (B). Since now λ → uλ (x) is increasing, λ ∗ the function u (x) := limλ λ∗ uλ (x) is well defined (in the pointwise sense), u∗ ∈ 1 3 ∗ H02 (B), 1−u is a weak solution of (P )λ∗ . Moreover u∗ is the ∗ ∈ L (B) and u unique weak solution of (P )λ∗ . The second result we list from [4] is critical in identifying the extremal solution. Theorem 5.7. If u ∈ H02 (B) is a singular weak solution of (P )λ , then u is semistable if and only if (u, λ) = (u∗ , λ∗ ).  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 )λ with non-homogeneous boundary conditions such as  2 λ in B  ∆ u = (1−u)2 α<u<1 in B (P )λ,α,β  u = α , ∂ν u = β on ∂B, where α, β are given. Notice first that some restrictions on α and β are necessary. Indeed, letting Φ(x) := (α − β2 ) + β2 |x|2 denote the unique solution of ∆2 Φ = 0 Φ = α , ∂ν Φ = β  in B on ∂B,  (5.8)  we infer immediately from Lemma 5.5 that the function u − Φ is positive in B, which yields to sup Φ < sup u ≤ 1. B  B  To insure that Φ is a classical sub-solution of (P )λ,α,β , we impose α = 1 and β ≤ 0, and condition sup Φ < 1 rewrites as α − β2 < 1. We will then say that the pair B  (α, β) is admissible if β ≤ 0, and α −  β 2  < 1.  This section will be devoted to obtaining results for (P )λ,α,β when (α, β) is an admissible pair, which are analogous to those for (P )λ . To cut down on notation, we shall sometimes drop α and β from our expressions whenever such an emphasis is not needed. For example in this section uλ and u∗ will denote the minimal and extremal solution of (P )λ,α,β . We now introduce a notion of weak solution for (P )λ,α,β . Definition 5.9. We say that u is a weak solution of (P )λ,α,β if α ≤ u ≤ 1 a.e. in 1 1 B, (1−u) 2 ∈ L (B) and if (u − Φ)∆2 φ = λ B  B  φ , (1 − u)2  ¯ ∩ H02 (B), ∀φ ∈ C 4 (B)  where Φ is given in (5.8). We say that u is a weak super-solution (resp. weak sub-solution) of (P )λ,α,β if the equality is replaced with the inequality ≥ (resp. ≤) for φ ≥ 0. We now define as before λ∗ := sup{λ > 0 : (P )λ,α,β has a classical solution}  112  5.2. The effect of boundary conditions on the pull-in voltage and λ∗ := sup{λ > 0 : (P )λ,α,β has a weak solution}. Observe that by the Implicit Function Theorem, one can always solve (P )λ,α,β for small λ’s. Therefore, λ∗ (and also λ∗ ) is well defined. Let now U be a weak super-solution of (P )λ,α,β . Recall the following standard existence result. Theorem 5.10 ([2]). For every 0 ≤ f ∈ L1 (B), there exists a unique 0 ≤ u ∈ L1 (B) which satisfies u∆2 φ = B  fφ B  ¯ ∩ H 2 (B). for all φ ∈ C 4 (B) 0 We can now introduce the following “weak iterative scheme”: Start with u0 = U and (inductively) let un , n ≥ 1, be the solution of (un − Φ)∆2 φ = λ B  B  φ (1 − un−1 )2  ¯ ∩ H02 (B) ∀ φ ∈ C 4 (B)  given by Theorem 5.10. Since 0 is a sub-solution of (P )λ,α,β , one can easily show inductively by using Lemma 5.5 that α ≤ un+1 ≤ un ≤ U for every n ≥ 0. Since (1 − un )−2 ≤ (1 − U )−2 ∈ L1 (B), we get by Lebesgue Theorem, that the function u = lim un is a weak solution of n→+∞  (P )λ,α,β such that α ≤ u ≤ U . In other words, the following result holds. Proposition 5.2.1. Assume the existence of a weak super-solution U of (P )λ,α,β . Then there exists a weak solution u of (P )λ,α,β so that α ≤ u ≤ U a.e. in B. In particular, we can find a weak solution of (P )λ,α,β for every λ ∈ (0, λ∗ ). Now we show that this is still true for regular weak solutions. Proposition 5.2.2. Let (α, β) be an admissible pair and let u be a weak solution of (P )λ,α,β . Then for every 0 < µ < λ, there is a regular solution for (P )µ,α,β . Proof. Let ε ∈ (0, 1) be given and let u ¯ = (1 − ε)u + εΦ, where Φ is given in (5.8). We have that sup u ¯ ≤ (1 − ε) + ε sup Φ < 1 , B  B  inf u ¯ ≥ (1 − ε)α + ε inf Φ = α, B  B  ¯ ∩ H 2 (B) there holds: and for every 0 ≤ φ ∈ C 4 (B) 0 (¯ u − Φ)∆2 φ =  (u − Φ)∆2 φ = (1 − ε)λ  (1 − ε)  B  B  =  B  3  (1 − ε) λ B  φ (1 − u)2  φ ≥ (1 − ε)3 λ (1 − u ¯ + ε(Φ − 1))2  B  φ . (1 − u ¯)2  113  5.2. The effect of boundary conditions on the pull-in voltage Note that 0 ≤ (1−ε)(1−u) = 1− u ¯ +ε(Φ−1) < 1− u ¯. So u ¯ is a weak super-solution of (P )(1−ε)3 λ,α,β satisfying sup u ¯ < 1. From Proposition 5.2.1 we get the existence B  of a weak solution w of (P )(1−ε)3 λ,α,β so that α ≤ w ≤ u ¯. In particular, sup w < 1 B  and w is a regular weak solution. Since ε ∈ (0, 1) is arbitrarily chosen, the proof is complete. Proposition 5.2.2 implies in particular the existence of a regular weak solution Uλ for every λ ∈ (0, λ∗ ). Introduce now a “classical” iterative scheme: u0 = 0 and (inductively) un = vn + Φ, n ≥ 1, where vn ∈ H02 (B) is the (radial) solution of ∆2 vn = ∆2 (un − Φ) =  λ (1 − un−1 )2  in B.  (5.11)  Since vn ∈ H02 (B), un is also a weak solution of (5.11), and by Lemma 5.5 we know that α ≤ un ≤ un+1 ≤ Uλ for every n ≥ 0. Since sup un ≤ sup Uλ < 1 for n ≥ 0, we B  B  get that (1 − un−1 )−2 ∈ L2 (B) and the existence of vn is guaranteed. Since vn is easily seen to be uniformly bounded in H02 (B), we have that uλ := lim un does n→+∞  hold pointwise and weakly in H 2 (B). By Lebesgue Theorem, we have that uλ is a radial weak solution of (P )λ,α,β so that sup uλ ≤ sup Uλ < 1. By elliptic regularity B  B  ¯ and uλ − Φ = ∂ν (uλ − Φ) = 0 on ∂B. So we can integrate theory [1] uλ ∈ C ∞ (B) by parts to get ∆2 u λ φ = B  ∆2 (uλ − Φ)φ = B  (uλ − Φ)∆2 φ = λ B  B  φ (1 − uλ )2  ¯ ∩ H 2 (B). Hence, uλ is a radial classical solution of (P )λ,α,β for every φ ∈ C (B) 0 ∗ showing that λ = λ∗ . Moreover, since Φ and vλ := uλ − Φ are radially decreasing in view of [20], we get that uλ is radially decreasing too. Since the argument above shows that uλ < U for any other classical solution U of (P )µ,α,β with µ ≥ λ, we have that uλ is exactly the minimal solution and uλ is strictly increasing as λ ↑ λ∗ . In particular, we can define u∗ in the usual way: u∗ (x) = lim∗ uλ (x). 4  λ  λ  Finally, we show the finiteness of the pull-in voltage. Lemma 5.12. If (α, β) is an admissible pair, then λ∗ (α, β) < +∞. Proof. Let u be a classical solution of (P )λ,α,β and let (ψ, ν1 ) denote the first eigenpair of ∆2 in H02 (B) with ψ > 0. Now, let C be such that (β∆ψ − α∂ν ∆ψ) = C ∂B  ψ. B  Multiplying (P )λ,α,β by ψ and then integrating by parts one arrives at  B  λ − ν1 u − C ψ = 0. (1 − u)2  114  5.2. The effect of boundary conditions on the pull-in voltage λ Since ψ > 0 there must exist a point x ¯ ∈ B where (1−u(¯ x) − C ≤ 0. Since x))2 − ν1 u(¯ α < u(¯ x) < 1, one can conclude that λ ≤ supα<u<1 (ν1 u + C)(1 − u)2 , which shows ∗ that λ < +∞.  The following summarizes what we have shown so far. Theorem 5.13. If (α, β) is an admissible pair, then λ∗ := λ∗ (α, β) ∈ (0, +∞) and the following hold: 1. For each 0 < λ < λ∗ there exists a classical, minimal solution uλ of (P )λ,α,β . Moreover uλ is radial and radially decreasing. 2. For each x ∈ B the map λ → uλ (x) is strictly increasing on (0, λ∗ ). 3. For λ > λ∗ there are no weak solutions of (P )λ,α,β .  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 H 2 (B)−weak solutions, which is an intermediate class between classical and weak solutions. Definition 5.14. We say that u is a H 2 (B)−weak solution of (P )λ,α,β if u − Φ ∈ 1 1 H02 (B), α ≤ u ≤ 1 a.e. in B, (1−u) 2 ∈ L (B) and if ∆u∆φ = λ B  B  φ , (1 − u)2  ¯ ∩ H02 (B), ∀φ ∈ C 4 (B)  where Φ is given in (5.8). We say that u is a H 2 (B)−weak super-solution (resp. H 2 (B)−weak sub-solution) of (P )λ,α,β if for φ ≥ 0 the equality is replaced with ≥ (resp. ≤) and u ≥ α (resp. ≤), ∂ν u ≤ β (resp. ≥) on ∂B. Theorem 5.15. Suppose (α, β) is an admissible pair. 1. The minimal solution uλ is then stable and is the unique semi-stable H 2 (B)−weak solution of (P )λ,α,β . 2. The function u∗ := lim∗ uλ is a well-defined semi-stable H 2 (B)−weak solution of λ  λ  (P )λ∗ ,α,β . 3. When u∗ is classical solution, then µ1 (u∗ ) = 0 and u∗ is the unique H 2 (B)−weak solution of (P )λ∗ ,α,β . 4. If v is a singular, semi-stable H 2 (B)−weak solution of (P )λ,α,β , then v = u∗ and λ = λ∗ The crucial tool is a comparison result which is valid exactly in this class of solutions.  115  5.2. The effect of boundary conditions on the pull-in voltage Lemma 5.16. Let (α, β) be an admissible pair and u be a semi-stable H 2 (B)−weak solution of (P )λ,α,β . Assume U is a H 2 (B)−weak super-solution of (P )λ,α,β so that U − Φ ∈ H02 (B). Then 1. u ≤ U a.e. in B; 2. If u is a classical solution and µ1 (u) = 0 then U = u. Proof. (i) Define w := u − U . Then by the Moreau decomposition [17] for the biharmonic operator, there exist w1 , w2 ∈ H02 (B), with w = w1 + w2 , w1 ≥ 0 a.e., ∆2 w2 ≤ 0 in the H 2 (B)−weak sense and B ∆w1 ∆w2 = 0. By Lemma 5.5, we have that w2 ≤ 0 a.e. in B. Given now 0 ≤ φ ∈ Cc∞ (B), we have that ∆w∆φ ≤ λ B  (f (u) − f (U ))φ, B  where f (u) := (1 − u)−2 . Since u is semi-stable, one has f (u)w12 ≤  λ B  (∆w1 )2 = B  ∆w∆w1 ≤ λ B  (f (u) − f (U ))w1 . B  Since w1 ≥ w one also has f (u)ww1 ≤ B  (f (u) − f (U ))w1 , B  which once re-arranged gives f˜w1 ≥ 0, B  where f˜(u) = f (u) − f (U ) − f (u)(u − U ). The strict convexity of f gives f˜ ≤ 0 and f˜ < 0 whenever u = U . Since w1 ≥ 0 a.e. in B one sees that w ≤ 0 a.e. in B. The inequality u ≤ U a.e. in B is then established. (ii) Since u is a classical solution, it is easy to see that the infimum in µ1 (u) is 2λ attained at some φ. The function φ is then the first eigenfunction of ∆2 − (1−u) 3 2 in H0 (B). Now we show that φ is of fixed sign. Using the above decomposition, one has φ = φ1 + φ2 where φi ∈ H02 (B) for i = 1, 2, φ1 ≥ 0, B ∆φ1 ∆φ2 = 0 and ∆2 φ2 ≤ 0 in the H02 (B)−weak sense. If φ changes sign, then φ1 ≡ 0 and φ2 < 0 in B (recall that either φ2 < 0 or φ2 = 0 a.e. in B). We can write now: 0 = µ1 (u) ≤  B  <  B  (∆(φ1 − φ2 ))2 − λf (u)(φ1 − φ2 )2 (φ1 − φ2 )2 B (∆φ)2 − λf (u)φ2 = µ1 (u) φ2 B  116  5.2. The effect of boundary conditions on the pull-in voltage in view of φ1 φ2 < −φ1 φ2 in a set of positive measure, leading to a contradiction. So we can assume φ ≥ 0, and by the Boggi’s principle we have φ > 0 in B. For 0 ≤ t ≤ 1 define ∆ [tU + (1 − t)u] ∆φ − λ  g(t) = B  f (tU + (1 − t)u)φ, B  where φ is the above first eigenfunction. Since f is convex one sees that g(t) ≥ λ  [tf (U ) + (1 − t)f (u) − f (tU + (1 − t)u)] φ ≥ 0 B  for every t ≥ 0. Since g(0) = 0 and ∆(U − u)∆φ − λf (u)(U − u)φ = 0,  g (0) = B  we get that f (u)(U − u)2 φ ≥ 0.  g (0) = −λ B  Since f (u)φ > 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 (α, β) be an admissible pair and β ≤ 0. Let u be a semi-stable H 2 (B)−weak sub-solution of (P )λ,α,β with u = α, ∂ν u = β ≥ β on ∂B. Assume that U is a H 2 (B)−weak super-solution of (P )λ,α,β with U = α, ∂ν U = β on ∂B. Then U ≥ u a.e. in B. Proof. Let u ˜ ∈ H02 (B) denote a weak solution to ∆2 u ˜ = ∆2 (u − U ) in B. Since u ˜ − u + U = 0 and ∂ν (˜ u − u + U ) ≤ 0 on ∂B, by Lemma 5.5 one has that u ˜ ≥ u−U a.e. in B. Again by the Moreau decomposition [17], we may write u ˜ as u ˜ = w + v, where w, v ∈ H02 (B), w ≥ 0 a.e. in B, ∆2 v ≤ 0 in a H 2 (B)−weak sense and ¯ ∩ H 2 (B) one has ∆w∆v = 0. Then for 0 ≤ φ ∈ C 4 (B) 0 B ∆(u − U )∆φ ≤ λ  ∆˜ u∆φ = B  B  (f (u) − f (U ))φ. B  In particular, we have that ∆˜ u∆w ≤ λ B  (f (u) − f (U ))w. B  Since by semi-stability of u f (u)w2 ≤  λ B  (∆w)2 = B  ∆˜ u∆w, B  117  5.2. The effect of boundary conditions on the pull-in voltage we get that f (u)w2 ≤ B  (f (u) − f (U ))w. B  By Lemma 5.5 we have v ≤ 0 and then w ≥ u ˜ ≥ u − U a.e. in B. So we see that 0≤  (f (u) − f (U ) − f (u)(u − U )) w. B  The strict convexity of f implies as in Lemma 5.16 that U ≥ u a.e. in B. We shall need the following a-priori estimates along the minimal branch uλ . Lemma 5.18. Let (α, β) be an admissible pair. Then one has 2 B  (uλ − Φ)2 ≤ (1 − uλ )3  B  uλ − Φ , (1 − uλ )2  where Φ is given in (5.8). In particular, there is a constant C > 0 so that for every λ ∈ (0, λ∗ ), we have 1 ≤ C. (5.19) (∆uλ )2 + (1 − uλ )3 B B ¯ ∩ H 2 (B), we see that Proof. Testing (P )λ,α,β on uλ − Φ ∈ C 4 (B) 0  λ B  uλ − Φ = (1 − uλ )2  (∆(uλ − Φ))2 ≥ 2λ  ∆uλ ∆(uλ − Φ) = B  B  B  (uλ − Φ)2 (1 − uλ )3  2  in view of ∆ Φ = 0. In particular, for δ > 0 small we have that  {|uλ −Φ|≥δ}  1 (1 − uλ )3  (uλ − Φ)2 1 ≤ 2 3 δ {|uλ −Φ|≥δ} (1 − uλ ) 1 ≤ δ + Cδ 3 {|uλ −Φ|≥δ} (1 − uλ )  ≤  1 δ2  B  1 (1 − uλ )2  by means of Young’s inequality. Since for δ small,  {|uλ −Φ|≤δ}  1 ≤C (1 − uλ )3  for some C > 0, we can deduce that for every λ ∈ (0, λ∗ ),  B  1 ≤C (1 − uλ )3  for some C > 0. By Young’s and H¨older’s inequalities, we now have (∆uλ )2 B  =  ∆uλ ∆Φ B  + λ B  uλ − Φ ≤δ (1 − uλ )2  2  (∆uλ ) + Cδ + C B  B  1 (1 − uλ )3  2 3  and estimate (5.19) is therefore established.  118  5.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 We are now ready to establish Theorem 5.15. Proof (of Theorem 5.15): (1) Since uλ ∞ < 1, the infimum defining µ1 (uλ ) is achieved at a first eigenfunction for every λ ∈ (0, λ∗ ). Since λ → uλ (x) is increasing for every x ∈ B, it is easily seen that λ → µ1 (uλ ) is an increasing, continuous function on (0, λ∗ ). Define λ∗∗ := sup{0 < λ < λ∗ : µ1 (uλ ) > 0}. We have that λ∗∗ = λ∗ . Indeed, otherwise we would have that µ1 (uλ∗∗ ) = 0, and for every µ ∈ (λ∗∗ , λ∗ ) uµ would be a classical super-solution of (P )λ∗∗ ,α,β . A contradiction arises since Lemma 5.16 implies uµ = uλ∗∗ . Finally, Lemma 5.16 guarantees uniqueness in the class of semi-stable H 2 (B)−weak solutions. (2) By estimate (5.19) it follows that uλ → u∗ in a pointwise sense and weakly in 1 3 ∗ 2 H 2 (B), and 1−u ∗ ∈ L (B). In particular, u is a H (B)−weak solution of (P )λ∗ ,α,β which is also semi-stable as limiting function of the semi-stable solutions {uλ }. (3) Whenever u∗ ∞ < 1, the function u∗ is a classical solution, and by the Implicit Function Theorem we have that µ1 (u∗ ) = 0 to prevent the continuation of the minimal branch beyond λ∗ . By Lemma 5.16 u∗ is then the unique H 2 (B)−weak solution of (P )λ∗ ,α,β . An alternative approach –which we do not pursue here– based on the very definition of the extremal solution u∗ is available in [4] when α = β = 0 (see also [15]) to show that u∗ is the unique weak solution of (P )λ∗ , regardless of whether u∗ is regular or not. (4) If λ < λ∗ , by uniqueness v = uλ . So v is not singular and a contradiction arises. By Theorem 5.13(3) we have that λ = λ∗ . Since v is a semi-stable H 2 (B)−weak solution of (P )λ∗ ,α,β and u∗ is a H 2 (B)−weak super-solution of (P )λ∗ ,α,β , we can apply Lemma 5.16 to get v ≤ u∗ a.e. in B. Since u∗ is a semi-stable solution too, we can reverse the roles of v and u∗ in Lemma 5.16 to see that v ≥ u∗ a.e. in B. So equality v = u∗ holds and the proof is done.  5.3  Regularity of the extremal solution for 1≤N ≤8  We now return to the issue of the regularity of the extremal solution in problem (P )λ . Unless stated otherwise, uλ and u∗ refer to the minimal and extremal solutions of (P )λ . We shall show that the extremal solution u∗ is regular provided 1 ≤ N ≤ 8. We first begin by showing that it is indeed the case in small dimensions: Theorem 5.20. u∗ is regular in dimensions 1 ≤ N ≤ 4. Proof. As already observed, estimate (5.19) implies that f (u∗ ) = (1 − u∗ )−2 ∈ 3 L 2 (B). Since u∗ is radial and radially decreasing, we need to show that u∗ (0) < 1  119  5.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 to get the regularity of u∗ . The integrability of f (u∗ ) along with elliptic regularity 3 theory shows that u∗ ∈ W 4, 2 (B). By the Sobolev imbedding Theorem we get that ∗ u is a Lipschitz function in B. Now suppose u∗ (0) = 1 and 1 ≤ N ≤ 3. Since C 1 ≥ 1−u |x|  in B  for some C > 0, one sees that +∞ = C 3 B  1 ≤ |x|3  B  1 < +∞. (1 − u∗ )3  A contradiction arises and hence u∗ is regular for 1 ≤ N ≤ 3. 1 ¯ by the For N = 4 we need to be more careful and observe that u∗ ∈ C 1, 3 (B) ∗ ∗ Sobolev Imbedding Theorem. If u (0) = 1, then ∇u (0) = 0 and 1 C ≥ 4 1 − u∗ |x| 3  in B  for some C > 0. We now obtain a contradiction exactly as above. We now tackle the regularity of u∗ for 5 ≤ N ≤ 8. We start with the following crucial result: Theorem 5.21. Let N ≥ 5 and (u∗ , λ∗ ) be the extremal pair of (P )λ . When u∗ is singular, then 4 1 − u∗ (x) ≤ C0 |x| 3 in B, where C0 :=  λ∗ λ  1 3  ¯=λ ¯ N := 8 (N − 2 )(N − 8 ). and λ 9 3 3  Proof. First note that Theorem 5.6(4) gives the lower bound: ¯= λ∗ ≥ λ  8 2 8 (N − )(N − ). 9 3 3 4  For δ > 0, we define uδ (x) := 1 − Cδ |x| 3 with Cδ := we have that uδ ∈  1 2 Hloc (RN ), 1−u δ  ∆2 u δ =  ∈  L3loc (RN )  ¯ λ∗ + δ λ (1 − uδ )2  λ∗ ¯ λ  (5.22) 1 3  +δ  > 1. Since N ≥ 5,  2  and uδ is a H −weak solution of in RN .  We claim that uδ ≤ u∗ in B, which will finish the proof by just letting δ → 0. Assume by contradiction that the set Γ := {r ∈ (0, 1) : uδ (r) > u∗ (r)} is non-empty, and let r1 = sup Γ. Since uδ (1) = 1 − Cδ < 0 = u∗ (1),  120  5.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 we have that 0 < r1 < 1 and one infers that α := u∗ (r1 ) = uδ (r1 ) ,  β := (u∗ ) (r1 ) ≥ uδ (r1 ).  −4  Setting uδ,r1 (r) = r1 3 (uδ (r1 r) − 1) + 1, we easily see that uδ,r1 is a H 2 (B)−weak super-solution of (P )λ∗ +δλ¯ N ,α ,β , where −4  α := r1 3 (α − 1) + 1 , − 43  Similarly, let us define u∗r1 (r) = r1  −1  β := r1 3 β.  (u∗ (r1 r) − 1) + 1. The dilation map − 34  w → wr1 (r) = r1  (w(r1 r) − 1) + 1  (5.23)  is a correspondence between solutions of (P )λ on B and of (P ) (u∗r1 , λ∗ )  2  −4 3  λ,1−r1  ,0  on Br−1 1  is the extremal pair which preserves the H −integrability. In particular, ∗ −1 (defined in the obvious way). Moreover, u of (P ) on B −4 r1 is a singular r 3 λ,1−r1  ,0 2  1  semi-stable H (B)− weak solution of (P )λ∗ ,α ,β . Since u∗ is radially decreasing, we have that β ≤ 0. Define the function w as w(x) := (α − β2 ) + β2 |x|2 + γ(x), where γ is a solution of ∆2 γ = λ∗ in B with γ = ∂ν γ = 0 on ∂B. Then w is a classical solution of ∆2 w = λ∗ w = α , ∂ν w = β Since  λ∗ 2 (1−u∗ r1 )  in B on ∂B.  ≥ λ∗ , by Lemma 5.5 we have u∗r1 ≥ w a.e. in B. Since w(0) =  α − β2 + γ(0) and γ(0) > 0, the bound u∗r1 ≤ 1 a.e. in B yields to α − β2 < 1. Namely, (α , β ) is an admissible pair and by Theorem 5.15(4) we get that (u∗r1 , λ∗ ) coincides with the extremal pair of (P )λ,α ,β in B. Since (α , β ) is an admissible pair and uδ,r1 is a H 2 (B)−weak super-solution of (P )λ∗ +δλ¯ N ,α ,β , by Proposition 5.2.1 we get the existence of a weak solution of ¯ N > λ∗ , we contradict the fact that λ∗ is the extremal (P )λ∗ +δλ¯ N ,α ,β . Since λ∗ +δ λ parameter of (P )λ,α ,β . Thanks to this lower estimate on u∗ , we get the following result. Theorem 5.24. If 5 ≤ N ≤ 8, then the extremal solution u∗ of (P )λ is regular. Proof. Assume that u∗ is singular. For ε > 0 set ψ(x) := |x|  4−N 2  +ε  and note that  (∆ψ)2 = (HN + O(ε))|x|−N +2ε where HN :=  N 2 (N − 4)2 . 16  121  5.4. The extremal solution is singular for N ≥ 9 Given η ∈ C0∞ (B), and since N ≥ 5, we can use the test function ηψ ∈ H02 (B) into the stability inequality to obtain 2λ B  ψ2 ≤ (1 − u∗ )3  where O(1) is a bounded function as ε ¯N 2λ B  ψ2 ≤ |x|4  (∆ψ)2 + O(1), B  0. By Theorem 5.21 we find that (∆ψ)2 + O(1), B  and then ¯N 2λ  |x|−N +2ε + O(1).  |x|−N +2ε ≤ (HN + O(ε)) B  B  Computing the integrals one arrives at ¯ N ≤ HN + O(ε). 2λ ¯ N ≤ HN . Graphing this relation one sees that As ε → 0 finally we obtain 2λ N ≥ 9. We can now slightly improve the lower bound (5.22). Corollary 5.25. In any dimension N ≥ 1, we have ¯N = λ∗ > λ  8 2 8 (N − )(N − ). 9 3 3  (5.26)  4  Proof. The function u ¯ := 1 − |x| 3 is a H 2 (B)− weak solution of (P )λ¯ N ,0,− 34 . If by ¯ N , then u contradiction λ∗ = λ ¯ is a H 2 (B)−weak super-solution of (P )λ for every ∗ λ ∈ (0, λ ). By Lemma 5.16 we get that uλ ≤ u ¯ for all λ < λ∗ , and then u∗ ≤ u ¯ a.e. in B. If 1 ≤ N ≤ 8, u∗ is then regular by Theorems 5.20 and 5.24. By Theorem 5.15(3) there holds µ1 (u∗ ) = 0. Lemma 5.16 then yields that u∗ = u ¯, which is a contradiction since then u∗ will not satisfy the boundary conditions. ¯ = λ∗ , then C0 = 1 in Theorem 5.21, and we then have u∗ ≥ u If now N ≥ 9 and λ ¯. It means again that u∗ = u ¯, a contradiction that completes the proof.  5.4  The extremal solution is singular for N ≥ 9  We prove in this section that the extremal solution is singular for N ≥ 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∗ , λ∗ ) denotes the extremal pair of (P )λ .  122  5.4. The extremal solution is singular for N ≥ 9 • Case N ≥ 17: To establish the singularity of u∗ for these dimensions we shall need the following well known improved Hardy-Rellich inequality, which is valid for N ≥ 5. There exists C > 0, such that for all φ ∈ H02 (B) (∆φ)2 dx ≥ B  N 2 (N − 4)2 16  B  φ2 dx + C |x|4  φ2 dx.  (5.27)  B  • Case 10 ≤ N ≤ 16: For this case, we shall need the following inequality valid for all φ ∈ H02 (B) (∆φ)2  ≥  B  (N − 2)2 (N − 4)2 16 +  φ2 B  (N − 1)(N − 4)2 4  (|x|2 − |x| φ2  N 2  +1 )(|x|2  N  B  |x|2 (|x|2 − |x| 2 )  N  − |x| 2 )  (5.28)  .  • Case N = 9: This case is the trickiest and will require the following inequality for all φ ∈ H02 (B) (∆φ)2 ≥ B  Q(|x|) P (|x|) + B  N −1 |x|2  φ2 ,  (5.29)  where P (r) =  ∆N ϕ ϕ  and  Q(r) =  ∆N −2 ψ , ψ  with ϕ and ψ being two appropriately chosen polynomials, namely N  ϕ(r) := r− 2 +1 + r − 1.9 and  N  ψ(r) := r− 2 +2 + 20r−1.69 + 10r−1 + 10r + 7r2 − 48.  Recall that for a radial function ϕ, we set ∆N ϕ(r) = ϕ (r) + We shall first show the following upper bound on u∗ .  (N −1) ϕ r  (r).  4  Lemma 5.29. If N ≥ 9, then u∗ ≤ 1 − |x| 3 in B. ¯ := 8 (N − 2 )(N − 8 ) < λ∗ . We now claim Proof. Recall from Corollary 5.25 that λ 9 3 3 ∗ ¯ that uλ ≤ u ¯ for all λ ∈ (λ, λ ). Indeed, fix such a λ and assume by contradiction that R1 := inf{0 ≤ R ≤ 1 : uλ < u ¯ in (R, 1)} > 0. From the boundary conditions, one has that uλ (r) < u ¯(r) as r → 1− . Hence, 0 < R1 < 1, α := uλ (R1 ) = u ¯(R1 ) and β := uλ (R1 ) ≤ u ¯ (R1 ). Introduce, as in the proof of Theorem 5.21, the functions uλ,R1 and u ¯R1 . We have that uλ,R1 is a classical super-solution of (P )λ¯ N ,α ,β , where −4  α := R1 3 (α − 1) + 1 ,  −1  β := R1 3 β.  123  5.4. The extremal solution is singular for N ≥ 9 Note that u ¯R1 is a H 2 (B)−weak sub-solution of (P )λ¯ N ,α ,β which is also semi-stable in view of the Hardy-Rellich inequality (5.27) and the fact that ¯ N ≤ HN := 2λ  N 2 (N − 4)2 . 16  By Lemma 5.17, we deduce that uλ,R1 ≥ u ¯R1 in B. Note that, arguing as in the proof of Theorem 5.21, (α , β ) is an admissible pair. We have therefore shown that uλ ≥ u ¯ in BR1 and a contradiction arises in view of the fact that lim u ¯(x) = 1 and x→0 ∗ ¯ uλ ∞ < 1. It follows that uλ ≤ u ¯ in B for every λ ∈ (λN , λ ), and in particular u∗ ≤ u ¯ in B. The following lemma is the key for the proof of the of u∗ in higher dimensions. Lemma 5.30. Let N ≥ 9. Suppose there exist λ > 0, β > 0 and a singular radial 1 ¯ function w ∈ H 2 (B)with 1−w ∈ L∞ loc (B \ {0}) such that λ ∆2 w ≤ (1−w) for 0 < r < 1, 2 w(1) = 0, w (1) = 0,  (5.31)  φ2 ≤ (1 − w)3  (5.32)  and 2β B  (∆φ)2 for all φ ∈ H02 (B), B  1. If β ≥ λ , then λ∗ ≤ λ . 2. If either β > λ or if β = λ = singular.  HN 2  , then the extremal solution u∗ is necessarily  1 1 1 ¯ Proof: 1) First, note that (5.32) and 1−w ∈ L∞ loc (B \ {0}) yield to (1−w)2 ∈ L (B). By a density argument, (5.31) implies now that w is a H 2 (B)−weak sub-solution of (P )λ whenever N ≥ 4. If now λ < λ∗ , then by Lemma 5.17 w would necessarily be below the minimal solution uλ , which is a contradiction since w is singular while uλ is regular. 2) Suppose first that β = λ = H2N and that N ≥ 9. Since by part 1) we have λ∗ ≤ H2N , we get from Lemma 5.29 and the improved Hardy-Rellich inequality (5.27) that there exists C > 0 so that for all φ ∈ H02 (B)  (∆φ)2 − 2λ∗ B  B  φ2 ≥ (1 − u∗ )3  (∆φ)2 − HN B  B  φ2 ≥C |x|4  φ2 . B  It follows that µ1 (u∗ ) > 0 and u∗ must therefore be singular since otherwise, one could use the Implicit Function Theorem to continue the minimal branch beyond λ∗ . Suppose now that β > λ , and let λβ < γ < 1 in such a way that α := (  γλ∗ 1/3 ) < 1. λ  (5.33)  124  5.4. The extremal solution is singular for N ≥ 9 Setting w ¯ := 1 − α(1 − w), we claim that u∗ ≤ w ¯ in B.  (5.34)  Note that by the choice of α we have α3 λ < λ∗ , and therefore to prove (5.34) it suffices to show that for α3 λ ≤ λ < λ∗ , we have uλ ≤ w ¯ in B. Indeed, fix such λ and note that ∆2 w ¯ = α∆2 w ≤  α3 λ λ αλ = ≤ . 2 2 (1 − w) (1 − w) ¯ (1 − w) ¯ 2  Assume that uλ ≤ w ¯ does not hold in B, and consider R1 := sup{0 ≤ R ≤ 1 | uλ (R) > w(R)} ¯ > 0. Since w(1) ¯ = 1 − α > 0 = uλ (1), we then have R1 < 1, uλ (R1 ) = w(R ¯ 1 ) and (uλ ) (R1 ) ≤ (w) ¯ (R1 ). Introduce, as in the proof of Theorem 5.21, the functions uλ,R1 and w ¯R1 . We have that uλ,R1 is a classical solution of (P )λ,α ,β , where −4  α := R1 3 (uλ (R1 ) − 1) + 1 ,  −1  β := R1 3 (uλ ) (R1 ).  Since λ < λ∗ and then 2λ∗ 2λ 2β 2λ ≤ 3 = < , 3 (1 − w) ¯ α (1 − w)3 γ(1 − w)3 (1 − w)3 by (5.32) w ¯R1 is a stable H 2 (B)−weak sub-solution of (P )λ,α ,β . By Lemma 5.17, we deduce that uλ ≥ w ¯ in BR1 which is impossible, since w ¯ is singular while uλ is regular. Note that, arguing as in the proof of Theorem 5.21, (α , β ) is an admissible pair. This establishes claim (5.34) which, combined with the above inequality, yields 2λ∗ 2λ∗ 2β ≤ < , (1 − u∗ )3 α3 (1 − w)3 (1 − w)3 and therefore inf  φ∈H02 (B)  B  2λ∗ φ2 (1−u∗ )3 φ2  (∆φ)2 − B  > 0.  It follows that again µ1 (u∗ ) > 0 and u∗ must be singular, since otherwise, one could use the Implicit Function Theorem to continue the minimal branch beyond λ∗ . Consider for any m > 0 the following function: wm := 1 −  4 3m 4/3 r + rm , 3m − 4 3m − 4  (5.35)  which satisfies the right boundary conditions: wm (1) = wm (1) = 0. We can now prove that the extremal solution is singular for N ≥ 9. Theorem 5.36. Let N ≥ 9. The following upper bounds on λ∗ hold:  125  5.4. The extremal solution is singular for N ≥ 9 ¯ N and β = 1. If N ≥ 31, then Lemma 5.30 holds with w := w2 , λ = 27λ ∗ ¯ therefore λ (N ) ≤ 27λN . 2. If 17 ≤ N ≤ 30, then Lemma 5.30 holds with w := w3 , λ = β = λ∗ (N ) ≤ H2N .  HN 2  HN 2  , and  , and therefore  3. If 10 ≤ N ≤ 16, then Lemma 5.30 holds with w := w3 , λN < βN given in Table 5.1, and therefore λ∗ (N ) ≤ λN . 4. If N = 9, then Lemma 5.30 holds with w := w2.8 , λ9 := 366 < β9 := 368.5, and therefore λ∗ (9) ≤ 366. The extremal solution is therefore singular for dimension N ≥ 9.  Table 5.1: Summary 2 βN N w λN 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 ≤ N ≤ 30 w3 HN /2 HN /2 ¯ N HN /2 N ≥ 31 w2 27λ  ¯ ≤ HN . We shall show that w2 is Proof. 1) Assume first that N ≥ 31, then 27λ 2 a singular H 2 (B)−weak sub-solution of (P )27λ¯ so that (5.32) holds with β = H2N . Indeed, write 4 4 w2 := 1 − |x| 3 − 2(|x| 3 − |x|2 ) = u ¯ − φ0 , 4  1 where φ0 := 2(|x| 3 − |x|2 ), and note that w2 ∈ H02 (B), 1−w ∈ L3 (B), 0 ≤ w2 ≤ 1 2 in B, and ¯ ¯ 27λ 3λ ∆2 w2 = 8 ≤ in B \ {0}. (1 − w2 )2 r3  So w2 is H 2 (B)−weak sub-solution of (P )27λ¯ . Moreover, by φ0 ≥ 0 and (5.27) we get that HN B  φ2 = HN (1 − w2 )3  φ2 4  B  (|x| 3 + φ0 )3  ≤ HN B  φ2 ≤ |x|4  (∆φ)2 B  126  5.4. The extremal solution is singular for N ≥ 9 for all φ ∈ H02 (B). It follows from Lemma 5.30 that u∗ is singular and that λ∗ ≤ ¯ ≤ HN . 27λ 2 2) Assume 17 ≤ N ≤ 30 and consider the function 9 4 4 w3 := 1 − r 3 + r3 . 5 5 We show that w3 is a semi-stable singular H 2 (B)−weak sub-solution of (P ) HN . Indeed, we clearly have that 0 ≤ w3 ≤ 1 in B, w3 ∈ H02 (B) and show the stability condition, we consider φ ∈ H02 (B) and write HN B  φ2 (1 − w3 )3  =  φ2  125HN B 2  = HN B  4 3  (9r − 4r3 )3  φ ≤ r4  ≤ 125HN sup 0<r<1  1 1−w3  2  ∈ L3 (B). To φ2 r4  1 5 3  (9 − 4r )3  B  (∆φ)2 B  by virtue of (5.27). An easy computation shows that HN − ∆2 w3 2(1 − w3 )2  =  25HN 4 3  2(9r − 4r3 )2  ¯ 9λ 8 3  −  12 N 2 − 1 5 r  5r 8(N − 32 )(N − 83 ) 12 N 2 − 1 25N (N − 4) . − − 4 8 5 r 32(9r 3 − 4r3 )2 5r 3 2  =  −  2  By using Maple one can verify that this final quantity is nonnegative on (0, 1) whenever 17 ≤ N ≤ 30, and hence w3 is a H 2 (B)−weak sub-solution of (P ) HN . It 2  follows from Lemma 5.30 that u∗ is singular and that λ∗ ≤ H2N . 3) Assume 10 ≤ N ≤ 16. We shall prove that again w := w3 satisfies the assumptions of Lemma 5.30. Indeed, using Maple, we show that for each dimension 10 ≤ N ≤ 16, inequality (5.31) holds with λN given by Table 5.1. Then, by using Maple again, we show that for each dimension 10 ≤ N ≤ 16, the following inequality holds (N − 2)2 (N − 4)2 1 N N 16 (|x|2 − |x| 2 +1 )(|x|2 − |x| 2 ) +  (N − 1)(N − 4)2 1 2βN ≥ . N 2 2 4 (1 − w3 )3 |x| (|x| − |x| 2 )  where βN is again given by Table 5.1. The above inequality and the Hardy-Rellich inequality (5.28) guarantee that the stability condition (5.32) holds with β := βN . Since βN > λN , we deduce from Lemma 5.30 that the extremal solution is singular for 10 ≤ N ≤ 16. 4) Suppose now N = 9 and consider w := w2.8 . Using Maple on can see that ∆2 w ≤  366 in B (1 − w)2  127  5.5. Improved Hardy-Rellich Inequalities and  N −1 723 ≤ Q(r) P (r) + 3 (1 − w) r2  for all r ∈ (0, 1),  where P and Q are given in (5.29). Since 723 > 2 × 366, by Lemma 5.30 the extremal solution u∗ 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 sufficient 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 ∈ C 1 (0, 1), R and 0 rN −11V (r) dr = +∞. Say that the couple (V, W ) is a Bessel pair on (0, R) if the ordinary differential equation y (r) + ( N r−1 +  (BV,W )  Vr (r) V (r) )y  (r) +  W (r) V (r) y(r)  =0  has a positive solution on the interval (0, R). ∞ The space of radial functions in C0∞ (B) will be denoted by C0,r (B). The needed inequalities will follow from the following result.  Theorem 5.38. (Ghoussoub-Moradifam [10]) Let V and W be positive radial C 1 -functions on B\{0}, where B is a ball centered at zero with radius R in RN R R (N ≥ 1) such that 0 rN −11V (r) dr = +∞ and 0 rN −1 V (r)dr < +∞. The following statements are then equivalent: 1. (V, W ) is a Bessel pair on (0, R). 2.  B  V (|x|)|∇φ|2 dx ≥  B  W (|x|)φ2 dx for all φ ∈ C0∞ (B).  3. If limr→0 rα V (r) = 0 for some α < N − 2, then the above are equivalent to V (|x|)(∆φ)2 dx ≥ B  W (|x|)|∇φ|2 dx + (N − 1) B  for all φ ∈  ( B  V (|x|) Vr (|x|) )|∇φ|2 dx − |x|2 |x|  ∞ C0,r (B).  4. If in addition, W (r) − equivalent to  2V (r) r2  V (|x|)(∆φ)2 dx ≥ B  +  2Vr (r) r  − Vrr (r) ≥ 0 on (0, R), then the above are  W (|x|)|∇φ|2 dx + (N − 1) B  ( B  V (|x|) Vr (|x|) )|∇φ|2 dx − |x|2 |x|  for all φ ∈ C0∞ (B).  128  5.5. Improved Hardy-Rellich Inequalities We shall now deduce the following corollary. Corollary 5.39. Let N ≥ 5 and B be the unit ball in RN . Then the following improved Hardy-Rellich inequality holds for all φ ∈ C0∞ (B): (∆φ)2  φ2  (N − 2)2 (N − 4)2 16  ≥  B  +  (|x|2 − |x| φ2  B  (N − 1)(N − 4)2 4  N 2  +1 )(|x|2  N  − |x| 2 )  (5.40)  N  B  |x|2 (|x|2 − |x| 2 )  N  Proof. Let 0 < α < 1 and define y(r) := r− 2 +1 − α. Since − the couple  1, (N −2) 4  y +  (N −1) y r  y  2  1 r 2 −αr  =  (N − 2)2 1 , N 4 r2 − αr 2 +1  is a Bessel pair on (0, 1). By Theorem 5.38(4)  N +1 2  the following inequality then holds: (∆φ)2 dx ≥ B  (N − 2)2 4  |∇φ|2 B  for all φ ∈ C0∞ (B). Set V (r) :=  |x|2 − α|x| 1  r 2 −αr  N +1 2  N 2  +1  + (N − 1) B  |∇φ|2 |x|2  (5.41)  and note that N  2 α(N − 2) r 2 −2 2 Vr =− + ≥− . N −1 V r 2 r 1 − αr 2 N  The function y(r) = r− 2 +2 − 1 is decreasing and is then a positive super-solution on (0, 1) for the ODE y +(  N − 1 Vr W1 (r) + )y (r) + y = 0, r V V (r)  where W1 (r) =  (N − 4)2 N  N  4(r2 − r 2 )(r2 − αr 2 +1 )  .  Hence, by Theorem 5.38(2) we deduce |∇φ|2 B  |x|2 − α|x|  N 2  +1  ≥(  N −4 2 ) 2  for all φ ∈ C0∞ (B). Similarly, for V (r) =  B  φ2 B 1 r2  |∇φ|2 N −4 2 ≥( ) |x|2 2  (|x|2 − α|x|  N 2  +1 )(|x|2  N  − |x| 2 )  we have that φ2 N  B  |x|2 (|x|2 − |x| 2 )  for all φ ∈ C0∞ (B). Combining the above two inequalities with (5.41) and letting α → 1 we get inequality (5.40).  129  5.5. Improved Hardy-Rellich Inequalities N  Corollary 5.42. Let N = 9 and B be the unit ball in RN . Define ϕ(r) := r− 2 +1 + N r − 1.9 and ψ(r) := r− 2 +2 + 20r−1.69 + 10r−1 + 10r + 7r2 − 48. Then the following improved Hardy-Rellich inequality holds for all φ ∈ C0∞ (B): (∆φ)2 ≥ B  Q(|x|) P (|x|) + B  N −1 |x|2  φ2 ,  (5.43)  where P (r) := −  ϕ (r) + N r−1 ϕ (r) ψ (r) + N r−3 ψ (r) and Q(r) := − . ϕ(r) ψ(r)  Proof. By definition (1, P (r)) is a Bessel pair on (0, 1). One can easily see that P (r) ≥ r22 . Hence, by Theorem 5.38(4) the following inequality holds: (∆φ)2 dx ≥ B  P (|x|)|∇φ|2 + (N − 1) B  B  |∇φ|2 |x|2  (5.44)  for all φ ∈ C0∞ (B). Using Maple it is easy to see that Pr 2 ≥ − in (0, 1), P r and therefore ψ(r) is a positive super-solution for the ODE y +(  P (r)Q(r) N − 1 Pr (r) + )y (r) + y = 0, r P (r) P (r)  on (0, 1). Hence, by Theorem 5.38(2) we have for all φ ∈ C0∞ (B) P (|x|)|∇φ|2 ≥ B  P (|x|)Q(|x|)φ2 , B  and similarly B  |∇φ|2 ≥ |x|2  B  Q(|x|) 2 φ , |x|2  since ψ(r) is a positive solution for the ODE y +  N −3 y (r) + Q(r)y = 0. r  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. 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Systems Appl. 3 (1994), no. 4, 465–487.  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  λ  β∆2 u − τ ∆u = (1−u)2 0<u≤1  u = ∆u = 0  in Ω, in Ω, on ∂Ω,  (Gλ )  where λ > 0 is a parameter, τ > 0, β > 0 are fixed constants, and Ω ⊂N (N ≥ 2) is a bounded smooth domain. This problem with β = 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 (Gλ ) in the study of the charged plates in electrostatic actuators. They showed that there exists 0 < λ∗ < ∞ such that for λ ∈ (0, λ∗ ) (Gλ ) has a minimal regular solutions uλ (supB uλ < 1) while for λ > λ∗ , (Gλ ) does not have any regular solution. Moreover, the branch λ → uλ (x) is increasing for each x ∈ B, and therefore the function u∗ = limλ λ∗ uλ can be considered as a generalized solution that corresponds to the pull-in voltage λ∗ . Now the important question is whether the extremal solution u∗ is regular or not. In a recent paper Guo and Wei [17] proved that the extremal solution u∗ is regular for dimensions N ≤ 4. In this paper we consider the problem (Gλ ) on the unit ball in N :  λ in B,  β∆2 u − τ ∆u = (1−u)2 (Pλ ) 0<u≤1 in B,  u = ∆u = 0 on ∂B, 5  A 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 (Pλ ) is N = 9. Indeed we prove that the extremal solution of (Pλ ) is regular (supB u∗ < 1) for N ≤ 8 and β, τ > 0 and it is singular (supB u∗ = 1) for N ≥ 9, β > 0, and τ > 0 with βτ small. Our proof of regularity of the extremal solution in dimensions 5 ≤ N ≤ 8 is heavily inspired by [4] and [6]. On the other hand we shall use certain improved HardyRellich inequalities to prove that the extremal solution is singular in dimensions N ≥ 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 (Pλ ) that will be needed in the sequel. Define λ∗ (B) := sup{λ > 0 : (Pλ ) has a classical solution}. We now introduce the following notion of solution. We say that u is a weak solution of (Gλ ), if 0 ≤ u ≤ 1 a.e. in Ω, and if u(β∆2 φ − τ ∆φ) dx = λ Ω  Ω  φ dx, (1 − u)2  1 (1−u)2  ∈ L1 (Ω)  ∀φ ∈ W 4,2 (Ω) ∩ H01 (Ω),  Say that u is a weak super-solution (resp. weak sub-solution) of (Gλ ), if the equality is replaced with ≥ (resp. ≤) for φ ≥ 0. We now introduce the notion of stability. First, we equip the function space H := H 2 (Ω) ∩ H01 (Ω) = W 2,2 (Ω) ∩ H01 (Ω) with the norm [τ |∇ψ|2 + β|∆ψ|2 ]dx  ψ =  1/2  .  Ω  We say that a weak solution uλ of (Gλ ) is stable (respectively semi-stable) if the first eigenvalue µ1,λ (uλ ) of the problem −τ ∆h + β∆2 h −  2λ h = µh in Ω, (1 − uλ )3  h = ∆h = 0 on ∂Ω  (6.1)  is positive (resp., nonnegative). The operator β∆2 u − τ ∆u satisfies the following maximum principle which will be frequently used in the sequel. Lemma 6.2. ([17]) Let u ∈ L1 (Ω). Then u ≥ 0 a.e. in Ω, provided one of the following conditions hold: 1. u ∈ C 4 (Ω), β∆2 u − τ ∆u ≥ 0 on Ω, and u = ∆u = 0 on ∂Ω. 2.  Ω  u(β∆2 φ − τ ∆φ) dx ≥ 0 for all 0 ≤ φ ∈ W 4,2 (Ω) ∩ H01 (Ω).  3. u ∈ W 2,2 (Ω), u = 0, ∆u ≤ 0 on ∂B, and 0 ≤ φ ∈ W 2,2 (Ω) ∩ H01 (Ω).  Ω  β∆u∆φ + τ ∇u∇φ dx ≥ 0 for all  Moreover, either u ≡ 0 or u > 0 a.e. in Ω.  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 (Pλ ) with non-homogeneous boundary conditions such as  λ in B,  β∆2 u − τ ∆u = (1−u)2 (Pλ , α, γ) α<u≤1 in B,  u = α, ∆u = γ on ∂B, where α, γ are given. Whenever we need to emphasis the parameters β and τ we will refer to problem (Pλ,α,γ ) as (Pλ,β,τ,α,γ ). In this section and Section 3 we will obtain several results for the following general form of (Pλ , α, γ)  λ in Ω,  β∆2 u − τ ∆u = (1−u)2 (Gλ , α, γ) α<u≤1 in Ω,  u = α, ∆u = γ on ∂Ω, which are analogous to the results obtained by Gui and Wei for (Gλ ) in [17]. Let Φ denote the unique solution of β∆2 Φ − τ ∆Φ = 0 in Ω, Φ = α , ∆Φ = γ on ∂Ω.  (6.3)  We will say that the pair (α, γ) is admissible if γ ≤ 0, α < 1, and supΩ Φ < 1. We now introduce a notion of weak solution. We say that u is a weak solution of 1 1 (Pλ,α,γ ), if α ≤ u ≤ 1 a.e. in Ω, (1−u) 2 ∈ L (Ω) and if (u − Φ)(β∆2 φ − τ ∆φ) = λ Ω  where Φ is solution) of say a weak u ∞ = 1).  Ω  φ (1 − u)2  ∀φ ∈ W 4,2 (Ω) ∩ H01 (Ω),  given in (6.3). We say u is a weak super-solution (resp. weak sub(Pλ,α,γ ), if the equality is replaced with ≥ (resp. ≤) for φ ≥ 0. We solution u of (Pλ,α,γ ) is regular (resp. singular) if u ∞ < 1 (resp. We now define λ∗ (α, γ) := sup λ > 0 : (Pλ,α,γ ) has a classical solution  and λ∗ (α, γ) := sup λ > 0 : (Pλ,α,γ ) has a weak solution . Observe that by the Implicit Function Theorem, we can classically solve (Pλ,α,γ ) for small λ’s. Therefore, λ∗ (α, γ) and λ∗ (α, γ) are well defined for any admissible pair (α, γ). To cut down on notations we won’t always indicate α and γ. For example, λ∗ and λ∗ will denote the “weak and strong critical voltages” of (Pλ,α,γ ). Now let U be a weak super-solution of (Pλ,α,γ ) and recall the following existence result. ([17]) For every 0 ≤ f ∈ L1 (Ω) there exists a unique 0 ≤ u ∈ L1 (Ω) which satisfies u(β∆2 φ − τ ∆φ) dx = f φ dx, Ω  Ω  135  6.2. The pull-in voltage for all φ ∈ W 4,2 (Ω) ∩ H01 (Ω). We can introduce the following “weak” iterative scheme: u0 = U and (inductively) let un , n ≥ 1, be the solution of (un − Φ)(β∆2 φ − τ ∆φ) = λ Ω  Ω  φ (1 − un−1 )2  ¯ ∩ H01 (Ω) ∀ φ ∈ W 4,2 (Ω)  given by Theorem 6.2. Since 0 is a sub-solution of (Pλ,α,γ ), inductively it is easily shown by Lemma 6.2 that α ≤ un+1 ≤ un ≤ U for every n ≥ 0. Since (1 − un )−2 ≤ (1 − U )−2 ∈ L1 (Ω), by Lebesgue Theorem the function u = lim un is a weak solution of (Pλ,α,γ ) so n→+∞  that α ≤ u ≤ U . We therefore have the following result. Lemma 6.4. Assume the existence of a weak super-solution U of (Pλ,α,γ ). Then there exists a weak solution u of (Pλ,α,γ ) so that α ≤ u ≤ U a.e. in Ω. In particular, for every λ ∈ (0, λ∗ ), we can find a weak solution of (Pλ,α,γ ). In the same range of λ s, this is still true for regular weak solutions as shown in the following lemma. Lemma 6.5. Let (α, γ) be an admissible pair and u be a weak solution of (Pλ,α,γ ). Then, there exists a regular solution for every 0 < µ < λ. Proof: Let ∈ (0, 1) be given and let u ¯ = (1 − )u + Φ, where Φ is given in (6.3). By Lemma 6.2 supΩ Φ < supΩ u ≤ 1. Hence sup u ¯ ≤ (1 − ) + sup Φ < 1 , Ω  Ω  inf u ¯ ≥ (1 − )α + inf Φ = α, Ω  Ω  ¯ ∩ H 1 (Ω) there holds: and for every 0 ≤ φ ∈ W 4,2 (Ω) 0 (¯ u − Φ)(β∆2 φ − τ ∆φ)  =  (u − Φ)(β∆2 φ − τ ∆φ)  (1 − )  Ω  Ω  =  (1 − )λ  =  3  Ω  φ (1 − u)2  (1 − ) λ Ω 3  ≥ (1 − ) λ Ω  φ (1 − u ¯ + (Φ − 1))2 φ . (1 − u ¯)2  Note that 0 ≤ (1− )(1−u) = 1− u ¯ + (Φ−1) < 1− u ¯. So u ¯ is a weak super-solution of (P(1− )3 λ,α,γ ) so that sup u ¯ < 1. By Lemma 6.4 we get the existence of a weak Ω  solution w of (P(1−  )3 λ,α,γ )  so that α ≤ w ≤ u ¯. In particular, sup w < 1 and w is a Ω  regular weak solution. Since  ∈ (0, 1) is arbitrarily chosen, the proof is done.  136  6.2. The pull-in voltage Lemma 6.5 implies the existence of a regular weak solution Uλ for every λ ∈ (0, λ∗ ). Introduce now a “classical” iterative scheme: u0 = 0 and (inductively) un = vn + Φ, n ≥ 1, where vn ∈ W 4,2 (Ω) ∩ H01 (Ω) is the solution of β∆2 vn − τ ∆vn = β∆2 un − τ ∆un =  λ (1 − un−1 )2  in Ω and ∆vn = 0  on ∂Ω.  (6.6) Since vn ∈ W 4,2 (Ω) ∩ H01 (Ω), un is also a weak solution of (6.6), and by Lemma 6.2 we know that α ≤ un ≤ un+1 ≤ Uλ for every n ≥ 0. Since sup un ≤ sup Uλ < 1 for Ω  Ω  n ≥ 0, we get that (1−un−1 )−2 ∈ L2 (Ω) and the existence of vn is guaranteed. Since vn is easily seen to be uniformly bounded in H 2 (Ω), we have that uλ := lim un n→+∞  does hold pointwise and weakly in H 2 (Ω). By Lebesgue theorem, we have that uλ is a radial weak solution of (Pλ ) so that sup uλ ≤ sup Uλ < 1. By elliptic regularity Ω  Ω  ¯ and uλ = ∆uλ = 0 on ∂Ω. So we can integrate by parts to theory [1], uλ ∈ C ∞ (Ω) get β(∆2 uλ − τ ∆uλ )φ dx = Ω  uλ (β∆2 φ − τ ∆φ) dx = λ Ω  Ω  φ (1 − uλ )2  for every φ ∈ W 4,2 (Ω) ∩ H01 (Ω). Hence, uλ is a classical solution of (Pλ ) showing that λ∗ = λ∗ . Since the argument above shows that uλ < U for any other classical solution U of (Pµ , α, γ) with µ ≥ λ, we have that uλ is exactly the minimal solution and uλ is strictly increasing as λ ↑ λ∗ . In particular, we can define u∗ in the usual way: u∗ (x) = lim∗ uλ (x). λ  λ  Lemma 6.7. λ∗ (Ω) < +∞. Proof: Let u be a classical solution of (Pλ,α,γ ) and let (ψ, µ1 ) with ∆ψ = 0 on ∂Ω denote the first eigenpair of β∆2 − τ ∆ in H 2 (Ω) ∩ H01 (Ω) with ψ > 0. Now let C be such that ((τ α − βγ)∂ν ψ − βα∂ν (∆ψ)) = C ψ. ∂Ω  Ω  Multiplying (Pλ,α,γ ) by ψ and then integrating by parts one arrives at  Ω  λ − µ1 u − C ψ = 0. (1 − u)2  λ Since ψ > 0 there must exist a point x ¯ ∈ Ω where (1−u(¯ x) − C ≤ 0. Since x))2 − µ1 u(¯ α < u(¯ x) < 1, hence one can conclude that λ ≤ sup0<u<1 (µ1 u + C)(1 − u)2 , which shows that λ∗ < +∞.  In conclusion, we have shown the following description of the minimal branch. λ∗ ∈ (0, +∞) and the following holds:  137  6.3. Stability of the minimal solutions 1. For each 0 < λ < λ∗ there exists a regular and minimal solution uλ of (Pλ,α,γ ). 2. For each x ∈ Ω the map λ → uλ (x) is strictly increasing on (0, λ∗ ). 3. For λ > λ∗ there are no weak solutions of (Pλ,α,γ ).  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−weak solutions, which is an intermediate class between classical and weak solutions. We say that u is an H−weak solution of (Pλ,α,γ ) if u − Φ ∈ H 2 (Ω) ∩ H01 (Ω), 1 1 0 ≤ u ≤ 1 a.e. in Ω, (1−u) 2 ∈ L (Ω) and β∆u∆φ + τ ∇u∇φ dx = λ Ω  Ω  φ , (1 − u)2  ∀φ ∈ W 2,2 (Ω) ∩ H01 (Ω).  where Φ is given by (6.3). We say that u is an H−weak super-solution (resp. an H−weak sub-solution) of (Pλ,α,γ ) if for φ ≥ 0 the equality is replaced with ≥ (resp. ≤) and u ≥ 0 (resp. ≤), ∆u ≤ 0 (resp. ≥) on ∂Ω. Suppose that (α, γ) is an admissible pair. 1. The minimal solution uλ is stable, and is the unique semi-stable H−weak solution of (Pλ,α,γ ). 2. The function u∗ := lim∗ uλ is a well-defined semi-stable H−weak solution of λ  λ  (Pλ∗ ,α,γ ). 3. u∗ is the unique H−weak solution of (Pλ∗ ,α,γ ), and when u∗ is classical solution, then µ1 (u∗ ) = 0. 4. If v is a singular, semi-stable H−weak solution of (Pλ,α,γ ), then v = u∗ and λ = λ∗ The main tool is the following comparison lemma which is valid exactly in the class H. Lemma 6.8. Let (α, γ) be an admissible pair and u be a semi-stable H−weak solution of (Pλ,α,γ ). Assume U is a H−weak super-solution of (Pλ,α,γ ). Then 1. u ≤ U a.e. in Ω; 2. If u is a classical solution and µ1 (u) = 0 then U = u.  138  6.3. Stability of the minimal solutions Proof: (i) Define w := u − U . Then by means of the Moreau decomposition for the biharmonic operator (see [19] and [2]), there exist w1 and w2 ∈ H 2 (Ω) ∩ H01 (Ω), with w = w1 + w2 , w1 ≥ 0 a.e., β∆2 w2 − τ ∆w2 ≤ 0 in the H−weak sense and β∆w1 ∆w2 + τ ∇w1 .∇w2 = 0. Lemma 6.2 gives that w2 ≤ 0 a.e. in Ω. Ω Given 0 ≤ φ ∈ Cc∞ (Ω), we have β∆w∆φ + τ ∇w.∇φ ≤ λ Ω  (f (u) − f (U ))φ, Ω  where f (u) := (1 − u)−2 . Since u is semi-stable, one has f (u)w12 ≤  λ Ω  β(∆w1 )2 + τ |∇w1 |2  β∆w∆w1 + τ ∇w.∇w1  =  Ω  Ω  ≤ λ  (f (u) − f (U ))w1 . Ω  Since w1 ≥ w one has f (u)ww1 ≤ Ω  (f (u) − f (U ))w1 , Ω  which re-arranged gives f˜w1 ≥ 0, Ω  where f˜(u) = f (u) − f (U ) − f (u)(u − U ). The strict convexity of f gives f˜ ≤ 0 and f˜ < 0 whenever u = U . Since w1 ≥ 0 a.e. in Ω, one sees that w ≤ 0 a.e. in Ω. The inequality u ≤ U a.e. in Ω is then established. (ii) Since u is a classical solution, it is easy to see that the infimum of µ1 (u) is attained at some φ. The function φ is then the first eigenfunction of β∆2 − τ ∆ − 2λ 2 1 (1−u)3 in H (Ω) ∩ H0 (Ω). Now we show that φ is of fixed sign. Using the above decomposition, one has φ = φ1 + φ2 where φi ∈ H 2 (Ω) ∩ H01 (Ω) for i = 1, 2, φ1 ≥ 0, β∆φ1 ∆φ2 + τ ∇φ1 .∇φ2 = 0 and β∆2 φ2 − τ ∆φ2 ≤ 0 in the H−weak sense. If φ Ω changes sign, then φ1 ≡ 0 and φ2 < 0 in Ω (recall that either φ2 < 0 or φ2 = 0 a.e. in Ω). We can write now 0 = µ1 (u) ≤  Ω  <  Ω  β(∆(φ1 − φ2 ))2 + τ |∇(φ1 − φ2 )|2 − λf (u)(φ1 − φ2 )2 (φ − φ2 )2 Ω 1 β(∆φ)2 + τ |∇φ|2 − λf (u)φ2 = µ1 (u), φ2 Ω  in view of φ1 φ2 < −φ1 φ2 in a set of positive measure, leading to a contradiction. So we can assume φ ≥ 0, and by Lemma 6.2 we have φ > 0 in Ω. For 0 ≤ t ≤ 1, define β∆ [tU + (1 − t)u] ∆φ + τ ∇ [tU + (1 − t)u] .∇φ − λ  g(t) = Ω  f (tU + (1 − t)u)φ, Ω  139  6.3. Stability of the minimal solutions where φ is the above first eigenfunction. Since f is convex one sees that g(t) ≥ λ  [tf (U ) + (1 − t)f (u) − f (tU + (1 − t)u)] φ ≥ 0 Ω  for every t ≥ 0. Since g(0) = 0 and β∆(U − u)∆φ + τ ∇(U − u).∇φ − λf (u)(U − u)φ = 0,  g (0) = Ω  we get that f (u)(U − u)2 φ ≥ 0.  g (0) = −λ Ω  Since f (u)φ > 0 in Ω, we finally get that U = u a.e. in Ω. A more general version of Lemma 6.8 is available in the following. Lemma 6.9. Let (α, γ) be an admissible pair and γ ≤ 0. Let u be a semi-stable H−weak sub-solution of (Pλ,α,γ ) with u = α ≤ α, ∆u = β ≥ β on ∂Ω. Assume that U is a H−weak super-solution of (Pλ,α,γ ) with U = α, ∆U = β on ∂Ω. Then U ≥ u a.e. in Ω. Proof: Let u ˜ ∈ H 2 (Ω) ∩ H01 (Ω) denote a weak solution of β∆2 u ˜ − τ ∆˜ u = β∆2 (u − U )−τ ∆(u−U ) in Ω and u ˜ = ∆˜ u = 0 on ∂Ω. Since u ˜−u+U ≥ 0 and ∆(˜ u−u+U ) ≤ 0 on ∂Ω, by Lemma 6.2 one has that u ˜ ≥ u − U a.e. in Ω. By means of the Moreau decomposition (see [19] and [2]) we write u ˜ as u ˜ = w+v, where w, v ∈ H02 (Ω), w ≥ 0 a.e. in Ω, β∆2 v − τ ∆v ≤ 0 in a H−weak sense and Ω β∆w∆v + τ ∇w.∇v = 0. ¯ ∩ H 1 (Ω), one has Then for 0 ≤ φ ∈ W 4,2 (Ω) 0 β∆˜ u∆φ + τ ∇˜ u.∇φ ≤ λ Ω  (f (u) − f (U ))φ. Ω  In particular, we have β∆˜ u∆w + τ ∇˜ u.∇w ≤ λ Ω  (f (u) − f (U ))w. Ω  Since the semi-stability of u gives that f (u)w2 ≤  λ Ω  β(∆w)2 + τ |∇w|2 = Ω  β∆˜ u∆w + τ ∇˜ u.∇w, Ω  we get that f (u)w2 ≤ Ω  (f (u) − f (U ))w. Ω  By Lemma 6.2 we have v ≤ 0 and then w ≥ u ˜ ≥ u − U a.e. in Ω. So we obtain that 0≤  (f (u) − f (U ) − f (u)(u − U )) w. Ω  The strict convexity of f implies that U ≥ u a.e. in Ω.  140  6.3. Stability of the minimal solutions We need also some a-priori estimates along the minimal branch uλ . Lemma 6.10. Let (α, γ) be an admissible pair. Then for every λ ∈ (0, λ∗ ), we have (uλ − Φ)2 uλ − Φ 2 ≤ , 3 2 Ω (1 − uλ ) Ω (1 − uλ ) where Φ is given by (6.3). In particular, there is a constant C > 0 independent of λ so that 1 ≤ C, (6.11) (τ |∇uλ |2 + β|∆uλ |2 )dx + (1 − uλ )3 Ω Ω for every λ ∈ (0, λ∗ ). Proof: Testing (Pλ,α,γ ) on uλ − Φ ∈ W 4,2 (Ω) ∩ H01 (Ω), we see that λ Ω  uλ − Φ = (1 − uλ )2  (τ |∇(uλ − Φ)|2 + β(∆(uλ − Φ))2 )dx ≥ 2λ Ω  Ω  (uλ − Φ)2 . (1 − uλ )3  In the view of β∆2 Φ − τ ∆Φ = 0. In particular, for δ > 0 small we have that  {|uλ |≥δ}  1 (1 − uλ )3  (uλ − Φ)2 1 ≤ 2 3 (1 − u ) δ λ {|uλ −Φ|≥δ} 1 + Cδ ≤ δ 3 {|uλ −Φ|≥δ} (1 − uλ ) ≤  1 δ2  Ω  1 (1 − uλ )2  by means of Young’s inequality. Since for δ small  {|uλ −Φ|≤δ}  1 ≤ C, (1 − uλ )3  for some C > 0, we get that  Ω  1 ≤ C, (1 − uλ )3  for some C > 0 and for every λ ∈ (0, λ∗ ). Since (τ |∇uλ |2 + β|∆uλ |2 )dx = Ω  (β∆uλ ∆Φ + τ ∇uλ .∇Φ) + λ Ω  Ω  uλ − Φ (1 − uλ )2  (τ |∇uλ |2 + β|∆uλ |2 )dx  ≤ δ Ω  + Cδ + C Ω  1 (1 − uλ )3  2 3  in view of Young’s and H¨ older’s inequalities, estimate (6.11) is finally established. Proof of Theorem 6.3: (1) Since uλ ∞ < 1, the infimum defining µ1 (uλ ) is achieved at a first eigenfunction for every λ ∈ (0, λ∗ ). Since λ → uλ (x) is increasing  141  6.4. Regularity of the extremal solutions in dimensions N ≤ 8 for every x ∈ Ω, it is easily seen that λ → µ1 (uλ ) is a decreasing and continuous function on (0, λ∗ ). Define λ∗∗ := sup{0 < λ < λ∗ : µ1 (uλ ) > 0}. We have that λ∗∗ = λ∗ . Indeed, otherwise we would have µ1 (uλ∗∗ ) = 0, and for every µ ∈ (λ∗∗ , λ∗ ), uµ would be a classical super-solution of (Pλ∗∗ ,α,γ ). A contradiction arises since Lemma 6.8 implies uµ = uλ∗∗ . Finally, Lemma 6.8 guarantees the uniqueness in the class of semi-stable H−weak solutions. (2) It follows from (6.11) that uλ → u∗ in a pointwise sense and weakly in 1 3 ∗ is a H 2 −weak solution of (Pλ∗ ,α,γ ) H (Ω), and 1−u ∗ ∈ L (Ω). In particular, u which is also semi-stable as the limiting function of the semi-stable solutions {uλ }. 2  (3) Whenever u∗ ∞ < 1, the function u∗ is a classical solution, and by the Implicit Function Theorem we have that µ1 (u∗ ) = 0 to prevent the continuation of the minimal branch beyond λ∗ . By Lemma 6.8, u∗ is then the unique H−weak solution of (Pλ∗ ,α,γ ). (4) If λ < λ∗ , we get by uniqueness that v = uλ . So v is not singular and a contradiction arises. Now, by Theorem 6.2(3) we have that λ = λ∗ . Since v is a semi-stable H− weak solution of (Pλ∗ ,α,γ ) and u∗ is a H− weak super-solution of (Pλ∗ ,α,γ ), we can apply Lemma 6.8 to get v ≤ u∗ a.e. in Ω. Since u∗ is also a semi-stable solution, we can reverse the roles of v and u∗ in Lemma 6.8 to see that v ≥ u∗ a.e. in Ω. So equality v = u∗ holds and the proof is done.  6.4  Regularity of the extremal solutions in dimensions N ≤ 8  In this section we shall show that the extremal solution is regular in small dimensions. Let us begin with the following lemma. Lemma 6.12. Let N ≥ 5 and (u∗ , λ∗ ) be the extremal pair of (Pλ ). If u∗ is singular, and he set Γ := {r ∈ (0, 1) : uδ (r) > u∗ (r)} (6.13) 4  is non-empty, where uδ (x) := 1 − Cδ |x| 3 and Cδ > 1 is a constant. Then there exists r1 ∈ (0, 1) such that uδ (r1 ) ≥ u∗ (r1 ) and ∆uδ (r1 ) ≤ ∆u∗ (r1 ). Proof. Assume by contradiction that for every r with uδ (r1 ) ≥ u∗ (r1 ) one has ∆uδ (r1 ) > ∆u∗ (r1 ). Since Γ is non-empty and uδ (1) = 1 − Cδ < 0 = u∗ (1), there exists s1 ∈ (0, 1) such that uδ (s1 ) = u∗ (s1 ). We claim that uδ (s) > u∗ (s),  142  6.4. Regularity of the extremal solutions in dimensions N ≤ 8 for 0 < s < s1 . Assume that there exist s3 < s2 ≤ s1 such that u∗ (s2 ) = uδ (s2 ), u∗ (s3 ) = uδ (s3 ) and uδ (s) ≥ u∗ (s) for s ∈ (s3 , s2 ). By our assumption ∆us > ∆u∗ (s) for s ∈ (s3 , s2 ) which contradicts the maximum principle and justifies the claim. Therefore uδ (s) > u∗ (s) for 0 < s < s1 . Now set w := uδ − u∗ . Then w ≥ 0 on Bs1 and ∆w ≤ 0 in Bs1 . Since w(0) = 0, by strong maximum principle we get w ≡ 0 on Bs1 . This is a contradiction and completes the proof. Let N ≥ 5 and (u∗ , λ∗ ) be the extremal pair of (Pλ ). When u∗ is singular, then 4  1 − u∗ ≤ C|x| 3 in B, 8(N − 32 )(N − 83 ) . 9  ∗ 1 ¯ := where C := ( βλλ¯ ) 3 and λ ∗ ( βλλ¯  4 3  Proof. For δ > 0, define uδ (x) :=  1 3  2 N ( ) and 1 − Cδ |x| with Cδ := + δ) > 1. Since N ≥ 5, we have that uδ ∈ Hloc uδ is a H−weak solution of  β∆2 uδ − τ ∆uδ =  ¯ 2 λ∗ + βδ λ 4 2 + τ Cδ (N − )|x|− 3 in 2 (1 − uδ ) 3 3  N  .  We claim that uδ ≤ u∗ in B, which will finish the proof by just letting δ → 0. Assume by contradiction that the set Γ := {r ∈ (0, 1) : uδ (r) > u∗ (r)} is non-empty. By Lemma 6.12 the set Λ := {r ∈ (0, 1) : uδ (r) ≥ u∗ (r) and ∆uδ (r) ≤ ∆u∗ (r)} is non-empty. Let r1 ∈ Λ. Since uδ (1) = 1 − Cδ < 0 = u∗ (1), we have that 0 < r1 < 1. Define α := u∗ (r1 ) ≤ uδ (r1 ), γ := ∆u∗ (r1 ) ≥ ∆uδ (r1 ). −4  Setting uδ,r1 = r1 3 (uδ (r1 r) − 1) + 1, we see that uδ,r1 is a H−weak super-solution of (Pλ∗ +δλ,β,r−2 τ,α ,γ ), where 1  −4  2  α := r1 3 (α − 1) + 1, γ = r13 γ. −4  Similarly, define u∗r1 (r) = r1 3 (u∗ (r1 r) − 1) + 1. Note that ∆2 u∗ − α∆u∗ ≥ 0 in B and ∆u∗ = 0 on ∂B. Hence by maximum principle we have ∆u∗ ≤ 0 in B and therefore γ ≤ 0. Also obviously α < 1. So, (α , γ ) is an admissible pair and by Theorem 6.3(4) we get that (u∗r1 , λ∗ ) coincides with the extremal pair of (Pλ,β,r−2 τ,α ,γ ) in B. Also by Lemma 6.4 we get the existence of a week solution 1 of (Pλ∗ +δλ,β,r−2 τ,α ,γ ). Since λ∗ + δλ > λ∗ , we contradict the fact that λ∗ is the 1 extremal parameter of (Pλ,β,r−2 τ,α ,γ ). 1  Now we are ready to prove the following result. If 5 ≤ N ≤ 8, then the extremal solution u∗ of (P )λ is regular.  143  6.5. The extremal solution is singular in dimensions N ≥ 9 Proof. Assume that u∗ is singular. For note that  > 0 define ϕ(x) := |x|  (∆ϕ)2 = (HN + O( ))|x|−N +2 , where HN :=  4−N 2  +  and  N 2 (N − 4)2 . 16  Given η ∈ C0∞ (B), and since N ≥ 5, we can use the test function ηϕ ∈ H02 (B) into the stability inequality to obtain 2λ∗ B  ϕ2 ≤β (1 − u∗ )3  (∆ϕ)2 + τ B  → 0. By Theorem 6.4 we find  where O(1) is a bounded function as ¯ 2λ B  |∇ϕ|2 + O(1), B  ϕ2 ≤ |x|4  (∆ϕ)2 + O(1), B  and then ¯ 2λ  |x|−N +2 ≤ (HN + O( )) B  |x|−N +2 + O(1). B  Computing the integrals on obtains ¯ ≤ HN + O( ). 2λ Letting  6.5  ¯ ≤ HN . Graphing this relation we see that N ≥ 9. → 0 we get 2λ  The extremal solution is singular in dimensions N ≥ 9  In this section we will show that the extremal solution u∗ of (Pλ,β,τ,0,0 ) in dimensions N ≥ 9 is singular for τ > 0 sufficiently small. To do this, first we shall show that the extremal solution of (Pλ,1,0,0,0 ) is singular in dimensions N ≥ 9. Again to cut down the notation we won’t always indicate that β = 1 and τ = 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 established in the last section, by using the recent results of Ghoussoub-Moradifam [12]. • Case N ≥ 16: To establish the singularity of u∗ for these dimensions we shall need the classical Hardy-Rellich inequality, which is valid for all φ ∈ H 2 (B)∩H01 (B): (∆φ)2 dx ≥ B  N 2 (N − 4)2 16  B  φ2 dx. |x|4  (6.14)  144  6.5. The extremal solution is singular in dimensions N ≥ 9 • Case 10 ≤ N ≤ 16: For this case, we shall need the following inequality valid for all φ ∈ H 2 (B) ∩ H01 (B) (∆φ)2  ≥  B  φ2  (N − 2)2 (N − 4)2 16 +  (|x|2 −  B  (N − 1)(N − 4)2 4  N N 2 +1 )(|x|2 2(N −1) |x| 2  φ  N  B  |x|2 (|x|2 − |x| 2 )  N  − |x| 2 )  .  (6.15)  • Case N = 9: This case is the trickiest and will require the following inequality for all φ ∈ H 2 (B) ∩ H01 (B), which is valid for N ≥ 7 W (|x|)u2 .  |∆u|2 ≥  (6.16)  B  B  where where  (N − 2)2  W (r) = K(r)(  4(r2  K(r) = − and  −  N N 2 +1 ) 2(N −1) r  +  (N − 1) ), r2  ϕ (r) + (n−3) r ϕ (r) , ϕ(r)  N  ϕ(r) = r− 2 +2 + 9r−2 + 10r − 20. The next lemma will be our main tool to guarantee that u∗ is singular for N ≥ 9. The proof is based on an upper estimate by a singular stable sub-solution. Lemma 6.17. Suppose there exist λ > 0 and a radial function u ∈ H 2 (B) ∩ 4,∞ Wloc (B \ {0}) such that ∆2 u ≤  λ for 0 < r < 1, (1 − u)2  u(1) = 0,  ∆u|r=1 = 0,  u is singular, and 2β B  ϕ2 ≤ (1 − u)3  (6.18) (6.19) (6.20)  (∆ϕ)2 for all ϕ ∈ H 2 (B) ∩ H01 (B),  (6.21)  B  for some β > λ . Then u∗ is singular and λ∗ ≤ λ Proof. By Lemma 6.9 we have (6.22). Let α := (  (6.22) λ β  γλ∗ 1/3 ) , λ  < γ < 1 and (6.23)  145  6.5. The extremal solution is singular in dimensions N ≥ 9 and define u ¯ := 1 − α(1 − u). We claim that u∗ ≤ u ¯ in B.  (6.24)  To prove this, we shall show that for λ < λ∗ uλ ≤ u ¯ in B.  (6.25)  Indeed, we have ∆2 (¯ u) = α∆2 (¯ u) ≤  αλ α3 λ = . 2 (1 − u) (1 − u ¯)2  By (6.22) and the choice of α α3 λ < λ∗ . To prove (6.24) it suffices to prove it for α3 λ < λ < λ∗ . Fix such λ and assume that (6.24) is not true. Then Λ = {0 ≤ R ≤ 1 | uλ (R) > u ¯(R)}, in non-empty. There exists 0 < R1 < 1, such that uλ (R1 ) ≥ u∗ (R1 ) and ∆uλ (R1 ) ≤ ∆u∗ (R1 ), since otherwise we can find 0 < s1 < s2 < 1 so that uλ (s1 ) = u ¯(s1 ), uλ (s2 ) = u ¯(s2 ), uλ (R) > u ¯(R), and ∆uλ (R1 ) > ∆u∗ (R1 ) which contradict the maximum principle. Now consider the following problem ∆2 u u ∆u  λ in B (1 − u)2 = uλ (R1 ) on ∂B = ∆uλ on ∂B. =  Then uλ is a solution to the above problem while u ¯ is a sub-solution to the same problem. Moreover u ¯ is stable since, λ < λ∗ and hence  2λ 2λ∗ 2λ 2β ≤ = < . (1 − u ¯)3 α3 (1 − u)3 γ(1 − u)3 (1 − u)3  We deduce u ¯ ≤ uλ in BR1 which is impossible, since u ¯ is singular while uλ is smooth. This establishes (6.24). From (6.24) and the above two inequalities we have 2λ∗ 2λ β ≤ < . ∗ 3 3 (1 − u ) γ(1 − u) (1 − u)3 Thus inf (B)  ϕ∈C0∞  B  2λ∗ ϕ2 (1−u∗ )3 ϕ2  (∆ϕ)2 − B  > 0.  146  6.5. The extremal solution is singular in dimensions N ≥ 9 This is not possible if u∗ is a smooth solution. For any m > 43 define 4  wm := 1 − aN,m r 3 + bN,m rm , where aN,m :=  m(N + m − 2) , m(N + m − 2) − 43 (N − 2/3)  and bN,m :=  4 3 (N  − 2/3) . m(N + m − 2) − 43 (N − 2/3)  Now we are ready to prove the main result of this section. The following upper bounds on λ∗ hold in large dimensions. ¯ and β = 1. If N ≥ 31, then Lemma 6.17 holds with u := w2 , λN = 27λ ¯ 27λ. 2. If 16 ≤ N ≤ 30, then Lemma 6.17 holds with u := w3 , λN = HN 2 .  HN 2  HN 2  >  − 1, βN =  3. If 10 ≤ N ≤ 15, then Lemma 6.17 holds with u := w3 , λN < βN given in Table 6.1. 4. If N = 9, then Lemma 6.17 holds with u := w2.8 , λ9 := 249 < β9 := 251. The extremal solution is therefore singular for dimensions N ≥ 9. Proof. 1) Assume first that N ≥ 31, then it is easy to see that aN,2 < 3 and ¯ ≤ 27λ ¯ < HN . We shall show that w2 is a singular H−weak sub-solution of a3N,2 λ 2 1 ∈ L3 (B), 0 ≤ w2 ≤ 1 in B, (P )a3N,2 λ¯ which is stable. Note that w2 ∈ H 2 (B), 1−w 2 and ¯ a3N,2 λ ∆2 w2 ≤ in B \ {0}. (1 − w2 )2 So w2 is a H−weak sub-solution of (P )27λ¯ . Moreover, 4  4  4  w2 = 1 − |x| 3 + (aN,2 − 1)(|x| 3 − |x|2 ) ≤ 1 − |x| 3 . ¯≤ Since 27λ  HN 2  ¯ 54λ B  , we get that ϕ2 ≤ HN (1 − w2 )3  B  ϕ2 ≤ HN (1 − w2 )3  B  ϕ2 ≤ |x|4  (∆ϕ)2 B  for all ϕ ∈ C0∞ (B). Hence, w2 is stable. Thus it follows from Lemma 6.17 that u∗ ¯ is singular and λ∗ ≤ 27λ. 2) Assume 16 ≤ N ≤ 30 and consider 4  w3 := 1 − aN,3 r 3 + bN,3 r3 .  147  6.5. The extremal solution is singular in dimensions N ≥ 9 We show that it is a singular H−weak sub-solution of (P HN −1 ) which is stable. 2  Indeed, we clearly have 0 ≤ w3 ≤ 1 a.e. in B, w3 ∈ H 2 (B) and Note that HN B  ϕ2 (1 − w3 )3  ∈ L3 (B).  ϕ2  = HN  4 3  (aN,m r − bN,m rm )3  B  ≤  1 1−w3  HN  sup  4  0<r<1  (aN,m − bN,m rm− 3 )3  B  ϕ2 r4  2  ϕ ≤ r4  = HN B  (∆ϕ)2 . B  Using maple one can verify that for 16 ≤ N ≤ 31 ∆2 w3 ≤  HN 2  −1 on (0, 1). (1 − w3 )2  Hence w3 is a sub-solution of (P HN −1 ). By Lemma 6.17 u∗ is singular and λ∗ ≤ 2  HN 2  − 1. 3) Assume 10 ≤ N ≤ 15. We shall show that w3 satisfies the assumptions of Lemma 6.17 for each dimension 10 ≤ N ≤ 15. Using maple, for each dimension 10 ≤ N ≤ 15, one can verify that inequality (6.26) holds for λN given by Table 6.1. Then, by using maple again, we show that there exists βN > λN such that (N − 2)2 (N − 4)2 16 (|x|2 − +  1 N N 2 +1 )(|x|2 2(N −1) |x|  N  − |x| 2 )  (N − 1)(N − 4)2 2βN 1 ≥ . N 2 2 4 (1 − w3 )3 |x| (|x| − |x| 2 )  The above inequality and improved Hardy-Rellich inequality (6.40) guarantee that the stability condition (6.29) holds for βN > λ . Hence by Lemma 6.17 the extremal solution is singular for 10 ≤ N ≤ 15. The values of λN and βN are shown in Table 6.1. 4) Let u:=w2.8 . Using Maple on can see that ∆2 u ≤  249 in B (1 − u)2  and  502 ≤ W (r) for all r ∈ (0, 1), (1 − u(r))3 where W is given by (6.42). Since, 502 > 2 × 249, by Lemma 6.17 the extremal solution u∗ is singular in dimension N = 9. Remark 6.5.1. It follows from the proof of Theorem 6.5 that for N ≥ 9 and 2  sufficiently small, there exists u ∈ H (B) ∩ ∆2 u −  4,∞ Wloc (B  τ β  \ {0}) such that  τ λN ∆u ≤ for 0 < r < 1, β (1 − u)2  (6.26)  148  6.6. Improved Hardy-Rellich Inequalities  Table 6.1: Summary 1 N λN βN 9 249 251 10 320 367 11 405 574 12 502 851 13 610 1211 14 730 1668 15 860 2235 HN HN 16 ≤ N ≤ 30 2 −1 2 HN ¯ N ≥ 31 27λ 2  u(1) = 0,  ∆u|r=1 = 0,  (6.27)  u is singular,  (6.28)  τ |∇ϕ|2 for all ϕ ∈ H 2 (B) ∩ H01 (B), β  (6.29)  and 2βN B  ϕ2 ≤ (1 − u)3  (∆ϕ)2 + B  where βN > λN > 0 are constants. Indeed, for each dimension N ≥ 9, it is enough to take u to be the sub-solution we constructed in the proof of Theorem λ −λ 6.5, βN := βN , λ < λ < β. If βτ is sufficiently small so that − βτ ∆u < (1−u) 2 on (0, 1), then with an argument similar to that of Lemma 6.17 we deduce that the extremal solution u∗ of (Pλ,β,τ,0,0 ) is singular. We believe that the extremal solution of (Pλ,β,τ,0,0 ) is singular for all β, τ > 0 in dimensions N ≥ 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 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 R radius R in N , V, W ∈ C 1 (0, 1), and 0 rN −11V (r) dr = +∞. Say that the couple (V, W ) is a Bessel pair on (0, R) if the ordinary differential equation (BV,W )  y (r) + ( N r−1 +  Vr (r) V (r) )y  (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 C 1 -functions on B\{0}, where B is a ball centered at zero with R R radius R in N (N ≥ 1) such that 0 rN −11V (r) dr = +∞ and 0 rN −1 V (r)dr < +∞. The following statements are then equivalent: 1. (V, W ) is a Bessel pair on (0, R). 2.  B  V (|x|)|∇φ|2 dx ≥  B  Let B be the unit ball in B  |∆u|2 dx ≥  W (|x|)φ2 dx for all φ ∈ C0∞ (B). N  (N ≥ 5). Then the inequality  |∇u|2 dx N B |x|2 − N |x| 2 +1 2(N −1)  + (N − 1)  B  |∇u|2 |x|2 dx,  (6.30) ¯ holds for all u ∈ C0∞ (B). We shall need the following result to prove (6.30). Lemma 6.31. For every u ∈ C 1 ([0, 1]) the following inequality holds 1  1  |u (r)|2 rN −1 dr ≥ 0  u2 r2 −  0 N  Proof. Let ϕ := r− 2 +1 − r ∈ [0, 1]. Then 1  N N 2 +1 2(N −1) r  N 2(N −1)  rN −1 dr − (N − 1)(u(1))2 .  (6.32)  and k(r) := rN −1 . Define ψ(r) = u(r)/ϕ(r),  1  |u (r)|2 k(r)dr  |ψ(r)|2 |ϕ (r)|2 k(r)dr  =  0  0 1  +  1  |ϕ(r)|2 |ψ (r)|2 k(r)dr  2ϕ(r)ϕ (r)ψ(r)ψ (r)k(r)dr + 0  0 1  |ψ(r)|2 (|ϕ (r)|2 k(r) − (kϕϕ ) (r))dr  = 0 1  |ϕ(r)|2 |ψ (r)|2 k(r)dr + ψ 2 (1)ϕ (1)ϕ(1)  + 0 1  |ψ(r)|2 (|ϕ (r)|2 k(r) − (kϕϕ ) (r))dr + ψ 2 (1)ϕ (1)ϕ(1)  ≥ 0  (1) = −(N − 1)u2 (1). Hence, we have Note that ψ 2 (1)ϕ (1)ϕ(1) = u2 (1) ϕϕ(1) 1  1  |u (r)|2 k(r)dr 0  −u2 (r)  ≥ 0  k (r)ϕ (r) + k(r)ϕ (r) )dr − (N − 1)u2 (1) ϕ  Simplifying the above inequality we get (6.32).  150  6.6. Improved Hardy-Rellich Inequalities The decomposition of a function into its spherical harmonics will be one of our ¯ By decomposing u into spherical tools to prove Theorem 6.6. Let u ∈ C0∞ (B). harmonics we get u = Σ∞ k=0 uk where uk = fk (|x|)ϕk (x) and (ϕk (x))k are the orthonormal eigenfunctions of the Laplace-Beltrami operator with corresponding eigenvalues ck = k(N + k − 2), k ≥ 0. The functions fk belong to u ∈ C ∞ ([0, 1]), fk (1) = 0, and satisfy fk (r) = O(rk ) and f (r) = O(rk−1 ) as r → 0. In particular, ϕ0 = 1 and f0 =  1 N ωN r N −1  ∂Br  uds =  1 N ωN  |x|=1  u(rx)ds.  (6.33)  We also have for any k ≥ 0, and any continuous real valued W on (0, 1), |∆uk |2 dx =  ∆fk (|x|) − ck  B  B  fk (|x|) |x|2  2  dx,  (6.34)  and W (|x|)|∇uk |2 dx = B  W (|x|)|x|−2 fk2 dx.  W (|x|)|∇fk |2 dx + ck B  (6.35)  B  Now we are ready to prove Theorem 6.6. We shall use the inequality 1 0  |x (r)|2 rN −1 dr ≥  1 x2 (r) rN −1 dr N 0 r2 − N r 2 +1 2(N −1)  (N −2)2 4  for all x ∈ C 1 ([0, 1]),  with x(1) = 0. Proof of Theorem 6.6: For all N ≥ 5 and k ≥ 0 we have 1 N wN  |∆uk |2 dx = B  1 N wN  ∆fk (|x|) − ck B  1  =  fk (r) + 0  fk (|x|) |x|2  2  dx  fk (r) N −1 fk (r) − ck 2 r r  1  r  dr  1  (fk (r))2 rN −1 dr + (N − 1)2  =  2 N −1  0  0 1  1  fk2 (r)rN −5  +c2k  (fk (r))2 rN −3 dr fk (r)fk (r)rN −2  + 2(N − 1)  0  0 1  fk (r)fk (r)rN −3 dr  −2ck 0  1  fk (r)fk (r)rN −4 dr.  −2ck (N − 1) 0  151  6.6. Improved Hardy-Rellich Inequalities Integrate by parts and use (6.33) for k = 0 to get 1  1 N ωN  1  (fk (r))2 rN −1 dr + (N − 1 + 2ck )  |∆uk |2 dx ≥ B  (fk (r))2 rN −3 dr  0  0 1  +  (2ck (n − 4) + c2k )  rn−5 fk2 (r)dr 0  + Now define gk (r) =  fk (r) r  (N − 1)(fk (1))2  (6.36)  and note that gk (r) = O(rk−1 ) for all k ≥ 1. We have 1  1  (fk (r))2 rN −3  (gk (r))2 rN −1 dr  = 0  0  1  1  2gk (r)gk (r)rN −2 dr +  + 0  gk2 (r)rN −3 dr 0  1  1  (gk (r))2 rN −1 dr − (N − 3)  = 0  gk2 (r)rN −3 dr 0  Thus, 1  (fk (r))2 rN −3  ≥  0  −  (N − 2)2 4  1 0  fk2 (r) r2  N N r 2 +1 rN −3 dr − (N − 3) 2(N − 1)  (6.37) 1  fk2 (r)rN −5 dr 0  (6.39)  152  6.6. Improved Hardy-Rellich Inequalities 1  Substituting 2ck 0 (fk (r))2 rN −3 in (6.36) by its lower estimate in the last inequality (6.37), and using Lemma 6.31 we get 1 N ωN  |∆uk |2 dx ≥ B  +  1  (N − 2)2 4 2ck  (fk (r))2 N N 2 +1 2(N −1) r  r2 −  0  1  (N − 2)2 4  rN −1 dr  fk2 (r) r2 −  0  N N 2 +1 2(N −1) r  rn−3 dr  1  +  (fk (r))2 rN −3 dr  (N − 1) 0  1  (fk (r))2 rN −5 dr  + ck (N − 1) 0  1  rN −5 fk2 (r)dr  + ck (ck − (N − 1)) 0 1  + ck 0  ≥  (N − 2)2 4(r2  N N 2 +1 ) 2(N −1) r  − 1  (N − 2)2 4  r2  0  −  N N 2 +1 2(N −1) r  1  (N − 2) 4  2 )dr. r2  (fk (r))2  2  + ck  −  fk2 (r) r2 −  0  rN −1 dr  N N 2 +1 2(N −1) r  rn−3 dr  1  +  (fk (r))2 rN −3 dr  (N − 1) 0  1  (fk (r))2 rN −5 dr  + ck (N − 1) 0  The proof is complete in the view of (6.35). We shall now deduce the following corollary. Let N ≥ 5 and B be the unit ball in N . Then the following improved Hardy-Rellich inequality holds for all φ ∈ H 2 (B) ∩ H01 (B): (∆φ)2  ≥  B  (N − 2)2 (N − 4)2 16 +  Proof. Let α :=  (N − 1)(N − 4)2 4  N 2(N −1)  and V (r) :=  φ2 B  (|x|2 −  N N 2 +1 )(|x|2 2(N −1) |x| 2  φ  B  |x|2 (|x|2 1  r 2 −αr  N +1 2  N  − |x| 2 )  N  − |x| 2 ) (6.40)  and note that  N  2 Vr 2 α(N − 2) r 2 −2 ≥− . =− + N V r 2 r 1 − αr 2 −1  153  6.6. Improved Hardy-Rellich Inequalities N  The function y(r) = r− 2 +2 − 1 is decreasing and is then a positive super-solution on (0, 1) for the ODE y +(  W1 (r) N − 1 Vr + )y (r) + y = 0, r V V (r)  where W1 (r) =  (N − 4)2 N  N  4(r2 − r 2 )(r2 − αr 2 +1 )  .  Hence, by Theorem 6.6 we deduce |∇φ|2 B  |x|2 − α|x|  N 2  +1  N −4 2 ) 2  ≥(  φ2 B  (|x|2 − α|x|  for all φ ∈ H 2 (B) ∩ H01 (B). Similarly, for V (r) =  B  |∇φ|2 N −4 2 ≥( ) 2 |x| 2  1 r2  N 2  +1 )(|x|2  N  − |x| 2 )  we have that φ2 N  B  |x|2 (|x|2 − |x| 2 )  for all φ ∈ H 2 (B) ∩ H01 (B). Combining the above two inequalities with (6.32) we get (6.40). Let N ≥ 7 and B be the unit ball in N . Then the following improved HardyRellich inequality holds for all φ ∈ H 2 (B) ∩ H01 (B): |∆u|2 ≥ B  where W (r) = K(r)(  (N − 2)2 4(r2 −  K(r) = − and  W (|x|)u2 .  (6.41)  B  N N 2 +1 ) 2(N −1) r  +  (N − 1) ), r2  (6.42)  ϕ (r) + (n−3) r ϕ (r) , ϕ(r)  N  ϕ(r) = r− 2 +2 + 9r−2 + 10r − 20. Proof. Let α :=  N 2(N −1)  and V (r) :=  ODE y +(  1 r 2 −αr  . Then ϕ is a sub-solution for the  N +1 2  N − 1 Vr W2 (r) + )y (r) + y = 0, r V V (r)  where W2 (r) =  K(r) r2  N  − αr 2 +1  ,  Hence by Theorem 6.6 we have |∇u|2 N  B  |x|2 − α|x| 2 +1  W2 (|x|)u2 .  ≥  (6.43)  B  154  6.6. Improved Hardy-Rellich Inequalities Similarly B  |∇u|2 ≥ |x|2  W3 (|x|)u2 .  (6.44)  B  where  K(r) . r2 Combining the above two inequalities with (6.32) we get improved Hardy-Rellich inequality (6.41). W3 (r) =  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 application to higher order Sobolev spaces, preprint. [3] C. Cowan and N. Ghoussoub: Regularity of the extremal solution in a MEMS model with advection, Methods and Applications of Analysis, Vol. 15, No 3 (2008) p. 355-362. [4] 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. [5] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), pp. 207-218. [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] P. Esposito, N. Ghoussoub, and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2008), pp. 1731-1768. [8] F. Gazzola and H.-Ch. Grunau, Critical dimensions and higher order Sobolev inequalities with remainder terms, NoDEA Nonlinear Differential Equations Appl., 8 (2001), pp. 35-44. [9] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal., 38 (2007), pp. 14231449. [10] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl., 15 (2008), pp. 115145.  156  Chapter 6. Bibliography [11] Y. Guo, On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior, J. Differential Equations, 244 (2008), pp. 2277-2309. [12] N. Ghoussoub, A. Moradifam, Bessel pairs and optimal Hardy and HardyRellich inequalities, submitted. [13] 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. [14] Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120 (2006), pp. 193-209. [15] Z. M. Guo and J. C. Wei, Symmetry of nonnegative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), pp. 963-994. [16] Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular nonlinearity, J. London Math. Soc., 78 (2008), pp. 21-35. [17] Z. Gui, J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2009), 2034-2054. [18] F. Lin and Y. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A (2007) 463, 1323-1337. [19] J.-J. Moreau, Decomposition orthogonale dun espace hilbertien selon deux cones mutuellement polaires, C.R. Acad. Sci. 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 ∆2 u = λeu in B ∂u = 0 on ∂B, u = ∂n  (7.1)  where B is the unit ball in N , N ≥ 1, n is the exterior unit normal vector and λ ≥ 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 ≥ 1 there exists a λ∗ > 0 such that for every 0 < λ < λ∗ , there exists a smooth minimal (smallest) solution of (7.1), while for λ > λ∗ there is no solution even in a weak sense. Moreover, the branch λ → uλ (x) is increasing for each x ∈ B, and therefore the function u∗ (x) := limλ λ∗ uλ (x) can be considered as a generalized solution that corresponds to λ∗ . Now the important question is whether u∗ is regular (u∗ ∈ L∞ (B)) or singular (u∗ ∈ / L∞ (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 ≤ 12, u∗ is regular. But unlike the Gelfand problem the natural candidate u = −4 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 6  A 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. D´avila et al. [5] used a computer assisted proof to show that the extremal solution is singular in dimensions 13 ≤ N ≤ 31 while they gave a mathematical proof in dimensions N ≥ 32. In this paper we introduce a unified mathematical approach to deal with this problem and show that for N ≥ 13, the extremal solution is singular. One of our main tools is an improved Hardy-Rellich inequality that follows from the recent result of GhoussoubMoradifam 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 , R V, W ∈ C 1 (0, 1), and 0 rN −11V (r) dr = +∞. Say that the couple (V, W ) is a Bessel pair on (0, R) if the ordinary differential equation y (r) + ( N r−1 +  (BV,W )  Vr (r) V (r) )y  (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 C 1 -functions on B\{0}, where B is a ball centered at zero with radius R in N (N ≥ 1) such that R R 1 dr = +∞ and 0 rN −1 V (r)dr < +∞. The following statements are 0 r N −1 V (r) then equivalent: 1. (V, W ) is a Bessel pair on (0, R) and β(V, W ; R) ≥ 1. 2.  B  V (x)|∇u|2 dx ≥  B  W (x)u2 dx for all u ∈ C0∞ (B).  3. If limr→0 rα V (r) = 0 for some α < N −2 and W (r)− 2Vr2(r) + 2Vrr(r) −Vrr (r) ≥ 0 on (0, R), then the above are equivalent to B  V (x)|∆u|2 dx ≥  B  W (x)|∇u|2 dx + (N − 1)  B  ( V|x|(x) 2 −  Vr (|x|) 2 |x| )|∇u| dx,  for all u ∈ C0∞ (B). As an application we have the following improvement of the classical HardyRellich inequality. Let N ≥ 5 and B be the unit ball in N . Then the following improved HardyRellich inequality holds for all u ∈ C0∞ (B). |∆u|2  ≥  B  +  (N − 2)2 (N − 4)2 16 (N − 1)(N − 4)2 4  u2 B  (|x|2 − u  B  |x|2 (|x|2  N 9 2 +1 )(|x|2 10 |x| 2 N  − |x| 2 )  .  N  − |x| 2 ) (7.2)  159  7.2. An improved Hardy-Rellich inequality As a consequence the following improvement of classical Hardy-Rellich inequality holds: N 2 (N − 4)2 u2 (7.3) |∆u|2 ≥ N . 2 2 16 B B |x| (|x| − |x| 2 ) N  Proof. Let ϕ := r− 2 +1 − − (1, (N −2) 4  2  1 9 r 2 − 10 r  N +1 2  ϕ +  9 10 .  Since  (N −1) ϕ r  ϕ  =  (N − 2)2 1 , . N 9 2 4 r − 10 r 2 +1  ) is a bessel pair on (0, 1). By Theorem 7.2 the following  inequality holds for all u ∈ C0∞ (B). |∆u|2 dx ≥ B  Let V (r) :=  (N − 2)2 4  1 9 r 2 − 10 r  N +1 2  |∇u|2 B  |x|2 −  N 9 2 +1 10 |x|  + (N − 1) B  |∇u|2 . |x|2  (7.4)  . Then N  2 9 N −2 r 2 −2 2 Vr =− + ( ) ≥− , N 9 −1 V r 10 2 r 1− r2  (7.5)  10  N  and ψ(r) = r− 2 +2 − 1 is a positive super-solution for the ODE y +(  W1 (r) N − 1 Vr + )y (r) + y = 0, r V V (r)  where W1 (r) =  (N − 4)2 N  4(r2 − r 2 )(r2 −  9 N 2 +1 ) 10 r  (7.6)  .  Hence the ODE (7.6) has actually a positive solution and by Theorem 7.2 we have |∇u|2 B  |x|2 −  9 10 |x|  N 2  +1  ≥(  N −4 2 ) 2  u2 B  (|x|2 −  9 10 |x|  N 2  +1 )(|x|2  N  − |x| 2 )  .  (7.7)  Similarly B  |∇u|2 N −4 2 ) ≥( |x|2 2  u2 N  B  |x|2 (|x|2 − |x| 2 )  .  (7.8)  Combining the above two inequalities with (7.4) we get (7.2).  160  7.3. Main results  7.3  Main results  In this section we shall prove that the extremal solution u∗ of the problem (7.1) is singular in dimensions N ≥ 13. The next lemma will be our main tool to guarantee that u∗ is singular for N ≥ 13. The proof is based on an upper estimate by a singular stable sub-solution. Lemma 7.9. Suppose there exist λ > 0 and a radial function u ∈ H 2 (B) ∩ 4,∞ Wloc (B \ {0}) such that ∆2 u ≤ λ eu for all 0 < r < 1, ∂u (1) = 0, ∂n u∈ / L∞ (B),  u(1) = 0,  (7.10)  (7.11) (7.12)  and (∆ϕ)2 for all ϕ ∈ C0∞ (B),  eu ϕ2 ≤  β B  (7.13)  B  for some β > λ . Then u∗ is singular and λ∗ ≤ λ  (7.14)  Proof. By Lemma 2.6 in [5] we have (7.14). Define α := ln( where  λ β  λ ), γλ∗  (7.15)  < γ < 1 and let u ¯ := u + α. We claim that u∗ ≤ u ¯ in B.  (7.16)  To prove this, we shall show that for λ < λ∗ uλ ≤ u ¯ in B.  (7.17)  Indeed, we have ∆2 (¯ u) = ∆2 (u) ≤ λ eu = λ e−α eu¯ = γλ∗ eu¯ . To prove (7.16) it suffices to prove it for γλ∗ < λ < λ∗ . Fix such λ and assume that (7.16) is not true. Let R1 := sup{0 ≤ R ≤ 1 | uλ (R) = u ¯(R)}. Since u ¯(1) = α > 0 = uλ (1), we have 0 < R1 < 1, uλ (R1 ) = u ¯(R1 ), and uλ (R1 ) ≤ u ¯ (R1 ). Now consider the following problem  2 in Ω  ∆ u = λeu u = uλ (R1 ) on ∂Ω  ∂u = u (R ) on ∂Ω. 1 λ ∂n  161  7.3. Main results Obviously uλ is a solution to the above problem while u ¯ is a sub-solution to the same problem. Moreover u ¯ is stable since, λ < λ∗ and hence λeu¯ ≤ λ∗ eα eu =  λ u e < βeu . γ  We deduce u ¯ ≤ uλ in BR1 which is impossible, since u ¯ is singular while uλ is smooth. This establishes (7.16). From (7.16) and the above two inequalities we have ∗  λ∗ eu ≤ λ∗ ea eu = Since  λ γ  λ u e . γ  < β, ∗  inf ∞  ϕ∈C0 (B)  B  (∆ϕ)2 − λ∗ eu > 0. ϕ2 B  ∗  This is not possible if u is a smooth solution. In the following, for each dimension N ≥ 13, we shall construct u satisfying all the assumptions of Lemma 7.9. Define wm := −4 ln(r) − 2  4 4 + rm , m > 0, m m  2  −4) and let HN := N (N . Now we are ready to prove our main result. 16 The following upper bounds on λ∗ hold in large dimensions.  1. If N ≥ 32, then Lemma 7.9 holds with u := w2 , λN = 8(N − 2)(N − 4)e2 and β = HN > λN . 2. If 13 ≤ N ≤ 31, then Lemma 7.9 holds with u := w3.5 and λN < βN given in Table 1. The extremal solution is therefore singular for dimensions N ≥ 13. Proof. 1) Assume first that N ≥ 32, then 8(N − 2)(N − 4)e2 < and ∆2 w2 = 8(N − 2)(N − 4)  N 2 (N − 4)2 , 16  1 ≤ 8(N − 2)(N − 4)e2 ew2 . r4  Moreover, 8(N − 2)(N − 4)e2  e−4 ln(|x|) ϕ2 = Hn  ew2 ϕ2 ≤ Hn B  B  B  ϕ2 ≤ |x|2  |∆ϕ|2 . B  162  7.3. Main results Thus it follows from Lemma 7.9 that u∗ is singular and λ∗ ≤ 8(N − 2)(N − 4)e2 . 2) Assume 13 ≤ N ≤ 31. We shall show that u = w3.5 satisfies the assumptions of Lemma 7.9 for each dimension 13 ≤ N ≤ 31. Using Maple, for each dimension 13 ≤ N ≤ 31, one can verify that inequality (7.10) holds for λN given by Table 7.1. Then, by using Maple again, we show that there exists βN > λN such that 1 (N − 2)2 (N − 4)2 N N 16 (|x|2 − 0.9|x| 2 +1 )(|x|2 − |x| 2 ) +  (N − 1)(N − 4)2 1 N 2 2 4 |x| (|x| − |x| 2 )  ≥ βN ew3.5 . The above inequality and improved Hardy-Rellich inequality (7.2) guarantee that the stability condition (7.13) holds for βN > λ . Hence by Lemma 7.9 the extremal solution is singular for 13 ≤ N ≤ 31. The values of λN and βN are shown in Table 7.1.  Table 7.1: Summary 3 N λN N ≥ 32 8(N − 2)(N − 4)e2 31 20000 30 18500 29 17000 28 16000 27 14500 26 13500 25 12200 24 11100 23 10100 22 9050 21 8150 20 7250 19 6400 18 5650 17 4900 16 4230 15 3610 14 3050 13 2525  βN Hn 86900 76500 67100 58500 50800 43870 37630 32050 27100 22730 18890 15540 12645 10155 8035 6250 4765 3545 2560  163  7.3. Main results Remark 7.3.1. The values of λN and βN in Table 7.1 are not optimal. Remark 7.3.2. The improved Hardy-Rellich inequality (7.2) is crucial to prove that u∗ is singular in dimensions N ≥ 13. Indeed by the classical Hardy-Rellich inequality and u := w3.5 , Lemma 7.9 only implies that u∗ is singular in dimensions N ≥ 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 methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), pp. 207-218. [5] J. D´ avila, 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 HardyRellich 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 −∆u = λeu . 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 ∈ Rn×n 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 feature 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 methods 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 problem 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 AT Ax = AT b  or AAT y = b, x = AT y.  (8.2)  7 A 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 CGNR methods 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 κ(A) = A · A−1 ,  (8.3)  and in the case where A is positive definite and symmetric, the condition number is then equal to λmax (A) , (8.4) κ(A) = λmin (A) where λmin (A) (resp., λmax (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’s statement [9] that numerical analysts cringe at the thought of solving these normal equations because the condition number (see below) of the new matrix AT A is the square of the condition number of the original matrix A. In this paper, we shall follow a similar approach that consists of symmetrizing 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 AT M Ax = AT M b,  (8.5)  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 AT M A. 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 by minimizing the condition number κ(AT M A), 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 selfadjoint partial differential equations that do not normally fit in the standard EulerLagrangian theory. Applied to the linear system (8.1), the new principle yields the following procedure. Split the matrix A into its symmetric Aa (resp., antisymmetric part Aa ) A = As + Aa , (8.6) where  1 1 (A + AT ) and Aa := (A − AT ). (8.7) 2 2 Proposition 8.8. (Selfdual symmetrization) Assume the matrix A is positive definite, i.e., for some δ > 0, As :=  Ax, x ≥ δ|x|2 for all x ∈ Rn .  (8.9)  168  8.1. Introduction and main results The convex continuous functional I(x) =  1 1 Ax, x + A−1 (b − Aa x), b − Aa x − b, x 2 2 s  (8.10)  then attains its minimum at some x ¯ in Rn , in such a way that I(¯ x)  =  inf I(x) = 0  x∈Rn  A¯ x = b.  (8.11) (8.12)  Here x, y = xT y and |x|2 = x, x . Symmetrization and preconditioning via selfduality: Note that the functional I can be written as I(x) =  1 −1 1 ˜ Ax, x + Aa A−1 A b, b , s b − b, x + 2 2 s  (8.13)  T −1 A˜ := As − Aa A−1 s Aa = A As A.  (8.14)  where By writing that DI(¯ x) = 0 (DI is the subdifferential of the functional I), one gets the following equivalent way of solving (8.1). If both A ∈ Rn×n 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 −1 −1 T −1 AT A−1 s Ax = (As − Aa As Aa )x = b − Aa As b = A As b.  (8.15)  One can therefore apply to (8.15) all known iterative methods for symmetric systems 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., AT A−1 s 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 ˜ is our observation important and less obvious than the symmetrization effect of A, 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 –which can ˜ has the potential of providing an now be applied to the symmetrized matrix A– 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 ˜ of A˜ = AT A−1 on the condition number κ(A) s A. We observe in section 2.3 that for ˜ may be very small and hence SD-CGN a large class of ill-conditioned matrices, κ(A) can be very efficient. In other words, the inverse C of AT A−1 can be an efficient s 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 ∗ = −J 2 = I). Consider then a matrix A = As + 1 J which has an arbitrarily large anti-symmetric part. One can show that κ(A˜ ) ≤ κ(As ) +  2  λmax (As )2 ,  (8.16)  which means that the larger the anti-symmetric part, the smaller our upper bound for κ(A˜ ) and consequently the more efficient is our proposed selfdual preconditioning. Needless to say that this method is of practical interest only when the equation As x = 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 calculated 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 AT A−1 s 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 preconditioner, is by noting that the convergence of “simple iteration” As xk = −Aa xk−1 + b  (8.17)  applied to the decomposition of A into its symmetric and anti-symmetric parts, −1 requires that the spectral radius ρ(I − A−1 s A) = ρ(As Aa ) < 1. By multiplying (8.17) by A−1 , we see that this is equivalent to the process of applying simple s iteration to the original system (8.1) conditioned by A−1 s , i.e., to the system −1 A−1 s Ax = As b.  (8.18)  On the other hand, “simple iteration” applied to the decomposition of A˜ into As and Aa A−1 s Aa is given by −1 As xk = Aa A−1 s Aa xk−1 + b − Aa As b.  (8.19)  −1 2 −1 2 ˜ Its convergence is controlled by ρ(I − A−1 s A) = ρ((As Aa ) ) = ρ(As Aa ) which −1 is strictly less than ρ(As 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 ∈ Rn×n 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 T −1 ¯Ax := A−1 s A As Ax = =  2 [I − (A−1 s Aa ) ]x −1 −1 T −1 (I − A−1 s Aa )As b = As A As 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. Anti-symmetry in transport problems: Numerical experiments on standard linear ODEs (Example 3.1) and PDEs (Example 3.2), show the efficiency of SDCGN for non-selfadjoint equations. Roughly speeking, discretization of differential equations normally leads to a symmetric component coming from the Laplace operator, 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( h12 ), while the anti-symmetric part is of order O( h1 ), 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 1 1 1 1 2 A˜ = As − Aa A−1 s Aa = O( 2 ) − O( )O(h )O( ) = O( 2 ) − O(1), h h h h  (8.21)  making the matrix A˜ an O(1) perturbation of As , and therefore a matrix of the form As + αI becomes a natural candidate to precondition the new system (8.15). Resolvents of As as preconditioners: One may therefore consider preconditioned equations of the form AT M Ax = AT M b, where M is of the form Mα = αAs + (1 − α)I  −1  or  Nβ = βA−1 s + (1 − β)I,  (8.22)  for some 0 ≤ α, β ∈ R, and where I is the unit matrix. Note that we obviously recover (8.2) when α = 0, and (8.15) when α = 1. As α → 0 the matrix αAs + (1 − α)I becomes easier to invert, but the matrix A1,α = AT (αAs + (1 − α)I)−1 A  (8.23)  may become more ill conditioned, eventually leading (for α = 0) to AT Ax = 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 αAs + (1 − α)I, and so by an appropriate choice of the parameter α 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 –and it is sometimes preferable as shown in example (3.4)– α > 1, as long as α < 1−λ1s , min where λsmin 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 α small enough, so that the matrix Mα (and hence A1,α ) becomes positive definite, and therefore making CG applicable (See example 3.4). Similarly, the matrix Nβ = βA−1 s + (1 − β)I provides another choice for the matrix M in (8.5), λs for β < λs max−1 where λsmax is the largest eigenvalue of As . Again we may choose max α close to zero to make the matrix Nβ positive definite. As we will see in the last section, appropriate choices of β, can lead to better convergence of CG for equation (8.5). One can also combine both effects by considering matrices of the form Lα,β = αAs + (1 − α)I  −1  + βI,  (8.24)  171  8.2. Selfdual methods for non-symmetric systems as is done in example (3.4). We also note that the matrices Mα := (αAs +(1−α)I)−1 and Nβ := β(As )−1 + (1−β)I can be other options for the matrix M , where As is a suitable approximation of As , chosen is such a way that Mα q and Nβ q 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 ∈ Rn×n , where B and (B − C) are both invertible. In this case, B(B − C)−1 can be a preconditioner for the equation (8.1). Indeed, since B − CB −1 C = (B − C)B −1 A, x is a solution of (8.1) if and only of it is a solution of the system (B − C)B −1 Ax = (B − CB −1 C)x = b − CB −1 b.  (8.25)  In the next section, we shall describe a general framework based on the ideas explained above for the use of iterative methods for solving non-symmetric linear systems. In section 3 we present various numerical experiments to test the effectiveness 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 matrix M is chosen to be the inverse of the symmetric part of A. If M 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 M q for certain vectors q. Since selfdual methods call for a preconditioner matrix M that involves inverting another one, the computation of M q 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 M q 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 matrix M , in order to compute M q. 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 As y = b Compute b = b − Aa y. Solve As y0 = Aa x0 Compute r0 = b − As x0 + Aa y0 and set p0 = r0 . For k=1,2, . . . , Solve As z = Aa pk−1 Compute w = As pk−1 − Aa z . <rk−1 ,rk−1 > . Set xk = xk−1 + αk−1 pk−1 , where αk−1 = <p k−1 ,w> Cpmpute rk = rk−1 − αk−1 w. k ,rk > Set pk = rk + bk−1 pk−1 , where bk−1 = <r<r . k−1 ,rk−1 > Check convergence; continue if necessary.  Table 8.1: GCGN where the coefficient matrix A in (8.1) is positive definite, and when AT A−1 s 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 SDMINRESN 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 As x = q (8.26) can be solved with less computational effort than the original equation (8.1). Actually, 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 –which is a direct consequence of Theorem 4.4.3 in [9]– shows that if we solve the inner equations (8.26) “accurately enough” then ISD-CGN and ISD-MINRESN can be used to solve (8.1) with a pre-determined accuracy. Indeed, given > 0, we assume that in each iteration of ISD-CGN or ISD-MINRESN, we can solve the inner equation –corresponding to As – accurately  173  8.2. Selfdual methods for non-symmetric systems enough in such a way that −1 (As − Aa A−1 s Aa )p − (As p − Aa y) = Aa As Aa p − Aa y < ,  (8.27)  where y is the (inexact) solution of the equation As y = Aa p.  (8.28)  In other words, we assume CG and MINRES are implemented on (8.28) in a finite precision arithmetic with machine precision . Set 0  := 2(n + 4) ,  1  := 2(7 + n  |As − Aa A−1 s Aa | | ) , As − Aa A−1 s Aa  (8.29)  where |D| denotes the matrix whose terms are the absolute values of the corresponding terms in the matrix D. Let λ1 ≤ ... ≤ λn be the eigenvalues of (As − Aa A−1 s Aa ) and let Tk+1,k be the (k + 1) × 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) × k block, whose eigenvalues all lie in the intervals S = ∪ki=1 [λi − δ, λi + δ], (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 rkIM then satisfies ||rkIM || ≤ ||r0 ||  √ λn (1 + 2 0 )(k + 1) min max |pk (z)| + 2 k( ) 1 . pk z=S d  (8.32)  If A is positive definite, then the ISD-CGN residual rIC satisfies ||rkIC || ≤ ||r0 ||  (1 + 2 0 )(λn + δ)/d min max |pk (z)| + pk  z=S  √ k(  λn ) 1. d  (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 − Aa A−1 s 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 techniques 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 −1 Ax = C −1 b, in such a way that it has the same solution, but hopefully with more favorable spectral properties. As such the reformulation of (1) as T −1 AT A−1 s Ax = A As b,  (8.34)  can be seen as a preconditioning procedure with C being the inverse of AT A−1 s . 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˜ = AT A−1 s A. Proposition 8.35. Assume A is an invertible positive definite matrix, then ˜ ≤ min{κ1 , κ2 }, κ(A)  (8.36)  where κ1 := κ(As ) +  Aa 2 λmin (As )2  and  κ2 := κ(As )κ(−A2a ) +  λmax (As )2 λmin (−A2a ) .  (8.37)  Proof: We have ˜ = λmin (As − Aa A−1 λmin (A) s Aa ) ≥ λmin (As ). We also have ˜ λmax (A)  =  sup x=0  ˜ xt (As − Aa A−1 xT Ax s Aa )x = sup Tx xT x x x=0  ≤ λmax (As ) +  ||Aa ||2 . λmin (As )  ˜ ˜ = λmax (A) ˜ Since κ(A) ˜ , it follows that κ(A) ≤ κ1 . λmin (A) To obtain the second estimate, observe that  ˜ λmin (A)  −1 = λmin (As − Aa A−1 s Aa ) > λmin (−Aa As Aa ) −xT Aa A−1 s Aa x = inf T x=0 x x (Aa x)T A−1 (Aa x)T (Aa x) s (Aa x) = inf { × } x=0 (Aa x)T (Aa x) xT x (Aa x)T A−1 xT (Aa )T (Aa )x s (Aa x) ≥ inf × inf x=0 x=0 (Aa x)T (Aa x) xT x 1 = × λmin ((Aa )T Aa ) λmax (As ) 1 = × λmin (−A2a ) λmax (As )  ˜ we get κ(A) ˜ ≤ κ2 . With the same estimate for λmax (A)  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 κ1 or κ2 be small. Indeed, ˜ ≤ κ1 shows that the condition number κ(A) ˜ is reasonable as • The inequality κ(A) long as the anti-symmetric part Aa is not too large. On the other hand, even if ˜ is then as large as κ(As )2 , it may still Aa is of the order of λmax (As ), and κ(A) be an improved situation, since this can happen for cases when κ(A) is exceedingly large. This can be seen in example 2.2 below. ˜ ≤ κ2 is even more interesting especially in situations when • The inequality κ(A) 2 λmin (−Aa ) is arbitrarily large while remaining of the same order as ||Aa ||2 . This ˜ can remain of the same order as κ(As ) regardless how large is Aa . means that κ(A) A typical example is when the anti-symmetric matrix Aa is a multiple of the symplectic matrix J (i.e. JJ ∗ = −J 2 = I). Consider then a matrix A = As + 1 J ˜ ≤ κ2 , one which has an arbitrarily large anti-symmetric part. By using that κ(A) gets κ(A˜ ) ≤ κ(As ) + 2 λmax (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 =  1 1  −1 −1 +  (8.39)  which is a typical example of an ill-conditioned non-symmetric matrix. One can actually show that κ(A ) = O( 1 ) → ∞ as → 0 with respect to any norm. However, the condition number of the associated selfdual coefficient matrix A˜ = As − Aa (As )−1 Aa =  −1  0  0  1 is κ(A˜ ) = 1−ε , and therefore goes to 1 as ε → 0. Note also that the condition number of the symmetric part of A goes to one as → 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 κ(A−1 goes to zero, which means that s A) goes to ∞ as besides making the problem symmetric, our proposed conditioned matrix AT A−1 s A has a much smaller condition number than the matrix A−1 A, which uses A s as a s preconditioner. Similarly, consider the non-symmetric linear system with coefficient matrix  A =  1 1  −1 + −1  .  (8.40)  As → 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˜ also converges to 1 as 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 − y + y = 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 . We use ESD-CGN and ISD-CGN (with relative residual 10−7 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(πx). N=64 ESD-CGN ISD-CGN GCNE QMR PQMR BiCGSTAB PBiCGSTAB BiCG PBiCG CGS PCGS  = 10−2 22 24 88 114 34 63.5 26.5 125 31 > 1000 27  = 10−3 8 9 64 > 1000 51 78.5 46.5 > 1000 44 > 1000 51  = 10−4 5 6 64 > 1000 50 92.5 50.5 > 1000 50 > 1000 46  = 10−6 4 4 64 > 1000 52 98.5 50 > 1000 50 > 1000 46  = 10−10 3 3 64 > 1000 52 100.5 51.5 > 1000 52 > 1000 46  = 10−16 2 2 64 > 1000 52 103.5 51.5 > 1000 52 > 1000 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 ( 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 ∂u −∆u + a = f (x, y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, (8.42) ∂x  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 10−6 , 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−x) cos(x) . N=128 ESD-CGN ISD-CGN(10−7 ) GCNE QMR PQMR BiCGSTAB PBiCGSTAB BiCG PBiCG CGS PCGS  = 10−2 37 38 266 > 1000 40 136.5 35.5 > 1000 37 > 1000 34  = 10−3 11 12 140 > 1000 77 167.5 87.5 > 1000 76 > 1000 80  = 10−4 6 7 128 > 1000 87 241 106.5 > 1000 84 > 1000 96  = 10−6 4 4 128 > 1000 92 226.5 109 > 1000 89 > 1000 91  = 10−10 3 3 128 > 1000 90 233.5 110.5 > 1000 85 > 1000 94  = 10−16 2 2 128 > 1000 85 237.5 110.5 > 1000 91 > 1000 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’s method is more efficient than SD-CGN, but for large values of a, SD-CGN turns out to be more efficient than Widlund’s Lanczos method. Remark 8.3.1. As we see in Tables 2,3, and 4, the number of iterations for ESDCGN and ISD-CGN (with relative residual 10−7 for the solutions of the inner equations) are almost the same One might choose dynamic relative residuals for the solutions 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 −∆u + 10  ∂(exp(3.5(x2 + y 2 )u) ∂u + 10 exp(3.5(x2 + y 2 )) = f (x), ∂x ∂x  (8.43)  on [0, 1] × [0, 1] with Dirichlet boundary condition, and choose f so that sin(πx) sin(πy) 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 100 100 100 100 1000 1000 1000 1000 106 106 106 106 1016  N 49 225 961 961 49 225 961 961 49 225 961 961 961  I (ESD-CGN) 18 40 44 52 10 31 36 31 4 6 6 6 2  I (ISD-CGN) 18 37 46 51 10 31 37 39 4 6 6 6 2  Solution random random random sin πx sin πy. exp((x/2 + y)3 ) random random random sin πx sin πy. exp((x/2 + y)3 ) random random random sin πx sin πy. exp((x/2 + y)3 ) sin πx sin πy. exp((x/2 + y)3 )  which CG needs to converge to a solution with relative residual 10−6 when applied to the preconditioned matrix AT (αA−1 s + (1 − α)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 λsmax ( 1−α α ) = −.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 α with −1 < λsmax ( 1−α α ) ≤ 0. Our experiments show that for a well chosen α > 1, one may considerably decrease the number of iterations. Obtaining theoretical results on how to choose parameter α 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 − αλsmin I)−1 A = AT (As − αλsmin I)−1 b,  (8.45)  where λsmin is the smallest eigenvalue of As and α < 1. The number of iterations function of α, that CG needs to converges to a solution with relative residual 10−6 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 1 1 1 1 10 10 10 10 100 100 100 100 1000 1000 1000 100  N 49 225 961 961 49 225 961 961 49 225 961 961 49 225 961 961  I (ESD-CGN) 21 73 91 72 18 65 78 65 31 42 43 38 65 130 140 150  Solution random random random sin πx sin πy. exp((x/2 + y)3 ) random random random sin πx sin πy. exp((x/2 + y)3 ) random random random sin πx sin πy. exp((x/2 + y)3 ) random random random sin πx sin πy. exp((x/2 + y)3 )  Relative Residoual 6.71 × 10−6 9.95 × 10−6 8.09 × 10−6 9.70 × 10−6 9.97 × 10−6 5.90 × 10−6 8.95 × 10−6 7.78 × 10−6 6.07 × 10−6 5.20 × 10−6 5.03 × 10−6 4.69 × 10−6 4.54 × 10−6 8.66 × 10−6 2.12 × 10−6 5.98 × 10−6  Remark 8.3.3. As we see in the above table, for α = 0.99 in (8.45) we have the minimum number of iterations. Obtaining theoretical results on how to choose the parameter α seems to be an interesting problem to study. We also repeat the experiment by applying CG to the system of equations 0.99 I b. λsmax (8.46) Then CG needs 131 iterations to converge to a solution with relative residual 10−6 . As another experiment we apply CG to the preconditioned linear system AT  As − 0.99λsmin I)−1 −  0.99 I A = AT λsmax  (As − o.99λsmin I)−1 −  T −1 −1 T −1 A−1 s A As Ax = As A As b,  to solve the non-symmetric linear system obtained from discritization of the Equation (8.43). The CG converges in 31 iterations to a solution with relative residual less than 10−6 . 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 −∆u+10  ∂(exp(3.5(x2 + y 2 )u) ∂u +10 exp(3.5(x2 +y 2 )) −200u = f (x), ∂x ∂x  on [0, 1]×[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 − αλsmin I)−1 + βI)A = AT ((As − αλsmin I)−1 + βI)b,  (8.48)  with α < 1. For different values of α the number of iterations which CG needs to converge to a solution with the relative residual 10−6 are presented in Table 8. We  Table 8.6: Number of iterations for SD-CGN with different values of α. λsmax ( 1−α ) α ∞(α = 0) 0(α = 1) -0.1 -0.25 -0.5 -0.7 -0.8 -0.9 -0.95 -0.99 -0.999 -0.9999  I > 5000 229 221 216 201 191 186 180 179 177 180 234  λsmax ( 1−α ) α 0.1 0.2 0.4 0.8 1 5 10 20 50 100 1000 10000  I 232 237 249 263 272 384 474 642 890 1170 2790 4807  repeat our experiment with stepsize 1/61 and get a system with 3600 unknowns. With α = −1.00000001 and β = 0, CG converges in one single iteration to a solution with relative residual less than 10−6 . 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 10−6 are presented in Table 9. Acknowledgments: This paper wouldn’t 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 α. α 0 0.5 0.9 0.99 0.999 0.9999 0.99999 0.999999 0.9999999 0.99999999  I 229 204 177 166 168 181 194 222 248 257  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. Golub, A generalized conjugate gradient method for nonsymmetric systems of linear equations, Computing Methods in Applied Sciences and Engineering, Lecture Notes in Econom. and Math. Systems 134, R. Glowinski and J.L. Lions, eds., Springer-Verlag, Berlin, 1976, pp. 56-65; also available online fromh ttp://www.sccm. stanford.edu. [4] M. Eiermann, W. Niethammer, and R. S. Varga, Acceleration of relaxation methods for non-Hermitian linear systems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 979-991. [5] R. Fletcher, Conjugate gradient methods for indefinite systems, Lecture Notes in Math., 506 (1976), pp. 73-89. [6] N. Ghoussoub, Anti-selfdual Lagrangians: Variational resolutions of non selfadjoint equations and dissipative evolutions, AIHP-Analyse non lin´eaire, 24 (2007), 171-205. [7] N. Ghoussoub, Selfdual partial differential systems and their variational principles, Springer-Verlag, Universitext Series, In press (2007) 350 pp. [8] G. H. Golub and D. Vanderstraeten, On the preconditioning of matrices with a dominant skew-symmetric component, Numer. Algorithms, 25 (2000), pp. 223-239. [9] A. Greenbaum, Iterative Methods for Solving Linear Systems, Frontiers Appl. Math. 17, SIAM, Philadelphia, 1997. [10] J. A. Meijerink and H. A. Van Der Vorst, An iterative solution method for linear systems of which the coeJficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.  183  Chapter 8. Bibliography [11] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003. [12] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869. [13] P. Sonneveld, CGS: a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36-52. [14] H. A. 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 H 2 ∩ H01 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 dimension 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 ≤ N ≤ 8 does not work. So determining the critical 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 important 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 ≥ 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 eigenvalue 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, HardyPoincar´e inequalities and applications to nonlinear diffusions, C. R. Acad. Sci. Paris, Ser. I 344 (2007), 431-436. [3] H. Brezis and J. L. V´ azquez, Blowup solutions of some nonlinear elliptic problems, 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 Inequalities, Math. Annalen, Published Online (2010) 57 pp. [10] N. Ghoussoub, A. Moradifam,N. Ghoussoub and A. Moradifam, On simultaneous 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 pLaplace equations with Hardy potential in exterior domains, Journal of Differential 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 exponential nonlinearity, Proc. Amer. Math Soc., 138 (2010), 1287-1293. [14] A. Moradifam, Optimal weighted Hardy-Rellich inequalities on H 2 ∩ H01 , submitted. [15] A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Advances in Mathematics, 209 (2007) 407-459.  187  

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