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Regularity in second and fourth order nonlinear elliptic problems Cowan, Craig 2010

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Regularity in second and fourth order nonlinear elliptic problems by Craig Cowan  M.Sc., Simon Fraser University, 2006 B.Sc., Simon Fraser University, 2004  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 Craig Cowan 2010  Abstract This thesis consists of six research papers. In “Regularity of the extremal solution in a MEMS model with advection,” we examine the equation given by −∆u + c(x) · ∇u = λf (u) in Ω with Dirichlet boundary conditions and where f (u) = (1 − u)−2 or f (u) = eu . Our main result is that the associated extremal solution is smooth provided this is the case for the advection free case; c(x) = 0. In “Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems” we prove some results, which were observed numerically, regarding equations of the form −∆u = λ|x|α F (u) in B where B is the unit ball in RN . In addition we obtain upper and lower estimates on the extremal solutions associated with various nonlinear eigenvalue problems. In “The critical dimension for a fourth order elliptic problem with singular nonlinearity,” we examine the equation given by ∆2 u = λ(1 − u)−2 in B with Dirichlet boundary conditions where B is the unit ball in RN . Our main result is that the extremal solution u∗ is smooth if and only if N ≤ 8. In “Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains” we examine the equation ∆2 u = λf (u) in Ω with Navier boundary conditions where Ω is a general bounded domain in RN . We obtain various results concerning the regularity of the associated extremal solution. In “Regularity of the extremal solutions in elliptic systems” we examine the elliptic system given by −∆u = λev , −∆v = γeu in Ω where λ and γ are positive constants and we obtain results concering the regularity of the extremal solutions. In “Optimal Hardy inequalities for general elliptic operators with improvements” we examine some very general Hardy inequalities. Optimal constants are obtained and we characterize the improvements of these general Hardy inequalities. In addition we prove various weighted versions of these inequalities with improvements and many other results.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Second order nonlinear eigenvalue problems . . . . . . 1.1.1 Introduction to second order problems . . . . 1.1.2 Second order models with advection . . . . . . 1.1.3 Pull in distance in the MEMS model . . . . . 1.2 Fourth order nonlinear eigenvalue problems . . . . . . 1.2.1 Introduction to fourth order problems . . . . . 1.2.2 The critical dimension in a fourth order MEMS 1.2.3 Fourth order models on general domains . . . 1.2.4 Elliptic systems . . . . . . . . . . . . . . . . . 1.3 Hardy inequalities . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . model . . . . . . . . . . . .  1 1 1 3 4 5 5 6 6 7 9  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  2 Regularity of the extremal solution in a MEMS model with advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A general Hardy inequality and non-selfadjoint eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof of theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  16 16 19 20 24  iii  Table of Contents 3 Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lower estimates for the L∞ −norm of the extremal solution . 3.3 Upper estimates for the L∞ −norm of the extremal solution . 3.3.1 Upper estimates on general domains . . . . . . . . . . 3.3.2 Upper estimates on radial domains . . . . . . . . . . 3.4 Effect of power-law profiles on pull-in distances . . . . . . . . 3.5 Asymptotic behavior of stable solutions near the pull-in voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26 26 30 33 33 38 42 46  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  4 The critical dimension for a fourth order elliptic problem with singular nonlinearity . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The effect of boundary conditions on the pull-in voltage . . . 4.2.1 Stability of the minimal branch of solutions . . . . . . 4.3 Regularity of the extremal solution for 1 ≤ N ≤ 8 . . . . . . 4.4 The extremal solution is singular for N ≥ 9 . . . . . . . . . . 4.5 Improved Hardy-Rellich inequalities . . . . . . . . . . . . . .  53 53 57 61 66 70 76  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  80  5 Regularity of extremal solutions in fourth order eigenvalue problems on general domains . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.1.1 The second order case . . . . . . . . . . . . 5.1.2 The fourth order case . . . . . . . . . . . . 5.2 Sufficient Lq -estimates for regularity . . . . . . . . 5.3 A general regularity result for low dimensions . . 5.4 Regularity in higher dimension (I) . . . . . . . . . 5.5 Regularity in higher dimension (II) . . . . . . . . 5.6 Singular nonlinearities . . . . . . . . . . . . . . . .  nonlinear . . . . . . 82 . . . . . . 82 . . . . . . 83 . . . . . . 84 . . . . . . 86 . . . . . . 91 . . . . . . 96 . . . . . . 98 . . . . . . 101  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Regularity of the extremal solutions in elliptic systems . . 108 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 110  iv  Table of Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7 Optimal Hardy inequalities for general elliptic operators with improvements . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.1.1 Outline and approach . . . . . . . . . . . . . . . . . . 116 7.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.2.1 Weighted versions . . . . . . . . . . . . . . . . . . . . 130 7.2.2 More general weighted inequalities . . . . . . . . . . . 134 7.2.3 Improvements . . . . . . . . . . . . . . . . . . . . . . 135 7.2.4 Hardy inequalities valid for u ∈ H 1 (Ω) . . . . . . . . 144 7.2.5 H 1 (Ω) inequalities for exterior and annular domains . 148 7.2.6 The non-quadratic case . . . . . . . . . . . . . . . . . 151 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . 160 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161  v  Acknowledgements I am indebted to my supervisor Nassif Ghoussoub for his guidance and constant support. He introduced me to many interesting problems and was always available for help despite his busy schedule. He took me on as a student when many others would not. For all of this I am truly grateful.  vi  Dedication To my brother and my parents who have shown me immense support during my academic pursuits.  vii  Statement of Co-Authorship Chapters 2 and 3 were jointly authored by Craig Cowan and Nassif Ghoussoub. Chapter 4 was jointly authored by Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub and Amir Moradifam. Chapter 5 was jointly authored by Craig Cowan, Pierpaolo Esposito and Nassif Ghoussoub. Chapter 6 was authored by Craig Cowan. Chapter 7 was authored Craig Cowan. 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.  viii  Chapter 1  Introduction The main focus of research in this thesis is the study of the regularity of the extremal solution associated with various semi-linear elliptic problems and also generalized Hardy inequalities. This thesis consists of six papers which have either been published, accepted or submitted: [17, 19, 20, 18, 16, 15].  1.1 1.1.1  Second order nonlinear eigenvalue problems Introduction to second order problems  The following equation  v vt − ∆v = λ(1 − εv)m e 1+εv v = 0  in Ω, on ∂Ω,  is used to model a combustion process. The model is known as the solid fuel ignition model see [26]. It is a model for the thermal reaction process in a combustible, nondeformable material of constant density during the ignition period. Here λ is known as the Frank-Kamenetskii parameter, v is a dimensionless temperature, and 1ε is the activation energy. The following model has been proposed, see [36] and [37], for the description of the steady state of a simple Electrostatic MEMS device:   R 2u = β 2 dx + γ ∆u +    λg(x) α∆ |∇u| in Ω R  Ω dx  2 (1−u)  0<u<1    u = α∂ u = 0 ν  1+χ  Ω (1−u)2  in Ω on ∂Ω,  (1.1) where α, β, γ, χ ≥ 0, g ∈ C(Ω, [0, 1]) are fixed, Ω is a bounded domain in RN and λ ≥ 0 is a varying parameter. 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.  1  1.1. Second order nonlinear eigenvalue problems A first step in understanding the above models is to examine a simplified and generalized stationary version of the above models given by  −∆u = λf (u) in Ω, (P )λ u = 0 on ∂Ω, where λ is a positive parameter and where Ω is a bounded domain in RN . One also generally restricts the nonlinearities f to one of two classes: (R): f is smooth, increasing, convex with f (0) = 1 and limu→∞  f (u) u  = ∞,  or (S): f is smooth, increasing, convex with f (0) = 1 and limu%1 f (u) = ∞. We now recall some well known facts, see [8, 11, 14, 21, 32, 23, 4, 31]. Define λ∗ := sup {λ ≥ 0 : there exists a smooth solution of (P )λ } . Then 0 < λ∗ < ∞ and for each 0 < λ < λ∗ there exists a smooth solution of (P )λ and for λ > λ∗ there are no solutions of (P )λ even in a very weak sense. For 0 < λ < λ∗ there exists a smooth minimal solution, which we denote by uλ , here minimal means that if v is a solution of (P )λ then uλ ≤ v a.e. in Ω. It is also known that the minimal solution uλ is semi-stable in the sense that the (possibly formal) energy associated with the problem (P )λ has nonnegative second variation at uλ . In other words Z Z 0 2 λf (uλ )ψ dx ≤ |∇ψ|2 dx, ∀ψ ∈ H01 (Ω). (1.2) Ω  Ω  For each x ∈ Ω, λ 7→ uλ is increasing in (0, λ∗ ). This allows one to define, at least pointwise, the extremal solution u∗ (x) := lim∗ uλ (x). λ%λ  The extremal solution is indeed a weak solution of (P )λ∗ and in fact can be shown to be the unique weak solution. Now one can ask whether the extremal solution is a classical solution which means bounded when f satisfies (R) (or bounded away from 1 in the case where f satisfies (S)) or is indeed a honest to goodness weak solution of (P )λ∗ . It turns out that both situations can occur. This issue of the regularity of the extremal solution is the main focus of this thesis, at least as far as nonlinear eigenvalue problems are concerned. We now list some of the known results concerning the regularity of the extremal solution. Recall again that Ω is a bounded domain in RN . 2  1.1. Second order nonlinear eigenvalue problems 1. [30] Suppose f (u) = eu and Ω is the unit ball in RN . For N ≥ 10, u∗ = −2 log(|x|). Hence u∗ is unbounded. 2. [21] Suppose f (u) = eu and N ≤ 9. Then u∗ is bounded. 3. [34] Suppose f satisfies (R) and N ≤ 3. Then u∗ is bounded. In addition Lp estimates are obtained on f (u∗ ) in the case N ≥ 4. . 4. [13] Suppose f satisfies (R) and Ω is the unit ball in RN with N < 10. Then u∗ is bounded. Considering the first result above shows this is optimal. 5. [12] Suppose f satisfies (R) and Ω is a bounded convex domain in R4 . Then u∗ is bounded. In [39] various results concerning the regularity of u∗ are also obtained. One may ask why is the regularity of the extremal solution important. The ultimate goal is to understand the full bifurcation diagram associated with the problem (P )λ and the first step is to understand whether u∗ is a classical solution of (P )λ∗ . If this is the case then the results of [21] allows one to start a second branch of solutions emanating from (λ∗ , u∗ ).  1.1.2  Second order models with advection  A natural generalization of the problem (P )λ (in the case of the MEMS nonlinearity) is to consider the problem ( λ −∆u + c(x) · ∇u = (1−u) in Ω, 2 (S)λ u = 0 on ∂Ω, where λ > 0 is a parameter, Ω is a bounded domain in RN and where c(x) is a smooth vector field defined in Ω. Modifying the proofs used in analyzing the advection free case (c = 0) one can again show the existence of a positive finite critical parameter λ∗ such that for 0 < λ < λ∗ there exists a smooth minimal solution uλ of (S)λ , while there are no smooth solutions of (S)λ for λ > λ∗ . Moreover, the minimal solutions are also semi-stable in the sense that the principal eigenvalue of the corresponding linearized operator Lu,λ,c := −∆ + c(x) · ∇ −  2λ (1 − uλ )3  3  1.1. Second order nonlinear eigenvalue problems in H01 (Ω) is non-negative. See [6] where these results are proved for general C 1 convex nonlinearities which are superlinear at ∞. In Chapter 2 we consider the question of the regularity of the extremal solution associated with (S)λ . Our main result is given by the following: Theorem 1. If 1 ≤ N ≤ 7, then the extremal solution u∗ of (S)λ∗ is smooth. It should be noted that N = 7 is optimal here after one considers the case of c(x) = 0 on the unit ball in R8 , see [23]. In this paper we also show that 1 u if (1−u) 2 is replaced with e then the extremal solution is bounded provided N ≤ 9, which, again is optimal. One should note that the interesting case of c(x) is given when c(x) is divergence free. If c(x) is given by the gradient of a scalar function then (S)λ is of variational nature and the standard approach (when c(x) = 0) is easily modified to handle this case.  1.1.3  Pull in distance in the MEMS model  One can study slight generalizations of (P )λ given by  −∆u = λg(x)f (u) in B, (P )λ,g(x) u = 0 on ∂B, where λ > 0 is a parameter, B is the unit ball in RN and where g(x) ≥ 0 is a nonzero Hölder continuous function. In [29], in the case of the MEMS nonlinearity f (u) = (1 − u)−2 , it was noticed numerically that the L∞ norm of the extremal solution associated with (P )λ,|x|α , on the unit ball in R2 , was independent of α. In Chapter 3 we show, using a change of variables, that this is true for any nonlinearity f . Using the results from [13] we show that the extremal solution associated with (P )λ,|x|α is bounded provided N < 10 + 4α, for any general nonlinearity f satisfying (R). It was also noted numerically in [37] that maximum stable deflection of the membrane was always at least one third of the distance from the undeflected membrane to the ground plate, in our setting this translates to: the extremal solution u∗ associated with (P )λ in the case where f (u) = (1 − u)−2 satisfies ku∗ kL∞ ≥ 31 . In Chapter 3 we show this is indeed the case. In fact this estimate is essentially independent of the operator −∆. The estimate holds for any reasonable second or fourth order linear operator which satisfies a weak maximum principle. The initial motivation of writing Chapter 3 was to obtain upper estimates on the pull in distance in the case of the MEMS model, here we are defining the pull in distance as ku∗ kL∞ . Before our work generally one was only 4  1.2. Fourth order nonlinear eigenvalue problems interested in whether ku∗ kL∞ < 1 or = 1. We obtain various upper estimates and we also examine some asymptotics of the minimal solution uλ for λ close to λ∗ .  1.2 1.2.1  Fourth order nonlinear eigenvalue problems Introduction to fourth order problems  We now turn our attention to fourth order problems of the form ∆2 u = λf (u)  in Ω,  (1.3)  where, for the time being, we won’t specify the boundary conditions. If one examines the second order case (P )λ and in particular the proofs showing: the extremal parameter is finite, the existence of minimal solutions, the monotonicity of the minimal solutions in λ, they will realize that the availability of a weak maximum principle is crucial. By a weak maximum principle we mean that if f ≥ 0 is any reasonable function defined in Ω and ∆2 v = f in Ω with the imposed boundary conditions then v ≥ 0 a.e. in Ω. In other words the associated Greens function is nonnegative. Generally there are two popular boundary conditions for which one does indeed have the weak maximum principle: • (Dirichlet) The Dirichlet boundary conditions u = ∂ν u = 0 on ∂Ω where Ω is the unit ball in RN . The positivity of the Greens function here is due to [7] and is called Boggio’s Principle. • (Navier) The Navier boundary conditions u = ∆u = 0 on ∂Ω where Ω is a general domain. Here two applications of the second order maximum principle shows the desired result. The basic properties, not including the regularity of the extremal solution, associated with the fourth order problems were established in [2, 5]. In addition some results concerning the regularity of the extremal solution were obtained but these results can be considered as subcritical and critical results. A major breakthrough in the regularity of the extremal solution in fourth order problems is due to [22] where they examined the fourth order analog of the Gelfand Problem given by  in B,  −∆u = λeu u = 0 on ∂B, (D)λ  ∂ν u = 0 on ∂B 5  1.2. Fourth order nonlinear eigenvalue problems where B is the unit ball in RN . They showed that the extremal solution u∗ is bounded if and only if N ≤ 12. Their proof is very particular to the unit ball but can be extended to domains with enough symmetry, we omit the details. It is important to note that their proof does not use the usual method of energy estimates. Since this work there has been a flurry of results concerning the regularity of the extremal solution in fourth order problems with either Dirichlet or Navier boundary conditions but all are restricted to the unit ball.  1.2.2  The critical dimension in a fourth order MEMS model  In Chapter 4 we examine the fourth order MEMS model given by  (D)λ   2 λ  ∆ u = (1−u)2 0<u<1  u = ∂ν u = 0  in B in B on ∂B,  where B is the unit ball in RN . Our main result is given by the following theorem: Theorem 2. Let u∗ denote the extremal solution associated with (D)λ . Then supB u∗ < 1 if and only if N ≤ 8. Our approach is heavily inspired by the approach taken in [22]. We remark that we did not need to resort to a computer assisted proof to show that the extremal solution is singular in the intermediate dimensions as was the case in [22].  1.2.3  Fourth order models on general domains  In Chapter 5 we are interested in the regularity of the extremal solution u∗ associated with  2 ∆ u = λf (u) in Ω (N )λ u = ∆u = 0 on ∂Ω, where λ ≥ 0 is a parameter, Ω is a bounded domain in RN , N ≥ 2, and where f satisfies (R). Our main result is given by the following theorem: Theorem 3. Suppose f ,N and Ω is as above. The extremal solution u∗ is bounded in all of the following cases: 6  1.2. Fourth order nonlinear eigenvalue problems 1. N ≤ 5. (For N ≥ 6 we show that f (u∗ ) ∈ Lp (Ω) for all p <  N N −2 .)  2. f (t) = et or f (t) = (t + 1)p where p > 1 for N ≤ 8. 3. N ≥ 9 and f (t) = (t + 1)p where 1 < p < 4. lim inf t→∞  f (t)f 00 (t) f 0 (t)2  5. γ := lim supt→∞  N N −8 .  > 0 and N ≤ 7.  f (t)f 00 (t) f 0 (t)2  and N < γ8 .  Remark 1.2.1. Recall that in [22] they show that provided Ω is the unit ball in RN then the extremal solution associated with ∆2 u = λeu  Ω,  with Dirichlet boundary conditions is bounded if and only if N ≤ 12. With this in mind one fully expects that the extremal solution associated with (N )λ , in the case where f (t) = et , is bounded on a general domain provided N ≤ 12. One should note that we have vastly improved on the best result to date: u∗ is bounded for N ≤ 4 [5], but there is still much room for improvement. We also examine the case of singular nonlinearities of the form f (t) = where p > 1. Now the question of the regularity of the extremal solution is whether supΩ u∗ < 1 or supΩ u∗ = 1. Our result with respect to these singular nonlinearities is given by: 1 (1−t)p  Theorem 4. Suppose f (t) = then supΩ u∗ < 1.  1.2.4  1 (1−t)p  where p > 1 and p 6= 3. If N ≤  8p p+1  Elliptic systems  Closely related to fourth order models with Navier boundary conditions are elliptic systems. One can examine systems of nonlinear eigenvalue problems given by  −∆u = λf (v) Ω    −∆v = γg(u) Ω (P )λ,γ u = 0 ∂Ω    v = 0 ∂Ω where λ, γ are positive parameters, f and g satisfy (R) and where Ω is a bounded domain in RN . We now follow the work of M. Montenegro [33], 7  1.2. Fourth order nonlinear eigenvalue problems where all of the following results are taken from. We also mention that he obtains many more results and also that he studies a much more general system then (P )λ,γ . We let Q = {(λ, γ) : λ, γ > 0} and we define U := {(λ, γ) ∈ Q : there exists a smooth solution (u, v) of (P )λ,γ } . We set Υ := ∂U ∩ Q. The curve Υ is well defined and separates Q into two connected components Q and V. We omit the various properties of Υ but the interested reader should consult [33]. One point we mention is that if for x, y ∈ R2 we say x ≤ y provided xi ≤ yi for i = 1, 2 then it is easily seen, using the method of sub/supersolutions, that if (0, 0) < (λ0 , γ0 ) ≤ (λ, γ) ∈ U then (λ0 , γ0 ) ∈ U. Now it can be shown that for each (λ, γ) ∈ U there exists a smooth minimal solution (uλ,γ , vλ,γ ) of (P )λ,γ and if (0, 0) < (λ1 , γ1 ) ≤ (λ2 , γ2 ) ∈ U then (uλ1 ,γ1 , vλ1 ,γ1 ) ≤ (uλ2 ,γ2 , vλ2 ,γ2 ). Now for (λ∗ , γ ∗ ) ∈ Υ there is some 0 < σ < ∞ such that γ ∗ = σλ∗ and we can define the extremal solution (u∗ , v ∗ ) at (λ∗ , γ ∗ ) by passing to the limit along the ray given by γ = σλ for 0 < λ < λ∗ . Moreover it can be shown that (u∗ , v ∗ ) is indeed a weak solution of (P )λ∗ ,γ ∗ . We now come to the issue of stability. Theorem 5. ([33]) Let (λ, γ) ∈ U and let (u, v) denote the minimal solution of (P )λ,γ . Then (u, v) is semi-stable in the sense that there is some smooth 0 < φ, ψ ∈ H01 (Ω) and 0 ≤ K such that −∆φ = λev ψ + Kφ,  −∆ψ = γeu φ + Kψ,  In Chapter 6 we examine the system given by  −∆u = λev    −∆v = γeu (P )λ,γ u = 0    v = 0  Ω.  Ω Ω ∂Ω ∂Ω  where λ, γ are positive parameters and where Ω is a smooth bounded domain in RN . As before we let Q = {(λ, γ) : λ, γ > 0} and we define U := {(λ, γ) ∈ Q : there exists a smooth solution (u, v) of (P )λ,γ } . We set Υ := ∂U ∩ Q. Our main result is given by the following theorem. 8  1.3. Hardy inequalities Theorem 1.4. Let 3 ≤ N ≤ 9 and suppose that (λ, γ) ∈ Υ with N −2 γ 8 < < . 8 λ N −2 Then the associated extremal solution (u∗ , v ∗ ) is smooth.  1.3  Hardy inequalities  We begin by recalling the various Hardy inequalities. Let Ω be a bounded domain in Rn containing the origin and where n ≥ 3. Then Hardy’s inequality (see [35]) asserts that Z  2  |∇u| dx ≥ Ω    n−2 2  2 Z Ω  u2 dx, |x|2  (1.5)  2 for all u ∈ H01 (Ω). Moreover the constant n−2 is optimal and not at2 tained. An analogous result asserts that for any bounded convex domain Ω ⊂ Rn with smooth boundary and δ(x) := dist(x, ∂Ω) (the euclidean distance from x to ∂Ω), there holds (see [9]) Z 2 Z u 1 2 dx, (1.6) |∇u| dx ≥ 2 4 Ω δ Ω for all u ∈ H01 (Ω). Moreover the constant 41 is optimal and not attained. We will refer to this inequality as Hardy’s boundary inequality. Recently Hardy inequalities involving more general distance functions than the distance to the origin or distance to the boundary have been studied (see [3]). Suppose Ω is a domain in Rn and M a piecewise smooth surface of co-dimension k, k = 1, ..., n. In case k = n we adopt the convention that M is a point, say, the origin. Set d(x) := dist(x, M ). Suppose k 6= 2 and −∆d2−k ≥ 0 in Ω\M then Z Z 2 (k − 2)2 u 2 |∇u| dx ≥ dx, (1.7) 2 4 Ω Ω d for all u ∈ H01 (Ω\M ). We comment that the above inequalities all have Lp analogs. In the last few years improved versions of the above inequalities have been obtained, in the sense that non-negative terms are added to the right  9  1.3. Hardy inequalities hand sides of the inequalities; see [11], [9], [3], [10],[24], [25],[38]. One common type of improvement for the above Hardy inequalities are the so called potentials; we call 0 ≤ V (x), defined in Ω, a potential for (1.5) provided   Z Z Z n−2 2 u2 |∇u|2 dx − dx ≥ V (x)u2 dx, u ∈ H01 (Ω). 2 2 |x| Ω Ω Ω Most of the results in this direction are explicit examples of potentials V where, in the best results, V is an infinite series involving complicated inductively defined functions. Recently Ghoussoub and Moradifam [27] gave necessary and sufficient conditions for a nonnegative function V to be a potential for an improved Hardy inequality on a radial domain, which allowed them to unify all of the known results concerning explicit potentials, and to prove new ones, including Hardy-Rellich type inequalities [28]. In another direction people have considered Hardy inequalities for operators more general than the Laplacian. One case of this is the results obtained by Adimurthi and A. Sekar [1]: Suppose Ω is a smooth domain in Rn which contains the origin, A(x) = ((ai,j (x))) denotes a symmetric, uniformly positive definite matrix with suitably smooth coefficients and for ξ ∈ Rn we define |ξ|2A := |ξ|2A(x) := A(x)ξ · ξ. Now suppose E is a solution   A∇E = δ0 in Ω with E = 0 on ∂Ω where δ0 of LA,p (E) := −div |∇E|p−2 A is the Dirac mass at 0. Then for all u ∈ W01,p (Ω)   Z Z |∇E|pA p p−1 p 2 |∇u|A dx − |u| dx ≥ 0. p Ep Ω Ω Improvements of this were also obtained and they posed the folp  inequality p−1 lowing question: Is optimal? p In Chapter 7 we establish Hardy inequalities of the form Z Z |∇E|2A 2 1 2 |∇u|A dx ≥ u dx, u ∈ H01 (Ω) (1.8) 4 Ω E2 Ω where E is a positive function defined in Ω, −div(A∇E) is a nonnegative nonzero finite measure in Ω which we denote by µ and where A(x) is a n × n symmetric, uniformly positive definite matrix defined in Ω with |ξ|2A := A(x)ξ · ξ for ξ ∈ Rn . We show that (1.8) is optimal if E = 0 on ∂Ω or E = ∞ on the support of µ and is not attained in either case. When E = 0 on ∂Ω we show Z Z 2 Z |∇E|2A 2 1 1 u 2 |∇u|A dx ≥ u dx + dµ, u ∈ H01 (Ω) (1.9) 2 4 E 2 E Ω Ω Ω 10  1.3. Hardy inequalities is optimal and not attained. Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1.8) and (1.9) are established for p 6= 2 which, in the case that µ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar, see [1]. Since the above inequalities do not attain a standard question to ask is for what functions 0 ≤ V (x) do we have Z Ω  |∇u|2A dx  1 ≥ 4  Z Ω  |∇E|2A 2 u dx + E2  Z  V (x)u2 dx,  u ∈ H01 (Ω).  (1.10)  Ω  Necessary and sufficient conditions on V are obtained (in terms of the solvability of a linear pde) for (1.10) to hold. Analogous results involving improvements are obtained for the weighted versions. We establish optimal inequalities which are similar to (1.8) and are valid for u ∈ H 1 (Ω). We obtain results on improvements of this inequality which are similar to the above results on improvements. In addition weighted versions of this inequality are also obtained. Provided E satisfies various properties we show (1.8) holds and is optimal on exterior and annular domains for u non-zero on the inner boundary. We remark that most of the known Hardy inequalities (in the case where p = 2) can be obtained, via the above approach, by making suitable choices for E and A(x).  11  Bibliography [1] Adimurthi, Anusha Sekar, Role of the fundamental solution in HardySobolev-type inequalities, Proceedings of the Royal Society of Edinburgh, 136A, 1111-1130, 2006 [2] G. Arioli, F. Gazzola, H.C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36, 1226-1258 (2005). [3] B. Barbatis, S. Filippas, A. Tertikas, Series expansions for Lp Hardy inequalities, (August 20, 2007) preprint [4] L. Ben Chaabane,On the extremal solutions of semilinear elliptic problems, Abstr. Appl. Anal. 2005, no. 1, 1-9. [5] E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic Journal of Differential Equations, Vol. 2005(2005), No. 34, pp. 120. [6] H. Berestycki, A. Kiselev, A. Novikov and L. Ryzhik, The explosion problem in a flow, preprint (2008). [7] T. Boggio, Sulle funzioni di Green dordine m, Rend. Circ. Mat. Palermo (1905), 97-135. [8] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa Blow up for ut − ∆u = g(u) revisited, Ad. Diff. Eq. 1 (1996), 73-90. [9] H. Brezis, M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 217–237 (1998). [10] H. Brezis, M. Marcus, I.Shafrir, Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000), 177-191 [11] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469.  12  Chapter 1. Bibliography [12] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, (preprint) (2009). [13] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), no. 2, 709–733. [14] T. Cazenave, M. Escobedo and M. Assunta Pozio Some stability properties for minimal solutions of −∆u = λg(u). Port. Math. (N.S.) 59 (2002), no. 4, 373-391. [15] C. Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal. 9 (2010), no. 1, 109–140. [16] C. Cowan, Regularity of the extremal solutions in elliptic systems, submitted (2010). [17] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains, Discrete Contin. Dyn. Syst., In press (2010). [18] C. Cowan, P. Esposito, N. Ghoussoub, A. Mordifam, The critical dimension for a fourth order elliptical problem with singular nonlinearity, Arch. Ration. Mech. Anal., In press (2009) 19 pp [19] 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. [20] C. Cowan and N. Ghoussoub, Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems, Submitted (2009). [21] M.G. Crandall and P.H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), pp.207-218. [22] 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. [23] P. Esposito, N. Ghoussoub, Y. J. Guo: Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Research Monograph, In press (2009) 260 p.  13  Chapter 1. Bibliography [24] S. Filippas, A. Tertika, Optimizing improved Hardy inequalities, J.Funct. Anal. 192 (2002), n0.1, 186-233 [25] J.Fleckinger, E.M. Harrell II, F. Thelin, Boundary behaviour and estimates for solutions of equations containing the p-Laplacian, Electron. J. Differential Equations 38 (1999), 1-19 [26] G. R. Gavalas, “Nonlinear Differential Equations of Chemically Reacting Systems,” Springer-Verlag, New York Inc., New York, 1968. [27] 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. [28] N. Ghoussoub and A. Moradifam, Bessel Pairs and Optimal Hardy and Hardy-Rellich Inequalities, (preprint) 2008 [29] Y. Guo, Z. Pan and M.J. Ward, Touchdown and pull-in voltage behavior of a mems device with varying dielectric properties, SIAM J. Appl. Math 66 (2005), 309-338. [30] D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive source, Arch. Rational Mech. Anal. 49 (1973), 241269. [31] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, J. Diff. Eq. 16 (1974), 103-125. [32] F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), 791-836. [33] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc.37 (2005) 405-416. [34] G., Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sr. I Math. 330 (2000), no. 11, 997– 1002. [35] A. Kufner and B. Opic, Hardy Type Inequalities, Pitman Research Notes in Mathematics, Vol. 219, Longman, New York, 1990 [36] J.A. Pelesko, Mathematical modeling of electrostatic mems with tailored dielectric properties, SIAM J. Appl. Math. 62 (2002), 888-908.  14  Chapter 1. Bibliography [37] J.A. Pelesko and A.A. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002. [38] L. Vazquez and E. Zuazua, The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential, J. Funct. Anal. 173, 103-153 (2000). [39] D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math. 4 (2002), no. 3, p. 547-558.  15  Chapter 2  Regularity of the extremal solution in a MEMS model with advection1 2.1  Introduction  The following equation has often been used to model a simple Micro-ElectroMechanical System (MEMS) device: ( λ −∆u = (1−u) in Ω, 2 (P )λ u = 0 on ∂Ω, where Ω is a smooth bounded domain in RN , λ > 0 is proportional to the applied voltage and 0 < u(x) < 1 denotes the deflection of the membrane. This model has been extensively studied, see [10], [11] in regards to the model and [7], [6], [8] for mathematical aspects of (P )λ . It is well known (see above references) that there exists some positive finite critical parameter λ∗ such that for all 0 < λ < λ∗ , the equation (P )λ has a smooth minimal stable (see below) solution uλ , while for λ > λ∗ there are no weak solutions of (P )λ (see [7] for a precise definition of weak solution). Standard elliptic regularity theory yields that a solution u of (P )λ is smooth if and only if supΩ u < 1. One can also show that λ 7→ uλ (x) is increasing and hence one can define the extremal solution u∗ (x) := lim∗ uλ (x), λ%λ  which can be shown to be a weak solution of (P )λ∗ . Recall that a smooth solution u of (P )λ is said to be minimal if any other solution v of (P )λ satisfies u ≤ v a.e. in Ω. Such solutions are then 1  A version of this chapter has been accepted for publication: 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)  16  2.1. Introduction semi-stable meaning that the principal eigenvalue of the linearized operator Lu,λ := −∆ −  2λ (1 − u)3  in H01 (Ω) is nonnegative. This property can be expressed variationally by the inequality Z Z ψ2 |∇ψ|2 , ∀ψ ∈ H01 (Ω). (2.1) ≤ 2λ 3 (1 − u) Ω Ω which can be viewed as the nonnegativeness of the second variation of the energy functional associated with (P )λ at u. Now a question of interest is whether u∗ is a smooth solution of (P )λ∗ . It is shown in [7] that this is indeed the case provided N ≤ 7. This result is optimal in the sense that u∗ is singular in dimension N ≥ 8 with Ω taken to be the unit ball. Our main interest here will be in the regularity of the extremal solution associated with ( λ −∆u + c(x) · ∇u = (1−u) in Ω, 2 (S)λ u = 0 on ∂Ω, where c ∈ C ∞ (Ω, RN ) and where again Ω is a smooth bounded domain in RN . Modifying the proofs used in analyzing (P )λ one can again show the existence of a positive finite critical parameter λ∗ such that for 0 < λ < λ∗ there exists a smooth minimal solution uλ of (S)λ , while there are no smooth solutions of (S)λ for λ > λ∗ . Moreover, the minimal solutions are also semi-stable in the sense that the principal eigenvalue of the corresponding linearized operator Lu,λ,c := −∆ + c(x) · ∇ −  2λ (1 − uλ )3  in H01 (Ω) is non-negative. See [3] where these results are proved for general C 1 convex nonlinearities which are superlinear at ∞. Our main result concerns the regularity of the extremal solution of (S)λ . Theorem 2.2. If 1 ≤ N ≤ 7, then the extremal solution u∗ of (S)λ∗ is smooth.  17  2.1. Introduction Remark 2.1.1. A crucial (in fact the main) ingredient in proving the regularity of u∗ in (P )λ , is the energy inequality (2.1) which is used in conjunction with the equation (P )λ , to obtain uniform (in λ) Lp -estimates on (1 − uλ )−2 whenever uλ is the minimal solution (See [7]). However, the semi-stability of uλ in the case of (S)λ , does not translate into an energy inequality which allows the use of arbitrary test functions. Overcoming this will be the major hurdle in proving Theorem 6.2. We point out, however, that if c(x) = ∇γ for some smooth function γ on Ω̄, then the semi-stability condition on the minimal solution uλ of (S)λ translates into Z Z e−γ ψ 2 2λ ≤ e−γ |∇ψ|2 , ∀ψ ∈ H01 (Ω). (2.3) 3 Ω (1 − uλ ) Ω Then, with slight modifications, one can use the standard approach for (P )λ to obtain the analogous result for (S)λ stated in Theorem 6.2. The novel case is therefore when c is a divergence free vector field. Actually, we shall use the following version of the Hodge decomposition, in order to deal with general vector fields c. Lemma 2.4. Any vector field c ∈ C ∞ (Ω, RN ) can be decomposed as c(x) = −∇γ+a(x) where γ is a smooth scalar function and a(x) is a smooth bounded vector field such that div(eγ a) = 0. Proof. By the Krein-Rutman theory, the linear eigenvalue problem  ∆α + div(αc) = µα Ω, (∇α + αc) · n = 0 ∂Ω,  (2.5)  where n is the unit outer normal on ∂Ω, has a positive solution α in Ω when µ is the principal eigenvalue. Integrating the equation over Ω, one sees that µ = 0. The positivity of α on the boundary follows from the boundary condition and the maximum principle. In other words, we have that ∆α + div(αc) = 0 on Ω, and α > 0 on Ω̄. Now define γ := log(α) and a := c + ∇γ. An easy computation shows that div(eγ a) = 0. Throughout the rest of this note c, a, γ will be defined as above.  18  2.2. A general Hardy inequality and non-selfadjoint eigenvalue problems  2.2  A general Hardy inequality and non-selfadjoint eigenvalue problems  Consider the linear eigenvalue problem  −∆φ + c · ∇φ − ρφ = Kφ φ = 0  Ω, ∂Ω,  (2.6)  where c is a smooth bounded vector field on Ω, ρ ∈ C ∞ (Ω) and K is a scalar. We assume that (φ, K) is the principal eigenpair for (2.6) and that φ > 0 in Ω, and K ≥ 0. Note that elliptic regularity theory shows that φ is then smooth. We shall now use a general Hardy inequality to make up for the lack of a variational characterization for the pair (φ, K). The following result is taken from [4], which we duplicate here for the convenience of the reader. For a complete discussion on general Hardy inequalities including best constants, attainability and improvements of, see [4]. We should point out that this approach to Hardy inequalities is not new, but it is generally restricted to specific functions E which yield known versions of Hardy inequalities; see [1] and reference within. Lemma 2.7. Let A(x) denote a uniformly positive definite N × N matrix with smooth coefficients defined on Ω. Suppose E is a smooth positive function on Ω and fix a constant β with 1 ≤ β ≤ 2. Then, for all ψ ∈ H01 (Ω) we have Z Z Z |∇E|2A 2 β β(2 − β) −div(A∇E) 2 2 |∇ψ|A ≥ ψ + ψ , (2.8) 2 4 E 2 E Ω Ω Ω R R where Ω |∇ψ|2A = Ω A(x)∇ψ · ∇ψ. Proof. For simplicity we prove the case where A(x) is given by the identity matrix. For the general case, we refer to [4]. Let E0 denote a smooth positive function defined in Ω and let ψ ∈ Cc∞ (Ω). Set v := √ψE . Then 0  |∇ψ|2 = E0 |∇v|2 +  |∇E0 |2 2 ψ + v∇v · ∇E0 . 4E02  (2.9)  Integrating the last term by parts gives Z Z 1 −∆E0 2 v∇v · ∇E0 = ψ 2 Ω E0 Ω 19  2.3. Proof of theorem 2.2 and so integrating (6.8) gives Z Z Z 1 |∇E0 |2 2 1 −∆E0 2 2 |∇ψ| ≥ ψ , ψ + 2 4 Ω E0 2 Ω E0 Ω  (2.10)  where we dropped a nonnegative term. So we have the desired result for β = 1. When β 6= 1 one puts E0 := E β into (2.10) and collects like terms to obtain the desired result. We now use the above lemma to obtain an energy inequality valid for the principal eigenpair of (2.6). Theorem 2.11. Suppose that the principal eigenpair (φ, K) of (2.6) are such that φ > 0 and K ≥ 0. Then, for 1 ≤ β ≤ 2 we have for all ψ ∈ H01 (Ω), β(2 − β) e |∇ψ| ≥ 4 Ω  Z  γ  2  Z Ω  eγ |∇φ|2 2 β ψ + φ2 2  Z  β e ρ(x)ψ − 2 Ω γ  2  Z Ω  eγ a · ∇φ 2 ψ . φ (2.12)  Proof. Note that (2.6) can be rewritten as −div(eγ ∇φ) + eγ a · ∇φ = eγ (ρ(x) + K) φ  in Ω,  where as mentioned above we are using the decomposition c = −∇γ + a. We now set E := φ and A(x) = eγ I (where I is the identity matrix) and use (2.8) along with the above equation to obtain the desired result. Note that we have dropped the nonnegative term involving K.  2.3  Proof of theorem 2.2  For 0 < λ < λ∗ , we denote by uλ the smooth minimal semi-stable solution of (S)λ . Let (φ, K) denote the principal eigenpair associated with the linearization of (S)λ at uλ . Then 0 < φ in Ω, 0 ≤ K and (φ, K) satisfy ( 2λ −∆φ + c · ∇φ = ( (1−u Ω, 3 + K)φ λ) (2.13) φ = 0 ∂Ω. Again, elliptic regularity theory shows that φ is smooth. Consider c = −∇γ + a to be the decomposition of c described in Lemma 1. We now obtain the main estimate.  20  2.3. Proof of theorem 2.2 Theorem 2.14. For 0 < λ < λ∗ , 1 < β < 2 and 0 < t < β + have the following estimate:  λ β−  t2 2t + 1  Z Ω  p β 2 + β, we  eγ t+3 Ω (1 − uλ ) Z βkak2L∞ eγ + . 4(2 − β) Ω (1 − uλ )2t  eγ (1 − uλ )2t+3  Z  ≤ 2βλ  Proof. Fix 0 < β < 2, let 0 < t and u denote the minimal solution associated 2λ with (S)λ . We shall use Theorem 2.11 with ρ(x) = (1−u . Put ψ := )3 λ  1 (1−u)t  − 1 into (2.12) to obtain 2  Z  t  Ω  eγ |∇u|2 (1 − u)2t+2   2 eγ 1 ≥ βλ −1 3 (1 − u)t Ω (1 − u)   Z (2 − β) |∇φ|2 a · ∇φ β γ e − ψ2. + 2 Ω 2 φ2 φ Z  Now note that (S)λ can be rewritten as −div(eγ ∇u) + eγ a · ∇u = and test this on φ̄ := Z (2t + 1) Ω  1 (1−u)2t+1  Z  γ  e a · ∇u  H := Ω  in Ω,  − 1 to obtain  eγ |∇u|2 +H =λ (1 − u)2t+2  where  λeγ (1 − u)2  Z    Ω  eγ (1 − u)2     1 −1 , (1 − u)2t+1   1 −1 . (1 − u)2t+1  One easily sees R that H = 0 after considering the fact H can be rewritten in the form Ω (eγ a) · ∇G(u) for an appropriately chosen function G with G(0) = 0. Combining the above two inequalities and dropping some positive terms gives  Z Z eγ eγ t2 ≤ 2βλ λ β− 2t+3 3+t 2t + 1 Ω (1 − u) Ω (1 − u)  2 Z β 1 γ e Λ(x) −1 + 2 Ω (1 − u)t 21  2.3. Proof of theorem 2.2 where Λ(x) :=  a · ∇φ (2 − β) |∇φ|2 − . φ 2 φ2  Simple calculus shows that sup Λ(x) ≤ Ω  kak2L∞ , 2(2 − β)  which, after substituting into the above inequality, completes the proof of the main estimate. p Note now that the restriction t < β + β 2 + β is needed to ensure that t2 is positive. It follows then that (1−u1 )2 is uniformly the coefficient β − 2t+1 λ √ bounded (in λ) in Lp (Ω) for all p < p0 := 27 + 6 ≈ 5.94... and after passing to limits we have the same result for the extremal solution u∗ . To conclude the proof of Theorem 6.2, it suffices to note the following result. Lemma 2.15. Suppose 3 ≤ N ≤ 7 and the extremal solution u∗ satisfies 3N 1 ∈ L 4 (Ω), then u∗ is smooth. (1−u∗ )2 3N  Proof. First note that by elliptic regularity one has u∗ ∈ W 2, 4 (Ω) and 2 after applying the Sobolev embedding theorem one has u∗ ∈ C 0, 3 (Ω). Now suppose kukL∞ = 1 so that there is some x0 ∈ Ω such that u(x0 ) = 1. Then 1 C ≥ 2 1 − u(x) |x − x0 | 3 and hence  Z ∞> Ω  Z  1 ((1 − u∗ )2 )  3N 4  which is a contradiction. It follows that smooth.  ≥C Ω 1 (1−u∗ )2  1 = ∞, |x|N  ∈ L∞ (Ω), and u∗ is therefore  1 ∗ Using this lemma and the above Lp -bound on (1−u ∗ )2 , one sees that u is smooth for 3 ≤ N ≤ 7. To show the result in dimensions N = 1, 2, one needs a slight variation of the above argument. We omit the details, and the interested reader can consult [7] for the proof when c(x) = 0.  22  2.3. Proof of theorem 2.2 Remark 2.3.1. As mentioned in the abstract, this method applies to most non-selfadjoint eigenvalue problems of the form  −∆u + c(x) · ∇u = λf (u) in Ω, (S)λ u = 0 on ∂Ω, where f (u) is an appropriate convex nonlinearity such as f (u) = eu , and f (u) = (1 + u)p . It shows in particular that the presence of an advection does not change the critical dimension of the problem, hence addressing an issue raised recently by Berestycki et al [3]. One can also extend the general regularity results of Nedev [9] (for general convex f in dimensions 2 and 3) and those of Cabre and Capella [2] (for general radially symmetric f on a ball, and up to dimension 9). All these questions are the subject of a forthcoming paper [5].  23  Bibliography [1] Adimurthi and Anusha Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities, Proceedings of the Royal Society of Edinburgh, 136A, 1111-1130, 2006 [2] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), no. 2, 709–733. [3] H. Berestycki, A. Kiselev, A. Novikov and L. Ryzhik, The explosion problem in a flow, preprint (2008). [4] C. Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, (preprint). [5] C. Cowan, N. Ghoussoub, Regularity and stability of solutions in nonlinear eigenvalue problems with advection, (In preparation) (2008). [6] 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 (2007), 1731–1768. [7] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2007), 1423-1449. [8] Y. Guo, Z. Pan and M.J. Ward, Touchdown and pull-in voltage behavior of a mems device with varying dielectric properties, SIAM J. Appl. Math 66 (2005), 309-338. [9] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations. C. R. Math. Acad. Sci. Paris 330 (2000), 997–1002. [10] J.A. Pelesko, Mathematical modeling of electrostatic mems with tailored dielectric properties, SIAM J. Appl. Math. 62 (2002), 888-908.  24  Chapter 2. Bibliography [11] J.A. Pelesko and A.A. Berstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002.  25  Chapter 3  Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems2 3.1  Introduction  We examine problems of the form ( −∆u = λf (x)F (u) u=0  in Ω, on ∂Ω,  (Pλ,f )  where Ω is a bounded domain in RN , 0 < λ, f is a nonnegative nonzero bounded Hölder continuous function, usually dubbed as the permittivity profile, and where F is a suitably defined nonlinearity. We have in mind the examples F (u) = eu , F (u) = (u + 1)p where p > 1 and F (u) = (1 − u)−p where p > 0. We now formalize the classes of of nonlinearities we examine. Suppose F is a smooth, increasing, convex nonlinearity on its domain 0 ∈ DF ⊂ R, such that F (0) = 1. If DF = [0, ∞) and F is superlinear at ∞, then F is said to be a regular nonlinearity. When DF := [0, 1) and limu%1 F (u) = +∞ then we say that F is a singular nonlinearity. We say that a solution u of (Pλ,f ) is classical provided kukL∞ < ∞ (resp., kukL∞ < 1) if F is a regular (resp., singular) nonlinearity. Note that by elliptic regularity theory, this is equivalent to saying that a classical solution is in C 2,α for some α > 0. It is by now well-known (see [6], [25], [7], [19], [26], [3]) that – regardless whether F is a regular or singular nonlinearity – there exists an extremal 2  A version of this chapter has been submitted for publication: C. Cowan and N. Ghoussoub Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems  26  3.1. Introduction parameter λ∗ ∈ (0, +∞) depending on Ω, f and N , and which can be defined as λ∗ (Ω, f ) := sup{λ > 0 : (Pλ,f ) has a classical solution}, such that (Pλ,f ) has a minimal classical solution uλ for every λ ∈ (0, λ∗ ), and no solution for λ > λ∗ . By a “minimal solution” u, we mean one such that any other solution v of (Pλ,f ) satisfies v ≥ u a.e. in Ω. One can then also show that λ 7→ uλ (x) is increasing on (0, λ∗ ) for each x ∈ Ω. This allows us to define the extremal solution by u∗ (x) := lim∗ uλ (x), λ%λ  which can be seen as some kind of a “weak solution” for (Pλ∗ ,f ), see [25], [3]. We shall also need the notion of stability. Given a solution u of (Pλ,f ), we say that u is stable (resp., semi-stable) provided µ1 (λ, u) > 0, (resp., µ1 (λ, u) ≥ 0) where Z  Z 2 0 2 ∞ 2 µ1 (λ, u) := inf (|∇ψ| − λf (x)F (u)ψ )dx : ψ ∈ Cc (Ω), ψ = 1 . Ω  Ω  Under our assumptions on the nonlinearity F , and whether it is regular or singular, one can show that for all 0 < λ < λ∗ the minimal solution uλ is stable. If in addition, u∗ is a classical solution of (Pλ∗ ,f ), then necessarily µ1 (λ∗ , u∗ ) = 0, since otherwise one could use the Implicit Function Theorem, in a suitable function space, to obtain solutions to (Pλ,f ) for λ > λ∗ , which would be a contradiction. The question of the regularity of the extremal solution has attracted a lot of attention in the last decade. For general regular nonlinearities, with f (x) = 1, the extremal solution is classical provided one of the following holds: • Ω is contained in RN with N ≤ 3 [27]. This has recently been improved to N ≤ 4, see [8]. • Ω is a ball in RN with N ≤ 9, see [9]. • Ω is contained in RN where N ≤ 9 under various weak assumptions on F (u), see [31]. The second result is optimal after one considers F (u) = eu on the unit ball in R10 . It is an open question as to whether for 4 ≤ N ≤ 9, there is a 27  3.1. Introduction regular nonlinearity F and a domain Ω ⊂ RN on which the corresponding extremal solution is unbounded. In the case of the MEMS model, where F (u) = (1−u)−2 , it is known that the extremal solution is classical provided N ≤ 7 and that this result is optimal (see [19]). On the other hand, for any dimension N > 2, there exists a singular nonlinearity, namely F (u) = (1 − u)−p for some p := p(N ) > 0, such that the corresponding extremal solution is not classical (see Chapter 3 of [16]). In this paper, we are mostly interested in the quantitative aspects of the regularity of the extremal solution u∗ , which were initially motivated by the equation ( λf (x) −∆u = (1−u) in Ω, 2 (Mλ,f ) u=0 on ∂Ω. In dimension N = 2 this equation models a simple Micro-ElectromechanicalSystems MEMS device, which roughly consists of a dielectric elastic membrane that is attached to the boundary of Ω, by which upper surface has a thin conducting film. At a distance of 1 above the undeflected membrane sits a grounded plate, i.e., a plate held at zero voltage. When a voltage V > 0 is applied to the thin film of the membrane, it deflects towards the ground plate. After various physical limits of the parameters involved, a dimensional argument and a simplification, ones arrives at (Mλ,f ) for the steady state of the membrane. Here λ is proportional to the applied voltage V and the permittivity profile f (x) allows for varying dielectric properties of the membrane. As seen above (see [26] or [19]), one expects the extremal solution u∗ in small dimension N to be bounded away from 1, hence to be a classical solution. Since the parameter λ∗ corresponds to the critical voltage beyond which there is a snap-through, and since u∗ is the optimal deflection of the membrane, it is therefore important for the design of MEMS devices to know how the critical voltage λ∗ and the pull-in-distance – defined as ku∗ kL∞ – depend on the geometry of the membrane and on the permittivity profile. Several analytical and numerical estimates on λ∗ have been derived by Pelesko [28], Guo-Pan-Ward [22], Guo-Ghoussoub [19] and others in the 1 case of the MEMS non-linearity F (u) = (1−u) 2 . On the other hand, only numerical estimates have been obtained for the pull-in distance in the case of power-law (resp., exponential) permittivity profiles f (x) = |x|α (resp., f (x) = eαx ). In this paper, we shall see that one can give rigorous proofs and estimates for phenomena, which so far have only been observed numerically by various authors. We shall also include corresponding results for general – not necessarily MEMS-type – nonlinearities. 28  3.1. Introduction Here is a brief description of the paper. In section 2, we give upper estimates on the pull-in voltage λ∗ (Ω, f ) in fairly general situations, which will in turn yield lower bounds on ku∗ kL∞ . What is remarkable here is that the estimates – which are valid for general nonlinearities – turn out to only depend on the permittivity profiles and not on the domain, nor on the dimension. Actually, they also apply to any reasonable differential operator, see Remark 3.2.1. In section 3, we give upper estimates on ku∗ kL∞ which are computationally friendly. Just as in the proof of the regularity of u∗ in low dimensions, we use the energy estimates on the minimal solutions coupled with Lp to L∞ Sobolev-type constants related to corresponding linear equations. While the result is satisfactory for exponential nonlinearity, it is not so for the MEMS model, which led us to reconsider this nonlinearity in the case of the ball where more precise Lp to Hölder estimates can be used. We stress here that we are not interested in optimal upper estimates but rather estimates for which, if given a specific domain Ω and a nonlinearity F , one can easily obtain some numerical parameters by plotting a function of a single variable – possibly – using a Computer Algebra System. Section 4 was motivated by an intriguing phenomenon observed numerically by Guo-Pan-Ward [22], namely that on a two dimensional disc, the pull-in distance does not depend on the power of the permittivity profile f (x) = |x|α . We prove that this is indeed the case by a simple scaling argument which relates the problem (Pλ,|x|α ) on the unit ball of RN to (Pλ,1 ) (which for simplicity we denote by (Pλ )) on a ball in a fractional dimension N (α). (Note that when f is radial and Ω is the unit ball in RN , all stable solutions of (Pλ,f ) are then radial and hence we can examine the problem in fractional dimensions). One can then easily transfer many results established for (Pλ ) to (Pλ,|x|α ). This observation, combined for example with the results of Cabre and Capella [9], leads to new regularity results for the extremal solution associated with (Pλ,|x|α ). In section 5, we study the asymptotics in λ, and we obtain upper and lower pointwise bounds on the minimal solutions uλ , in the case where u∗ is singular. The upper estimates are valid on arbitrary domains and we restrict ourselves to radial domains for the lower estimates since more explicit bounds can then be found. For that we exploit the fact that both u∗ d and dλ uλ |λ=λ∗ are explicitly known in the case where Ω is a ball and u∗ is singular. We now list our main notation. For a nonlinearity F , we denote by aF the upper bound of the domain DF , which means that aF := ∞ if F is regular, and aF := 1 if F is singular, in such a way that DF := [0, aF ). 29  3.2. Lower estimates for the L∞ −norm of the extremal solution We shall also associate to F the numbers BF :=  sup τ ∈(0,aF )  τ F (τ )  and  CF :=  R aF 0  dτ F (τ ) .  (3.1)  The ball of radius R centred at x0 in RN will be denoted by BR (x0 ). If x0 = 0 then we omit x0 and if R = 1 then we just write B. Given a set Ω in RN we let |Ω| denote its N -dimensional Lebesgue measure, while ωN denotes the volume of the unit ball B in RN . The conjugate index of p will be denoted by p0 in such a way that p1 + p10 = 1. For a radial function u we write u(r) = u(|x|). The first eigenvalue of −∆ in H01 (Ω) will be denoted by λ1 (Ω) and the corresponding positive eigenfunction will be φΩ , assuming R the normalization Ω φΩ = 1. Before we proceed further, we would like to express our gratitude to the referees of this paper for their very pertinent comments and suggestions.  3.2  Lower estimates for the L∞ −norm of the extremal solution  This section is devoted to the proof of the following result. Theorem 3.2. Suppose F is either a regular or singular nonlinearity and that u∗ is the extremal solution of (Pλ,f ), which we assume to be classical. Then, R    1 inf Ω f 1 Ω f φΩ dx ∗ ∞ 0 −1 ku kL ≥ (F ) max , , (3.3) BF supΩ f CF supΩ f where we define (F 0 )−1 (z) = 0 for z < F 0 (0). Before proceeding with the proof, we give some applications. Corollary 3.4. Suppose f is a non-negative bounded Hölder continuous permittivity profile and that the extremal solution u∗ of (Pλ,f ) on a bounded domain Ω is regular. 1. If F (u) = ∗  1 (1−u)p ,  p > 0, then  ku kL∞ ≥ 1 − min    1 1  p p  supΩ f  p+1 supΩ f  p+1 R , p + 1 inf Ω f p + 1 Ω f φΩ dx   . (3.5) 30  3.2. Lower estimates for the L∞ −norm of the extremal solution In particular, when the qpermittivity f ≡ 1, then for any dimension 4p p 1 ≤ N ≤ 2 + p+1 + 4 p+1 , and any bounded domain Ω ⊂ RN , we have ku∗ kL∞ ≥  1 . p+1  (3.6)  2. If F (u) = (u + 1)p , p > 1, then  1  p − 1 R f φ dx  1  p  inf Ω f  p−1 Ω p−1 ∗ ∞ Ω ku kL ≥ max , −1 p − 1 supΩ f p supΩ f (3.7) In particular, when f ≡ 1, then for any dimension 1 ≤ N ≤ 10, and 1 . any bounded domain Ω ⊂ RN , we have ku∗ kL∞ ≥ p−1 3. If F (u) = eu , then ∗  +∞ > ku kL∞   inf f   R f φ dx  Ω Ω ≥ max 1 + log , log Ω . (3.8) supΩ f supΩ f   In particular, when the permittivity f ≡ 1, then for any dimension 1 ≤ N ≤ 9, and any bounded domain Ω ⊂ RN , we have +∞ > ku∗ kL∞ ≥ 1. Note that the dimension restrictions above (which are actually sharp, see [26] and [23]) are exactly what is needed to ensure that the corresponding extremal solution is regular. The proof of Theorem 3.2 follows immediately from the combination of the following two propositions. The first provides upper estimates on λ∗ , in terms of F , Ω and f . Proposition 3.2.1. Suppose F is either a regular or singular nonlinearity. Then   BF CF ∗ R λ (Ω, f ) ≤ λ1 (Ω) min , (3.9) , inf Ω f Ω f φΩ dx where BF and CF are given in (3.1). Proof. Supposing u is a classical solution of (Pλ,f ), we multiply both sides of the equation by φΩ and integrate to obtain Z (λF (u)f − λ1 (Ω)u) φΩ dx = 0. Ω  Since φΩ > 0 we must have λ ≤ λ1 (Ω) sup Ω  u λ1 (Ω) z λ1 (Ω)BF ≤ sup = . f F (u) inf Ω f z∈DF F (z) inf Ω f 31  3.2. Lower estimates for the L∞ −norm of the extremal solution For the second bound, multiply (Pλ,f ) by Z  φΩ F (u)  and integrate to obtain  Z  λf φΩ dx = Ω  = ≤ = =  φΩ dx F (u) Ω Z Z ∇u · ∇φΩ φΩ F 0 (u)|∇u|2 dx dx − F (u) F (u)2 Ω Ω Z ∇u · ∇φΩ dx F (u) Ω ! Z Z u(x) 1 ∇φΩ · ∇ dτ dx, F (τ ) Ω 0 ! Z Z u(x) 1 dτ dx λ1 (Ω) φΩ F (τ ) Ω 0 (−∆u)  ≤ λ1 (Ω)CF , after recalling the normalization of φΩ . Proposition 3.2.2. Suppose u∗ is the extremal solution of (Pλ,f ) which we assume to be classical. Then λ1 (Ω) ≤ λ∗ kf F 0 (u∗ )kL∞ .  (3.10)  Proof. Since u∗ is a classical solution we have µ1 (λ∗ , u∗ ) = 0 and hence there is some 0 < ψ ∈ H01 (Ω) such that −∆ψ = λ∗ f (x)F 0 (u∗ )ψ in Ω. Multiplying this by ψ gives Z Z Z ∗ 0 ∗ 2 2 λ f (x)F (u )ψ dx = |∇ψ| dx ≥ λ1 (Ω)ψ 2 dx, Ω  Ω  Ω  which shows the desired result. Remark 3.2.1. Note that the lower bounds (when f = 1) are independent of the domain. It is also fairly easy to adapt the proof below to show that they are not particularly exclusive to the Laplacian −∆. Indeed, the same lower bounds can be obtained if we replace it by any operator of the form L(u) := −div(A(x)∇u) where A(x) is a symmetric uniformly positive definite N ×N matrix defined in Ω. Moreover, the same arguments show that the extremal solution associated with ∆2 u = λF (u) Ω, 32  3.3. Upper estimates for the L∞ −norm of the extremal solution also satisfies the same lower bound, where for general domains Ω we restrict our attention to the Navier boundary conditions: u = ∆u = 0 on ∂Ω, while in the case of Ω being a ball we can use the Dirichlet boundary conditions: u = ∂ν u = 0 on ∂B. For recent advances on fourth order nonlinear eigenvalue problems, we refer to [10], [13], [15], [12], [4] and [2].  Upper estimates for the L∞ −norm of the extremal solution  3.3  In this section we look for upper estimates on the extremal solution u∗ associated with (Pλ ), where F is one of the three linearities considered in Corollary 3.4, and where we take f (x) = 1 for simplicity. The methods consist of combining the energy estimates – which are critical in showing that the extremal solution is regular in low dimension – with various L∞ and Hölder estimates for linear equations. We point out that our main interest here is to show how one can obtain explicit upper estimates on u∗ using rigorous analysis. We are not interested in (or claim to have) sharp estimates. The following simple observation can be useful when looking for upper estimates. Observation 3.3.1. Suppose u∗ is the extremal solution associated with (Pλ ) in Ω with extremal parameter λ∗ . Then the extremal solution associated with (Pλ ) in the domain Ωρ := ρΩ (where ρ > 0) is given by vρ∗ (x) := u∗ ( xρ ) with extremal parameter λ∗ (Ωρ ) =  3.3.1  λ∗ (Ω) . ρ2  Upper estimates on general domains  We begin with the case of exponential nonlinearities. Theorem 3.11. Suppose F (u) = eu , Ω is a bounded domain in RN and u∗ is the extremal solution associated with (Pλ ). 1. If 3 ≤ N ≤ 9, then ku∗ kL∞ ≤  λ1 (Ω)βN  |Ω|  N2 , e(N − 2) ωN  (3.12)  where βN    −1 := inf N 2t+1   4 1 2t 2t t ) 2t+1 ; 4t + 2 − N 2−t   N −2 <t<2 . 4 33  3.3. Upper estimates for the L∞ −norm of the extremal solution 2. If Ω ⊂ B 1 ⊂ R2 , then 2  ku∗ kL∞ ≤  λ1 (Ω) inf {M ; e  0 < t < 2} ,  where  2t  4  1 |Ω|  1  2t + 1 |Ω| 1  2t+1 t 2t+1 2 Λ M := ,( ) 2−t 2π 2t π and where we define for p > 1 and 0 < R < 1,  Z  R  (− log(r))p rdr.  Λ(p, R) := 0  We also consider the case of a MEMS nonlinearity. Theorem 3.13. Suppose F (u) = (1 − u)−2 , Ω is a bounded domain in RN and u∗ is the extremal solution associated with (Mλ ) in Ω. If 3 ≤ N ≤ 7, then 2 −  ku∗ kL∞ ≤ 1 − e  λ1 (Ω)γN 2(N −2)  |Ω| ωN  N  ,  (3.14)  where ( γN := inf  −3  16N 2t+3 27    2t 4t + 6 − 3N    2t 2t+3    4(2t + 1) 4t + 2 − t2  2 t  ;   √ 3(N − 2) <t<2+ 6 . 4 Remark 3.3.1. Using a similar approach one can show that if u∗ is the extremal solution associated with (Pλ ), in the case where F (u) = (u + 1)p , p > 1, and N = 3 or N = 4 then ku∗ kL∞ ≤  (p − 1)p−1 λ1 (Ω)βN,p  |Ω|  N2 , pp (N − 2) ωN  where ( βN,p = inf  (2tp − p − t2 )  −p t  2t−1  (2t − 1) 2t+p−1  p  + pt  p  (2p) t  2t−1  :  N 2t+p−1 (4t + 2p − 2 − N p) 2t+p−1 + max{t− p , tN,p } < t < tp ,  34  3.3. Upper estimates for the L∞ −norm of the extremal solution and where t− p := p −  p p2 − p,  t+ p := p +  p p2 − p,  pN p 1 − + . 4 2 2  tN,p :=  We have omitted N = 2 just for simplicity. To obtain estimates for N ≤ 10 one has to perform a bootstrap argument or restrict the range of values for p. For proving the above theorems we shall need the following easy lemmas. Lemma 3.15. Let Ω be a smooth bounded domain in RN . 1. If N ≥ 3 and τ > Z Ω  N 2,  then for all x ∈ Ω,  1 dy |y − x|(N −2)τ 0  2 1− N   10 τ  ≤  ωN  1  N 1− τ (τ − 1) (2τ − N )  τ −1 τ  τ −1 τ  2  1  |Ω| N − τ  .  2. If N = 2, τ > 1 and Ω ⊂ B 1 ⊂ R2 , then for all x ∈ Ω, 2  Z   10  τ0  (− log(|y − x|)) dy  τ  ≤ (2π)  τ −1 τ  Λ  Ω  1  τ |Ω| 2  τ −1 τ , 1 . τ − 1 π2  We shall need L∞ bounds on the solutions of Poisson linear equations. We settle here for the following elementary ones, though the constants involved could possibly be improved by using the approach of A. Brandt in [5]. Lemma 3.16. Suppose −∆u = g(x) ≥ 0 in Ω with u = 0 on ∂Ω where Ω a bounded domain in RN and g is smooth. 1. If N ≥ 3, then for all τ > kuk  L∞  N 2,  kgkLτ (τ − 1)  ≤  1  τ −1 τ  2  1  |Ω| N − τ  2  N τ (N − 2)ωNN (2τ − N )  τ −1 τ  .  2. If N = 2 and Ω ⊂ B 1 , then for all τ > 1, 2  kukL∞ ≤   1  τ −1 τ 2 τ kgkLτ Λ τ −1 , |Ω|1 π2  (2π)  1 τ  .  35  3.3. Upper estimates for the L∞ −norm of the extremal solution Proof. In both cases, we let v(x) denote the Newtonian potential of g, i.e., Z g(y) 1 v(x) := dy, N (N − 2)ωN Ω |y − x|N −2 for N ≥ 3 and 1 2π  v(x) :=  Z (− log(|y − x|))g(y)dy, Ω  for N = 2. Since 0 ≤ u(x) ≤ v(x) in Ω, it suffices to show the desired L∞ estimate on v. To do this, one uses (for N ≥ 3) Hölder’s inequality to write v(x) ≤  1 kgkLτ N (N − 2)ωN  Z Ω  1 dy |y − x|(N −2)τ 0   10 τ  .  and then use the integral estimate in the previous lemma. For the convenience of the reader we now derive some standard energy estimates for stable solutions of (Pλ ). Lemma 3.17. ([14], [26]) Suppose u is a classical semi-stable solution of (Pλ ). 1. If F (u) = eu , then for all 0 < t < 2, we have keu kL2t+1 ≤   4 1 1 t |Ω| 2t+1 . 2−t  2. If F (u) = (1 − u)−2 , then for all 0 < t < 2 + −2  k(1 − u)   k  3 t+ 2  L  ≤  4(2t + 1) 4t + 2 − t2  √  6, we have  2 t  2  |Ω| 2t+3 .  Proof. 1) Using the test function ψ := etu − 1, where 0 < t < 2, in the stability conditions gives Z Z λ u tu 2 e (e − 1) ≤ e2tu |∇u|2 . t2 Ω Ω Now testing (Pλ ) on φ = e2tu − 1 and rearranging, gives Z Z λ 2tu 2 e |∇u| = eu (e2tu − 1). 2t Ω Ω 36  3.3. Upper estimates for the L∞ −norm of the extremal solution Comparing the last two inequalities and dropping some positive terms gives Z  Z 2 1 1 (2t+1)u e ≤ e(t+1)u , − t 2 t Ω Ω and after an application of Hölder’s inequality on the right one obtains 1  4t  u  ke kL2t+1 ≤  (2 − t)  1  1 t  |Ω| 2t+1 .  (3.18)  2) Take ψ := (1 − u)−2 − 1, φ := (1 − u)−2t−1 − 1 and proceed as in 1) by putting ψ into the stability condition and testing (Mλ ) on φ. We obtain Z  Z 1 1 4 1 2 − ≤ 2 , 2 2t+3 t 2t + 1 t Ω (1 − u)t+3 Ω (1 − u) after dropping a couple of positive terms. Hölder’s inequality then yields   t t 2 1 1 4 − (3.19) ≤ 2 |Ω| 2t+3 . 2 2t+3 t 2t + 1 1−u L t  We now combine the energy estimates with the linear estimates to obtain upper estimates on u∗ . ∗  Proof of Theorem 3.11: Use Lemma 3.16 with g(x) := λ∗ eu and τ = 2t + 1 along with the estimate λ∗ ≤ λ1 e(Ω) to arrive at an estimate of the form λ1 (Ω) u∗ ku∗ kL∞ ≤ C(t, N, |Ω|) ke kL2t+1 , e where C(t, N, |Ω|) is provided by Lemma 3.16. Now replace the Lp −norm on the right using the energy estimates from Lemma 3.17 to arrive at the desired result. The restrictions on t are a result of the restrictions on τ in the linear estimates along with the restrictions on t from the energy estimates.  Proof of Theorem 3.13: Let Ω denote a bounded domain in RN where 3 ≤ N ≤ 7 and let u∗ denote the extremal solution associated with (Mλ ) in Ω. Since the reasoning works for any log-convex nonlinearity F (i.e., u 7→ log(F (u)) is convex), we define v := log(F (u∗ )), and so −∆v = −  d2 log(F (u)) du2  u=u∗  |∇u∗ |2 + λ∗ F 0 (u∗ )  in Ω, 37  3.3. Upper estimates for the L∞ −norm of the extremal solution with v = 0 on ∂Ω. Since F is log convex, the first term on the right is negative. We now define w by −∆w = λ∗ F 0 (u∗ ) w = 0  in Ω, on ∂Ω,  and so 0 ≤ v(x) ≤ w(x) a.e. in Ω by the maximum principle. Using the linear estimates from Lemma 3.16 with g(x) := λ∗ F 0 (u∗ ) one has k log  1 1 kL∞ = kvkL∞ ≤ kwkL∞ ≤ C̃τ λ∗ kF 0 (u∗ )kLτ = C̃τ λ∗ (1 − u)2 1 − u∗  3 L3τ  N τ Taking now τ = 2t 3 + 1 > 2 , we can then replace the L −norm on the right by using the energy estimates from Lemma 3.17, which will give the desired conclusion.   3.3.2  Upper estimates on radial domains  While the upper estimate on general domains obtained in the last subsection is quite satisfactory for the exponential nonlinearity, it is not so for the case of the MEMS nonlinearity. Indeed, using Maple one sees that if Ω := (0, 1)3 the unit cube in R3 (and so λ1 (Ω) = 3π 2 ), Formula (3.14) would then give that ku∗ kL∞ ≤ .993..., (3.20) which is clearly not a very good upper estimate. This is mainly due to the fact that we drop a potentially large term in the proof of Theorem 3.2, when we replaced v by w in order to apply the linear estimate of Lemma 3.2. Note that this was not needed for the exponential nonlinearity in the proof of Theorem 3.1. In this section we examine radial domains, where better results are available on u∗ , at least in the case of F (u) = (1 − u)−2 . One can also examine the exponential nonlinearity using this approach but we won’t do this since the last section seems to give satisfactory results. For simplicity, we shall also restrict our attention to the case of f ≡ 1. The main difference is that we use here Hölder estimates on linear equations versus the L∞ estimates of the last subsection. For the remainder of this section we assume that Ω is the unit ball B in N R and F (u) = (1 − u)−2 . We define the following parameter:  38  .  3.3. Upper estimates for the L∞ −norm of the extremal solution  γ(τ, N ) :=                  τ 2τ −1  N =1  τ 4(τ −1)  N =2  (τ −1) 1 τ  τ −1 τ  (N −2)N (2τ −N )  N ≥ 3.  τ −1 τ  Lemma 3.21. Let u denote a smooth radially decreasing solution of −∆u = g(r) ≥ 0 in the unit ball B of RN . If max{1, N2 } < τ < ∞, then one has the estimate: u(0) ≥ u(R) ≥ u(0) −  γ(τ,N )kgkLτ 1 τ ωN  N  R2− τ  for all R ∈ (0, 1).  (3.22)  Proof. When N = 1, we integrate the equation between 0 and r, and apply 1 τ0  Hölder’s inequality to obtain −u0 (r) ≤ kgkτ2r . Now integrate both terms between 0 and R, and use again Hölder’s inequality to obtain the desired result. When N ≥ 2, we multiply the equation by r and integrate over (0, R) to arrive at Z R 0 R(−u (R)) + (N − 2)(u(0) − u(R)) = rg(r)dr. 0  If now N = 2, then one has 0  Z  2  R  rg(r)dr ≤  R(−u (R)) = 0  kgkLτ R τ 0 1  2π 1− τ 0  .  Dividing by R and integrating the result over (0, R) gives the claim. Now take N ≥ 3. Since −u0 (R) ≥ 0 we can drop a term to arrive at Z Z R 1 g(x) (N − 2)(u(0) − u(R)) ≤ rg(r)dr = dx N ωN BR |x|N −2 0 Z  10 τ 1 kgkLτ ≤ , 0 dx (N −2)τ N ωN BR |x| and then use Lemma 3.15 to evaluate the integral on the right and finish the proof. We now come to the result which will yield our upper estimates on u∗ . 39  3.3. Upper estimates for the L∞ −norm of the extremal solution Theorem 3.23. Suppose u is a smooth semi-stable  solution of (Pλ ) on√the N unit ball B in R , where 1 ≤ N ≤ 11. Then, for max 0, N 2−3 < t < 2+ 6, we have RN −1 dR  3  1  Z 0   1 − kukL∞ +  4λ1 (B)γ(t+ 2 ,N ) 27  4(2t+1) 4t+2−t2  2  where 1 M := N    4(2t + 1) 4t + 2 − t2  t  R  4t+6−2N 2t+3  2t+3 ≤ M, (3.24)   2t+3 t  .  Remark 3.3.2. Note that the above theorem only shows that kukL∞ is bounded away from 1 if 4t + 6 − 2N ≥ 2N − 1 which, once coupled with the other condition on t cannot be satisfied in the higher dimensions. This is to be expected since the extremal solution u∗ satisfies u∗ (0) = 1 for N ≥ 8. Proof. Suppose u is a smooth semi-stable (so radial) solution of (Pλ ). Then, the above linear estimate applied with g(r) := λ(1 − u)−2 , gives that for all R ∈ (0, 1), N  1 − u(R) ≤ 1 − u(0) +  λγ(τ, N )k(1 − u)−2 kLτ R2− 2 1  .  ωNτ 1 (Ω) Now use the upper bound λ∗ ≤ 4λ27 from Proposition 3.2.1, take τ = t+ 23 , and replace the Lτ −norm on the right via the energy estimate from Lemma 3.17, to obtain  4λ1 (B)γ(t + 32 , N ) 1 − u(R) ≤ 1 − u(0) + 27 This yields the inequality Z 1 N ωN  0  1 − kukL∞ +  RN −1 dR  3  4λ1 (B)γ(t+ 2 ,N ) 27  where Z M0 := N ωN 0  1    4(2t + 1) 4t + 2 − t2  4(2t+1) 4t+2−t2  2 t  R  2 t  4t+6−2N 2t+3  R  4t+6−2N 2t+3  .  2t+3 ≤ M0  RN −1 dR . (1 − u(R))2t+3 40  3.3. Upper estimates for the L∞ −norm of the extremal solution t+ 32  But the right hand side is actually equal to k(1 − u)−2 k  3  Lt+ 2  , hence we can  again use the energy estimate from Lemma 3.17 to majorize it and complete the proof. Remark 3.3.3. Using Maple to approximate the integral in (3.24) while optimizing over t, we get the following estimates on the extremal solution u∗ of (Pλ ) on the unit ball in RN . 1. If N = 1, then ku∗ kL∞ ≤ .49... 2. If N = 2, then ku∗ kL∞ ≤ .55... We note here that even though these estimates are not better than those obtained numerically (for example in [22] where kukL∞ = .4444...), the idea here is that we only use elementary analysis that is also applicable to other domains and nonlinearities. We now obtain some explicit upper bounds on u∗ in dimensions N = 1, 2. For that, we define  2 4λ1 (B)γ(t + 23 , N ) 4(2t + 1) t C(t, N ) := . 27 4t + 2 − t2 Corollary 3.25. Suppose u∗ is the extremal solution of (Pλ ) on the unit ball in RN . 1. If N = 1, then n √ o ku∗ kL∞ ≤ 1 − sup M1 : 0 < t < 2 + 6 , where  M1 :=  2C(t, 1)(t + 1)  4(2t + 1) 4t + 2 − t2   2t+3 t  1 + C(t, 1)2+2t  −1 ! 2t+2  .  2. If N = 2, then   √ 1 ku∗ kL∞ ≤ 1 − sup M2 : ≤ t < 2 + 6 , 2 where M2 :=  C(t, 2)2 (t + 1)    4(2t + 1) 4t + 2 − t2   2t+3 t  2t + 2 + C(t, 2)2t+1  −1 ! 2t+1  .  41  3.4. Effect of power-law profiles on pull-in distances √ Proof. 1) For 0 < t < 2+ 6 one has 4t+6−2N ≥ 1 and so we can replace the 2t+3 power on R in (3.24) by 1, so as to be able to explicitly calculate the integral in (3.24). One then drops√a few positive terms to arrive at the desired result. 2) For 12 ≤ t < 2 + 6 one has 4t+6−2N ≥ 1, so again we replace the 2t+3 power on R in (3.24) by 1 and carry on as in the first part.  3.4  Effect of power-law profiles on pull-in distances  Our goal in this section is to study the effect of power-like permittivity profiles f (x) = |x|α on the problem (Pλ,α ) (our notation for (Pλ,|x|α )) on the unit ball B = B1 (0). Numerical results – in particular those obtained by Guo, Pan and Ward in [22] for MEMS nonlinearities– give lots of information, but the most intriguing one is their observation that on a 2-dimensional disc, the pull-in distance does not depend on α, at least in the case where F (u) = (1 − u)−2 , and that the solution develops a boundary-layer structure near the boundary of the domain as α is increased. In other words, the L∞ −norm of the extremal solution of (Mλ,α ) is independent of α ≥ 0. In this section, we shall give a simple proof of this observation and other interesting phenomena, which actually holds true for more general nonlinearities. We first observe that since r 7→ rα is increasing, the moving plane method of Gidas, Ni and Nirenberg [21] does not guarantee the radial symmetry of all solutions to (Sλ,f ). However if ones assumes the solution is minimal or semi-stable then the solution is indeed radial, see for instance [19]. Similar results are proven in [1]. We include the proof from [19] for the readers convenience. Proposition 3.4.1. Let Ω be a radially symmetric domain and assume f is a radial profile on Ω. Then, the minimal solutions of (Pλ,f ) on Ω are necessarily radially symmetric and consequently  λ∗ (Ω, f ) = λ∗r (Ω, f ) = sup λ; (Pλ,f ) has a radial solution . Moreover, if Ω is a ball, then any radial solution of (Pλ,f ) attains its maximum at 0. Proof. It is clear that λ∗r (Ω, f ) ≤ λ∗ (Ω, f ), and the reverse will be proved if we establish that every minimal solution of (Pλ,f ) with 0 < λ < λ∗ (Ω, f ) is radially symmetric. The recursive linear scheme that is used to construct the minimal solutions, gives a radial function at each step, and the resulting limiting function is therefore radially symmetric. 42  3.4. Effect of power-law profiles on pull-in distances For a solution u(r) on the ball of radius R, we have ur (0) = 0 and −urr −  N −1 ur = λf (r)F (u) r  in  (0, R) .  N −1  Hence, − d(r dr ur ) = λf (r)rN −1 F (u) ≥ 0, and therefore ur < 0 in (0, R) since ur (0) = 0. This shows that u(r) attains its maximum at 0, and that – just as in the case where f ≡ 1 – we have ku∗ k∞ = u∗ (0). It follows from this proposition that for radially symmetric domains Ω and profiles f , the extremal solution u∗ is necessarily radially symmetric and that the pull-in distance is nothing but u∗ (0). We shall denote by λ∗α (N ) (resp., u∗α ) the pull-in voltage (resp., the extremal solution) of (Pλ,f ) when f (x) = |x|α , and Ω is the unit ball in RN . Given a radial function u we define, 0 the possibly, fractional n dimensional Laplacian of u by ∆n u = u00 + n−1 r u. We now make the following crucial observation. α  Proposition 3.4.2. For any α > −2, the change of variable u(r) = w(r1+ 2 ) gives a correspondence between the radially symmetric solutions of the equation  −∆N u = λ(1 + α2 )2 |x|α F (u) in B, (3.26) u=0 on ∂B, in dimension N and those of the equation ( −∆ 2(N +α) w = λF (w) in B,  (3.27)  2+α  w=0  on ∂B,  in – the potentially fractional – dimension N (α) = have λ∗α (N ) = (1 + α2 )2 λ∗0 (N (α))  and  2(N +α) 2+α .  α  Moreover, we  u∗α (r) = w∗ (r1+ 2 ),  (3.28)  where u∗α is the extremal solution for (3.26) and w∗ is the extremal solution of (3.27). Proof: A straightforward calculation gives that   α α α α ∆N u(r) + (1 + )2 λrα F (u(r)) = (1 + )2 rα ∆N (α) w(r1+ 2 ) + λF (w(r1+ 2 , 2 2 +α) where N (α) = 2(N 2+α . To finish the proof it suffices to use the fact that the extremal solution is unique in either problem, including the case of a fractional dimension. The change of variables will therefore take the extremal  43  3.4. Effect of power-law profiles on pull-in distances solution to the extremal solution. We leave the details to the interested reader.  The above transformation allows us to deduce many results for the case of a power-law profile, from corresponding ones associated to constant profiles. The fact that it preserves the L∞ -norm has consequences on the pull-in distance and on the role of the profile in the critical dimension. It does also give proofs for various intriguing phenomena displayed by the numerical results below, especially in the case of a two dimensional disc, where the transformation does not alter the dimension since then N (α) = 2. The following corollary summarizes these consequences. Corollary 3.29. With the above notations, the following hold: 1. For any dimension N ≥ 1, we have for α >> 1, λ∗α (N ) ∼ (1 +  α 2 ∗ ) λ0 (2). 2  (3.30)  2. If N = 2, then λ∗α (2) = (1 + α2 )2 λ∗0 (2)  and  ku∗α kL∞ = ku∗0 kL∞ for all α > −2. (3.31)  +2α ), Proof. 1) From the above proposition, we have λ∗α (N ) = (1 + α2 )2 λ∗0 ( 2Nα+2 2N +2α ∗ ∗ and λ0 ( α+2 ) ∼ λ0 (2) whenever α is large. 2) follows from the fact that for N = 2, we then have Nα = 2 for each α which means that (α + 2)2 ∗ (3.32) λ∗α (2) = λ0 (2), 4 and the pull-in distance in dimension 2 on the ball is ku∗α kL∞ = kw∗ kL∞ , α where u∗α (r) = w∗ (r1+ 2 ). The pull-in distance is therefore independent of α.  Corollary 3.33. The following estimates hold in a MEMS model with a power-law permittivity profile, i.e., if F (u) = (1 − u)−2 and f (x) = |x|α . 1. For any dimension N ≥ 1, we have for α >> 1, λ∗α (N ) ∼ 0.789(1 +  α 2 ) . 2  (3.34)  44  3.4. Effect of power-law profiles on pull-in distances 2. If N = 2, then λ∗α (2) = 0.789(1 + α2 )2  and  ku∗α kL∞ = 0.445 for all α > −2. (3.35) √  −14−4 √ 6 , then the 3. If 1 ≤ N ≤ 7 or if N ≥ 8 and α > αN := 3N 4+2 6 extremal solution u∗α of (Mλ,α ) on the ball is classical and the pull-in distance ku∗α kL∞ < 1.  4. If the dimension N ≥ 8, and 0 ≤ α ≤ αN := extremal solution is exactly λ∗α (N ) =  u∗α (x)  = 1 − |x|  (2+α)(3N +α−4) 9  and  2+α 3  √ 3N −14−4 √ 6, 4+2 6  then the  , which means that  ku∗α kL∞ = 1.  (3.36)  Proof. 1) and 2) follow from the above proposition and the fact that λ∗0 (2) = 0.789 and ku∗0 kL∞ = 0.445. 3) The extremal solution u∗α of (Mλ,α ) is regular if and only if ku∗α kL∞ = kw∗ kL∞ < 1, where w∗ is the extremal solution for (Mq λ ) in dimension N (α). According to [19], this happens if  α > αN :=  √ 3N −14−4 √ 6. 4+2 6  N (α) 2  < 1+  4 3  +2  2 3  which means that  2+α  4) Note first that u∗ (x) = 1 − |x| 3 is a H01 (B)−weak solution of +α−4) (Mλ,|x|α ) for any α > 4 − 3N . The voltage is then λα (N ) = (2+α)(3N . 9 ∗ Since now ku kL∞ = 1, then by the above proposition, it remains only to show that for all φ ∈ H01 (B), Z Z 2λ|x|α 2 2 |∇φ| ≥ φ . (3.37) ∗ 3 B B (1 − u ) But Hardy’s inequality gives for N ≥ 3 that  R B  |∇φ|2 ≥  (N −2)2 4  any φ ∈ H01 (B), which means that (3.37) holds whenever 2λα (N ) or, equivalently, if N ≥ 8 and 0 ≤ α ≤ αN =  √ 3N −14−4 √ 6. 4+2 6  φ2 B |x|2 for 2 ≤ (N −2) 4  R  The above scaling has also the following direct consequences. Corollary 3.38. Suppose F is a regular nonlinearity and N < 10 + 4α, then the extremal functional u∗α of (Pλ,α ) on the ball is classical. Proof. Cabre and Capella [9] showed that the extremal solution on the ball is always bounded for N ≤ 9. They were only interested in integer dimensions, but an inspection of their proof indicates that the same result holds for 45  3.5. Asymptotic behavior of stable solutions near the pull-in voltage any fractional dimensions N < 10. Combining this with our observation in Proposition 3.4.2 completes the proof. To see that this is optimal one recalls that when F (v) = ev the extremal solution is unbounded in N = 10. Using this fact and the change of variables above yields the optimality of this result. Remark 3.4.1. One can also use this change of variables to study permittivity profiles with negative powers (i.e., f (x) = |x|α for 0 > α > −2. For example suppose F (u) = eu , then using the above change of variables, one can show that for a fixed N (3 ≤ N ≤ 9), the extremal solution associated with −∆u = |x|α eu on B, is singular for α ∈ (−2, 10−N 4 ], while it is a classical solution for α ∈ 10−N ( 4 , 0).  3.5  Asymptotic behavior of stable solutions near the pull-in voltage  We now establish pointwise upper and lower estimates on the minimal sod lutions uλ in terms of λ, λ∗ , the extremal solution u∗ and dλ uλ λ=λ∗ . For u simplicity we restrict our attention to F (u) = e and F (u) = (1 − u)−2 . In addition we allow fractional dimensions for results on the unit ball since then, one can apply the results of the previous section to deal with powerlaw profiles (Pλ,α ). We first recall that on a ball, one can explicitly write the extremal solutions whenever they are singular. See for example [7, 9, 19]. • If F (u) = eu , then u∗ (x) = log( |x|1 2 ) is an extremal solution on the unit ball in RN at λ∗ = 2N − 4, provided N ≥ 10. • If F (u) =  1 , (1−u)2  2  then u∗ (x) = u∗ (x) = 1−|x| 3 is an extremal solution  on the unit ball in RN at λ∗ =  6N −8 9 ,  provided N ≥  √ 14+ 6 . 3  Theorem 3.39. Let u∗ denote the extremal solution of (Pλ ) on a smooth bounded domain Ω in RN . 1. If F (u) = (1 − u)−2 , then for 0 < λ < λ∗ , we have  uλ (x) ≤  λ λ∗  1 3  u∗ (x)  for a.e. x ∈ Ω.  (3.40)  46  3.5. Asymptotic behavior of stable solutions near the pull-in voltage √  6 Moreover, if Ω is the unit ball in RN with N ≥ 14+4 = 7.93..., then 3 6N −8 ∗ for 0 < λ < λ = 9 we have   √ 9N 2 −84N +100 −N 2 3(λ∗ − λ) +1+ 6 |x| 2 1 − |x| 3 − − 1 ≤ uλ (x) (6N − 8)  1 2 λ 3 uλ (x) ≤ (1 − |x| 3 λ∗  for a.e. x ∈ Ω. 2. If F (u) = eu , then for 0 < λ < λ∗ ,   λ∗ uλ (x) ≤ log λ∗ − λ + λe−u∗  for a.e. x ∈ Ω.  (3.41)  Moreover, if Ω is the unit ball in RN with N ≥ 10, then for 0 < λ < λ∗ = 2N − 4 we have   √ 2 +20 −N 1 (λ∗ − λ) +1+ N −12N 2 2 log( 2 ) − |x| − 1 ≤ uλ (x) |x| (2N − 4)   λ∗ uλ (x) ≤ log , λ∗ − λ + λ|x|2 for a.e. x ∈ Ω. Proof. The upper estimates follow easily from of uλ and the   the ∗minimality 1 λ 3 ∗ λ fact that x 7→ λ∗ u (x) (resp., x 7→ log λ∗ −λ+λe−u∗ is a supersolution of (Pλ ) in the case that F (u) = (1 − u)−2 (resp., F (u) = eu ). For the lower bound, we shall proceed as follows: First, recall that λ 7→ uλ is differentiable (this is a result of the Implicit Function Theorem) and d increasing on (0, λ∗ ), and so if one defines vλ := dλ uλ , then vλ is positive and solves the linear equation  −∆v = F (uλ ) + λF 0 (uλ )v in Ω, (Qλ ) v=0 on ∂Ω, where, F is given by either eu or (1 − u)−2 . We shall need the following notion. Definition 3.42. An extremal solution u∗ associated with (Pλ ) is said to be super-stable provided there exists ε > 0 such that R R (λ∗ + ε) Ω F 0 (u∗ )ψ 2 ≤ Ω |∇ψ|2 for all ψ ∈ H01 (Ω). 47  3.5. Asymptotic behavior of stable solutions near the pull-in voltage Note that if u∗ is a super-stable extremal solution then µ1 (λ∗ , u∗ ) > 0. We shall see at the end of this section that the converse is however not true. We shall need the following result, a part of which is contained in [11], where many other interesting properties of the mapping λ 7→ uλ (x) are exhibited. Lemma 3.43. Assume Ω is a smooth bounded domain in RN . Then, 1. For 0 < λ < λ∗ , vλ is the unique H01 −weak solution of (Qλ ). 2. λ 7→ vλ is increasing on (0, λ∗ ), and therefore v ∗ (x) := limλ→λ∗ vλ (x) is defined for a.e. x ∈ Ω. 3. λ 7→ uλ is convex on (0, λ∗ ), and therefore for 0 < λ < λ∗ we have for a.e. x ∈ Ω, uλ (x) ≥ u∗ (x) + (λ − λ∗ )v ∗ (x). (3.44) 4. If u∗ is super-stable, then v ∗ is the unique H01 −weak solution of (Q)λ∗ . Proof. (1) One can use the fact that µ1 (λ, uλ ) ≥ 0, and a standard minimization argument to show the existence of an H01 −solution to (Qλ ). Using the fact that µ1 (λ, uλ ) > 0 one can see that the solution is unique. (2) Let 0 < λ < λ∗ and ε > 0 small. Note first that −∆(vλ+ε − vλ ) = F (uλ+ε ) − F (uλ ) + εF 0 (uλ+ε )vλ+ε +λF 0 (uλ+ε )vλ+ε − λF 0 (uλ )vλ = g(x) + λF 0 (uλ )(vλ+ε − vλ ), where g(x) := F (uλ+ε ) − F (uλ ) + εF 0 (uλ+ε )vλ+ε + λ F 0 (uλ+ε )vλ+ε − F 0 (uλ )vλ+ε    is in H 1 (Ω) and is positive. Now set w := vλ+ε − vλ in such a way that w solves −∆w = g(x) + λF 0 (uλ )w w = 0  on Ω, on ∂Ω.  Testing this equation on w− gives Z Z − gw− ≥ µ1 (λ, uλ ) (w− )2 , Ω  Ω  and hence w− = 0 a.e. in Ω. By the maximum principle one then get that w > 0 in Ω and hence that λ → vλ is increasing. We can therefore define 48  3.5. Asymptotic behavior of stable solutions near the pull-in voltage the limit v ∗ (x) := limλ→λ∗ vλ (x), which exists a.e. x in Ω, though it might be infinite on a large set. (3) The convexity of λ 7→ uλ follows from the fact that λ 7→ vλ is increasing. We can therefore write uλ ≥ ut + (λ − t)vt for 0 < λ, t < λ∗ and a.e. x ∈ Ω. The claim now follows by letting t go to λ∗ . (4) Since u∗ is super-stable one has Z Z |∇ψ|2 ∀ψ ∈ H01 . (λ + ε) F 0 (uλ )ψ 2 ≤ Ω  Ω  Using this and testing (Qλ ) on vλ gives Z Z F (uλ )vλ . ε F 0 (uλ )vλ2 ≤ Ω  Ω  Since F is either F (u) = eu or F (u) = (1 − u)−2 , the left hand side is necessarily bounded. From this and again by testing (Qλ ) on vλ one sees that vλ is bounded in H01 . Passing to limits, one sees that v ∗ is a H01 −weak solution of (Qλ∗ ). The uniqueness follows from the fact that µ1 (λ∗ , u∗ ) > 0. We now complete the proof of Theorem 3.39. For that we assume that Ω is the unit ball in RN . 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Math. 4 (2002), no. 3, p. 547-558.  52  Chapter 4  The critical dimension for a fourth order elliptic problem with singular nonlinearity3 4.1  Introduction  The following model has been proposed for the description of the steadystate of a simple Electrostatic MEMS device:   R 2 2   λf (x)  in Ω R  dx  α∆ u = β Ω |∇u| dx + γ ∆u + 2 (1−u)  0<u<1    u = α∂ u = 0 ν  1+χ  Ω (1−u)2  in Ω on ∂Ω,  (4.1) 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 [20]). 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 (4.1) that has been studied extensively so far is the equation  f (x)   −∆u = λ (1−u)2 in Ω (S)λ,f 0<u<1 in Ω   u=0 on ∂Ω, where we have set α = β = χ = 0 and γ = 1 (see for example [7, 9, 10] and the monograph [8]). This simple model, which lends itself to the 3  A version of this chapter has been accepted for publication: C. Cowan, P. Esposito, N. Ghoussoub and A. Mordifam: The critical dimension for a fourth order elliptical problem with singular nonlinearity, Arch. Ration. Mech. Anal., In press (2009).  53  4.1. Introduction 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 [17], following the pioneering work of Joseph and Lundgren [14] who had considered the radially symmetric case. The case for a non constant permittivity profile f was advocated by Pelesko [19], taken up by [12], and studied in depth in [7, 9, 10]. The starting point of the analysis is the existence of a pull-in voltage λ∗ (Ω, f ), defined as n o 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 7→ 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 [10] 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 [9] 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 (P )λ 0<u<1 in Ω  u = ∂ν u = 0 on ∂Ω, was also considered by [4, 15] 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 54  4.1. Introduction 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. Theorem 4.2. 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 Dávila 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 HardyRellich inequalities recently established by Ghoussoub-Moradifam [11]. We remark that very recently, see [6], Dávila et al. examined the question of the multiplicity of solutions to the problem  2 ∆ u = λ(1 + sign(p)u)p in B u = ∂ν u = 0 on ∂B. In our case, where p = −2, they have shown that for 3 ≤ N ≤ 8 there exists a critical parameter 0 < λS < λ∗ where the problem has an infinite number of smooth solutions and also a singular solution. 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 , 55  4.1. Introduction that will be needed in the sequel. We define n o 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 defined as follows. Definition 4.3. We say that u is a weak solution of (P )λ if 0 ≤ u ≤ 1 a.e. 1 1 in B, (1−u) 2 ∈ L (B) and Z B  u∆2 φ = λ  Z B  φ , (1 − u)2  ∀φ ∈ C 4 (B̄) ∩ H02 (B).  We say that u is a weak super-solution (resp. weak sub-solution) of (P )λ if the equality is replaced with the inequality ≥ (resp. ≤) for all φ ∈ C 4 (B̄) ∩ H02 (B) with φ ≥ 0. We also recall the notions of regularity and stability. Definition 4.4. Say that a weak solution u of (P )λ is regular (resp. singular) if kuk∞ < 1 (resp. =) and stable (resp. semi-stable) if Z  Z φ2 2 2 µ1 (u) = inf (∆φ) − 2λ : φ ∈ H0 (B), kφkL2 = 1 3 B B (1 − u) is positive (resp. non-negative). The following extension of Boggio’s principle will be frequently used in the sequel (see [2, Lemma 16] and [5, Lemma 2.4]). Lemma 4.5 (Boggio’s Principle). Let u ∈ L1 (B). Then u ≥ 0 a.e. in B, provided one of the following conditions hold: ∂u 1. u ∈ C 4 (B), ∆2 u ≥ 0 on B, and u = ∂n = 0 on ∂B. R 2. B u∆2 φ dx ≥ 0 for all 0 ≤ φ ∈ C 4 (B) ∩ H02 (B). R ∂u 3. u ∈ H 2 (B), u = 0 and ∂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 4.6. The following assertions hold: 56  4.2. The effect of boundary conditions on the pull-in voltage 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 7→ uλ (x) is strictly increasing on (0, λ∗ ). 4. The pull-in voltage λ∗ satisfies the following bounds:   32(10N − N 2 − 12) 8 2 8 4ν1 max , (N − )(N − ) ≤ λ∗ ≤ 27 9 3 3 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 1 H02 (B) and that 1−u is uniformly bounded in L3 (B). Since now λ 7→ uλ (x) λ is increasing, the function u∗ (x) := limλ%λ∗ uλ (x) is well defined (in the 1 3 ∗ pointwise sense), u∗ ∈ H02 (B), 1−u ∗ ∈ L (B) and u is a weak solution of (P )λ∗ . Moreover u∗ is the unique weak solution of (P )λ∗ .  4.2  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 (P )λ,α,β α<u<1 in B  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 in B (4.7) Φ = α , ∂ν Φ = β on ∂B, we infer immediately from Lemma 4.5 that the function u − Φ is positive in B, which yields to sup Φ < sup u ≤ 1. B  B  57  4.2. The effect of boundary conditions on the pull-in voltage To insure that Φ is a classical sub-solution of (P )λ,α,β , we impose α 6= 1 and β ≤ 0, and condition sup Φ < 1 rewrites as α − β2 < 1. We will then say B  that the pair (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 . The notion of weak solution for (P )λ,α,β is defined as follows. Definition 4.8. We say that u is a weak solution of (P )λ,α,β if α ≤ u ≤ 1 1 1 a.e. in B, (1−u) 2 ∈ L (B) and if Z  2  Z  (u − Φ)∆ φ = λ B  B  φ , (1 − u)2  ∀φ ∈ C 4 (B̄) ∩ H02 (B),  where Φ is given in (4.7). 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∗ (α, β) := sup{λ > 0 : (P )λ,α,β has a classical solution} and λ∗ (α, β) := 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 4.9 ([2]). For every 0 ≤ f ∈ L1 (B), there exists a unique 0 ≤ u ∈ L1 (B) which satisfies R R 2 for all φ ∈ C 4 (B̄) ∩ H02 (B). B u∆ φ = B f φ, We can now introduce the following “weak iterative scheme”: Start with u0 = U and (inductively) let un , n ≥ 1, be the solution of Z Z φ 2 (un − Φ)∆ φ = λ ∀ φ ∈ C 4 (B̄) ∩ H02 (B) 2 (1 − u ) n−1 B B 58  4.2. The effect of boundary conditions on the pull-in voltage given by Theorem 4.9. Since 0 is a sub-solution of (P )λ,α,β , one can easily show inductively by using Lemma 4.5 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  n→+∞  solution of (P )λ,α,β such that α ≤ u ≤ U . In other words, the following result holds. Proposition 4.2.1. 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 4.2.2. 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 ū = (1 − ε)u + εΦ, where Φ is given in (4.7). We have that sup ū ≤ (1 − ε) + ε sup Φ < 1 , B  inf ū ≥ (1 − ε)α + ε inf Φ = α, B  B  B  and for every 0 ≤ φ ∈ C 4 (B̄) ∩ H02 (B) there holds: Z Z Z φ 2 2 (ū − Φ)∆ φ = (1 − ε) (u − Φ)∆ φ = (1 − ε)λ 2 B B (1 − u) B Z φ = (1 − ε)3 λ (1 − ū + ε(Φ − 1))2 ZB φ ≥ (1 − ε)3 λ . 2 B (1 − ū) Note that 0 ≤ (1 − ε)(1 − u) = 1 − ū + ε(Φ − 1) < 1 − ū. So ū is a weak supersolution of (P )(1−ε)3 λ,α,β satisfying sup ū < 1. From Proposition 4.2.1 we B  get the existence of a weak solution w of (P )(1−ε)3 λ,α,β so that α ≤ w ≤ ū. In particular, sup w < 1 and w is a regular weak solution. Since ε ∈ (0, 1) is B  arbitrarily chosen, the proof is complete.  59  4.2. The effect of boundary conditions on the pull-in voltage Proposition 4.2.2 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.  (4.10)  Since vn ∈ H02 (B), un is also a weak solution of (4.10), and by Lemma 4.5 we know that α ≤ un ≤ un+1 ≤ Uλ for every n ≥ 0. Since sup un ≤ sup Uλ < 1 B  B  for n ≥ 0, we 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 hold pointwise and weakly in H 2 (B). By n→+∞  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 theory [1], we have B  B  uλ ∈ C ∞ (B̄) and uλ − Φ = ∂ν (uλ − Φ) = 0 on ∂B. So we can integrate by parts to get Z Z Z Z φ 2 2 2 ∆ uλ φ = ∆ (uλ − Φ)φ = (uλ − Φ)∆ φ = λ , 2 B B B B (1 − uλ ) for every φ ∈ C 4 (B̄) ∩ H02 (B). Hence, uλ is a radial classical solution of (P )λ,α,β showing that λ∗ = λ∗ . Moreover, since Φ and vλ := uλ − Φ are radially decreasing in view of [21], 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). λ%λ  Finally, we show the finiteness of the pull-in voltage. Lemma 4.11. 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 Z Z (β∆ψ − α∂ν ∆ψ) = C ψ. ∂B  B  Multiplying (P )λ,α,β by ψ and then integrating by parts one arrives at  Z  λ − ν1 u − C ψ = 0. 2 B (1 − u) 60  4.2. The effect of boundary conditions on the pull-in voltage λ Since ψ > 0 there must exist a point x̄ ∈ B where (1−u(x̄)) 2 −ν1 u(x̄)−C ≤ 0. Since α < u(x̄) < 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 4.12. 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 7→ uλ (x) is strictly increasing on (0, λ∗ ). 3. For λ > λ∗ there are no weak solutions of (P )λ,α,β .  4.2.1  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 4.13. We say that u is a H 2 (B)−weak solution of (P )λ,α,β if 1 1 u − Φ ∈ H02 (B), α ≤ u ≤ 1 a.e. in B, (1−u) 2 ∈ L (B) and if Z  Z ∆u∆φ = λ  B  B  φ , (1 − u)2  ∀φ ∈ C 4 (B̄) ∩ H02 (B),  where Φ is given in (4.7). 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 4.14. 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 λ%λ  solution of (P )λ∗ ,α,β . 3. When u∗ is a classical solution, then µ1 (u∗ ) = 0 and u∗ is the unique H 2 (B)−weak solution of (P )λ∗ ,α,β .  61  4.2. The effect of boundary conditions on the pull-in voltage 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. Lemma 4.15. 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 supersolution 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. (1) Define w := u − U . Then by the Moreau decomposition [18] for the biharmonic operator, there exist w1 , w2 ∈ H02 (B),R 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 4.5, we have that w2 ≤ 0 a.e. in B. Given now 0 ≤ φ ∈ Cc∞ (B), we have that Z Z ∆w∆φ ≤ λ (f (u) − f (U ))φ, B  B  where f (u) := (1 − u)−2 . Since u is semi-stable, one has Z Z Z Z 0 2 2 λ f (u)w1 ≤ (∆w1 ) = ∆w∆w1 ≤ λ (f (u) − f (U ))w1 . B  B  B  B  Since w1 ≥ w one also has Z Z f 0 (u)ww1 ≤ (f (u) − f (U ))w1 , B  B  which once re-arranged gives Z  f˜w1 ≥ 0,  B  where f˜(u) = f (u) − f (U ) − f 0 (u)(u − U ). The strict convexity of f gives f˜ ≤ 0 and f˜ < 0 whenever u 6= 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. (2) Since u is a classical solution, it is easy to see that the infimum in µ1 (u) is attained at some φ. The function φ is then the first eigenfunction of 2λ 2 ∆2 − (1−u) 3 in H0 (B). Now we show that φ is of fixed sign. Using the 62  4.2. The effect of boundary conditions on the pull-in voltage Moreau decomposition, one has φ = φ1 + φ2 where φi ∈ H02 (B) for i = 1, 2, R φ1 ≥ 0, B ∆φ1 ∆φ2 = 0 and ∆2 φ2 ≤ 0 in the H02 (B)−weak sense. If φ changes sign, then φ1 6≡ 0 and φ2 < 0 in B (recall that either φ2 < 0 or φ2 = 0 a.e. in B). We can write now: R R 2 0 2 2 0 2 2 )) − λf (u)(φ1 − φ2 ) B (∆(φ1 − φ B (∆φ)R − λf (u)φ R 0 = µ1 (u) ≤ < 2 2 B (φ1 − φ2 ) Bφ but the right hand is equal to µ1 (u) and so 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 Boggio’s principle we have φ > 0 in B. For 0 ≤ t ≤ 1 define Z Z g(t) = ∆ [tU + (1 − t)u] ∆φ − λ f (tU + (1 − t)u)φ, B  B  where φ is the above first eigenfunction. Since f is convex one sees that Z 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 Z 0 g (0) = ∆(U − u)∆φ − λf 0 (u)(U − u)φ = 0, B  we get that 00  Z  g (0) = −λ  f 00 (u)(U − u)2 φ ≥ 0.  B  Since f 00 (u)φ > 0 in B, we finally get that U = u a.e. in B. Based again on Lemma 4.5(3), we can show a more general version of Lemma 4.15. Lemma 4.16. Let (α, β) be an admissible pair and β 0 ≤ 0. Let u be a semi-stable H 2 (B)−weak sub-solution of (P )λ,α,β with u = α, ∂ν u = β 0 ≥ 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 ũ ∈ H02 (B) denote a weak solution to ∆2 ũ = ∆2 (u − U ) in B. Since ũ − u + U = 0 and ∂ν (ũ − u + U ) ≤ 0 on ∂B, by Lemma 4.5 one has that ũ ≥ u − U a.e. in B. Again by the Moreau decomposition [18], we may write ũ as ũ = w + v, where w, v ∈ H02 (B), w ≥ 0 a.e. in B, ∆2 v ≤ 0 in a 63  4.2. The effect of boundary conditions on the pull-in voltage R H 2 (B)−weak sense and B ∆w∆v = 0. Then for 0 ≤ φ ∈ C 4 (B̄) ∩ H02 (B) one has Z Z Z ∆(u − U )∆φ ≤ λ (f (u) − f (U ))φ. ∆ũ∆φ = B  B  B  In particular, we have that Z Z ∆ũ∆w ≤ λ (f (u) − f (U ))w. B  B  Since by semi-stability of u Z Z Z λ f 0 (u)w2 ≤ (∆w)2 = ∆ũ∆w, B  B  B  we get that Z  f 0 (u)w2 ≤  B  Z (f (u) − f (U ))w. B  By Lemma 4.5 we have v ≤ 0 and then w ≥ ũ ≥ u − U a.e. in B. So we see that Z  0≤ f (u) − f (U ) − f 0 (u)(u − U ) w. B  The strict convexity of f implies as in Lemma 4.15 that U ≥ u a.e. in B. We shall need the following a-priori estimates along the minimal branch uλ . Lemma 4.17. Let (α, β) be an admissible pair. Then one has Z Z (uλ − Φ)2 uλ − Φ 2 ≤ , 3 (1 − u ) (1 − uλ )2 λ B B where Φ is given in (4.7). In particular, there is a constant C > 0 so that for every λ ∈ (0, λ∗ ), we have Z Z 1 2 ≤ C. (4.18) (∆uλ ) + 3 B B (1 − uλ ) Proof. Testing (P )λ,α,β on uλ − Φ ∈ C 4 (B̄) ∩ H02 (B), we see that Z λ B  uλ − Φ = (1 − uλ )2  Z  Z ∆uλ ∆(uλ − Φ) =  B  B  (∆(uλ − Φ))2 ≥ 2λ  Z B  (uλ − Φ)2 (1 − uλ )3  64  4.2. The effect of boundary conditions on the pull-in voltage in view of ∆2 Φ = 0. In particular, for δ > 0 small we have that Z Z Z 1 (uλ − Φ)2 1 1 1 ≤ ≤ 2 3 2 3 δ {|uλ −Φ|≥δ} (1 − uλ ) δ B (1 − uλ )2 {|uλ −Φ|≥δ} (1 − uλ ) Z 1 ≤ δ + Cδ , 3 {|uλ −Φ|≥δ} (1 − uλ ) by means of Young’s inequality. Since for δ small, Z 1 ≤ C 0, 3 (1 − u ) λ {|uλ −Φ|≤δ} for some C 0 > 0, we can deduce that for every λ ∈ (0, λ∗ ), Z 1 ≤ C, 3 B (1 − uλ ) for some C > 0. By Young’s and Hölder’s inequalities, we now have Z Z Z uλ − Φ 2 (∆uλ ) = ∆uλ ∆Φ + λ (1 − uλ )2 B B B Z ≤ δ (∆uλ )2 + Cδ B  Z +C B  1 (1 − uλ )3  2  3  and estimate (4.18) is therefore established. We are now ready to establish Theorem 5.2. Proof of Theorem 5.2: (1) Since kuλ k∞ < 1, the infimum defining µ1 (uλ ) is achieved at a first eigenfunction for every λ ∈ (0, λ∗ ). Since λ 7→ uλ (x) is increasing for every x ∈ B, it is easily seen that λ 7→ µ1 (uλ ) is an increasing, continuous function on (0, λ∗ ). Define λ∗∗ := sup{0 < λ < λ∗ : µ1 (uλ ) > 0}. We claim that λ∗∗ = λ∗ . Indeed, otherwise we would have that µ1 (uλ∗∗ ) = 0, and for every µ ∈ (λ∗∗ , λ∗ ) the function uµ would be a classical super-solution of (P )λ∗∗ ,α,β . A contradiction arises since Lemma 4.15 implies uµ = uλ∗∗ . Finally, Lemma 4.15 guarantees uniqueness in the class of semi-stable H 2 (B)−weak solutions. (2) By estimate (4.18) it follows that uλ → u∗ in a 65  4.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 1 3 ∗ pointwise sense and weakly in H 2 (B), and 1−u ∗ ∈ L (B). In particular, u is a H 2 (B)−weak solution of (P )λ∗ ,α,β which is also semi-stable as limiting function of the semi-stable solutions {uλ }. (3) Whenever ku∗ k∞ < 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 4.15, 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 [16]) to show that u∗ is the unique weak solution of (P )λ∗ , regardless of whether u∗ is regular or not. (4) Assume now that v is a singular semi-stable H 2 (B)−weak solution of (P )λ,α,β . If λ < λ∗ , then by the uniqueness of the semi-stable solution, we have v = uλ . So v is not singular and a contradiction arises.  By Theorem 4.12(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 4.15 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 4.15 to see that v ≥ u∗ a.e. in B. So equality v = u∗ holds and the proof is complete.  4.3  Regularity of the extremal solution for 1≤N ≤8  We now turn 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 4.19. u∗ is regular in dimensions 1 ≤ N ≤ 4. Proof. As already observed, estimate (4.18) implies that f (u∗ ) = (1 − 3 u∗ )−2 ∈ L 2 (B). Since u∗ is radial and radially decreasing, we need to show that u∗ (0) < 1 to get the regularity of u∗ . The integrability of f (u∗ ) along 3 with elliptic regularity 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 1 C ≥ 1−u |x|  in B, 66  4.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 for some C > 0, one sees that Z 3 +∞ = C B  1 ≤ |x|3  Z B  1 < +∞. (1 − u∗ )3  u∗  A contradiction arises and hence is regular for 1 ≤ N ≤ 3. 1 For N = 4 we need to be more careful and observe that u∗ ∈ C 1, 3 (B̄) by the 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 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 4.20. Assume N ≥ 5 and let (u∗ , λ∗ ) be the extremal pair of (P )λ . When u∗ is singular, then 4  1 − u∗ (x) ≤ C0 |x| 3 where C0 :=    λ∗ λ  1 3  in B,  and λ̄ = λ̄N := 89 (N − 32 )(N − 83 ).  Proof. First note that Theorem 4.6(4) gives the lower bound: 8 2 8 λ∗ ≥ λ̄N = (N − )(N − ). 9 3 3 4  For δ > 0, we define uδ (x) := 1 − Cδ |x| 3 with Cδ :=  (4.21)   λ∗ λ̄  +δ  1 3  > 1. Since  1 N 2 3 2 (RN ), N ≥ 5, we have that uδ ∈ Hloc 1−uδ ∈ Lloc (R ) and uδ is a H −weak solution of λ∗ + δ λ̄ ∆2 uδ = in RN . (1 − uδ )2  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), we have that 0 < r1 < 1 and one infers that α := u∗ (r1 ) = uδ (r1 ) ,  β := (u∗ )0 (r1 ) ≥ u0δ (r1 ). 67  4.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 −4  Setting uδ,r1 (r) = r1 3 (uδ (r1 r) − 1) + 1, we easily see that the function uδ,r1 is a H 2 (B)−weak super-solution of (P )λ∗ +δλ̄N ,α0 ,β 0 , where −4  −1  α0 := r1 3 (α − 1) + 1 , − 34  Similarly, define u∗r1 (r) = r1  β 0 := r1 3 β.  (u∗ (r1 r) − 1) + 1. The dilation map − 34  w → wr1 (r) = r1  (w(r1 r) − 1) + 1  (4.22)  is a correspondence between solutions of (P )λ on B and of (P )  −4  λ,1−r1 3 ,0 (u∗r1 , λ∗ ) is  on Br−1 which preserves the H 2 −integrability. In particular, 1 on Br−1 (defined in the obvious way). the extremal pair of (P ) 4 −3 λ,1−r1  ,0  1  Moreover, u∗r1 is a singular semi-stable H 2 (B)− weak solution of (P )λ∗ ,α0 ,β 0 . Since u∗ is radially decreasing, we have that β 0 ≤ 0. Define the function 0 0 w(x) := (α0 − 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∗ in B w = α0 , ∂ν w = β 0 on ∂B. λ∗ ≥ λ∗ , by Lemma 4.5 we have u∗r1 ≥ w a.e. in B. Since (1−u∗r1 )2 0 w(0) = α0 − β2 + γ(0) and γ(0) > 0, the bound u∗r1 ≤ 1 a.e. in B yields to 0 α0 − β2 < 1. Namely, (α0 , β 0 ) is an admissible pair and by Theorem 5.2(4) we get that (u∗r1 , λ∗ ) coincides with the extremal pair of (P )λ,α0 ,β 0 in B.  Since  Since (α0 , β 0 ) is an admissible pair and uδ,r1 is a H 2 (B)−weak super-solution of (P )λ∗ +δλ̄N ,α0 ,β 0 , we get from Proposition 4.2.1, the existence of a weak solution of (P )λ∗ +δλ̄N ,α0 ,β 0 . Since λ∗ + δ λ̄N > λ∗ , we contradict the fact that λ∗ is the extremal parameter of (P )λ,α0 ,β 0 . Thanks to this lower estimate on u∗ , we get the following result. Theorem 4.23. 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| that (∆ψ)2 = (HN + O(ε))|x|−N +2ε ,  4−N 2  +ε  and note  68  4.3. Regularity of the extremal solution for 1 ≤ N ≤ 8 where  N 2 (N − 4)2 . 16 Given η ∈ C0∞ (B), and since N ≥ 5, we can use the test function ηψ ∈ H02 (B) into the stability inequality to obtain Z Z ψ2 (∆ψ)2 + O(1), 2λ ≤ ∗ )3 (1 − u B B HN :=  where O(1) is a bounded function as ε & 0. By Theorem 4.20 we find that Z Z ψ2 2λ̄N ≤ (∆ψ)2 + O(1), 4 |x| B B and then Z  −N +2ε  |x|  2λ̄N  Z ≤ (HN + O(ε))  B  |x|−N +2ε + O(1).  B  Computing the integrals one arrives at 2λ̄N ≤ HN + O(ε). As ε → 0 finally we obtain 2λ̄N ≤ HN . Graphing this relation one sees that N ≥ 9. We can now slightly improve the lower bound (4.21). Corollary 4.24. In any dimension N ≥ 1, we have 8 2 8 λ∗ > λ̄N = (N − )(N − ). 9 3 3  (4.25)  4  Proof. The function ū := 1 − |x| 3 is a H 2 (B)− weak solution of (P )λ̄N ,0,− 4 . 3  If by contradiction λ∗ = λ̄N , then ū is a H 2 (B)−weak super-solution of (P )λ for every λ ∈ (0, λ∗ ). By Lemma 4.15 we get that uλ ≤ ū for all λ < λ∗ , and then u∗ ≤ ū a.e. in B. If 1 ≤ N ≤ 8, u∗ is then regular by Theorems 4.19 and 4.23. By Theorem 5.2(3) there holds µ1 (u∗ ) = 0. Lemma 4.15 then yields that u∗ = ū, which is a contradiction since then u∗ will not satisfy the boundary conditions. If now N ≥ 9 and λ̄ = λ∗ , then C0 = 1 in Theorem 4.20, and we then have u∗ ≥ ū. It means again that u∗ = ū, a contradiction that completes the proof. 69  4.4. The extremal solution is singular for N ≥ 9  4.4  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 [11]. As in the previous section (u∗ , λ∗ ) denotes the extremal pair of (P )λ . • 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) Z Z Z N 2 (N − 4)2 φ2 2 (∆φ) dx ≥ dx + C φ2 dx. (4.26) 4 16 |x| B B B • Case 10 ≤ N ≤ 16: For this case, we shall need the following inequality valid for all φ ∈ H02 (B) Z (N − 2)2 (N − 4)2 (∆φ)2 dx ≥ I (4.27) 16 B Z (N − 1)(N − 4)2 φ2 + dx. N 4 B |x|2 (|x|2 − |x| 2 ) where  φ2  Z I :=  N  B  N  (|x|2 − |x| 2 +1 )(|x|2 − |x| 2 )  .  • Case N = 9: This case is the trickiest and will require the following inequality for all φ ∈ H02 (B)   Z Z N −1 2 φ2 dx, (4.28) (∆φ) dx ≥ Q(|x|) P (|x|) + |x|2 B B where P (r) =  p00 (r)+  (N −1) 0 p (r) r  p  and  Q(r) =  q 00 (r)+  (N −3) 0 q (r) r  q  ,  with p and q being two appropriately chosen polynomials, namely N  p(r) := r− 2 +1 + r − 1.9, N  q(r) := r− 2 +2 + 20r−1.69 + 10r−1 + 10r + 7r2 − 48. 70  4.4. The extremal solution is singular for N ≥ 9 We shall first show the following upper bound on u∗ . 4  Lemma 4.29. If N ≥ 9, then u∗ ≤ 1 − |x| 3 =: ū in B. Proof. Recall from Corollary 4.24 that λ̄ := 98 (N − 23 )(N − 83 ) < λ∗ . We now claim that uλ ≤ ū for all λ ∈ (λ̄, λ∗ ). Indeed, fix such a λ and assume by contradiction that R1 := inf{0 ≤ R ≤ 1 : uλ < ū in the interval (R, 1)} > 0. From the boundary conditions, one has that uλ (r) < ū(r) as r → 1− . Hence, 0 < R1 < 1, α := uλ (R1 ) = ū(R1 ) and β := u0λ (R1 ) ≤ ū0 (R1 ). Introduce, as in the proof of Theorem 4.20, the functions uλ,R1 and ūR1 . We have that uλ,R1 is a classical super-solution of (P )λ̄N ,α0 ,β 0 , where −4  α0 := R1 3 (α − 1) + 1 ,  −1  β 0 := R1 3 β.  Note that ūR1 is a H 2 (B)−weak sub-solution of (P )λ̄N ,α0 ,β 0 which is also semi-stable in view of the Hardy-Rellich inequality (4.26) and the fact that 2λ̄N ≤ HN :=  N 2 (N − 4)2 . 16  By Lemma 4.16, we deduce that uλ,R1 ≥ ūR1 in B. Note that, arguing as in the proof of Theorem 4.20, (α0 , β 0 ) is an admissible pair. We have therefore shown that uλ ≥ ū in BR1 and a contradiction arises in view of the fact that lim ū(x) = 1 and kuλ k∞ < 1. It follows that uλ ≤ ū in B for every x→0  λ ∈ (λ̄N , λ∗ ), and in particular u∗ ≤ ū in B. The following lemma is the key for the proof of the singularity of u∗ in higher dimensions. Lemma 4.30. Let N ≥ 9. Suppose there exist λ0 > 0, β > 0 and a singular 1 radial function w ∈ H 2 (B)with 1−w ∈ L∞ loc (B̄ \ {0}) such that (  and Z 2β B  0  λ ∆2 w ≤ (1−w) for 0 < r < 1, 2 w(1) = 0, w0 (1) = 0,  (4.31)  φ2 ≤ (1 − w)3  (4.32)  Z  (∆φ)2 for all φ ∈ H02 (B).  B  1. If β ≥ λ0 , then λ∗ ≤ λ0 . 71  4.4. The extremal solution is singular for N ≥ 9 2. If either β > λ0 or if β = λ0 = necessarily singular.  HN 2 ,  then the extremal solution u∗ is  1 1 Proof. 1) First, note that (4.32) and 1−w ∈ L∞ loc (B̄ \ {0}) yield to (1−w)2 ∈ L1 (B). By a density argument, (4.31) implies that w is a H 2 (B)−weak sub-solution of (P )λ0 whenever N ≥ 4. If now λ0 < λ∗ , then by Lemma 4.16, w would necessarily be below the minimal solution uλ0 , which is a contradiction since w is singular while uλ0 is regular. 2) Suppose first that β = λ0 = H2N and that N ≥ 9. Since by part 1) we have λ∗ ≤ H2N , we get from Lemma 4.29 and the improved Hardy-Rellich inequality (4.26) that there exists C > 0 so that for all φ ∈ H02 (B) Z Z Z Z Z φ2 φ2 2 ∗ 2 (∆φ) − 2λ (∆φ) − HN φ2 . ≥ ≥C ∗ )3 4 (1 − u |x| B B B B 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∗ . 0 Suppose now that β > λ0 , and let λβ < γ < 1 in such a way that α := (  γλ∗ 1/3 ) < 1. λ0  (4.33)  Setting w̄ := 1 − α(1 − w), we claim that u∗ ≤ w̄ in B.  (4.34)  Note that by the choice of α we have α3 λ0 < λ∗ , and therefore to prove (4.34) it suffices to show that for α3 λ0 ≤ λ < λ∗ , we have uλ ≤ w̄ in B. Indeed, fix such λ and note that ∆2 w̄ = α∆2 w ≤  αλ0 α 3 λ0 λ = ≤ . 2 2 (1 − w) (1 − w̄) (1 − w̄)2  Assume that uλ ≤ w̄ does not hold in B, and consider R1 := sup{0 ≤ R ≤ 1 | uλ (R) > w̄(R)} > 0. Since w̄(1) = 1 − α > 0 = uλ (1), we then have R1 < 1, uλ (R1 ) = w̄(R1 ) and (uλ )0 (R1 ) ≤ (w̄)0 (R1 ). Introduce, as in the proof of Theorem 4.20, the functions uλ,R1 and w̄R1 . We have that uλ,R1 is a classical solution of (P )λ,α0 ,β 0 , where −4  α0 := R1 3 (uλ (R1 ) − 1) + 1 ,  −1  β 0 := R1 3 (uλ )0 (R1 ). 72  4.4. The extremal solution is singular for N ≥ 9 Since λ < λ∗ and then 2λ 2λ∗ 2λ0 2β ≤ = < , 3 3 3 3 (1 − w̄) α (1 − w) γ(1 − w) (1 − w)3 by (4.32) w̄R1 is a stable H 2 (B)−weak sub-solution of (P )λ,α0 ,β 0 . By Lemma 4.16, we deduce that uλ ≥ w̄ in BR1 which is impossible, since w̄ is singular while uλ is regular. Note that, arguing as in the proof of Theorem 4.20, (α0 , β 0 ) is an admissible pair. This establishes claim (4.34) which, combined with the above inequality, yields 2λ∗ 2λ∗ 2β ≤ < , (1 − u∗ )3 α3 (1 − w)3 (1 − w)3 and therefore R inf  φ∈H02 (B)  B (∆φ)  2  R B  2λ∗ φ2 (1−u∗ )3 φ2  −  > 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 −  3m 4/3 4 r + rm , 3m − 4 3m − 4  (4.35)  0 (1) = 0. We can which satisfies the right boundary conditions: wm (1) = wm now prove that the extremal solution is singular for N ≥ 9.  Theorem 4.36. Let N ≥ 9. The following upper bounds on λ∗ hold: 1. If N ≥ 31, then Lemma 4.30 holds with w := w2 , λ0 = 27λ̄N and β = H2N , and therefore λ∗ ≤ 27λ̄N . 2. If 17 ≤ N ≤ 30, then Lemma 4.30 holds with w := w3 , λ0 = β = and therefore λ∗ ≤ H2N .  HN 2 ,  3. If 10 ≤ N ≤ 16, then Lemma 4.30 holds with w := w3 , λ0N < βN given in Table 4.1, and therefore λ∗ ≤ λ0N . 4. If N = 9, then Lemma 4.30 holds with w := w2.8 , λ09 := 366 < β9 := 368.5, and therefore λ∗ ≤ 366. The extremal solution is therefore singular for dimension N ≥ 9. 73  4.4. The extremal solution is singular for N ≥ 9  Table 4.1: Summary N w λ0N 9 w2.8 366 10 w3 450 11 w3 560 12 w3 680 13 w3 802 14 w3 940 15 w3 1100 16 w3 1260 17 ≤ N ≤ 30 w3 HN /2 N ≥ 31 w2 27λ̄N  βN 366.5 487 739 1071 1495 2026 2678 3469 HN /2 HN /2  Proof. 1) Assume first that N ≥ 31, then 27λ̄ ≤ H2N . We shall show that w2 is a singular H 2 (B)−weak sub-solution of (P )27λ̄ so that (4.32) holds with β = H2N . Indeed, write 4  4  w2 := 1 − |x| 3 − 2(|x| 3 − |x|2 ) = ū − φ0 , 4  where φ0 := 2(|x| 3 − |x|2 ), and note that w2 ∈ H02 (B), 0 ≤ w2 ≤ 1 in B, and ∆ 2 w2 =  3λ̄ r  8 3  ≤  27λ̄ (1 − w2 )2  1 1−w2  ∈ L3 (B),  in B \ {0}.  So w2 is H 2 (B)−weak sub-solution of (P )27λ̄ . Moreover, by φ0 ≥ 0 and (4.26) we get that Z Z Z Z φ2 φ2 φ2 HN = HN ≤ HN ≤ (∆φ)2 4 3 4 3 (1 − w ) |x| 3 2 B (|x| + φ0 ) B B B for all φ ∈ H02 (B). It follows from Lemma 4.30 that u∗ is singular and that λ∗ ≤ 27λ̄ ≤ H2N . 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 2  74  4.4. The extremal solution is singular for N ≥ 9 1 1−w3  ∈ L3 (B). To show the stability condition, we consider φ ∈ H02 (B) and  write Z HN B  φ2 (1 − w3 )3  φ2  Z = 125HN  4 3  (9r − 4r3 )3 Z φ2 1 ≤ 125HN sup 5 4 0<r<1 (9 − 4r 3 )3 B r Z Z φ2 (∆φ)2 = HN ≤ 4 r B B B  by virtue of (4.26). An easy computation shows that HN − ∆ 2 w3 = 2(1 − w3 )2 =  25HN 4  2(9r 3 − 4r3 )2  −  9λ̄ 8  −  12 N 2 − 1 5 r  5r 3 8(N − 32 )(N − 83 ) 12 N 2 − 1 25N 2 (N − 4)2 − − . 4 8 5 r 32(9r 3 − 4r3 )2 5r 3  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 follows from Lemma 4.30 that u∗ is singular and that λ∗ ≤ H2N . 2  3) Assume 10 ≤ N ≤ 16. We shall prove that again w := w3 satisfies the assumptions of Lemma 4.30. Indeed, using Maple, we show that for each dimension 10 ≤ N ≤ 16, inequality (4.31) holds with λ0N given by Table 4.1. 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 N 4 |x|2 (|x|2 − |x| 2 ) ≥  2βN . (1 − w3 )3  where βN is again given by Table 4.1. The above inequality and the HardyRellich inequality (4.27) guarantee that the stability condition (4.32) holds with β := βN . Since βN > λ0N , we deduce from Lemma 4.30 that the extremal solution is singular for 10 ≤ N ≤ 16. 4) Suppose now N = 9 and consider w := w2.8 . Using Maple one can see that 366 in B ∆2 w ≤ (1 − w)2 75  4.5. Improved Hardy-Rellich inequalities and    N −1 723 ≤ Q(r) P (r) + for all r ∈ (0, 1), (1 − w)3 r2  where P and Q are given in (4.28). Since 723 > 2 × 366, by Lemma 4.30 the extremal solution u∗ is singular in dimension N = 9.  4.5  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 [11] 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 4.37. Say that a couple V, W of positive functions in C 1 (0, R) is a Bessel pair on the interval (0, R), if the ordinary differential equation y 00 (r) + (  N − 1 Vr (r) 0 W (r) + )y (r) + y(r) = 0, r V (r) V (r)  has a positive solution on the interval (0, R). Let BR denotes a ball centered at zero with radius R in RN (N ≥ 1). ∞ (B ). The The space of radial functions in C0∞ (BR ) will be denoted by C0,r R needed inequalities will follow from the following result. Theorem 4.38. (Ghoussoub-Moradifam [11]) Let V and W be posiRR tive C 1 -functions on the interval (0, R) such that 0 rN −11V (r) dr = +∞ and R R N −1 V (r)dr < +∞. The following statements are then equivalent: 0 r 1. (V, W ) is a Bessel pair on (0, R). R R 2. BR V (|x|)|∇φ|2 dx ≥ BR W (|x|)φ2 dx for all φ ∈ C0∞ (BR ). 3. If limr→0 rα V (r) = 0 for some α < N −2, then the above are equivalent to Z Z 2 V (|x|)(∆φ) dx ≥ W (|x|)|∇φ|2 dx BR BR Z V (|x|) Vr (|x|) +(N − 1) − )|∇φ|2 dx ( |x|2 |x| BR ∞ (B ). for all φ ∈ C0,r R  76  4.5. Improved Hardy-Rellich inequalities 4. If in addition, W (r) − 2Vr2(r) + 2Vrr(r) − Vrr (r) ≥ 0 on (0, R), then the above are equivalent to Z Z 2 V (|x|)(∆φ) dx ≥ W (|x|)|∇φ|2 dx BR BR Z V (|x|) Vr (|x|) − )|∇φ|2 dx +(N − 1) ( 2 |x| |x| BR for all φ ∈ C0∞ (BR ). We shall use the above theorem to deduce the following corollary. Corollary 4.39. Let N ≥ 5 and B be the unit ball in RN . Then the following improved Hardy-Rellich inequality holds for all φ ∈ C0∞ (B): Z Z φ2 dx (N − 2)2 (N − 4)2 (∆φ)2 dx ≥ N N 16 B (|x|2 − |x| 2 +1 )(|x|2 − |x| 2 ) B Z φ2 dx (N − 1)(N − 4)2 (4.40) + N . 4 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,  y 00 +  (N −2)2 4  (N −1) 0 r y  =  y   1 r2 −αr  (N − 2)2 1 , N 4 r2 − αr 2 +1  N +1 2  is a Bessel pair on (0, 1). By part (4) of  Theorem 4.38, the following inequality then holds: Z Z Z |∇φ|2 dx (N − 2)2 |∇φ|2 dx (∆φ)2 dx ≥ + (N − 1) , (4.41) N 4 |x|2 B |x|2 − α|x| 2 +1 B B for all φ ∈ C0∞ (B). Set V (r) :=  1 N r2 −αr 2 +1  and note that N  Vr 2 α(N − 2) r 2 −2 2 =− + ≥− . N V r 2 r 1 − αr 2 −1 N  The function y(r) = r− 2 +2 − 1 is decreasing and is then a positive supersolution on (0, 1) for the ODE y 00 + (  N − 1 Vr 0 W1 (r) + )y (r) + y = 0, r V V (r) 77  4.5. Improved Hardy-Rellich inequalities where W1 (r) =  (N − 4)2 N  N  4(r2 − r 2 )(r2 − αr 2 +1 )  .  Hence, by part 2) of Theorem 4.38, we can deduce that Z Z |∇φ|2 dx N −4 2 φ2 dx ≥ ( ) N N N 2 B |x|2 − α|x| 2 +1 B (|x|2 − α|x| 2 +1 )(|x|2 − |x| 2 ) for all φ ∈ C0∞ (B). Similarly, for V (r) = Z B  |∇φ|2 dx N −4 2 ≥( ) 2 |x| 2  1 r2  we have that φ2 dx  Z  N  B  |x|2 (|x|2 − |x| 2 )  for all φ ∈ C0∞ (B). Combining the above two inequalities with (4.41) and letting α → 1 we get inequality (4.40). Corollary 4.42. Let N = 9 and B be the unit ball in RN . Define p(r) := N N r− 2 +1 +r −1.9 and q(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):   Z Z N −1 2 (∆φ) dx ≥ Q(|x|) P (|x|) + φ2 dx, (4.43) 2 |x| B B where P (r) := −  q 00 (r) + N r−3 q 0 (r) p00 (r) + N r−1 p0 (r) and Q(r) := − . p(r) q(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 4.38(4) the following inequality holds: Z  2  Z  (∆φ) dx ≥ B  2  Z  P (|x|)|∇φ| dx + (N − 1) B  B  |∇φ|2 dx , |x|2  (4.44)  for all φ ∈ C0∞ (B). Using Maple it is easy to see that Pr 2 ≥ − in (0, 1), P r and therefore q(r) is a positive super-solution for the ODE y 00 + (  N − 1 Pr (r) 0 P (r)Q(r) + )y (r) + y = 0, r P (r) P (r) 78  4.5. Improved Hardy-Rellich inequalities on (0, 1). Hence, by Theorem 4.38(2) we have for all φ ∈ C0∞ (B) Z Z 2 P (|x|)Q(|x|)φ2 dx, P (|x|)|∇φ| dx ≥ B  B  and similarly Z B  |∇φ|2 dx ≥ |x|2  Z B  Q(|x|) 2 φ dx, |x|2  since q(r) is a positive solution for the ODE y 00 +  N −3 0 y (r) + Q(r)y = 0. r  Combining the above two inequalities with (4.44) we get (4.43).  79  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. 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Systems Appl. 3 (1994), no. 4, 465–487.  81  Chapter 5  Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains4 5.1  Introduction  We examine the problem   ∆2 u = λf (u) in Ω u = ∆u = 0 on ∂Ω,  (Nλ )  where λ ≥ 0 is a parameter, Ω is a bounded domain in RN , N ≥ 2, and where f satisfies one of the following two conditions: (R):  f is smooth, increasing, convex on R with f (0) = 1 and f is f (t) = ∞); superlinear at ∞ (i.e. lim t→∞ t (S): f is smooth, increasing, convex on [0, 1) with f (0) = 1 and lim f (t) = +∞.  t%1  Our main interest is in the regularity of the extremal solution u∗ associated with (Nλ ). Before we discuss some known results concerning the problem (Nλ ) we recall various facts concerning second order versions of the above problem. 4  A version of this chapter has been accepted for publication: C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst. (2010).  82  5.1. Introduction  5.1.1  The second order case  For a nonlinearity f of type (R) or (S), the following second order analog of (Nλ ) with Dirichlet boundary conditions  −∆u = λf (u) in Ω (Qλ ) u=0 on ∂Ω is by now quite well understood whenever Ω is a bounded smooth domain in RN . See, for instance, [4, 5, 6, 12, 15, 19, 20, 23]. We now list the properties one comes to expect when studying (Qλ ). • There exists a finite positive critical parameter λ∗ such that for all 0 < λ < λ∗ there exists a minimal solution uλ of (Qλ ). By minimal solution, we mean here that if v is another solution of (Qλ ) then v ≥ uλ a.e. in Ω. • For each 0 < λ < λ∗ the minimal solution uλ is semi-stable in the sense that Z Z 0 2 λf (uλ )ψ dx ≤ |∇ψ|2 dx, ∀ψ ∈ H01 (Ω), Ω  Ω  and is unique among all the weak semi-stable solutions. • The map λ 7→ uλ (x) is increasing on (0, λ∗ ) for each x ∈ Ω. This allows one to define u∗ (x) := limλ%λ∗ uλ (x), the so-called extremal solution, which can be shown to be a weak solution of (Qλ∗ ). In addition one can show that u∗ is the unique weak solution of (Qλ∗ ). See [19]. • There are no solutions of (Qλ ) (even in a very weak sense) for λ > λ∗ . A question which has attracted a lot of attention is whether the extremal function u∗ is a classical solution of (Qλ∗ ). This is of interest since one can then apply the results from [10] to start a second branch of solutions emanating from (λ∗ , u∗ ). Note that in the case where f satisfies (R) (resp. (S)) it is sufficient –in view of standard elliptic regularity theory– to show that u∗ is bounded (resp. supΩ u∗ < 1). This turned out to depend on the dimension, and so given a nonlinearity f , we say that N is the associated critical dimension provided the extremal solution u∗ associated with (Qλ∗ ) is a classical solution for any bounded smooth domain Ω ⊂ RM for any M ≤ N − 1, and if there exists a domain Ω ⊂ RN such that the associated extremal solution u∗ is not a classical solution. We now list some of the known results with regard to this question. 83  5.1. Introduction • For f (t) = et , the critical dimension is N = 10. For N ≥ 10, one can show that on the unit ball the extremal solution is explicitly given by u∗ (x) = −2 log(|x|). • For Ω = B the unit ball in RN , u∗ is bounded for any f satisfying (R) provided N ≤ 9, which –in view of the above– is optimal (see [6]). • On general domains, and if f satisfies (R), then u∗ is bounded for N ≤ 3 [23]. Recently this has been improved to N ≤ 4 provided the domain is convex [5]. 2  • For f (t) = (1 − t)−2 the critical dimension is N = 8 and u∗ = 1 − |x| 3 is the extremal solution on the unit ball for N ≥ 8. [15]. In the previous list, we have not considered the nonlinearity f (t) = (t + 1)p , p > 1, for which the critical dimension has been also computed but takes a complicated form. The general approach to showing N is the critical dimension for a particular nonlinearity f is to use the semi-stability of the minimal solutions uλ to obtain various estimates which translate to uniform L∞ bounds and then passing to the limit. These estimates generally depend on the ambient space dimension. On the other hand, in order to show the optimality of the regularity result one generally finds an explicit singular extremal solution u∗ on a radial domain. Here the crucial tool is the fact that if there exists a semi-stable singular solution in H01 (Ω), then it has to be the extremal solution. In practice one considers an explicit singular solution on the unit ball and applies Hardy-type inequalities to show its semi-stability in the right dimension. We also remark that one cannot remove the H01 (Ω) condition as counterexamples can be found.  5.1.2  The fourth order case  There are two obvious fourth order extensions of (Qλ ) namely the problem (Nλ ) mentioned above, and its Dirichlet counterpart  2 ∆ u = λf (u) in Ω (Dλ ) u = ∂ν u = 0 on ∂Ω, where ∂ν denote the normal derivative on ∂Ω. The problem (Qλ ) is heavily dependent on the maximum principle and hence this poses a major hurdle in the study of (Dλ ) since for general domains there is no maximum principle for ∆2 with Dirichlet boundary conditions. But if we restrict our attention to the unit ball then one does have a weak maximum principle [3]. The 84  5.1. Introduction problem (Dλ ) was studied in [1] and various results were obtained, but results concerning the boundedness of the extremal solution (for supercritical nonlinearities) were missing. The first (truly supercritical) results concerning the boundedness of the extremal solution in a fourth order problem are due to [11] where they examined the problem (Dλ ) on the unit ball in RN with f (t) = et . They showed that the extremal solution u∗ is bounded if and only if N ≤ 12. Their approach is heavily dependent on the fact that Ω is the unit ball. Even in this situation there are two main hurdles. The first is that the standard energy estimate approach, which was so successful in the second order case, does not appear to work in the fourth order case. The second is the fact that it is quite hard to construct explicit solutions of (Dλ ) on the unit ball that satisfy both boundary conditions, which is needed to show that the extremal solution is unbounded for N ≥ 13. So what one does is to find an explicit singular, semi-stable solution which satisfies the first boundary condition, and then to perturb it enough to satisfy the second boundary condition but not too much so as to lose the semi-stability. Davila et al. [11] succeeded in doing so for N ≥ 32, but they were forced to use a computer assisted proof to show that the extremal solution is unbounded for the intermediate dimensions 13 ≤ N ≤ 31. Using various improved Hardy-Rellich inequalities from [16] the need for the computer assisted proof was removed in [21]. The case where f (t) = (1 − t)−2 was settled at the same time in [9], where we used methods developed in 3DDGM to show that the extremal solution associated with (Dλ ) is a classical solution if and only if N ≤ 8. The problem (Nλ ) with Navier boundary conditions does not suffer from the lack of a maximum principle and the existence of the minimal branch has been shown in general [2, 7]. If the domain is the unit ball, then again one can use the methods of [11] and [9] to obtain optimal results in the case of f (t) = (1 − t)−2 (see for instance [13] and [22]). However, the case of a general domain is only understood in dimensions N ≤ 4 (See [17] and [13]). This paper is a first attempt at giving energy estimates on general domains, which –as mentioned above – while they do improve known results, they still fall short of the conjectured critical dimensions that were established when the domain is a ball. We now fix notation and some definitions associated with problem (Nλ ). Definition 5.1. Given a smooth solution u of (Nλ ), we say that u is a semi-stable solution of (Nλ ) if Z Z 0 2 λf (u)ψ dx ≤ (∆ψ)2 dx, ∀ψ ∈ H 2 (Ω) ∩ H01 (Ω). (5.2) Ω  Ω  85  5.2. Sufficient Lq -estimates for regularity Definition 5.3. We say a smooth solution u of (Nλ ) is minimal provided u ≤ v a.e. in Ω for any solution v of (Nλ ). We define the extremal parameter λ∗ as λ∗ := sup {0 < λ : there exists a smooth solution of (Nλ )} . It is known, see [2, 7, 17], that: 1. 0 < λ∗ < ∞. 2. For each 0 < λ < λ∗ there exists a smooth minimal solution uλ of (Nλ ). Moreover the minimal solution uλ is semi-stable and is unique among the semi-stable solutions. 3. For each x ∈ Ω, λ 7→ uλ (x) is strictly increasing on (0, λ∗ ), and it therefore makes sense to define u∗ (x) := limλ%λ∗ uλ (x), which we call the extremal solution. 4. There are no solutions for λ > λ∗ . It is standard to show that u∗ is a “weak solution” of (Nλ∗ ) in a suitable sense that we shall not define here since it will not be needed in the sequel. One can then proceed to show that u∗ it is the unique weak solution in a fairly broad class of solutions. Regularity results on u∗ translate into regularity properties for any weak semi-stable solution. Indeed, by points (2)-(4) above we see that a weak semi-stable solution is either the classical solution uλ or the extremal solution u∗ . Our preference for not stating the results in this generality is to avoid the technical details of defining precisely what we mean by a suitable weak solution.  5.2  Sufficient Lq -estimates for regularity  In this section, we address our attention to nonlinearities f of type (R). As mentioned above, since the extremal function u∗ is the pointwise monotone limit of the classical solutions uλ as λ % λ∗ , it suffices to consider a sequence (un )n of classical solutions of (Nλn ), (λn )n uniformly bounded, and try to show that sup kun k∞ < +∞. (5.4) n  By standard elliptic regularity theory (5.4) follows by a uniform bound of f (un ) in Lq (Ω), for some q > N4 . The following result provides a weakening of such a statement. 86  5.2. Sufficient Lq -estimates for regularity Theorem 5.5. Suppose that for some q ≥ 1 and 0 < β < α we have Z sup f q (un ) < +∞ (5.6) n  and  Ω  f α (un )  Z sup n  uβn + 1  Ω  < +∞.  (5.7)  Then: N 4  1. If 1 ≤ q ≤  and α ≤  N 4,  then sup kf (un )ks < +∞ for every s < n  max{ (α−β)N N −4β , q}. N 4  2. If either q >  or α >  N 4,  then sup kun k∞ < +∞. n  Proof. We shall first show that under assumption (5.7), the following holds: If sup kf (un )kq0 < +∞ for 1 ≤ q0 ≤ n  then  N 4  sup kf (un )ks < +∞ for every s < q1 ,  (5.8)  n  αN q0 where q1 := N q0 +β(N −4q0 ) . Indeed, for t > 0 set t  Ωn1,t := {x ∈ Ω : f (un (x)) ≤ (uβn (x) + 1) β } Since 1 ≤ q0 ≤  N 4  and  Ωn2,t = Ω\Ωn1,t .  and sup kf (un )kq0 < +∞, we have via the Sobolev n  embedding Theorem that sup kun ks < +∞ for every s < n  and hence on Ωn1,t we have Z sup f s (un ) < +∞ for all s < n  Ωn 1,t β  On Ωn2,t , we have f α− t (un ) ≤  f α (un ) , uβ n +1  Z sup n  Ωn 2,t  N q0 N −4q0 ,  N q0 t(N −4q0 ) .  (5.9)  therefore β  f α− t (un ) < +∞.  (5.10) 87  5.2. Sufficient Lq -estimates for regularity If q0 <  N 4,  then take t =  N q0 +β(N −4q0 ) α(N −4q0 )  α−  in such a way that  β N q0 = , t t(N − 4q0 )  to obtain that sup kf (un )ks < +∞ for all s < n  αN q0 N q0 +β(N −4q0 ) .  If q0 = N4 , then we can let t → +∞ in (5.10) and combine with (7.17) to obtain that sup kf (un )ks < +∞ for all s < α, n  and then (5.8) is proved. Note that for 1 < q0 ≤  N 4  (5.8) is equivalent to:  if sup kf (un )ks < +∞ for every 1 ≤ s < q0 ,  (5.11)  n  then  sup kf (un )ks < +∞ for every s < q1 , n  where q1 is as before. By elliptic regularity theory, assumption (5.6) implies for q > sup kun k∞ < +∞. When 1 ≤ q ≤ N4 , by (5.8) we can say that  N 4  that  n  sup kf (un )ks < +∞ n  for every s < q1 :=  αN q0 . N q0 + β(N − 4q0 )  If q1 > N4 we are done. Otherwise, thanks to (5.11) we can use an iteration argument to show that sup kf (un )ks < +∞ n  for every s < qn+1 :=  αN qn N qn + β(N − 4qn )  N for every n ≥ 1, as long as qn ≤ N4 . Since (α−β)N N −4β > 4 when α > 1 ≤ q ≤ N4 , an easy induction shows that the sequence qn is  N 4  and  • increasing to  (α−β)N N −4β  when α ≤  N 4  and 1 ≤ q <  • decreasing to  (α−β)N N −4β  when α ≤  N 4  and q >  N 4  after a finite number of steps when  • increasing and passes the value α > N4 .  (α−β)N N −4β ;  (α−β)N N −4β ;  88  5.2. Sufficient Lq -estimates for regularity Claims (1) and (2) are then established. We can now deduce the following. Corollary 5.12. Suppose (un )n is a sequence of solutions of (Nλn ) such that Z sup f q (un ) < +∞ (5.13) n  Ω  for q ≥ 1. Then sup kun k∞ < +∞, in either one of the following two cases: n  1. f (t) = et and q ≥  N 4;  2. f (t) = (t + 1)p and q >  N 4    1−  1 p    .  Proof. (1) For q > N4 it follows by standard regularity theory. The case q = N4 and f (t) = et can be treated in the following way. Since eun is N  uniformly bounded in L 4 (Ω), by elliptic regularity theory and the Sobolev embedding Theorem un is uniformly bounded in W01,N (Ω). The MoserTrudinger inequality states that, for suitable α > 0 and Ci > 0, there holds Z N N −1 C k∇ukNN L , eα|u| dx ≤ C0 + C1 e 2 ∀u ∈ W01,N (Ω). Ω  Now fix τ >  N 4  and pick C̃ big enough such that eτ z ≤ C̃eαz  N N −1  ,  for all z ≥ 0. Then we have Z 1 C k∇un kNN L ≤ C̄, eτ un dx ≤ C0 + C1 e 2 C̃ Ω and so we have eun uniformly bounded in Lτ (Ω) for some τ > N4 . By elliptic estimates, the validity of (5.4) follows also in this case.   (2) The case where f (t) = (t + 1)p and N4 1 − p1 < q ≤ N4 follows N from Theorem 5.5 with the choice α = q + 4p , β = N4 , since α > α > β. Note that Z α Z f (un ) sup ≤ C sup f q (un ) < +∞, β n n u + 1 Ω n Ω  N 4  and  for some C > 0. 89  5.2. Sufficient Lq -estimates for regularity We now show that the standard assumption supn kf (un )kq < +∞, q > which guarantees the uniform boundedness of un can be weakened in a different way, through a uniform integrability condition on f 0 (un ). Indeed, we have the following result. N 4,  Theorem 5.14. Suppose that for some q > N4 we have Z sup f s (un ) < +∞ for every 1 ≤ s < n  Ω  and  Z sup n  N N −2  (f 0 )q (un ) < +∞.  (5.15)  (5.16)  Ω  Then, sup kun k∞ < +∞.  (5.17)  n  Proof. Observe that ṽn = −∆un satisfies  2 ∆ ṽn ≤ λn f 0 (un )ṽn in Ω ṽn = 0, −∆ṽn = λn on ∂Ω. Introducing the function wn as the solution of  −∆wn = λn in Ω wn = 0 on ∂Ω, we are led to study uniform boundedness for vn = ṽn − wn , a solution of  2 ∆ vn ≤ λn f 0 (un )vn + λn f 0 (un )wn in Ω (5.18) vn = ∆vn = 0 on ∂Ω. Since λn is bounded, by elliptic regularity theory we deduce that sup kwn k∞ < +∞,  (5.19)  n  and then the uniform boundedness can be equivalently established on ṽn or vn . First we show that under assumption (5.16), the following hold: Nq If supn kvn ks < +∞ ∀ 1 ≤ s < q0 and q0 ≤ 4q−N , then sup kvn ks < +∞∀s < q1 .  (5.20)  n  If supn kvn ks < +∞ ∀ 1 ≤ s < q0 and q0 >  Nq 4q−N ,  then sup kvn k∞ < +∞  (5.21)  n  90  5.3. A general regularity result for low dimensions N qq0 N q0 +q(N −4q0 ) .  where q1 :=  Indeed, by (5.16) and (5.19) we get that  λn f 0 (un )vn + λn f 0 (un )wn uniformly bounded in Ls (Ω), ∀ 1 ≤ s <  qq0 . q + q0  Thanks to (5.18), by elliptic regularity theory and the maximum principle qq0 the previous estimate translates into: if q+q ≤ N4 0 sup kvn ks < +∞ n  and if  qq0 q+q0  >  for every 1 ≤ s < q1 ,  N 4  sup kvn k∞ < +∞. n  Therefore, (5.20)-(5.21) are established. Thanks to (5.19), by elliptic regularity theory assumption (5.15) reads on vn as sup kvn k∞ < +∞ if N = 2, 3 and n  Z sup n  (vn )s < +∞  Ω  if N ≥ 4. For N ≥ 4, set q0 = long as qi ≤  for every 1 ≤ s <  Nq 4q−N  N N −4  N N −4  and inductively qi+1 =  (5.22)  N qqi N qi +q(N −4qi )  as  so to get  sup kvn ks < +∞ n  in view of (5.11). Since q >  N 4,  for every 1 ≤ s < qn+1 ,  the sequence qi is increasing and passes  Nq 4q−N  Nq after a finite number of steps. As soon as qi becomes larger than 4q−N , we ∞ can use (5.19) and (5.21) to get an uniform L −bound on −∆un = vn +wn , and in turn the validity of (5.17) follows by elliptic estimates.  5.3  A general regularity result for low dimensions  To the best of our knowledge the only available energy estimates for smooth, semi-stable solutions u of (Nλ ) so far, are given by Z Z f 0 (u)u2 dx ≤ f (u)udx. (5.23) Ω  Ω  To see this, take ψ = u in (5.2) and integrate by parts (Nλ ) against u, and then equate. In view of Corollary 5.12, this yields the following 91  5.3. A general regularity result for low dimensions ∗  1. If f (t) = et , then eu (u∗ )2 ∈ L1 (Ω) and u∗ is then regular for N ≤ 4. p+1  2. If f (t) = (t + 1)p , then (u∗ + 1)p ∈ L p (Ω), therefore u∗ is a regular N +4 solution for 2 ≤ N < 4(p+1) p−1 (equivalently N ≤ 4 or 1 ≤ p < N −4 and N > 4) 3  3. If f (t) = (1 − t)−2 , then (1 − u∗ )−2 ∈ L 2 (Ω) and u∗ is regular for N ≤ 4. See Chapter 12 of [13]. We shall substantially improve on these results in the next sections. For now, we start by considering the case of a general superlinear f and establish a fourth order analogue of the results of Nedev [23] for N ≤ 3 and Cabre [5] for N = 4, regarding the regularity of the extremal solution of second order eigenvalue problems with a nonlinearity of type (R). Theorem 5.24. Let f be a nonlinearity of type (R). Then the extremal solution u∗ of (Nλ ) is regular for N ≤ 5, while f (u∗ ) ∈ Lq (Ω) for all q < NN−2 if N ≥ 6. We shall split the proof in several lemmas that may have their own interest, in particular the simple new energy estimate given in Lemma 5.27 below, coupled with a pointwise estimates on −∆u given in Lemma 5.25, and which was motivated by the proof of Souplet of the Lane-Emden conjecture in four space dimensions [24]. We start by the latter (next two lemmas) which does not require the stability of the solutions. Lemma 5.25. Suppose u is a solution of (Nλ ) and g is a smooth function defined on the range of u with f (t) ≥ g(t)g 0 (t) and g(t), g 0 (t), g 00 (t) ≥ 0 on the range of u with g(0) = 0. Then √ −∆u ≥ λg(u) in Ω. (5.26) √ Proof. Define v := −∆u − λg(u) and so v = 0 on ∂Ω and a computation shows that √ √ −∆v + λg 0 (u)v = λ[f (u) − g(u)g 0 (u)] + λg 00 (u)|∇u|2 in Ω. The assumptions on g allow one to apply the maximum principle and obtain that v ≥ 0 in Ω. Now we use the stability condition on the solution.  92  5.3. A general regularity result for low dimensions Lemma 5.27. Suppose u is a semi-stable solution of (Nλ ). Then Z Z 00 2 f (u)(−∆u)|∇u| dx ≤ λ f (u)dx.  (5.28)  Ω  Ω  Proof. Set ψ = ∆u in (5.2) to arrive at Z Z I := f 0 (u)(∆u)2 dx ≤ ∆2 uf (u)dx =: J. Ω  Ω  Now an integration by parts shows that Z Z 00 2 f (u)(−∆u)|∇u| dx − f 0 (u)∇u · ∇∆udx I = Ω Ω Z Z f 0 (u)∇u · ∇∆udx, J = λ f (u)dx − Ω  Ω  in view of f (0) = 1. Since I ≤ J one obtains the result. Lemma 5.29. Suppose u is a semi-stable solution R u of00 (Nλ ) and that g satisfies the conditions of Lemma 5.25. If H(u) := 0 f (τ )g(τ )dτ , then Z Z g(u)H(u)dx ≤ f (u)dx. (5.30) Ω  Ω  Proof. We rewrite the result from Lemma 5.27 as Z √ Z √ Z λ g(u)H(u)dx ≤ λ (−∆u)H(u)dx = λ ∇H(u) · ∇udx Ω Ω Z Ω √ Z λ H 0 (u)|∇u|2 dx ≤ λ f (u)dx, = Ω  Ω  where the two inequalities use the pointwise bound from Lemma 5.25. Lemma 5.31. Suppose u ≥ 0 is a semi-stable solution of (Nλ ). Then 3  R Ω  f (u) 2 √ dx u+1  ≤C  and  R  Ω f (u)dx  ≤ C,  (5.32)  for some constant C > 0 independent of λ and u. Proof. Define for u ≥ 0, the function g(u) :=  √  Z 2  u  1 2 (f (t) − 1)dt .  0  93  5.3. A general regularity result for low dimensions Clearly g(0) = 0 and g ≥ 0. Now square g and take a derivative to see that 2g(u)g 0 (u) = 2(f (u) − 1) and so we satisfy the requirement that f (u) ≥ g(u)g 0 (u). Also from this we see that g 0 (u) ≥ 0. We now show that g 00 (u) ≥ 0. Note that g 00 (u) has the same sign as Z u 1 0 γ(u) := f (u) (f (t) − 1)dt − (f (u) − 1)2 . 2 0 Now γ(0) = 0 and 0  0  Z  00  γ (u) = f (u)(f (u) − 1) + f (u)  u  (f (t) − 1)dt − (f (u) − 1)f 0 (u),  0  and so γ 0 (u) ≥ 0 and hence γ(u) ≥ 0. By Lemma 5.29 we have Z Z g(u)H(u)dx ≤ f (u)dx, Ω  (5.33)  Ω  where  Z H(u) :=  u  f 00 (τ )g(τ )dτ.  0  R1 Without loss of generality, we can assume that 0 (f (t)−1)dt > 0. For u > 1 we have Z τ 1 √ Z u 00 √ 2 H(u) ≥ 2 f (τ ) (f (t) − 1)dt dτ ≥ 2C0 (f 0 (u) − f 0 (1)) 1  0  1 R 2 1 where C0 := 0 (f (t) − 1)dt . Since by convexity f 0 (u) ≥ u → ∞, we can find M > 0 large so that H(u) ≥ C0 f 0 (u)  f (u)−1 u  → ∞ as  ∀ u ≥ M.  Since Z  u  Z (f (t) − 1) ≥  0  u  (f (t) − 1) ≥ (1 − 1  1 ) f (1)  Z  u  f (t)dt  (5.34)  1  (5.35)  94  5.3. A general regularity result for low dimensions for u ≥ 1, from (5.33) and the above estimate we see that Z  f 0 (u)  !1  u(x)  Z  2  f (t)dt  Ω  dx  0  Z  0  ≤  f (u)  0  Z  1 −1 ) f (1)  M  dx + f (M )  0  Z  0  2  f (t)dt  {u≥M }  ≤ C0−1 (1 −  !1  u(x)  Z   12 f (t)dt |Ω|  0  Z  Z  u(x)  2  (f (t) − 1)dt  H(u) Ω  !1  dx  0   21  M  |Ω|  +f (M ) f (t)dt 0 Z ≤ C1 f (u)dx Ω  R for some C1 > 0 (independent of λ and u), in view of |Ω| ≤ Ω f (u). Defining Z u 1 3 0 2 h(u) := u(f ) (u) f (t)dt − (f 2 (u) − 1)2 , 6 0 we have that h ≥ 0. Note h(0) = 0 and Z u 0 0 00 h (u) = 2uf (u)f (u) f (t)dt + u(f 0 )2 (u)f (u) + I, 0  where 0 2  u  Z  I = (f ) (u) 0  1 1 1 f (t)dt − f 2 (u)f 0 (u) + f 2 (u)f 0 (u). 2 2  Since 0  f (u)  2  u  Z  Z  0  f (t)dt ≥ f (u) 0  0  u  f 0 (t)f (t)dt = f 0 (u)  f (u)2 f 0 (u) − , 2 2  we have that I ≥ 0, and then h0 (u) ≥ 0. Hence, h(u) ≥ 0 leads to the fundamental estimate: Z u 1 3 2 f 2 (u) − 1 0 f (t)dt f (u) ≥√ √ ∀ u ≥ 0. (5.36) 6( u + 1) 0 So by (5.36) we get that Z Ω  3  f (u) 2 √ dx ≤ u+1  Z Ω  3  √ f (u) 2 − 1 √ dx + |Ω| ≤ ( 6C1 + 1) u+1  Z f (u)dx. Ω  95  5.4. Regularity in higher dimension (I) Since f is superlinear at ∞ this implies the validity of (5.32). For later purposes, note that from the above estimates there holds Z u 1 √ 2 1 1 f (t)dt g(u)H(u) ≥ 2C0 (1 − )2 f 0 (u) f (1) 0 3 √ 2 1 1 f (u) − 1 2C0 (1 − )2 √ √ ≥ f (1) 6( u + 1) for u ≥ M , and then by superlinearity of f at ∞ g(u)H(u) → +∞ f (u)  as u → +∞.  Hence, for general nonlinearities f of type (R) we can re-state Lemma 5.29 as Z g(u)H(u) ≤ C (5.37) Ω  for every semi-stable solution u of (Nλ ), where g(u) is exactly as before and C is independent of λ and u. Proof of Theorem 5.24: Recalling that u∗ is the limit of the classical solutions uλ as λ % λ∗ , it follows immediately from estimate (5.32) and Theorem 5.5. Indeed, in this case we can take q = 1, α = 32 and β = 12 to conclude that u∗ ∈ L∞ (Ω) when N ≤ 5, and f (u∗ ) ∈ Lq (Ω) for every q < NN−2 when N ≥ 6.  5.4  Regularity in higher dimension (I)  In this section, we will consider nonlinearities f of type (R) which satisfy the following growth condition: lim inf t→+∞  f (t)f 00 (t) > 0. (f 0 )2 (t)  (5.38)  The aim is to gain dimensions N = 6, 7 in Theorem 5.24 by showing that L2 −bounds on f (u) are still in order, as we state in the following theorem. Theorem 5.39. Let f be a nonlinearity of type (R) so that (5.38) holds. Let u be a semi-stable solution of (Nλ ). Then Z f 2 (u)dx ≤ C, Ω  where C > 0 is independent of λ and u. 96  5.4. Regularity in higher dimension (I) By standard elliptic regularity theory, Theorem 5.39 immediately yields the following improvement on Theorem 5.24. Theorem 5.40. Let f be a nonlinearity of type (R) so that (5.38) holds. Then the extremal solution u∗ of (Nλ ) is regular for N ≤ 7. Proof (of Theorem 5.39) Since   0 Z t 0 Z t 1 2 0 0 0 00 f = f (t) f + f (t)f (t) ≥ f (t)f (t) = f (t) f (t) , 2 0 0 one can integrate on [0, u] to get Z u 1 1 0 f (u) f ≥ f 2 (u) − . 2 2 0 Since f (u) → +∞ as u → +∞, we can find M ≥ 1 large so that Z u 1 0 f ≥ f 2 (u) f (u) ∀ u ≥ M. 4 0 Setting δ := lim inf t→+∞  (5.41)  f (t)f 00 (t) > 0, (f 0 )2 (t)  we can –modulo taking a larger M – also assume that δ f (u)f 00 (u) ≥ (f 0 )2 (u) 2  ∀ u ≥ M.  (5.42)  By (5.41)-(5.42) we get Z 1  u   Z Z t 1 f (t)( f ) 2 ( 00  0  u  f)  1 2  0  00  Z  u  ≥ f (u)  0  f Z δ (f 0 )2 (u) u f 2 f (u) 0 δ f (u)f 0 (u) 8 δ 2 (f (u))0 16 0  ≥ ≥ =  for all u ≥ M , which, integrated once more in [M, u], u ≥ M , yields to Z u  Z u Z t 1 1 δ δ 00 f (t)( f ) 2 ( f ) 2 ≥ f 2 (u) − f 2 (M ). 16 16 0 1 0 97  5.5. Regularity in higher dimension (II) Then we can find N ≥ M large so that for u ≥ N we have Z u  Z u Z t 1 1 δ f ) 2 ≥ f 2 (u). f 00 (t)( f ) 2 ( 32 0 0 1 Setting as always g(u) =  √  2 Z u  g(u)H(u) ≥ 2  1 (f (t) − 1)dt 2 , by (5.34) we can now deduce  Z u Z t 1 1 00 2 ( (f − 1)dt) 2 f (t)( (f − 1)ds)  Ru 0  0  1  ≥  0  δ 1 (1 − )f 2 (u) 16 f (1)  for u ≥ N . By Lemma 5.29 as re-stated in (5.37) we finally get that Z  2  Z  f 2 (u)dx + f 2 (N )|Ω| {u≥N } Z 16 1 −1 ≤ (1 − ) g(u)H(u) δ f (1) Ω +f 2 (N )|Ω| ≤ C  f (u)dx ≤ Ω  for some C > 0 independent of λ and u. Theorem 5.39 combined with Corollary 5.12 gives also immediately the following results. Corollary 5.43. When f (t) = (t + 1)p , p > 1, the extremal solution u∗ of (Nλ ) is regular if either N ≤ 8 or if N ≥ 9 and p < NN−8 . When f (t) = et , this is true for N ≤ 8.  5.5  Regularity in higher dimension (II)  We are still considering nonlinearities f of type (R). For N ≥ 6 we want to improve upon Theorem 5.24 under the following growth condition on f γ := lim sup t→+∞  f (t)f 00 (t) < +∞. (f 0 )2 (t)  (5.44)  Typical examples of such nonlinearities are again f (t) = et (with γ = 1) and f (t) = (t + 1)p (with γ = 1 − p1 ). The aim is to get the regularity of the extremal solution also in dimensions higher than 5 for values of γ not too large: 98  5.5. Regularity in higher dimension (II) Theorem 5.45. Let N ≥ 6 and f be a nonlinearity of type (R) satisfying (5.44). The extremal solution u∗ of (Nλ ) is regular for N < γ8 . The validity of Theorem 5.45 follows easily Theorem 5.24, Theorem 5.14 and the following crucial estimate for stable solutions. To apply Theorem 5.14, we need to require exactly γ < N8 when N ≥ 6, and (5.44) guarantees the validity of (5.47) with 0 < γ +  < 2 and M > 0 large enough. Theorem 5.46. Let f be a nonlinearity of type (R) so that f (u)f 00 (u) ≤ γ(f 0 )2 (u)  ∀ u ≥ M,  (5.47)  for some 0 < γ < 2 and M > 0. Let u be a semi-stable solution of (Nλ ). Then Z 2 (5.48) (f 0 ) γ (u)dx ≤ C, Ω  where C > 0 is a constant independent of u and λ. Proof. Re-write (5.47) as d d log(f 0 (t)) ≤ log(f γ (t)) dt dt  ∀t≥M  and integrate over [M, u] to deduce that f 0 (u) ≤  f 0 (M ) γ f (u) f γ (M )  ∀ u ≥ M.  (5.49)  Since f (u) ≥ f (0) = 1, we can write that f 0 (u) ≤ C0 f γ (u)  ∀ u ≥ 0,  (5.50)  where C0 is a suitable large constant. Setting Z u  1 Γ(u) := f (t)dt − f 2−γ (u) − 1 , (2 − γ)C0 0 1−γ  0  (u) ≥ 0. Hence, the one notes that Γ(0) = 0 and Γ0 (u) = f (u) − f (u)f C0 following estimate holds for every u ≥ 0: sZ u 1 1 f (t)dt ≥ p f 2−γ (u) − 1 2 . (5.51) (2 − γ)C0 0  99  5.5. Regularity in higher dimension (II) √ Ru 1 As in the previous section, set g(u) = 2 0 (f (t) − 1)dt 2 in such a way that it satisfies the assumptions of Lemma 5.29. By (5.34), (5.51) and the superlinearity of f at ∞, we can find N ≥ 1 large so that for all u ≥ N , Z u 1 2 1 1 g(u) ≥ )2 f (t)dt 2(1 − f (1) 0  1 √ 2 1 f (1) − 1 ≥ 2 (f 2−γ (u) − 1) 2 (2 − γ)C0 f (1)  1 2 γ f (1) − 1 ≥ f 1− 2 (u). (2 − γ)C0 f (1) √  Setting C1 :=    so that for u ≥  f (1)−1 (2−γ)C0 f (1) N0  Z H(u) := ≥  1 2  u  , we use (5.50) to find N 0 ≥ N sufficiently large  Z  00  u  γ  f (t)g(t)dt ≥ C1 f 00 (t)f 1− 2 (t)dt N 0 Z u 1 1 −1 −1 f 00 (t)(f 0 ) γ 2 (t)dt C1 C02 γ N  = ≥  1 − γ1 2   1 1 2γ  0 γ1 + 12 0 γ +2 C1 C0 (f ) (u) − (f ) (N ) γ+2 1 1 −1 γ +1 (f 0 ) γ 2 (u), C1 C02 γ γ+2  where we have used the convexity of f and the fact that f 0 (u) → +∞ as u → +∞. In conclusion, setting 1 − γ1 2  C2 := C1 C0  γ γ+2    f (1) − 1 (2 − γ)C0 f (1)  1  2  we have that Z Z 1 1 1 γ +1 −1 1− γ2 0 γ +2 (u)f (u) ≤ C2 H(u)g(u) + (f 0 ) γ 2 (N 0 )f 1− 2 (N 0 )|Ω|. (f ) Ω  Ω  To complete the proof of Theorem 5.46, it suffices to couple this lower bound with (5.37) to obtain Z 1 γ +1 (f 0 ) γ 2 (u)f 1− 2 (u) ≤ C, Ω  100  5.6. Singular nonlinearities and then by (5.49) to get Z  2  (f 0 ) γ (u) ≤ C 0  Ω  for any stable solution u, where C, C 0 are independent of λ and u. Remark 5.5.1. a) As a by-product of the above theorem, one obtains again the improvements for the exponential and power nonlinearities established in Corollary 5.43. Indeed, when f (t) = (t + 1)p , it turns out that γ = 1 − p1 , 8 and then u∗ is a regular solution whenever N ≤ 5 and 6 ≤ N < 8 + p−1 . N We can collect the two cases as N ≤ 8 or N ≥ 9 and p < N −8 . When f (t) = et , we have that γ = 1, and then u∗ is a regular solution for N < 8. The missing dimension N = 8 follows directly from Theorem 5.14 in view of the identity, Z Z Z Z 2 N 0 γ 0 2 2 (f ) (u)dx = (f ) (u)dx = f (u)dx = f 4 (u)dx. Ω  Ω  Ω  Ω  b) Recall from [11] that when f (t) = et on the unit ball in RN , the extremal solution associated with (Dλ ) is then smooth provided N ≤ 12. This suggests that our regularity results are not optimal, which likely is a product of the deficient energy estimate obtained in Lemma 5.27 above. c) Integrating once more (5.49) one sees that f 0 (M )  (u−M )  • f (u) ≤ f (M )e f (M ) for u ≥ M when γ = 1 h i 1 0 1−γ ) (u − M ) • f (u) ≤ f 1−γ (M ) + (1 − γ) ffγ(M for u ≥ M when 0 < (M ) γ < 1. This explains why (5.44) is sometimes referred to as a growth condition for f. d) For exponential and power nonlinearities, the uniform bound (5.48) can be re-formulated as an L2 −bound on f (u). Indeed, one can define g(u) as above, and use Lemma 5.29 to deduce directly such a bound, which in turn shows that the loss in optimality is not really coming from Theorem 5.46.  5.6  Singular nonlinearities  Nonlinearities of the form (1 − u)−p , p > 0, have recently attracted much attention, due to their connection with the so-called MEMS (Micro ElectroMechanical Systems) technology. Neglecting torsion effects, one is led to 101  5.6. Singular nonlinearities study the second-order nonlinear eigenvalue problem with p = 2. In this case, the picture is well understood [12, 15] and we refer the interested reader to the recent monograph [13]. The fourth-order case has been firstly addressed in [18] both in the form (Dλ ) and (Nλ ). As already mentioned in the introduction, for problem (Dλ ) the existence of the minimal branch on the unit ball has been proved in [7] along with its compactness for N ≤ 8 [9]. Since a maximum principle holds for (Nλ ), the existence of the minimal branch for (Nλ ) on a general domain follows by the same argument as in [7] for (Dλ ). We will consider now the question of regularity of u∗ = lim∗ uλ in terms λ%λ  of the dimension N . We will not consider general nonlinearities of type (S) in the sequel, but we shall restrict our attention to the interesting case (1 − u)−p . The general case has been addressed in [8] for the second-order case, and growth conditions as (5.38) and (5.44) are no longer sufficient for the analysis. We now establish the following result. Theorem 5.52. Suppose p > 1 and p 6= 3. Then the extremal solution u∗ 8p is regular (i.e. supΩ u∗ < 1) provided N ≤ p+1 . This will follow immediately from the following two theorems. Theorem 5.53. Let un denote a sequence of solutions of (Nλn ) such that there is some α > 1 and α ≥ (p+1)N such that supn kf (un )kα < ∞. Then 4p supn kun k∞ < 1. > 1, the lower Proof. We suppose that N is big enough so that (p+1)N 4p dimensional cases being similar we omit their details. If f (un ) is bounded (p+1)N  4,  (p+1)N 4p  in L 4p , then by elliptic regularity we have un bounded in W the Sobolev imbedding theorem we have un bounded in the space C  4−  h  4p p+1  i  −1,  h  4p p+1  i  4p +1− p+1  . By  (Ω).  This naturally breaks into the two cases: • 1 < p < 3 and then un is bounded in C • p > 3 and un is then bounded in C  1, 3−p p+1  4 0, p+1  .  We now let xn ∈ Ω be such that un (xn ) = maxΩ un . We claim that there exists some C > 0, independent of n, such that 4  |un (x) − un (xn )| ≤ C|x − xn | p+1 ,  x ∈ Ω. 102  5.6. Singular nonlinearities For the second case this is immediate, while for the first we use the fact that ∇un (xn ) = 0 and the fact that there is some 0 ≤ tn ≤ 1 such that un (x) − un (xn ) = ∇un (xn + tn (x − xn )) · (x − xn ) = (∇un (xn + tn (x − xn )) − ∇un (xn )) · (x − xn ) 0, 3−p  along with the fact that ∇un is bounded in C p+1 to show the claim. To complete the proof, we work towards a contradiction, and assume, after passing to a subsequence, that un (xn ) = 1 − εn → 1. By passing to another subsequence, we can assume that un converges in C(Ω) which along with the boundary conditions guarantees that xn → x0 ∈ Ω. Then one has 1 − un (x) = 1 − un (xn ) + un (xn ) − un (x) = εn + un (xn ) − un (x) 4  ≤ εn + C|x − xn | p+1 , and so there is some Cp > 0 such that  (p+1)N  (p+1)N N 4 (1 − un (x)) 4 ≤ Cp εn + |x − xn | . From this one sees that f (un (x))  (p+1)N 4p  ≥  Cp−1 (p+1)N 4  εn  + |x − xn  := hn (x). |N  R But since xn → x0 ∈ Ω and εn → 0, ones sees that Ω hn (x)dx → ∞ which contradicts the integrability condition on f (un ). Hence we must have supn kun k∞ < 1. We now obtain the familiar L2 bound on f (u) for semi-stable solutions. We prefer to prove this results using an explicit calculation, even if this result also follows from Theorem 4.1. Theorem 5.54. Suppose p > 1 and u ≥ 0 is a semi-stable solution of (Nλ ). Then kf (u)k2 ≤ C, where C is independent of u and λ.  103  5.6. Singular nonlinearities Proof. Define r g(u) :=  2 p−1  !  1 (1 − u)  p−1 2  −1 .  Note that this choice of g is different from the one used above, as it is easier to manage. It does verify the conditions of Lemma 3.2 and therefore one has −∆u ≥ g(u) a.e. in Ω, and by Lemma 3.4 we have Z Z g(u)H(u)dx ≤ f (u)dx, (5.55) Ω  Ω  Ru  f 00 (τ )g(τ )dτ. A computation shows that !   1 1 H(u) = Cp − 1 + C̃ 1 − p 3p+1 (1 − u)p+1 (1 − u) 2  where H(u) :=  0  where Cp , C̃p > 0. Now writing out (5.55) one obtains an estimate of the form Z Z Z 1 1 1 dx + C(u) dx ≤ C(p) dx. 3p+1 2p (1 − u) (1 − u)p Ω Ω Ω (1 − u) 2 Since p > 1, we have that desired result.  3p+1 2  < 2p, from which one easily obtains the  Acknowledgements: This work began while the second author was visiting the Pacific Institute for the Mathematical Sciences, for which he thanks Dr. Ghoussoub for the kind invitation and hospitality.  104  Bibliography [1] G. Arioli, F. Gazzola, H.-C. Grunau and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (2005), 1226–1258. [2] E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. Differential Equations 2005(2005), No. 34, 20 pp. [3] T. Boggio, Sulle funzioni di Green dordine m, Rend. Circ. Mat. Palermo (1905), 97–135. [4] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443–469. [5] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, preprint, arXiv:0909.4696v2. [6] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), 709–733. [7] D. Cassani, J. do O and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud. 9 (2009), 177197. [8] D. Castorina, P. Esposito and B. Sciunzi, Degenerate elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations 34 (2009), 279–306. [9] 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. [10] M.G. Crandall and P.H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. 58 (1975), 207–218. 105  Chapter 5. Bibliography [11] 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. [12] 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 (2007), 1731–1768. [13] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Research Monograph, Courant Lecture Notes, in press (2009) 332 pp. [14] L. Evans, Partial differential equations, Graduate Studies in Math. 19 AMS, Providence, RI, 1998. [15] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2007), 1423–1449. [16] N. Ghoussoub and A. Moradifam, Bessel pairs and optimal improved Hardy and Hardy-Rellich inequalities, Math. Ann., in press (Arxive 2007). [17] Z. Guo and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2008/09), 2034–2054. [18] F.H. Lin and Y.S. Yang, Nonlinear non-local elliptic equation modelling electrostatic acutation, Proc. R. Soc. London Ser. A 463 (2007), 1323– 1337. [19] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math. 23 (1997), 161-168. [20] F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), 791–836. [21] A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., in press (2009). [22] A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations 248 (2010), 594– 616.  106  Chapter 5. Bibliography [23] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sèr. I Math. 330 (2000), 997–1002. [24] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math. 221 (2009), 1409–1427.  107  Chapter 6  Regularity of the extremal solutions in elliptic systems5 6.1  Introduction  In this short note we are interested in solutions of the elliptic system given by  −∆u = λev Ω    u −∆v = γe Ω (P )λ,γ u = 0 ∂Ω    v = 0 ∂Ω where λ, γ are positive parameters and where Ω is a smooth bounded domain in RN . In particular we are interested in the regularity of the extremal solutions associated with (P )λ,γ , which we define more precisely later. Along the diagonal λ = γ the problem (P )λ,γ reduces to the scalar analog of (P )λ,γ , see below. Provided one stays sufficiently close to the diagonal we show that some basic maximum principle arguments coupled with a standard energy estimate approach (the familiar approach in the scalar case) shows the regularity of the extremal solutions in the expected dimensions. We now recall the well studied scalar version (with general nonlinearity f ) of (P )λ,γ given by  −∆u = λf (u) Ω (P )λ u = 0 ∂Ω where λ is a positive parameter and where Ω is a bounded domain in RN . See, for instance, [1], [2], [6], [7] and [8]. Here generally one assumes that f is a smooth, increasing, convex nonlinearity with f (0) = 1 and f superlinear at ∞, ie. limu→∞ f (u) u = ∞. It is known that there is an non degenerate finite interval U = (0, λ∗ ) such that for all 0 < λ < λ∗ there exists a smooth, 5 A version of this chapter has been submitted for publication: C. Cowan Regularity of the extremal solutions in elliptic systems  108  6.1. Introduction minimal solution uλ of (P )λ . By minimal we mean that any other solution v of (P )λ satisfies v ≥ uλ a.e. in Ω. In addition one can show that for each x ∈ Ω the map λ 7→ uλ (x) is increasing on (0, λ∗ ). This allows one to define the extremal solution u∗ (x) := lim∗ uλ (x), λ%λ  and it can be shown that u∗ is the unique weak solution of (P )λ∗ . Also it is known that for λ > λ∗ there are no weak solutions. One can also show that for each 0 < λ < λ∗ the minimal solution uλ is semi-stable in the sense that the principle eigenvalue of the linear operator Lλ,uλ := −∆ − λf 0 (uλ ), over H01 (Ω) is nonnegative. Using the variational structure this implies that Z Z 0 2 λf (uλ )ψ dx ≤ |∇ψ|2 dx, ∀ψ ∈ H01 (Ω). Ω  Ω  One can now ask the question whether u∗ is a classical solution of (P )λ∗ ? Elliptic regularity shows this is equivalent to the boundedness of u∗ . In the case where f (u) = eu one can show that u∗ is bounded provided N ≤ 9. Moreover this is optimal after one considers the fact that u∗ (x) = −2 log(|x|) provided Ω is the unit ball in RN where N ≥ 10. For more results concerning the regularity of the extremal solution u∗ the reader should see [10], [4], [3] and [11]. We mention that vital to all the results concerning the regularity of u∗ is to use the semi-stability of the minimal solutions uλ to obtain a priori estimates and then to pass to the limit. We now return to the system (P )λ,γ and we follow the work of M. Montenegro [9], where all of the following results are taken from. We also mention that he obtains many more results and also that he studies a much more general system then (P )λ,γ . We let Q = {(λ, γ) : λ, γ > 0} and we define U := {(λ, γ) ∈ Q : there exists a smooth solution (u, v) of (P )λ,γ } . We set Υ := ∂U ∩ Q. The curve Υ is well defined and separates Q into two connected components Q and V. We omit the various properties of Υ but the interested reader should consult [9]. One point we mention is that if for x, y ∈ R2 we say x ≤ y provided xi ≤ yi for i = 1, 2 then it is easily seen, using the method of sub/supersolutions, that if (0, 0) < (λ0 , γ0 ) ≤ (λ, γ) ∈ U then (λ0 , γ0 ) ∈ U. Now it can be shown that for each (λ, γ) ∈ U there exists 109  6.2. Main Results a smooth minimal solution (uλ,γ , vλ,γ ) of (P )λ,γ and if (0, 0) < (λ1 , γ1 ) ≤ (λ2 , γ2 ) ∈ U then (uλ1 ,γ1 , vλ1 ,γ1 ) ≤ (uλ2 ,γ2 , vλ2 ,γ2 ). Now for (λ∗ , γ ∗ ) ∈ Υ there is some 0 < σ < ∞ such that γ ∗ = σλ∗ and we can define the extremal solution (u∗ , v ∗ ) at (λ∗ , γ ∗ ) by passing to the limit along the ray given by γ = σλ for 0 < λ < λ∗ . Moreover it can be shown that (u∗ , v ∗ ) is indeed a weak solution of (P )λ∗ ,γ ∗ . We now come to the issue of stability. Theorem 6.1. ([9]) Let (λ, γ) ∈ U and let (u, v) denote the minimal solution of (P )λ,γ . Then (u, v) is semi-stable in the sense that there is some smooth 0 < φ, ψ ∈ H01 (Ω) and 0 ≤ K such that −∆φ = λev ψ + Kφ,  −∆ψ = γeu φ + Kψ,  Ω.  Now one should note that K < λ1 (Ω). To see this one multiplies either of these equations by the first positive eigenfunction of −∆ and integrates by parts.  6.2  Main Results  Our main result is given by the following theorem. Theorem 6.2. Let 3 ≤ N ≤ 9 and suppose that (λ, γ) ∈ Υ with N −2 γ 8 < < . 8 λ N −2 Then the associated extremal solution (u∗ , v ∗ ) is smooth. Now one should note that along the diagonal the problem reduces to the scalar problem. Also by symmetry it is enough to prove the result for 0 < γ ≤ λ. We prove the above Theorem in a series of lemma’s. Lemma 6.3. Suppose that (u, v) is a smooth solution of (P )λ,γ where 0 < γ ≤ λ. Then γ u≤v≤u a.e. in Ω. λ Proof. Taking the difference of the equations in (P )λ,γ we have −∆(u−v) = λev − γeu in Ω and multiplying this by (u − v)− and integrating by parts one arrives at Z Z 2 − |∇(u − v)− | dx = (λev − γeu )(u − v)− dx, Ω  Ω  110  6.2. Main Results and now note that the right hand side is nonnegative where as the left hand side is nonpositive. Hence we see that (u − v)− = 0 a.e. in Ω and so u ≥ v a.e. in Ω. Now note that γ −∆(v − u) = γ(eu − ev ) ≥ 0 Ω, λ since u ≥ v in Ω and so v ≥ λγ u in Ω. Lemma 6.4. Suppose that (λ, γ) ∈ U with 0 < γ ≤ λ and we let (u, v) denote the minimal solution of (P )λ,γ . Let K, φ, ψ be as in Theorem 6.1. Then ψ γ ≥ in Ω. φ λ Proof. First note that −∆(ψ − φ) = γeu φ − λev ψ + K(ψ − φ) ≥ γev (φ − ψ) + (γ − λ)ev ψ + K(ψ − φ) where we have used the fact that u ≥ v in Ω. Rearranging this we have −∆(ψ − φ) − K(ψ − φ) + γev (ψ − φ) ≥ (γ − λ)ev ψ  Ω.  (6.5)  We now define L := −∆ − K and H := L + γev . Now note that  H(ψ − φ +  (λ − γ) (λ − γ) φ) ≥ L(ψ − φ + φ) + γev (ψ − φ) γ γ (λ − γ) = H(ψ − φ) + L(φ) γ (λ − γ) v ≥ (γ − λ)ev ψ + λe ψ λ = 0  in Ω. Now since H satisfies the maximum principle we see that ψ − φ + ≥ 0 in Ω, which after re-arranging, gives the desired result.  λ−γ λ φ  Theorem 6.2 will easily follow from the following lemma. Lemma 6.6. Suppose 3 ≤ N ≤ 9 and that (um , vm ) denotes a sequence of smooth minimal solutions to (P )λm ,σλm where N 8−2 < σ ≤ 1. Then (um , vm ) is bounded in L∞ (Ω) × L∞ (Ω). 111  6.2. Main Results Proof. Fix N 8−2 < σ ≤ 1 and for notational simplicity we drop the subscript m from um , vm , φm , ψm and Km . From the previous lemma we have ψφ ≥ σ. Now for any smooth positive function E one has Z Z −∆E 2 |∇β|2 dx, ∀β ∈ H01 (Ω), β dx ≤ E Ω Ω see, for instance, [5]. Now note that −∆φ ψ = λev + K ≥ σλev φ φ  Ω.  Taking E = φ and β = etu − 1, where t is chosen such that gives Z Z 2 2 v tu e2tu |∇u|2 dx. σλ e e − 1 dx ≤ t  N −2 4  < t < 2σ, (6.7)  Ω  Ω  Now multiplying −∆u = λev by e2tu − 1 and integrating by parts gives Z Z 2tu 2 2t e |∇u| dx = λ ev (e2tu − 1)dx. (6.8) Ω  Ω  Now equating (6.7) and (6.8) gives, after some simplification, Z  Z σ 1 2σ v 2tu − e e dx ≤ etu ev dx. t 2 t Ω Ω Now note that since t < 2σ the coefficient on the left is positive. Now applying Holder’s inequality on the right and squaring gives   σ 1 − t 2  2 Z e Ω  4σ 2 e dx ≤ 2 t  2tu v  Z  ev dx,  Ω  and now since u ≥ v in Ω we see that this gives us an L2t+1 (Ω) bound for ev . We now return to the sequence notation. So we have that evm is bounded in L2t+1 (Ω) but note that 2t + 1 > N2 and also note that λm is bounded. Now since −∆um = λm evm in Ω with um = 0 on ∂Ω, and since λm is bounded one sees, using elliptic regularity, that um is bounded in L∞ (Ω). From this, and since σλm is bounded, we easily infer that vm is bounded in L∞ (Ω).  112  Bibliography [1] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa Blow up for ut − ∆u = g(u) revisited, Ad. Diff. Eq. 1 (1996), 73-90. [2] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. [3] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, (preprint) (2009). [4] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), no. 2, 709–733. [5] C. Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal. 9 (2010), no. 1, 109–140. [6] M.G. Crandall and P.H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), pp.207-218. [7] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, J. Diff. Eq. 16 (1974), 103-125. [8] F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), 791-836. [9] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc.37 (2005) 405-416. [10] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sr. I Math. 330 (2000), no. 11, 997–1002. [11] D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math. 4 (2002), no. 3, p. 547-558.  113  Chapter 7  Optimal Hardy inequalities for general elliptic operators with improvements6 7.1  Introduction  We begin by recalling the various Hardy inequalities. Let Ω be a bounded domain in Rn containing the origin and where n ≥ 3. Then Hardy’s inequality (see [11]) asserts that Z  2  |∇u| dx ≥ Ω    n−2 2  2 Z Ω  u2 dx, |x|2  (7.1)  2 for all u ∈ H01 (Ω). Moreover the constant n−2 is optimal and not at2 tained. An analogous result asserts that for any bounded convex domain Ω ⊂ Rn with smooth boundary and δ(x) := dist(x, ∂Ω) (the euclidean distance from x to ∂Ω), there holds (see [4]) Z 2 Z u 1 2 dx, (7.2) |∇u| dx ≥ 2 4 Ω δ Ω for all u ∈ H01 (Ω). Moreover the constant 41 is optimal and not attained. We will refer to this inequality as Hardy’s boundary inequality. Recently Hardy inequalities involving more general distance functions than the distance to the origin or distance to the boundary have been studied (see [3]). Suppose Ω is a domain in Rn and M a piecewise smooth surface of co-dimension k, k = 1, ..., n. In case k = n we adopt the convention that M is a point, say, the origin. Set d(x) := dist(x, M ). Suppose k 6= 2 and 6  A version of this chapter has been accepted for publication: C. Cowan: Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal. 9 (2010), no. 1, 109–140.  114  7.1. Introduction −∆d2−k ≥ 0 in Ω\M then Z Z 2 (k − 2)2 u 2 |∇u| dx ≥ dx, 2 4 Ω Ω d  (7.3)  for all u ∈ H01 (Ω\M ). We comment that the above inequalities all have Lp analogs. In the last few years improved versions of the above inequalities have been obtained, in the sense that non-negative terms are added to the right hand sides of the inequalities; see [6], [4], [3], [5],[8],[9],[16]. One common type of improvement for the above Hardy inequalities are the so called potentials; we call 0 ≤ V (x), defined in Ω, a potential for (7.1) provided Z  2  |∇u| dx − Ω    n−2 2  2 Z Ω  u2 dx ≥ |x|2  Z  V (x)u2 dx,  u ∈ H01 (Ω).  Ω  Most of the results in this direction are explicit examples of potentials V where, in the best results, V is an infinite series involving complicated inductively defined functions. Very recently Ghoussoub and Moradifam [10] gave the following necessary and sufficient conditions for a radial function V (x) = v(|x|) to be a potential in the case of Hardy’s inequality (7.1) on a radial domain Ω: V is a potential if and only if there exists a positive function y(r) which 0 solves y 00 + yr + vy = 0 in (0, supx∈Ω |x|). In another direction people have considered Hardy inequalities for operators more general than the Laplacian. One case of this is the results obtained by Adimurthi and A. Sekar [1]: Suppose Ω is a smooth domain in Rn which contains the origin, A(x) = ((ai,j (x))) denotes a symmetric, uniformly positive definite matrix with suitably smooth coefficients and for ξ ∈ Rn we define |ξ|2A := |ξ|2A(x) := A(x)ξ · ξ.   Now suppose E is a solution of LA,p (E) := −div |∇E|p−2 A∇E = δ0 in Ω A with E = 0 on ∂Ω where δ0 is the Dirac mass at 0. Then for all u ∈ W01,p (Ω)   Z Z |∇E|pA p p−1 p 2 |∇u|A dx − |u| dx ≥ 0. p Ep Ω Ω Improvements of this were also obtained and they posed the fol inequality p p−1 optimal? lowing question: Is p We show this is the case, even for a more general inequality.  115  7.1. Introduction  7.1.1  Outline and approach  Our approach will be similar to the one taken by Adimurthi and A. Sekar but we mostly concentrate on the quadratic case (p = 2) and for this we define LA (E) := −div(A∇E). We now motivate our main inequality. Suppose E is a smooth posi−1 tive function defined in Ω. Let u ∈ Cc∞ (Ω) and set v := E 2 u. Then a calculation shows that |∇u|2A −  |∇E|2A 2 A∇E · ∇(v 2 ) 2 u = E|∇v| + , A 4E 2 2  in Ω  and after integrating this over Ω we obtain Z Z Z 2 Z |∇E|2A 2 1 u 1 2 2 u dx = E|∇v|A dx + LA (E)dx. (7.4) |∇u|A dx − 2 4 Ω E 2 Ω E Ω Ω If we further assume that LA (E) ≥ 0 in Ω then Z Ω  |∇u|2A dx ≥  1 4  Z Ω  |∇E|2A 2 u dx, E2  u ∈ H01 (Ω).  (7.5)  From this we see that the optimal constant C(E)    R |∇u|2  1 C(E) := inf R Ω|∇E|2 A dx : u ∈ H01 (Ω)\{0} ≥ .   4 A 2 Ω E2 u It is possible to show that for all non-zero u ∈ H01 (Ω) we have Z E|∇v|2A dx > 0, Ω  where v is defined as above. Using this and (7.4) one sees that if C(E) = 14 then C(E) is not attained and hence if C(E) is attained then C(E) > 41 . This  |∇E|2A needs to be singular if we want C(E) = 14 . In fact one can E2 |∇E|2 |∇E|2 show that H01 (Ω) is compactly embedded in L2 (Ω, E 2 A dx) if E 2 A ∈ Lp (Ω) for some p > n2 and so one could then apply standard compactness arguments  shows that  to show that C(E) is attained. We are only interested in the case where |∇E|2  C(E) = 14 and hence we need to ensure E 2 A is singular and this can be done in two obvious ways. This naturally leads one to consider the following two classes of functions E (weights). 116  7.1. Introduction Definition 7.6. Suppose 0 < E in Ω and LA (E) is a nonnegative nonzero finite measure in Ω denoted by µ. 1) If in addition E ∈ H01 (Ω) then we call E a boundary weight on Ω. 2) If in addition E ∈ C ∞ (Ω\K) where K ⊂ Ω denotes the support of µ, E = ∞ on K and dimbox (K) < n − 2 (see below) then we call E an interior weight on Ω. Given a compact subset K of Rn we define the box-counting dimension (entropy dimension) of K by log(Hn (Kr )) r&0 log(r)  dimbox (K) := n − lim  provided this limit exists and where Kr := {x ∈ Ω : dist(x, K) < r} and Hα is the α- dimensional Hausdorff measure. Remark 7.1.1. It is possible to show that Cc0,1 (Ω\K) is dense in W01,p (Ω) provided K is compact and dimbox (K) < n − p (use appropriate Lipschitz cut off functions). From here on µ will denote the measure LA (E) and in the case where E is an interior weight on Ω, K will denote the support of µ. We now list the main results. We show that if E is either an interior or a boundary weight in Ω then we have the following inequality: Z Z |∇E|2A 2 1 2 |∇u|A dx − u dx ≥ 0, u ∈ H01 (Ω) (7.7) 4 Ω E2 Ω with optimal constant which is not attained. In the case that E is a boundary weight on Ω we obtain Z Z 2 Z |∇E|2A 2 1 1 u 2 |∇u|A dx − u dx ≥ dµ, u ∈ H01 (Ω). 2 4 E 2 E Ω Ω Ω  (7.8)  Moreover 12 is optimal (once one fixes 14 ) and is not attained. Using the methods developed in [10] we obtain necessary and sufficient conditions on 0 ≤ V (x) to be a potential for (7.7) in the case where E is an interior weight. We show that the following are equivalent: 1) For all u ∈ H01 (Ω) Z Ω  |∇u|2A dx −  1 4  Z Ω  |∇E|2A 2 u dx ≥ E2  Z  V (x)u2 dx.  Ω  117  7.1. Introduction 2)  There exists some 0 < θ ∈ C 2 (Ω\K) such that −LA (θ) |∇E|2A + +V ≤0 θ 4E 2  in Ω\K.  (7.9)  If we further assume that E = γ ≥ 0 (constant) on ∂Ω and if we are only interested in potentials of the form V (x) = f (E)|∇E|2A then we can replace 2) with 2’) There exists some 0 < h ∈ C 2 (γ, ∞) such that   1 00 h (t) + f (t) + 2 h(t) ≤ 0, in (γ, ∞). (7.10) 4t In practice this ode classification is more useful because of the shear abundance of ode results in the literature. We obtain weighted versions of (7.7) (respectively (7.8)) in the case that E is an interior weight (respectively boundary weight) on Ω which can be viewed as generalized versions of the Cafferelli-Kohn-Nirenberg inequality. To be more precise we obtain: Suppose E is an interior weight on Ω and t 6= 21 . Then Z Z 1 2 2t 2 E |∇u|A dx ≥ (t − ) |∇E|2A E 2t−2 u2 dx, (7.11) 2 Ω Ω for all u ∈ Cc0,1 (Ω\K). Moreover the constant is optimal and not attained in the naturally induced function space. Suppose E is a boundary weight on Ω and 0 6= t < 12 . Then (7.11) holds for all u ∈ Cc∞ (Ω) and is not attained in the naturally induced function space. Similarly Z Z Z 1 1 |∇E|2A E 2t−2 u2 dx ≥ ( − t) E 2t−1 u2 dµ, E 2t |∇u|2A dx − (t − )2 2 2 Ω Ω Ω (7.12) for all u ∈ Cc∞ (Ω). Moreover the constant on the right is optimal and not attained in the natural function space. In addition we show that the class of potentials for (7.11) is given by {E 2t V : V is a potential for (7.7)}. We also examine generalized Hardy inequalities which are valid for functions u ∈ H 1 (Ω). Suppose E a positive function with LA (E) + E a nonnegative nonzero finite measure denoted by µ, E = ∞ on the K (as before K denotes the support of µ) and where we assume that E satisfies a Neumann boundary condition. Then Z Z Z |∇E|2A 2 1 1 2 2 |∇u|A dx + u dx ≥ u dx, u ∈ H 1 (Ω). (7.13) 2 2 4 E Ω Ω Ω 118  7.1. Introduction Moreover these constants are optimal (in the sense that if one is fixed then the other is optimal). Improvements of (7.13) are also obtained. Assuming the same conditions on E we show that for 0 ≤ V we have Z Z Z Z |∇E|2A 2 1 1 V (x)u2 dx, u ∈ H 1 (Ω) |∇u|2A + u2 dx − u dx ≥ 2 2 4 E Ω Ω Ω Ω if and only if there exists some 0 < θ ∈ C ∞ (Ω\K) such that −LA (θ) −  θ |∇E|2A θ+Vθ ≤0 + 2 4E 2  in Ω\K,  (7.14)  with A∇θ · ν = 0 on ∂Ω. Weighted version of (7.13) are established. Assuming the same conditions on E we show that for t 6= 12 we have   Z Z 1 2 2t−2 1 2t 2 2t 2 E u dx ≥ t − E |∇E|2A u2 dx, E |∇u|A dx + 2 2 Ω Ω for all u ∈ Cc∞ (Ω\K). Moreover the constants are optimal and not obtained in the naturally induced function space. We establish optimal Hardy inequalities which are valid on exterior and annular domains. Suppose Ω is a exterior domain in Rn , E > 0 in Rn , lim|x|→∞ E = 0, −∆E = µ in Rn where µ is a nonzero nonnegative finite measure with compact support K. In addition we assume that dist(K, Ω) > 0 and ∂ν E ≥ 0 on ∂Ω. We define D1 (Ω ∪ ∂Ω) to be the completion of Cc∞ (Ω ∪ ∂Ω) with respect to the norm k∇ukL2 (Ω) . Then (i) For all u ∈ D1 (Ω ∪ ∂Ω) we have Z Z 1 |∇E|2 2 |∇u|2 dx ≥ u dx. (7.15) 4 Ω E2 Ω Moreover the constant is optimal and not attained. (ii) For all u ∈ D1 (Ω ∪ ∂Ω) we have Z Z Z 1 |∇E|2 2 1 u2 ∂ν E 2 |∇u| dx ≥ u dx + dS(x). 4 Ω E2 2 ∂Ω E Ω  (7.16)  Now suppose Ω = Ω2 \Ω1 where Ω1 ⊂⊂ Ω2 are both connected and Ω is connected. Suppose 0 < E in Ω2 and −∆E = µ in Ω2 where µ is a nonnegative nonzero finite measure compactly supported in Ω1 . In addition we assume that E = 0 on ∂Ω2 and ∂ν E ≤ 0 on ∂Ω1 . Then (7.15) is optimal and not attained over H01 (Ω ∪ ∂Ω1 ) := {u ∈ H 1 (Ω) : u = 0 on ∂Ω2 }. Optimal non-quadratic Hardy inequalities are also obtained in both the interior and boundary cases. 119  7.1. Introduction  7.1.2  Examples  We now look at various examples of Hardy inequalities (and applications of) which can be obtained after making suitable choices of weights E and matrices A. In most of the examples we will take A to be the identity matrix. 1. Hardy’s inequality: Let Ω denote a domain in Rn (n ≥ 3) which contains the origin and set E(x) := |x|2−n . Then −∆E = cδ0 where 2 1 2 c > 0 and δ0 is the Dirac mass at 0. Also |∇E| = n−2 and so 2 4E 2 |x|2 (7.7) gives the Hardy’s inequality. 2. Hardy’s inequality in dimension two: Now suppose Ω is a 2 domain in R which contains the origin. Put E(x) := − log(R−1 |x|) where R := supΩ |x|. Then −∆E = cδ0 where c > 0 and putting E into (7.7) gives Z Z u2 1 2 |∇u| dx ≥ dx, u ∈ Cc∞ (Ω). 2 log2 (R−1 |x|) 4 |x| Ω Ω 3. Hardy’s boundary inequality: Let Ω denote a bounded convex set in Rn and set E(x) := δ(x) := dist(x, ∂Ω). Since Ω is convex one can show δ is concave and hence −∆δ ≥ 0 in Ω. Putting E into (7.8) gives an improved version of (7.2). 4. Hardy’s boundary inequality in the unit ball: Let B denote the unit ball in Rn and set E(x) := 1 − |x|. Putting E into (7.8) gives Z Z Z 1 u2 n−1 u2 2 |∇u| dx ≥ dx + dx, 4 B (1 − |x|)2 2 B B |x|(1 − |x|) for all u ∈ Cc∞ (B). 5. Intermediate case: Set E(x) := d(x)2−k where d and k are as in (7.3). Since −∆E ≥ 0 we obtain (7.3) after subbing E into (7.7). 6. Hardy’s boundary inequality in the half space: Let Rn+ denote the half space and set E(x) := dist(x, Rn+ ) = xn . Then putting E into (7.7) gives Z Z 1 u2 2 |∇u| dx ≥ dx, u ∈ Cc∞ (Rn+ ). n x2 4 Rn R n + +  120  7.1. Introduction Maz’ja (see [12]) obtained the following improvement Z Z Z u2 1 u2 1 2 dx, dx ≥ |∇u| dx − 4 Rn+ x2n 16 Rn+ (x2n + x2 ) 21 xn Rn + n−1 for all u ∈ Cc∞ (Rn+ ). One might ask whether we can take a more symmetrical potential in the improvement, say something like V (x) = f (xn ) where f is strictly positive. Using our ode classification of potentials we will see that this is not possible. 7. Hardy’s inequality valid for u ∈ H 1 (Ω): Let B denote the unit ball in R3 and set E(x) := |x|−1 e|x| . Then a computation shows that −∆E + E = 4π 2 δ0  in B  and where ∂ν E = 0 on ∂B. Here δ0 is the Dirac mass at 0. Putting E into (7.13) we see that Z Z Z 1 1 (1 − |x|)2 2 2 2 |∇u| dx + u dx ≥ u dx, u ∈ H 1 (B). 2 B 4 B |x|2 B Also the constants are optimal (in the sense mentioned in (7.13)) and are not attained. 8. H 1 (Ω) Hardy inequalities in exterior domains: Let Ω denote an exterior domain in Rn wiith n ≥ 3, 0 ∈ / Ω and such that ν(x) · x ≥ 0 2−n for all x ∈ ∂Ω. Setting E := |x| in (7.15) we obtain   Z Z n−2 2 u2 2 |∇u| dx ≥ dx, 2 2 Ω Ω |x| for all u ∈ Cc∞ (Ω ∪ ∂Ω). Moreover the constant is optimal and not attained in the naturally induced function space. 9. Hardy’s inequality in a annular domain: Assume that 0 ∈ 2 Ω1 ⊂⊂ BR ⊂ R where Ω1 is connected and BR is the open ball centered at 0 with radius R. In addition we assume that x · ν(x) ≥ 0 on ∂Ω1 where ν is the outward pointing normal. Define Ω := BR \Ω1 , which we assume is connected, and set E(x) := − log(R−1 |x|). Then by the above mentioned results on annular domains one has Z Z 1 u2 2 |∇u| dx ≥ dx, 4 Ω |x|2 log2 (R−1 |x|) Ω for all u ∈ H01 (Ω ∪ ∂Ω1 ) := {u ∈ H 1 (Ω) : u constant is optimal and not attained.  ∂BR  = 0}. Moreover the  121  7.1. Introduction 10. Suppose E > 0 in Ω, let f : (0, ∞) → (0, ∞) and set Ẽ := f (E). Putting Ẽ into (7.4) for E gives  0  Z Z f (E)2 f 00 (E) 2 2 |∇u|A dx ≥ |∇E|A u2 dx − 2 4f (E) 2f (E) Ω Ω Z 0 f (E)LA (E) 2 1 + u dx, 2 Ω f (E) for all u ∈ Cc∞ (Ω). An important example will be when f (E) := E t where 0 < t < 1; in fact we will use E(x) := δ(x)t (δ(x) := dist(x, ∂Ω)) to show that if one drops the requirement that µ is a finite measure (and just assumes µ a locally finite measure) (7.7) need not be optimal. 11. Eigenvalue bound: Let Ω be a bounded subset of Rn and E > 0, LA (E) ≥ 0 in Ω with |∇E|2A = 1 a.e. in Ω. Let λA (Ω) denote the 2 first eigenvalue of LA in H01 (Ω). Then λA (Ω)kEk2L∞ ≥ π4 . To show πz this one puts f (z) := sin2 ( 2kEk ) into the above result and drops the L∞ term involving the measure. 12. Suppose E is an interior weight on Ω with E = 1 on ∂Ω. Then by using 1 the above result with f (E) := (log(E)) 2 one obtains the inequality Z Z 1 3 + 4 log(E) 2 |∇u|A dx ≥ |∇E|2A u2 dx, u ∈ H01 (Ω). 16 Ω E 2 log2 (E) Ω Taking instead f (E) := E log(E) gives Z Z 1 log2 (E) + 1 2 2 u dx, |∇u|A dx ≥ 4 Ω E 2 log2 (E) Ω  u ∈ H01 (Ω).  13. Poincare’s inequality in an unbounded slab: In general Z Z |∇u|2 dx ≥ C u2 dx, u ∈ Cc∞ (Ω) Ω  Ω  does not hold for unbounded domains. It is known that for certain unbounded domains the inequality does in fact hold. One example would be Ω := {x ∈ Rn : 0 < xn < π}. We now use (7.4) to show a slightly stronger result. Put E(x) := sin(xn ) into (7.4) and drop a term to arrive at Z Z Z 1 u(x)2 1 2 |∇u(x)| dx ≥ u(x)2 dx, u ∈ Cc∞ (Ω). dx + 4 Ω tan2 (xn ) 2 Ω Ω 122  7.1. Introduction 14. Hardy’s boundary inequality in a cone: Put Ω := (0, √∞)×(0, ∞) and E(x) := dist(x, ∂Ω) = min{x1 , x2 }. Then −∆E = 2σ where σ is the measure associated with the line Γ := {x : x2 = x1 }. Putting E into (7.4) gives Z Z Z u2 1 1 u2 2 √ |∇u| dx ≥ dx + dσ, 4 Ω (min{x1 , x2 })2 2 Γ min{x1 , x2 } Ω for all u ∈ Cc∞ (Ω). 15. Suppose −∆φ = 1 in Ω with φ = 0 on ∂Ω. Define E := etφ − 1. Then −∆E = tetφ (1 − t|∇φ|2 ) which is non-negative for sufficiently small t > 0. Then E is a boundary weight and hence putting E into (7.8) gives Z Z 2tφ Z tφ t2 e |∇φ|2 2 t e (1 − t|∇φ|2 ) 2 |∇u|2 dx ≥ u dx + u dx, 4 Ω (etφ − 1)2 2 Ω etφ − 1 Ω for all u ∈ H01 (Ω), which is optimal. Sending t & 0 recovers (7.8) with E = φ. 16. Trace theorem: Let Ω denote a domain in Rn where n ≥ 3 and such that B ⊂⊂ Ω (here B is the unit ball). Define ( 1 |x| < 1 E(x) := 1 |x| > 1. |x|n−2 A computation shows that −∆E = cσ where c > 0 and where σ is the surface measure associated with ∂B. Putting this E into (7.8) and dropping a couple of terms gives Z Z c 2 |∇u| dx ≥ u2 dσ, u ∈ Cc∞ (Ω). 2 Ω ∂B 17. Regularity: Suppose E ∈ L∞ loc (Ω) is a positive solution to LA (E) = µ in Ω where µ is locally finite measure. Then using (7.7) we see that 1 (Ω). E ∈ Hloc 18. Baouendi-Grushin operator: Here we mention that various operators can be put into the form we are interested in. Suppose Ω is an open subset of RN = Rn × Rk and ξ ∈ Ω is written ξ = (x, y) using the above decomposition of RN . For γ > 0 one defines the 123  7.2. Main Results vector field ∇γ := (∇x , |x|γ ∇y ) and the Baouendi-Grushin operator LA := −∆x − |x|2γ ∆y . Take   In 0 A(ξ) := 0 |x|2γ Ik where In , Ik are the identity matrices of size n and k. Then |∇γ E|2 = |∇E|2A and −div(A∇E) = LA (E).  7.2  Main Results  Throughout this article we shall assume that Ω is a bounded connected domain in Rn (unless otherwise mentioned) with smooth boundary and A(x) = ((ai,j (x))) is a n × n symmetric, uniformly positive definite matrix with ai,j ∈ C ∞ (Ω) and for ξ ∈ Rn we define |ξ|2A := |ξ|2A(x) := A(x)ξ · ξ. If E is an interior weight or a boundary weight on Ω we have, by the strong maximum principle (see [15]), E bounded away from zero on compact subsets of Ω. The following theorem gives the main inequalities. In addition we consider a slight generalization of the case where E is a boundary weight on Ω. Theorem 7.17. (i) weight on Ω. Then Z  Suppose E is either an interior or a boundary  |∇u|2A dx  Ω  1 − 4  Z Ω  |∇E|2A 2 u dx ≥ 0, E2  (7.18)  for all u ∈ H01 (Ω). Moreover 41 is optimal and not attained. (ii) Suppose E is a boundary weight on Ω. Then Z Ω  |∇u|2A dx  1 − 4  Z Ω  |∇E|2A 2 1 u dx ≥ 2 E 2  Z Ω  u2 dµ, E  (7.19)  for all u ∈ H01 (Ω). Moreover 21 is optimal (once one fixes 14 ) and is not attained. (iii) Suppose E ∈ C ∞ (Ω) with E > 0, LA (E) ≥ 0 in Ω and Γ := {x ∈ ∂Ω : E(x) = 0} contains B(x0 , r) ∩ ∂Ω for some x0 ∈ ∂Ω and r > 0. Then (7.18) is optimal. Remark 7.2.1. One can consider more general functions E. Most of the results (including the above one) concerning interior weights on Ω can be 124  7.2. Main Results generalized to the case where LA (E) = µ + h, here µ is again a nonnegative nonzero finite measure and h is a suitably smooth non-negative function. We begin by justifying (7.4). Lemma 7.20. (i) Z  Suppose E is an interior weight on Ω. Then  |∇u|2A dx  Ω  1 − 4  Z Ω  |∇E|2A 2 u dx ≥ E2  Z  E|∇v|2A dx,  (7.21)  Ω  −1  for all u ∈ Cc0,1 (Ω\K) and where v := E 2 u. (ii) Suppose E is a boundary weight on Ω. Then Z Ω  |∇u|2A dx  1 − 4  Z Ω  |∇E|2A 2 u dx ≥ E2  for all u ∈ H01 (Ω) and v := E  −1 2  Z Ω  E|∇v|2A dx  1 + 2  Z Ω  u2 dµ, E  (7.22)  u.  Proof. (i) Since E is smooth away from K and noting the supports of both u and v the integration by parts used in obtaining (7.4) is valid. (ii) Now suppose E in a boundary weight. Extend E to all of Rn by setting E = 0 outside of Ω and let Eε denote the ε mollification of E. Let −1  u ∈ Cc∞ (Ω), vε := Eε 2 u and define Fε := LA (Eε ). Now one easily obtains (7.4) but with E and v replaced with Eε , vε . Standard arguments show that uEε−1 → uE −1 in H01 (Ω), |∇Eε |2A Eε−2 → |∇E|2A E −2 , Eε |∇vε |2A → E|∇v|2A a.e. in Ω and uFε * uµ in H −1 (Ω). Using these results along with Fatou’s lemma allows us to pass to the limit. Remark 7.2.2. When we prove our various Hardy inequalities, which all stem from (7.4) we will generally drop the term   Z u 2 dx. E∇ √ E A Ω To show the given inequality does not attain we will generally just not drop this√term. This√term is positive for non-zero u provided u is not a multiple of E. Since E ∈ / H01 (Ω) this will not be an issue. In theorem 7.27 this will be a concern. As usual we will need an ample supply of test functions for best constant calculations. The next lemma provides this. When E is an interior weight we let g denote a solution to LA (g) = 0 in Ω with g = E on ∂Ω. 125  7.2. Main Results Lemma 7.23. Suppose E is an interior weight on Ω and 0 < γ := min∂Ω E. Then (i) ut := E t − g t ∈ H0R1 (Ω) for 0 < t < 12 . (ii) Define I(t) := Ω |∇E|2A E 2t−2 dx. Then I(t) is finite for t < 21 and I(t) → ∞ as t % 12 . (ii) Suppose E = γ > 0 on ∂Ω. Define vt,τ := E t logτ (γ −1 E) and Z Jt (τ ) := E 2t−2 |∇E|2A log2τ −2 (γ −1 E)dx. Ω  Then vt,τ ∈ H01 (Ω) for 0 < t < we have Jt (τ ) → ∞ as τ & 21 .  1 2  and τ > 12 . Moreover for each 0 < t <  1 2  Proof. We prove the results up to some unjustified integration by parts; which can be justified by regularizing the measure, integrating by parts and passing to limits. (i), (ii) Fix 0 < t < 12 and then note that |∇ut |2 ≤ CE 2t−2 |∇E|2A + Cg 2t−2 |∇g|2A where C is some uniform constant. The term involving g is harmless. Now multiply LA (E) = µ by E 2t−1 and integrate over Ω to obtain Z Z 2t−2 2 (1 − 2t) E |∇E|A dx = − g 2t−1 (A∇E) · ν Ω ∂Ω Z = ε(t) − (A∇E) · ν Z∂Ω = ε(t) − div(A∇E)dx Ω  = ε(t) + µ(Ω), R where ε(t) → 0 as t % 12 . Note RΩ E 2t−1 dµ = 0 since t < 12 and E = ∞ on K. From this we see that I(t) = Ω |∇E|2A E 2t−2 dx < ∞ and so ut ∈ H01 (Ω). We also see that limt% 1 I(t) = ∞. 2  (iii) Take 0 < t < 12 , τ > 21 and vt,τ defined as above. One easily sees that vt,τ is continuous near ∂Ω and vanishes on ∂Ω. So to show vt,τ ∈ H01 (Ω) it is sufficient to show w1 := E 2t−2 |∇E|2 log2τ (γ −1 E), w2 := E 2t−2 |∇E|2 log2τ −2 (γ −1 E) ∈ L1 (Ω). These functions are only singular near K and ∂Ω. Now set Wτ := E 2t−2 |∇E|2 log2τ −2 (γ −1 E) and so w2 = Wτ and w1 = Wτ +1 . Now suppose t0 ∈ (t, 21 ) and so 0  Wτ +1 = E 2t −2 |∇E|2  log2τ (γ −1 E) 0 ≤ CE 2t −2 |∇E|2 0 −2t 2t E  near K , 126  7.2. Main Results and so w1 = Wτ +1 ∈ L1 (Kε ) where Kε is a small neighborhood of K. Now note that w2 is better behaved than w1 near K and so we also have w2 ∈ L1 (Kε ). Define Ωε := {x ∈ Ω : E(x) < γ + ε} and take ε > 0 sufficiently small such that K ⊂ Ω\Ω2ε . Now using the co-area formula we have Z E 2t−2 |∇E|2 log2τ −2 (γ −1 E)dx ≤ sup |∇E|Λ Ωε  Ωε  Z ≤ C  1+ γε  s2t−2 log2τ −2 (s)ds,  1  where  Z Λ :=  E 2t−2 log2τ −2 (γ −1 E)|∇E|dx,  Ωε  which is finite for τ > 12 . So we see that w2 ∈ L1 (Ωε ) for sufficiently small ε > 0 and noting that w1 is better behaved near ∂Ω than w2 we have the same for w1 . Combining these results we see that vt,τ ∈ H01 (Ω). Fix 0 < t < 12 and τ > 12 . By Hopf’s lemma we have |∇E(x)| bounded away from zero on Ωε for ε > 0 sufficiently small; fix ε > 0 sufficiently small. Then Z Jt (τ ) ≥ C E 2t−2 log2τ −2 (γ −1 E)|∇E|dx Ωε 1+ γε  Z ≥ C̃  s2t−2 log2τ −2 (s)ds,  1  and a computation shows the last integral becomes unbounded as τ & 12 . Proof of theorem 7.17: (i) Using lemma 7.20 and, in the case where E is a interior weight on Ω, the fact that Cc0,1 (Ω\K) is dense in H01 (Ω) we obtain (7.18). We now show the constant is optimal. Suppose E is an interior weight Ron Ω and define Eε := ε + E, gε := ε + g where ε > 0. Define Iε (t) := Ω |∇Eε |2A Eε2t−2 dx. As in the proof of lemma 7.23 one can show that for each ε > 0 limt% 1 Iε (t) = ∞. We use ut,ε := Eεt − gεt as test functions. Let 0 < t < R Qt,ε := R  1 2  2  and ε > 0. Then  2 Ω |∇ut,ε |A dx  |∇Eε |2A 2 ut,ε dx Ω Eε2  ≤  p t2 Iε (t) + C0 + C1 Iε (t) , Iε (t) − C2 Iε ( 2t ) − C3 Iε (0)  127  7.2. Main Results where the constants Ck possibly depend on ε. From this we see that limt% 1 Qt,ε = 14 after recalling Qt,ε ≥ 41 . Now fix ε > 0 and let u ∈ Cc∞ (Ω) 2 be non-zero. Then a simple computation shows R R 2 2 Ω |∇u|A dx Ω |∇u|A dx , ≤ R |∇E|2A R |∇Eε |2A 2 dx 2 dx u u 2 2 Ω E Ω E ε  which, when combined with the above facts, gives the desired best constant result. To see 41 is not attained use (7.21). Now suppose E is a boundary weight on Ω, ε > 0 and t > 12 . Define fε (z) := z 2t−1 − ε2t−1 for z > ε and 0 otherwise. Using fε (E) ∈ H01 (Ω) as a test function in the pde associated with E one obtains, after sending ε & 0, Z Z 2t−2 2 (2t − 1) E |∇E|A dx = E 2t−1 dµ, (7.24) Ω  Ω  which shows that E t ∈ H01 (Ω) for 12 < t ≤ 1. To see 14 is optimal in (7.18) use E t (as t & 21 ) as a minimizing sequence. (ii) Suppose E is a boundary weight on Ω. Let 12 < t < 1 and so E t ∈ H01 (Ω). Using (7.24) we have R |∇E|2A t 2 1 4 Ω E 2 (E ) R (E t )2 Ω E dµ  t 2 Ω |∇E |A dx  R  −  =  t 1 + , 2 4  which shows that 12 is optimal. (iii) Suppose E is as in the hypothesis. The only issue is whether 14 is optimal. Without loss of generality assume that 0 ∈ ∂Ω and B(0, 2R)∩∂Ω ⊂ Γ. Suppose 0 < r < R and define  x ∈ Ω(r)  1 R−|x| φ(x) := x ∈ Ω(R)\Ω(r)  R−r 0 x ∈ Ω\Ω(R), where Ω(r) := B(0, r) ∩ Ω. Define ut := E t φ which can be shown to be an element of H01 (Ω) for t > 21 . One uses ut as t & 12 as a minimizing sequence along with arguments similar to the above to show 41 is optimal.  The following example shows that if we just assume that 0 < E ∈ H01 (Ω) with LA (E) a locally finite measure then (7.18) need not be optimal.  128  7.2. Main Results Example 7.25. Take Ω a bounded convex domain in Rn and set δ(x) := dist(x, ∂Ω). Fix 12 < t < 1 and set E := δ t ∈ H01 (Ω). Then |∇E|2 t2 = , E2 δ2  µ := −∆E = t(1 − t)δ t−2 + tδ t−1 (−∆δ) ≥ 0  in Ω,  and so putting E into (7.18) gives Z 2 Z u t2 |∇u|2 dx ≥ dx, 4 Ω δ2 Ω for u ∈ H01 (Ω). This shows that (7.18) was not optimal. This apparent failure of theorem 7.17 is due to the fact µ not a finite measure; use the co-area formula to show δ t−2 ∈ / L1 (Ω). We now give an alternate way to view best constants in (7.19). Define C to be the set of (β, α) ∈ R2 such that Z  |∇u|2A dx  Z ≥α  Ω  Ω  |∇E|2A 2 u dx + β E2  Z Ω  u2 dµ, E  u ∈ H01 (Ω).  (7.26)  Theorem 7.27. Suppose E ∈ L∞ (Ω) is a boundary weight on Ω. Then       1 1 1 2 C = (β, α) : β > , α ≤ β − β ∪ −∞, × −∞, =: C 0 . 2 2 4 Moreover (7.26) does attain on Γ := {(τ, τ − τ 2 ) : τ > 21 } ⊂ ∂C and does not attain on ∂C\Γ. Proof. Using similar arguments to the above one can show that E t ∈ H01 (Ω) for all t > 21 . Suppose (β, α) ∈ C. If β > 21 then testing (7.26) on u := E β shows that α ≤ β − β 2 . If β ≤ 21 then testing (7.26) on u := E t and sending t & 21 shows that α ≤ 14 . Now for the other inclusion. Fix t ≥ 1 and put E2 := E t . Then we have 2 |∇E2 |2A 2 |∇E|A = t , E2 E22  |∇E|2A LA (E2 ) LA (E) = t(1 − t) . +t E2 E2 E  Putting E = E2 into (7.8) we obtain Z Ω  |∇u|2A dx  t t2 ≥( − ) 2 4  Z Ω  |∇E|2A 2 t u dx + 2 E 2  Z Ω  u2 dµ, E  (7.28)  129  7.2. Main Results 2  and so we see that ( 2t , 2t − t4 ) ∈ C for all t ≥ 1. From this we see that the curve α = β − β 2 for β ≥ 12 is contained in C. It is straightforward to see the remaining portion of ∂C 0 is contained in C. To see the inequality does not attain when (β, α) ∈ ∂C\Γ use the fact that (7.7) does not attain in H01 and the fact that µ ≥ 0. To see the inequality does attain on the remaining portion of ∂C note that (7.28) attains at u := t E 2 ∈ H01 (Ω) for t > 1. We now give a result relating to the first eigenvalue of LA on subdomains of Ω. Suppose (E, λA (Ω)) is the first eigenpair (with E > 0) of LA on H01 (Ω) and for B ⊂ Ω we let λA (B) denote the first eigenvalue of LA on H01 (B). Corollary 7.29. Let E be as above. For B ⊂ Ω we set α(B) := inf B  (i)  |∇E|2A , E2  α(B) := sup B  |∇E|2A . E2  If α(B) > λA (Ω) then 4λA (B) ≥  (ii)  (α(B) + λA (Ω))2 . α(B)  If λA (Ω) > α(B) then  (α(B) + λA (Ω))2 . α(B) R Proof. Let B ⊂ Ω and let u ∈ Cc∞ (B) with B u2 = 1. Using (7.28) gives 4λA (B) ≥  Z 2  |∇u|2A dx ≥ (t −  B  |∇E|2A t2 ) inf + λA (Ω)t, 2 B E2  for 0 < t < 2. If t > 2 then we get the same expression but with the infimum replaced with supremum. Now take the infimum over u and in case (i) set A (Ω) A (Ω) < 2 and in case (ii) set t := 1 + λα(B) > 2 to see the result. t := 1 + λα(B)  7.2.1  Weighted versions  We now examine weighted versions of the above inequalities which, as mentioned earlier, can be seen as analogs of Cafferelli-Kohn-Nirenberg inequalities. We now introduce the spaces we work in.  130  7.2. Main Results R Definition 7.30. For t ∈ R we define kuk2t := Ω E 2t |∇u|2A dx. Suppose E is an interior weight on Ω. We define Xt to be the completion of Cc0,1 (Ω\K) with respect to k · kt . In the case that E is a boundary weight on Ω we define Xt to be the completion of Cc0,1 (Ω) with respect to the same norm. Remark 7.2.3. One should note that if E is an interior weight on Ω and t > 21 then Xt does not contain Cc∞ (Ω). To see this use (7.32) to see that 1 (Ω) which we know to be false. For t < 1 we if Cc∞ (Ω) ⊂ Xt then E t ∈ Hloc 2 do have Cc∞ (Ω) ⊂ Xt . Theorem 7.31. Suppose t 6= 12 and E an interior weight on Ω. Then Z Z 1 2 2t 2 |∇E|2A E 2t−2 u2 dx, (7.32) E |∇u|A dx ≥ (t − ) 2 Ω Ω for all u ∈ Xt . Moreover the constant is optimal and not attained. Proof. Let t 6= 0, 21 , u ∈ Cc0,1 (Ω\K) and define w := E t u ∈ Cc0,1 (Ω\K). Put w into Z Z |∇E|2A 2 1 2 |∇w|A dx ≥ w dx, 4 Ω E2 Ω and re-group to obtain (7.32). We now show the constant is optimal. Let vm ∈ Cc0,1 (Ω\K) be such that R |∇vm |2 dx 1 Dm := R Ω |∇E|2 A → . 4 A 2 Ω E 2 vm dx Define um := E −t vm ∈ Xt . A computation shows that R E 2t |∇um |2A dx 2 R Ω 2 E 2t−2 u2 dx = Dm + t − t, |∇E| m A Ω and since Dm → 14 we see that (t − 12 )2 is optimal. For the case γ := min∂Ω E > 0 we can show the constant is not obtained by using later results on improvements. If γ = 0 we then sub w into (7.4) instead of (7.18) and hold onto the extra term Z 1 E|∇(E t− 2 u)|2A dx Ω  to see the optimal constant is not attained.  131  7.2. Main Results Theorem 7.33. (i) Suppose 0 6= t < 12 and E is a boundary weight on Ω. Then Z Z 1 E 2t |∇u|2A dx − (t − )2 |∇E|2A E 2t−2 u2 dx ≥ 0, (7.34) 2 Ω Ω for all u ∈ Xt . Moreover the constant is optimal and not attained. (ii) Suppose 0 6= t < 21 and E is a boundary weight on Ω. Then Z Z Z 1 2 1 2t 2 2 2t−2 2 E |∇u|A dx − (t − ) |∇E|A E u dx ≥ ( − t) E 2t−1 u2 dµ, 2 2 Ω Ω Ω (7.35) for all u ∈ Xt . Moreover the constant on the right is optimal and not attained. (iii) Suppose t > 21 and E ∈ L∞ (Ω) is a boundary weight on Ω. Then  R  E 2t |∇u|2A dx Ω inf R 2 2t−2 u2 dx : u ∈ Xt \{0} = 0. Ω |∇E|A E Proof. We first prove (7.35) for u ∈ Cc0,1 (Ω) which then gives us (7.34) for the same class of u’s. Suppose 0 6= t < 12 and E is a boundary weight on Ω. We now use the notation introduced in the proof of lemma 7.20; namely Eε is the standard mollification of E and Fε := LA (Eε ). Recall that for any u ∈ Cc0,1 (Ω) we have uFε → uµ in H −1 (Ω) and that we have Z  |∇v|2A dx  Ω  1 ≥ 4  Z Ω  |∇Eε |2A 2 1 v dx + 2 Eε 2  Z Ω  v2 Fε dx, Eε  for all v ∈ H01 (Ω). Now let u ∈ Cc0,1 (Ω) and set v := Eεt u ∈ Cc0,1 (Ω). Putting v into the above gives Z Z Z 1 1 Eε2t |∇u|2A dx ≥ (t − )2 |∇Eε |2A Eε2t−2 u2 dx + ( − t) Eε2t−1 u2 Fε dx. 2 2 Ω Ω Ω (7.36) 2t 2t 1 Now since Eε → E in Lloc (Ω) we have Z Z 2t 2 Eε |∇u|A dx → E 2t |∇u|2A dx, Ω  Ω  and using similar ideas from the proof of lemma 7.20 one can show that Z Z 2t−1 2 Eε u Fε dx → E 2t−1 u2 dµ. Ω  Ω  132  7.2. Main Results So using these results, sending ε & 0 in (7.36) and after an application of Fatou’s lemma we arrive at (7.35) for u ∈ Cc0,1 (Ω). Now we show the constants are optimal. Recalling the proof of theorem 7.17 there exists vm ∈ Cc∞ (Ω) such that R R |∇E|2A 2 R 2 dx − 1 2 dx |∇v | |∇v | 1 1 m m A A 4 Ω Ω E 2 vm dx Dm := R Ω |∇E|2 → , Fm := → . R vm 2 4 2 A 2 dµ v dx Ω  E2  Ω E  m  Define um := E −t vm which one easily sees is an element of Xt . Then R R −1 E 2t |∇um |2A dx 2 Ω E vm ∇vm · A∇Edx Ω = D + t − 2t , Φm := R m R |∇E|2A 2 2t−2 u2 dx m Ω |∇E|A E v 2 dx Ω  m  R − (t − 12 )2 Ω |∇E|2A E 2t−2 u2m dx R := 2t−1 u2 dµ m ΩE R R |∇E|2A 2 −1 v ∇v · A∇Edx t m m 2 vm dx − 2t Ω E . = Fm + Ω E R vm 2 dµ Ω E R  Ψm  E2  and  ΩE  2t |∇u |2 dx m A  Using Eε , Fε as defined above one can show, using similar methods, that Z Z 2 Z |∇E|2A 2 vm −1 2 E vm ∇vm · A∇Edx = dµ + v dx. (7.37) E2 m Ω Ω E Ω So from this we see that 2 vm Ω E dµ t R |∇E|2 , A 2 v dx 2 m Ω E  R  2  Φm = Dm + t − t − and noting that 2 vm Ω E dµ R |∇E|2A 2 Ω E 2 vm dx  R  =  Dm − Fm  1 4  → 0,  we see that (7.34) is optimal. Similarly one sees using (7.37) that Ψm = Fm − t and hence (7.35) is optimal. To show the constants are not obtained we as usual Rhold on to the extra term that we dropped in the above calculations. Since Ω E −1 |∇E|2A dx = ∞ one can show this extra term is positive for u ∈ Xt \{0}. (iii) Now take t > 21 and E a boundary weight on Ω. For ε, τ > 0 but small define  0 E<ε uε,τ (x) := τ τ E −ε E > ε. 133  7.2. Main Results Then uε,τ ∈ Xt . Now use the sequence um where um := uεm ,τm to see desired result where εm := m−m and τm := m−1 .  7.2.2  More general weighted inequalities  We now investigate the possibility of inequalities of the form Z Z 2 U (x)u2 dx, u ∈ Cc0,1 (Ω\K). W (x)|∇u|A dx ≥ Ω  Ω  Theorem 7.38. Suppose E is an interior weight on Ω with γ := min∂Ω E and 0 < f ∈ C ∞ (γ, ∞). Then   Z Z f (E)2 2 2 2 00 f (E) |∇u|A dx ≥ |∇E|A + f (E)f (E) u2 dx, (7.39) 2 4E Ω Ω for all u ∈ Cc0,1 (Ω\K). In addition this is optimal (in the sense that the 2 00 optimal constant is 1) if lim inf z→∞ f 00 (z) > 0 or if limz→∞ z ff(z)(z) = 0. Proof. Let u ∈ Cc0,1 (Ω\K) and define w := f (E)u ∈ Cc0,1 (Ω\K). Putting w into (7.18), integrating by parts and re-grouping gives (7.39). Let vm ∈ Cc0,1 (Ω\K) be such that R |∇vm |2 dx 1 Dm := R Ω|∇E|2 → . 4 A 2 v dx Ω  E2  m  Without loss of generality we can assume the supports of vm concentrate on m K. Define um := fv(E) ∈ Cc0,1 (Ω\K). Then a computation shows that Qm :=  =  R f (E)2 |∇um |2A dx Ω  R f (E)2 2 00 (E) u2 dx |∇E| + f (E)f m A Ω 4E 2 2 00 R R |∇E|A f (E) 2 2 vm dx Ω |∇vm |A dx + Ω f (E)2 . R |∇E|2A f 00 (E) R |∇E|2A 2 dx + 2 dx v v 2 2 m m Ω 4E Ω f (E)  Now suppose lim inf z→∞ f 00 (z) > 0. Then using the monotonicity of x 7→ α+x β+x , where α and β are positive constants, shows Qm → 1. Now suppose  134  7.2. Main Results 2 00  limz→∞ z ff(z)(z) = 0. Using this and the fact that the vm ’s support concentrates on K one easily sees that R Ω  |∇E|2A f 00 (E) 2 vm dx f (E)2 R |∇E|2A 2 Ω 4E 2 vm dx  → 0.  Using this one sees that Qm → 1.  7.2.3  Improvements  We now investigate the possibility of improving (7.18) in the sense of potentials. The method we employ was first used by Ghoussoub and Moradifam (see [10]). We now define precisely what we mean by a potential. Suppose E is an interior weight on Ω and 0 ≤ V ∈ C ∞ (Ω\K) (recall K is the support of µ). We say V is a potential for E provided Z  |∇u|2A dx  Ω  1 − 4  Z Ω  |∇E|2A 2 u dx ≥ E2  Z  V (x)u2 dx,  (7.40)  Ω  for all u ∈ H01 (Ω). We analogously define a potential V for the case that E is a boundary weight on Ω except we restrict our attention to 0 ≤ V ∈ C ∞ (Ω). The next theorem gives necessary and sufficient conditions for V to be a potential of E in terms of solvability of a singular linear equation. For the necessary direction we will need to assume some conditions on Ω. (B1) Suppose E is an interior weight on Ω. We assume that that there exists a sequence (Ωm )m of non-empty subdomains of Ω which are connected, have a smooth boundary, Ωm ⊂⊂ Ω\K, Ωm ⊂⊂ Ωm+1 and Ω\K = ∪m Ωm . (B2) Suppose E is a boundary weight on Ω. We assume that there exists a sequence (Ωm )m of non-empty subdomains of Ω which are connected, have a smooth boundary, Ωm ⊂⊂ Ωm+1 and Ω = ∪m Ωm . Theorem 7.41. (interior improvements) Suppose E is an interior weight on Ω and 0 ≤ V ∈ C ∞ (Ω\K). (i)  Suppose there exists some 0 < φ ∈ C 2 (Ω\K) such that −LA (φ) +  A∇E · ∇φ +Vφ≤0 E  in Ω\K.  (7.42)  135  7.2. Main Results 1  Then V is a potential for E. After the change of variables θ := E 2 φ one sees that it is sufficient to find a 0 < θ ∈ C 2 (Ω\K) such that −LA (θ) |∇E|2A +V ≤0 + θ 4E 2  in Ω\K.  (7.43)  (ii) Suppose V is a potential for E and Ω satisfies (B1). Then there exists some 0 < θ ∈ C ∞ (\K) which satisfies (7.43). It is important to note that the above theorem can be used (in theory) for best constant calculations; without the need for constructing appropriate minimizing sequences. To see this suppose 0 ≤ V is a potential for the interior weight E and let C(V ) > 0 denote the associated best constant, ie R   |∇u|2 dx − 1 R |∇E|2A u2 dx  A R 4 Ω E2 1 Ω C(V ) := inf : u ∈ H (Ω)\{0} . 0 2   Ω V u dx Then one sees that  C(V ) = sup c > 0 : ∃0 < θ ∈ C 2 (Ω\K) s.t. −LA (θ) |∇E|2A + + cV ≤ 0 θ 4E 2   in Ω\K .  After theorem 7.53, which is analogous result to the above theorem but phrased in terms of solvability of a linear ode, this remark on best constants will be of more importance because of the shear magnitude of results concerning solvability of ode’s. Theorem 7.44. (boundary improvements) Suppose E is a boundary weight on Ω and 0 ≤ V ∈ C ∞ (Ω). (i) Suppose E ∈ C 0,1 (Ω), V is a potential for E and Ω satisfies (B2). Then there exists some 0 < θ ∈ C 1,α (Ω) for all α < 1 such that −LA (θ) |∇E|2A + +V ≤0 θ 4E 2 (ii)  in Ω.  (7.45)  Suppose there exists some 0 < φ ∈ C 2 (Ω) such that −LA (φ) A∇E · ∇φ µ + − +V ≤0 φ Eφ 2E  in Ω.  (7.46)  Then V is a potential for E. 136  7.2. Main Results 1  Remark 7.2.4. Note that putting θ := E 2 φ into (7.46) gives, at least formally, (7.45). Also one can replace µ by the absolutely continuous part of µ in (7.46). Proof of theorem 7.41. (i) Suppose V ∈ C ∞ (Ω\K) is non-negative and there exists some 0 < φ ∈ C 2 (Ω\K) which solves (7.42). Let u ∈ Cc0,1 (Ω\K) −1 and define v := E 2 u so by lemma 7.20 we have Z Z Z |∇E|2A 2 1 2 |∇u|A dx − E|∇v|2A dx. u dx = 2 4 E Ω Ω Ω Now define ψ ∈ Cc0,1 (Ω\K) by v := φψ. A calculation shows that E|∇v|2A = Eψ 2 |∇φ|2A + Eφ2 |∇ψ|2A + 2EφψA∇φ · ∇ψ,  (7.47)  and integrating, by parts, the last term over Ω we obtain Z Z 2 2 ψ E|∇φ|A dx + 2 φψEA∇φ · ∇ψdx Ω Z Ω = ψ 2 (LA (φ)φE − φA∇E · ∇φ) dx Ω ! Z A∇E·∇φ L (φ) − A E = u2 dx φ Ω =: Q, but by (7.42) Q ≥ Ω V (x)u2 dx and so we see Z Z Z Z |∇E|2A 2 1 2 2 2 Eφ |∇ψ|A dx + V u2 dx, u dx ≥ |∇u|A dx − 4 Ω E2 Ω Ω Ω R  for all u ∈ Cc0,1 (Ω\K). Now since Cc0,1 (Ω\K) is dense in H01 (Ω) and using Fatou’s lemma one can show (7.40) holds for all u ∈ H01 (Ω). (ii) Now suppose V ∈ C ∞ (Ω\K) is a potential for E and (Ωm )m is the sequence of domains from assumption (B1). Define the elliptic operator P by |∇E|2A P (u) := LA (u) − u − V u. 4E 2 Using a standard constrained minimization argument along with the strong maximum principle there exists some 0 < θm ∈ H01 (Ωm ) such that P (θm ) = λm θm θm = 0  in Ωm on ∂Ωm ,  (7.48) 137  7.2. Main Results where 0 ≤ λm , ie. (θm , λm ) is the first eigenpair of P in H01 (Ωm ). Since H01 (Ωm ) ⊂ H01 (Ωm+1 ) we see that λm is decreasing and hence there exists some 0 ≤ λ such that λm & λ. Let x0 ∈ ∩m Ωm and suitably scale θm such that θm (x0 ) = 1 for all m. Now fix k and let m > k + 1. Then P (θm ) − λm θm = 0  in Ωk+1 ,  and we now apply Harnacks inequality to the operator P − λm to see there exists some Ck such that sup(θm ) ≤ Ck inf (θm ) ≤ Ck . Ωk  Ωk  So we see that (θm ) is bounded in L∞ loc (Ω\K). Now applying elliptic regularity theory and a bootstrap argument one sees that (θm )m>k+1 is bounded in C 1,α (Ωk ) for α < 1 and after applying a diagonal argument one sees that there exists some non-zero 0 ≤ θ ∈ C 1,α (Ω\K) such that θm → θ in C 1,α (Ωk ) for all k. Using this convergence one can pass to the limit in (7.48) to see that P (θ) = λθ in Ω\K and after applying the strong maximum principle on Ωm one sees that θ > 0 in Ω\K. Now applying regularity theory one sees that θ ∈ C ∞ (Ω\K).  Proof of theorem 7.44. (i) The proof is essentially unchanged from the proof of theorem 7.41. (ii) Again the proof is the same as in theorem 7.41 except now the measure µ does not drop out.  The next theorem gives some explicit examples of potentials. Theorem 7.49. (i) Suppose E is an interior weight on Ω, 0 < γ := min∂Ω E and 0 < f ∈ C 2 ((γ, ∞)). Then for all u ∈ Cc0,1 (Ω\K) we have   Z Z Z |∇E|2A 2 |∇E|2A 1 f 0 (E) 2 00 |∇u|A dx − u dx ≥ −f (E) − u2 dx. 2 4 E f (E) E Ω Ω Ω p In particular by taking f (E) := log(γ −1 E) we obtain Z Ω  |∇u|2A dx  1 − 4  Z Ω  |∇E|2A 2 1 u dx ≥ 2 E 4  Z Ω  |∇E|2A u2 dx, E 2 log2 (γ −1 E)  for all u ∈ H01 (Ω). Now suppose 0 < γ = E on ∂Ω. Then hand side of (7.50)) is optimal.  1 4  (7.50)  (on the right  138  7.2. Main Results (ii)  Suppose E ∈ L∞ (Ω) is a boundary weight. Then Z Z Z |∇E|2A 2 |∇E|2A 1 1 2  |∇u|A dx − u dx ≥ E 4 Ω E2 4 Ω E 2 log2 Ω   u2 dx,  (7.51)  ekEkL∞  for all u ∈ H01 (Ω). Proof. (i) Let E be an interior weight on Ω, γ := min∂Ω E > 0 and suppose 0 < f ∈ C 2p ((γ, ∞)). Put φ := f (E) into (7.42) to obtain the result. Now take f (E) := log(γ −1 E) to obtain (7.50) for all u ∈ Cc0,1 (Ω\K) and extend to all of H01 (Ω) by density and by Fatou’s lemma. We now show 14 is optimal. Fix 0 < t < 21 and for τ > 12 define uτ := E t logτ (γ −1 E). By lemma 7.23 uτ ∈ H01 (Ω). A computation shows that R R |∇E|2A 2 1 2 Ω |∇uτ |A dx − 4 Ω E 2 uτ dx R |∇E|2A 2 Ω E 2 log2 (Eγ −1 ) uτ dx is equal to Jt (τ + 1/2) 1 Jt (τ + 1) + τ 2 + 2tτ , (t2 − ) 4 Jt (τ ) Jt (τ ) where Jt (τ ) is defined in lemma 7.23. Sending τ & 12 and using results from lemma 7.23 we see 41 is optimal. (ii) Suppose E ∈ L∞ (Ω) is a boundary weight on Ω. Here we use the notation from the proof of lemma 7.20; Eε := ηε ∗ E, Fε := LA (Eε ). Let 0 < f ∈ C 2 ((0, kEkL∞ ]). Then starting at (7.22) for Eε and decomposing v as usual one arrives at Z Z |∇Eε |2A 2 1 2 |∇u|A dx − u dx ≥ M + N (7.52) 4 Ω Eε2 Ω where    |∇Eε |2A f 0 (Eε ) 00 M := −f (Eε ) − u2 dx, f (E ) E ε ε Ω  Z  0 f (Eε ) 1 N := + u2 Fε dx f (E ) 2E ε ε Ω for all uq ∈ Cc∞ (Ω) after using methods similar to the proof of (i). Now take f (z) := − log( ekEkz L∞ ) and let u ∈ Cc∞ (Ω). Then one has Z  Z Ω  |∇u|2A dx  1 − 4  Z Ω  |∇Eε |2A 2 1 u dx ≥ 2 Eε 4  Z Ω  |∇E|2A u2 dx + Iε , Eε Eε2 log2 ( ekEk ) L∞ 139  7.2. Main Results where 1 Iε := 2  Z Ω  u2 Eε2  1+  1 Eε log( ekEk ) L∞  ! Fε dx.  Using methods similar to ones used in the proof of lemma 7.20 one easily sees that limε&0 Iε ≥ 0. Using this and standard results on convolutions and Fatou’s lemma we obtain the desired inequality for u ∈ Cc∞ (Ω) and we then extend to all of H01 (Ω). We now obtain a more useful (than (7.43)) necessary and sufficient condition for V to be a potential for E; at least in the case where E is an interior weight on Ω and E = γ ≥ 0 on ∂Ω. As in theorem 7.41 we assume some geometrical properties of Ω. Theorem 7.53. (Interior improvements using ode methods) Suppose E is an interior weight on Ω, E = γ ≥ 0 on ∂Ω, 0 ≤ f ∈ C ∞ (γ, ∞) and Ωt := {x ∈ Ω : γ + 1t < E(x) < t} is connected for sufficiently large t. Then the following are equivalent: (i) For all u ∈ H01 (Ω) Z Z Z |∇E|2A 2 1 2 |∇u|A dx − u dx ≥ f (E)|∇E|2A u2 dx. (7.54) 4 Ω E2 Ω Ω (ii)  There exists some 0 < h ∈ C 2 (γ, ∞) such that   1 00 h (t) + f (t) + 2 h(t) ≤ 0, 4t  (7.55)  in (γ, ∞). Proof. Let E be an interior weight on Ω, E = γ ≥ 0 on ∂Ω and 0 ≤ f ∈ C ∞ (γ, ∞). (ii) ⇒ (i) Setting θ := h(E) and using (ii) along with theorem 7.41 gives (i). (i) ⇒ (ii). The proof will be similar to theorem 7.41 (ii). Let γ < tm % ∞ and define Ωm := {x ∈ Ω : γ + t1m < E(x) < tm }. By hypothesis we can take Ωm to 1 (Ω ) := {φ ∈ be connected and non-empty for each m. Now define H0,E m 1 H0 (Ωm ) : φ is constant on level sets of E} and set Z Z 1 1 1 2 F (φ) := |∇φ|A dx, J(φ) := |∇E|2A (f (E) + )φ2 dx, 2 Ωm 2 Ωm 4E 2 140  7.2. Main Results 1 Mm := {φ ∈ H0,E (Ωm ) : J(φ) = 2−1 }. 1 (Ω ) such that Standard methods show the existence of 0 < φm ∈ H0,E m 2 λm := inf Mm F = F (φm ) and hence LA (φm ) = λm |∇E|A (f (E) + 4E1 2 )φm 1 (Ω ) ⊂ H 1 (Ω in Ωm with φm = 0 on ∂Ωm . Since H0,E m m+1 ) one sees that 0,E λm is decreasing and from (7.54) one sees that λm ≥ 1 and hence there exists some λ ≥ 1 such that λm & λ. By suitably scaling φm as before and after an application of Harnacks inequality we can assume that φm → φ in 1,α Cloc (Ω\K) where φ ≥ 0 is nonzero and constant on level sets of E. Passing to the limit shows that   1 2 φ in Ω\K, LA (φ) = λ|∇E|A f (E) + 4E 2  and a strong maximum principle argument shows that φ > 0 in Ω\K. Since φ constant on level sets of E we have φ = h(E) for some 0 < h in (γ, ∞) and since φ smooth on Ω\K we see that h is smooth on (γ, ∞). Writing the equation for φ in terms of h gives   1 00 2 −h (E)|∇E|A = λh(E) f (E) + |∇E|2A in Ω\K, 4E 2 and using Hopfs lemma we can cancel the gradients. Using the vast knowledge of ode’s one can use the above theorem to obtain various results concerning potentials of the form V (x) = |∇E|2A f (E). We don’t exploit this fact other than to look at one result. Corollary 7.56. Suppose E is an interior potential on Ω and E = 0 on ∂Ω. Then there no 0 < f ∈ C(0, ∞) such that Z Z Z |∇E|2A 2 1 2 |∇u|A dx − u dx ≥ f (E)|∇E|2A u2 dx, u ∈ H01 (Ω). 2 4 E Ω Ω Ω Proof. Suppose there is such a function f . Using the proof of theorem 7.53 one sees that there is some 0 < h ∈ C 2 (0, ∞) such that   1 00 h (t) + λ f (t) + 2 h(t) = 0, 4t √ in (0, ∞) where λ ≥ 1. Now set h(t) = ty(t) to see that   y 0 (t) λ−1 00 0 = y (t) + + y(t) λf (t) + , t 4t2 141  7.2. Main Results in (0, ∞) and y(t) > 0. But oscillation theory from ordinary differential equations shows this is impossible. Other than some regularity issues this ode approach extends immediately to the case where E is a boundary weight in Ω. Using this corollary (but in the boundary case) one can show the result mentioned in the examples section regarding improvements of Hardy’s boundary inequality in the half space; the regularity is not an issue in this example since δ(x) := dist(x, ∂Rn+ ) = xn is smooth. We now present a result obtained by Avkhadiev and Wirths (see [2]). Given a domain Ω in Rn we say it has finite inradius if δ(x) := dist(x, ∂Ω) is bounded in Ω. We let λ0 (Lambs constant) denote the first positive zero of J0 (t) − 2tJ1 (t) where Jn is the Bessel function of order n. Numerically one sees that λ0 = 0.940.... Now for their result. Theorem 7.57. (Avkhadiev, Wirths) Suppose Ω is a convex domain in Rn with finite inradius. Then Z Z 2 Z 1 u λ20 |∇u|2 dx ≥ dx + u2 dx, u ∈ H01 (Ω) 2 2 4 δ kδk Ω Ω L∞ Ω is optimal. This extends a result of H. Brezis and M. Marcus (see [4]) which said that if Ω is a convex subset of Rn then Z Z 2 Z 1 u 1 |∇u|2 dx ≥ dx + u2 dx, u ∈ H01 (Ω) 2 2 (Ω) 4 δ 4diam Ω Ω Ω where diam(Ω) denotes the diameter of Ω. Note that there are unbounded convex domains with infinite diameter but finite inradius; for example take a cylinder. We establish a generalized version of this result. Suppose µ is a nonnegative nonzero locally finite measure in Ω (possibly unbounded) and 0 < E ∈ L∞ (Ω) is a solution to LA (E) = µ  in Ω  |∇E|A = 1  a.e. in Ω  E = 0  on ∂Ω.  We then have the following theorem. 142  7.2. Main Results Theorem 7.58. Suppose E is as above. Then Z Z Z 1 u2 λ20 2 |∇u|A dx ≥ dx + u2 dx, 4 Ω E2 kEk2L∞ Ω Ω for all u ∈ Cc∞ (Ω). Proof. Let E be as above. Now extend E to all of Rn by setting E = 0 on Rn \Ω, let Eε denote the ε mollification of E and Fε := LA (Eε ). Returning to the proof of theorem 7.49 (ii) we have Z  1 |∇u|2A dx− 4  Ω  Z Ω  |∇Eε |2A 2 u dx ≥ Eε2  where  Z  Iε := Ω  |∇Eε |2A f (Eε )    f 0 (Eε ) 1 + f (Eε ) 2Eε    Z Ω  f 0 (Eε ) −f (Eε ) − Eε 00    u2 dx+Iε ,  u2 Fε dx, λ2  for u ∈ Cc∞ (Ω) and 0 < f ∈ C 2 ((0, kEkL∞ ]). Now set λ := kEk02 where λ0 L∞ √ is Lambs constant and define f (t) := J0 ( λt). It is possible to show that   f 0 (t) f 0 (t) 1 1 00 −f (t) − = λ, l(t) := + ≥0 f (t) > 0, f (t) t f (t) 2t in (0, kEkL∞ ). Fixing u ∈ Cc∞ (Ω) and subbing this f into the above gives Z Ω  |∇u|2A dx  1 − 4  Z Ω  |∇Eε |2A 2 λ20 u dx ≥ Eε2 kEk2L∞  Z  |∇Eε |2A u2 dx + Iε ,  Ω  R after noting that kEε kL∞ ≤ kEkL∞ and where Iε := Ω l(Eε )u2 Fε dx. It is possible to show that l ∈ C ∞ ((0, kEkL∞ ]). A standard argument shows that l(Eε )u → l(E)u in H01 (Ω) and uFε dx * uµ in H −1 (Ω) and hence one can conclude that lim inf ε&0 Iε ≥ 0. Passing to the limit (as ε & 0) in the remaining integrals gives the desired result. We now look at improvements of the weighted generalized Hardy inequalities. The next theorem allows us to transfer our knowledge of improvements from the non-weighted case to the weighted case, at least in the case that E is an interior weight.  143  7.2. Main Results Theorem 7.59. (Weighted interior improvements) Suppose E is an interior weight on Ω and 0 ≤ V ∈ C ∞ (Ω\K). Then the following are equivalent: (i) For all u ∈ H01 (Ω) Z  |∇u|2A dx  Ω  1 ≥ 4  Z Ω  |∇E|2A 2 u dx + E2  Z  V u2 dx.  (ii) For all t 6= 21 and u ∈ Xt Z Z Z 1 2 2 2t−2 2 2t 2 V E 2t u2 dx. |∇E|A E u dx + E |∇u|A dx ≥ (t − ) 2 Ω Ω Ω (iii)  (7.60)  Ω  (7.61)  For all u ∈ X 1 2  Z Ω  E|∇u|2A dx ≥  Z  V Eu2 dx.  (7.62)  Ω  Using similar arguments one can obtain a version of theorem 7.59 for the case when E is a boundary weight on Ω; we omit the details since the results is not as clean. Proof. Let E be an interior weight on Ω and 0 ≤ V ∈ C ∞ (Ω\K). (i) ⇒ (ii) Suppose (i) holds, t < 12 , u ∈ Cc0,1 (Ω\K) and define v := E t u ∈ Cc0,1 (Ω\K). Then putting v into (7.60) and performing some integration by parts gives (7.61). (ii) ⇒ (iii) Suppose (ii) holds. Let u ∈ Cc0,1 (Ω\K) which is an element of Xt for all t. Now using (7.61) for this u and sending t % 12 gives (7.62). (iii) ⇒(i) −1 Suppose (iii) holds, u ∈ Cc0,1 (Ω\K) and v := E 2 u ∈ Cc0,1 (Ω\K). Putting v into (7.62) and integrating by parts gives (7.60) for all u ∈ Cc0,1 (Ω\K).  7.2.4  Hardy inequalities valid for u ∈ H 1 (Ω)  Let K be a compact subset of Ω with dimbox (K) < n − 2. Standard arguments show that Cc0,1 (Ω\K) is dense in H 1 (Ω). Definition 7.63. We say E is a Neumann interior weight on Ω provided: there exists some compact K ⊂ Ω, dimbox (K) < n − 2, E ∈ C ∞ (Ω\K), 144  7.2. Main Results inf Ω E > 0, LA (E) + E is a nonnegative nonzero measure µ whose support is K, E = ∞ on K and A∇E · ν = 0 on ∂Ω where ν(x) denotes the outward normal vector at x ∈ ∂Ω. Theorem 7.64. Suppose E is a Neumann interior weight on Ω. Then For u ∈ Cc0,1 (Ω\K) and v := E  (i) Z  Ω  |∇u|2A dx  1 + 2  Z  1 u dx ≥ 4 Ω 2  Z Ω  −1 2  u we have  |∇E|2A 2 u dx + E2  Z  E|∇v|2A dx.  (7.65)  Ω  (ii) Z Ω  |∇u|2A dx +  1 2  Z  u2 dx ≥  Ω  1 4  Z Ω  |∇E|2A 2 u dx, E2  (7.66)  holds for all u ∈ H 1 (Ω). Moreover 41 and 12 are optimal in the sense that if one fixes 14 then you can do no better than 12 and vice-versa. Also the inequality is not attained. One can again view the best constants in a different manner which is analogous to theorem 7.27; we omit the details. Proof. Let E be a Neumann interior weight on Ω. −1 (i) Let u ∈ Cc0,1 (Ω\K) and define v := E 2 u. Then |∇u|2A = E|∇v|2A +  |∇E|2A 2 u + v∇v · A∇E, 4E 2  and integrating this over Ω gives Z Z Z Z |∇E|2A 2 1 1 2 2 |∇u|A dx + u dx = u dx + E|∇v|2A dx. 2 Ω 4 Ω E2 Ω Ω  (7.67)  (ii) Using (i) and the fact that Cc0,1 (Ω\K) is dense in H 1 (Ω) one obtains (7.66) for all u ∈ H 1 (Ω). We now show the constants are optimal. We first show that E t ∈ H 1 (Ω) for 0 < t < 21 . As in the proof of lemma 7.23 the following calculations are only formal but they can be justified as hinted at there; by first regularizing the measure, obtaining approximate solutions and passing to the limit. Fix 0 < t < 12 and multiply LA (E) + E = µ by E 2t−1 and integrate over Ω using integration by  145  7.2. Main Results parts and the fact that E = ∞ on K along with the boundary conditions of E to see that Z Z 2t (7.68) E dx = (1 − 2t) E 2t−2 |∇E|2A dx, Ω  Ω  which shows that ∈ for 0 < t < 21 . To show the constants are optimal we will use as a minimizing sequence E t as t % 12 . A computation shows R R t |2 dx + 1 2t |∇E 1 A 2 Ω E dx Ω = t2 + − t, R |∇E|2A 2 2t Ω E 2 E dx Et  H 1 (Ω)  and we see that 41 is optimal. One similarly shows 21 is optimal. To show the inequality does not attain we, as usual, just hold on to the extra term that we dropped in the above calculations. This term is positive 1 / H 1 (Ω) which is the case after one for non-zero u ∈ H 1 (Ω) provided E 2 ∈ considers (7.68). We now examine weighted versions of (7.66). Suppose E is a Neumann interior weight on Ω and as usual we let K denote the support of µ. For t 6= 12 and u ∈ Cc0,1 (Ω\K) we define kuk2t  R  R 2t |∇u|2 dx + 2t 2 E Ω Ω E u dx := R 2t 2 Ω E |∇u| dx  t < 12 t > 12 ,  and we let Yt denote the completion of Cc0,1 (Ω\K) with respect to this norm. We then have the following theorem. Theorem 7.69. Suppose E is a Neumann interior weight on Ω and t 6= 12 . Then  Z   Z Z 1 1 2 E 2t |∇u|2A dx + −t E 2t u2 dx ≥ t − E 2t−2 |∇E|2A u2 dx, 2 2 Ω Ω Ω for all u ∈ Yt . Moreover the constants are optimal and not attained. Note in particular that for t > 12 one only has a gradient term on the left hand side and so we can conclude that C ∞ (Ω) is not contained in Yt for t > 21 .  146  7.2. Main Results Proof. Suppose E is a Neumann interior weight on Ω, t 6= Cc0,1 (Ω\K). Putting E t u into (7.65) gives Z E  2t  |∇u|2A dx   +  Ω  1 −t 2  1 2  and let u ∈   Z  1 2 E 2t−2 |∇E|2A u2 dx E u dx ≥ t− 2 Ω Ω Z + E|∇w|2A dx  Z  2t 2  Ω 1  where w := E t− 2 u. To show the constants are optimal one takes the same approach as in theorem 7.31. We now show the optimal constants are not obtained. Suppose equality for some nonzero u ∈ Yt . Then it is √ we have 1 easily seen that E ∈ H (Ω) which we know is not the case. We now examine improvements of (7.66). Theorem 7.70. Suppose E is a Neumann interior weight on Ω. Then (i) Suppose V ∈ C ∞ (Ω\K) and there exists some 0 < φ ∈ C 2 (Ω\K) ∩ 1 C (Ω\K) such that −LA (φ) +  A∇E · ∇φ +Vφ≤0 E  in Ω\K,  (7.71)  with A∇φ · ν ≥ 0 on ∂Ω. Then Z Z Z Z |∇E|2A 2 1 1 u2 dx − u dx ≥ V (x)u2 dx, |∇u|2A + 2 2 4 E Ω Ω Ω Ω for all u ∈ H 1 (Ω). (ii) Suppose 0 ≤ V ∈ C ∞ (Ω\K) is such that Z Z Z Z |∇E|2A 2 1 1 2 2 |∇u|A + u dx − u dx ≥ V (x)u2 dx, 2 2 4 E Ω Ω Ω Ω holds for all u ∈ H 1 (Ω). In addition we assume that {x ∈ Ω : E(x) < t} is connected for sufficiently large t. Then there exists some 0 < θ ∈ C ∞ (Ω\K) such that θ |∇E|2A −LA (θ) − + θ+Vθ ≤0 in Ω\K, (7.72) 2 4E 2 with A∇θ · ν = 0 on ∂Ω. Note that one can go from (7.71) to (7.72) by using the change of vari1 ables θ = φE 2 in the case that A∇φ · ν = 0 on ∂Ω. 147  7.2. Main Results Proof. The proof is similar to the proof of theorem 7.41. Remark 7.2.5. One can obtain an analogous version of theorem 7.59 for the case where E is an interior weight on Ω satisfying a Neumann boundary condition.  7.2.5  H 1 (Ω) inequalities for exterior and annular domains  In this section we obtain optimal Hardy inequalities which are valid on exterior and annular domains. Moreover these inequalities will be valid for functions u which are nonzero on various portions of the boundary. For simplicity we only consider the case where A(x) is the identity matrix and hence LA = −∆; the results immediately generalize to the case where A(x) is not the identity matrix. We first examine the exterior domain case. Condition (Ext.): We suppose that E > 0 in Rn , −∆E is a nonnegative nonzero finite measure (which we denote by µ) with compact support K and we let Ω denote a connected exterior domain in Rn with dist(K, Ω) > 0. In addition we assume that the compliment of Ω denoted by Ωc is connected, lim|x|→∞ E = 0 and ∂ν E ≥ 0 on ∂Ω. We will work in the following function space. Let D1 (Ω ∪ ∂Ω) denote the completion of Cc∞ (Ω ∪ ∂Ω) with respect to the norm k∇ukL2 (Ω) . Note we don’t require u to be zero on the boundary of ∂Ω. We then have the following theorem. Theorem 7.73. Suppose E, µ, K, Ω are as in condition (Ext.). Then (i) For all u ∈ D1 (Ω ∪ ∂Ω) we have Z Z 1 |∇E|2 2 |∇u|2 dx ≥ u dx. (7.74) 4 Ω E2 Ω Moreover the constant is optimal and not attained. (ii) For all u ∈ D1 (Ω ∪ ∂Ω) we have Z Z Z 1 |∇E|2 2 u2 ∂ν E 1 |∇u|2 dx ≥ u dx + dS(x). 4 Ω E2 2 ∂Ω E Ω Proof. Let u ∈ Cc∞ (Ω ∪ ∂Ω) and set v := E |∇u|2 −  −1 2  (7.75)  u. Then as before we have  |∇E|2 u2 = E|∇v|2 + v∇v · ∇E, 4E 2  in Ω.  (7.76) 148  7.2. Main Results Integrating the last term by parts gives Z Z 1 u2 ∂ν E v∇v · ∇Edx = dS(x). 2 ∂Ω E Ω We obtain (7.75) by integrating (7.76) over Ω and since ∂ν E ≥ 0 on ∂Ω we obtain (7.74). We now show the constant is optimal. For big R we set ΩR := Ω ∩ BR where BR is the ball centered at 0 with radius R. Let 1 2t−1 and integrate over Ω to obtain R 2 < t < 1 and multiply −∆E = µ by E Z Z Z ∂ν EE 2t−1 dS(x) + ∂ν EE 2t−1 dS(x). (2t − 1) E 2t−2 |∇E|2 dx = ΩR  ∂BR  ∂Ω  Using a Newtonian potential argument one can show that as R → ∞ the surface integral over the ball BR goes to zero. So using this one sees that Z Z 2t−2 2 (2t − 1) E |∇E| dx = ∂ν EE 2t−1 dS(x), (7.77) Ω  R  ∂Ω  |∇E t |2 dx  and so Ω < ∞. With this along with a standard cut-off function argument one sees that E t ∈ D1 (Ω ∪ ∂Ω). Now one uses E t as t & 12 as a minimizing sequence to show that 14 is optimal. We now show the constant is not attained. Now assume that x0 ∈ ∂Ω is such that E(x0 ) = min∂Ω E. Then by Hopf’s lemma ∂ν E(x0 ) > 0 and so using this along with continuity 1 and (7.77) one sees that E 2 ∈ / D1 (Ω ∪ ∂Ω). Now to finish the proof it will be sufficient to show that Z E|∇v|2 dx > 0 Ω 1 D (Ω ∪ ∂Ω). The  for all nonzero u ∈ only nonzero u’s for which this integral 1 2 is zero are multiples of E which are not in D1 (Ω ∪ ∂Ω). Example 7.78. Take Ω a exterior domain in Rn where n ≥ 3, 0 ∈ / Ω, and such that ν(x) · x ≤ 0 on ∂Ω where ν(x) is the outward pointing normal. Define E(x) := |x|2−n and use theorem 7.73 to see that   Z Z u2 n−2 2 2 |∇u| dx ≥ dx, (7.79) 2 2 Ω |x| Ω for all u ∈ D1 (Ω ∪ ∂Ω). Moreover the constant is optimal and not attained. In fact using (ii) from the same theorem shows we can add the following nonnegative term to the right hand side of (7.79): Z (n − 2) u2 (−x · ν) dS(x). 2 |x|2 ∂Ω 149  7.2. Main Results We now examine the annular domain case. Condition (Annul.): We assume that Ω1 ⊂⊂ Ω2 are two bounded connected domains in Rn with smooth boundaries and Ω := Ω2 \Ω1 is connected. In addition we assume that E > 0 in Ω2 with −∆E = µ in Ω2 where µ is a nonnegative nonzero finite measure supported on K ⊂ Ω1 . We also assume that ∂ν E ≤ 0 on ∂Ω1 . We then have the following theorem. Theorem 7.80. Suppose Ω, K, E are as in condition (Annul.). Then (i) For all u ∈ H 1 (Ω) with u = 0 on ∂Ω2 we have Z Z 1 |∇E|2 2 2 |∇u| dx ≥ u dx. (7.81) 4 Ω E2 Ω Moreover the constant is optimal and not attained if we assume that E = 0 on ∂Ω2 . (ii) For all u ∈ H 1 (Ω) with u = 0 on ∂Ω2 we have Z Z Z 1 |∇E|2 2 u2 ∂ν E 1 2 |∇u| dx ≥ u dx + dS(x). (7.82) 4 Ω E2 2 ∂Ω E Ω Proof. The proof of (7.81) and (7.82) is similar to the previous theorem so we omit the details. We now show the constant is optimal. Let H01 (Ω ∪ ∂Ω1 ) denote {u ∈ H 1 (Ω) : u = 0 on ∂Ω2 }. Again we multiply −∆E = µ by E 2t−1 for 21 < t < 1 and integrate over Ω to obtain Z Z 2t−2 2 (2t − 1) E |∇E| dx = − ∂ν EE 2t−1 dS(x), Ω  ∂Ω1  which shows that E t ∈ H01 (Ω ∪ ∂Ω1 ). From this one obtains Z lim (2t − 1) E 2t−2 |∇E|2 dx = µ(Ω1 ) > 0, t& 21  Ω 1  which shows that E 2 ∈ / H01 (Ω ∪ ∂Ω1 ). To see the constant is optimal one uses the same minimizing sequence as in the previous theorem. To see the 1 constant is not attained one uses the fact that E 2 ∈ / H01 (Ω ∪ ∂Ω1 ). Remark 7.2.6. These inequalities have analogous weighted versions and using the methods developed earlier one easily obtains results concerning improvements. We leave this for the reader to develop. 150  7.2. Main Results  7.2.6  The non-quadratic case  For 1 < p ≤ n we define LA,p (E) := −div(|∇E|p−2 A A∇E). As mentioned earlier Adimurthi and Sekar [1] obtained generalized Hardy inequalities of the form  Z  Z |∇E|pA p p−1 p |∇u|pA dx − |u| dx ≥ 0, (7.83) p E Ω Ω where u ∈ W01,p (Ω). There approach (as their title suggests) was to look at functions E which solve LA,p (E) = δ0 E = 0  in Ω on ∂Ω,  where 0 ∈ Ω and where δ0 is again the Dirac mass at 0. p They posed the question (see [1]) as to whether ( p−1 p ) is optimal in (7.83)? The next theorem shows this is the case (at least for 1 < p < n); infact we show the result for a more general case. Interior case Suppose µ is a nonnegative nonzero finite measure supported on K ⊂ Ω, dimbox (K) < n − p (and hence Cc0,1 (Ω\K) is dense in W01,p (Ω)) and 0 < E is a solution of LA,p (E) = µ in Ω. (7.84) By regularity theory (see [7], [14]) there is some 0 < σ < 1 such that E ∈ C 1,σ (Ω\K) and by the maximum principle (see [15]) E > 0 in Ω\K. Now if we assume that µ = δ0 , as was the case in the question posed in [1], then one can show E(0) = ∞. Theorem 7.85. Suppose E is as above but we don’t assume that E = ∞ on K. (i) Then   Z Z |∇E|pA p p−1 p 2 |∇u|A dx ≥ |u| dx, (7.86) p Ep Ω Ω for all u ∈ W01,p (Ω). (ii) Suppose E = ∞ on K and E = γ on ∂Ω where γ is a non-negative constant. Then the constant in (7.86) is optimal.  151  7.2. Main Results Proof. (i) Let u ∈ Cc0,1 (Ω\K). Then ∇E 1−p = (1−p)E −p ∇E and dotting p both sides with |∇E|p−2 A A∇E|u| and integrating over Ω gives Z (1 − p) Ω  |∇E|pA p |u| dx = Ep  Z ZΩ  =    p−2 ∇E 1−p · |∇E|A A∇E|u|p dx E 1−p |u|p dµ  ΩZ  − ZΩ = − Ω  p−2 E 1−p |∇E|p−2 u∇udx A A∇E · p|u| p−2 E 1−p |∇E|p−2 u∇udx, A A∇E · p|u|  where we used the divergence theorem and also the fact that u = 0 on K. Now using the Cauchy-Schwarz inequality on the inner product induced by A(x) we see that p−1 p  Z Ω  |∇E|pA p |u| dx ≤ Ep  Z Ω  p−1 |∇E|p−1 A |u| |∇u|A dx, E p−1  and we now apply Holder’s inequality on the right after recalling (p−1)p0 = p where p0 is the conjugate of p. Now use density to extend to all of W01,p (Ω). (ii) We first consider the case γ > 0. We begin by showing that ut := p E t − γ t ∈ W01.p (Ω) for 0 < t < p−1 p . Fix 0 < t < p−1 and multiply (7.84) by E tp−p+1 and integrate over Ω to get Z Z tp−p+1 0= E dµ = (tp − p + 1) |∇E|2A E tp−p dx Ω Z Ω n−1 tp−p+1 −γ |∇E|p−2 A A∇E · νdH ∂Ω Z = (tp − p + 1) |∇E|2A E tp−p dx Z Ω p−2 −γ tp−p+1 div(|∇E|A A∇E)dx ΩZ = (tp − p + 1) |∇E|2A E tp−p dx + γ tp−p+1 µ(Ω), Ω  where the first integral is zero since E = ∞ on K and tp − p + 1 < 0. Re-arranging this we arrive at Z µ(Ω)γ tp−p+1 tp |∇E t |pA dx = , p − tp − 1 Ω 152  7.2. Main Results from which we see that E t ∈ W 1,p (Ω) for 0 < t < Z lim  t% p−1 p  Ω  p−1 p  and we also see that  |∇E|pA E tp−p dx = ∞.  Put t as above and set ut := E t − γ t ∈ W01,p (Ω). By the binomial theorem we have ∞ X p (1 + x) = (p, m)xm , m=0  for all |x| ≤ 1 where (p, m) are the binomial coefficients. One should note that (p, m) is eventually alternating and since we have convergence at x = P P m −1 we see that m (p, m)(−1) converges; which shows that m |(p, m)| < ∞. Now we have γt Et  |ut |p = E tp 1 − = E  tp  ∞ X  (p, m)  m=0  and we define R Qt :=  ΩR  (−1)m γ tm , E tm  |∇E|pA p E p |ut | dx . p Ω |∇ut |A dx  So   p tp−p P∞ m γ tm dx (p, m)(−1) E |∇E| tm m=1 A Ω E 1 R Qt − p = , p tp−p p t dx t Ω |∇E|A E R  and so 1 Qt − p t  ≤ =  ∞ 1 X (p, m) tp  R  1 tp  p − tp − 1 p − tp − 1 + tm  m=1 ∞ X  (p, m)  m=1  p tp−p tm −tm E γ E dx Ω |∇E| R A p tp−p dx Ω |∇E|A E  ≤  ∞ p − tp − 1 X (p, m) tp+1 m  =:  p − tp − 1 Cp , tp+1  m=1  153  7.2. Main Results and so we see that lim  t% p−1 p  Qt −  1 = 0, tp  which shows the constant in (7.86) is optimal. Now we handle the case γ = 0. Let LA,p (E) = µ in Ω and E = 0 on ∂Ω and define Eε := ε + E where ε > 0. Then LA,p (Eε ) = µ in Ω and Eε = ε on ∂Ω. For u ∈ Cc∞ (Ω) non-zero we have, after some simple algebra, R R p p Ω |∇u|A dx Ω |∇u|A dx ≤ , p p R |∇E|A R |∇Eε |A p dx p dx |u| |u| p p Ω E Ω E ε  which shows the constant is optimal in the case of γ = 0.  Boundary case Analogously to the quadratic case we will be interested in the validity of (7.86) when E is a solution to LA,p (E) = µ E = 0  in Ω, on ∂Ω  where µ is a nonnegative nonzero finite measure and where we impose some added regularity restrictions to E or µ. Recall in the quadratic case we added the condition that E ∈ H01 (Ω). For simplicity we will assume that µ is smooth; say dµ = f dx where 0 ≤ f ∈ C ∞ (Ω) is non-zero. One can show that E ∈ C 1,σ (Ω) for some 0 < σ < 1. Theorem 7.87. Suppose E is a positive solution to LA,p (E) = µ in Ω where µ is as above. (i) Then   Z   Z Z |∇E|pA p p−1 p p − 1 p−1 |u|p |u| dx + dµ ≤ |∇u|pA dx, p p−1 p E p E Ω Ω Ω (7.88) 1,p for all u ∈ W0 (Ω). Since µ is a measure we have  Z  Z |∇E|pA p p−1 p |u| dx ≤ |∇u|pA dx, (7.89) p p E Ω Ω for all u ∈ W01,p (Ω). (ii) Suppose E = 0 on ∂Ω. Then (7.89) is optimal. 154  7.2. Main Results (iii) Suppose E = 0 on ∂Ω. If one fixes the optimal constant from part (ii) then the other constant is also optimal in (7.88) ie.  p R R  |∇E|pA p dx  Ω |∇u|pA dx − p−1   p − 1 p−1 |u| p Ω Ep 1,p inf : u ∈ W0 (Ω) = . R |u|p   p Ω E p−1 dµ Proof. (i) Suppose E is a positive solution to LA,p (E) = µ in Ω and let u ∈ Cc∞ (Ω). From the proof of theorem 7.85 we have Z (p − 1) Ω  |∇E|pA p |u| dx + Ep  Z  Now let q denote p0 and Z |∇E|pA p B := |u| dx, Ep Ω  Ω  |∇E|p−2 A A∇E · ∇u|u|p−2 udx p−1 E Ω  10 Z p |∇E|pA p k|∇u|A kLp . |u| dx ≤ p Ep Ω  |u|p dµ = p E p−1  Z C := Ω  Z  |u|p dµ, E p−1  Z D := Ω  |∇u|pA dx.  Using Young’s inequality with t > 0 we arrive at 1 1 (p − 1) C B+ ≤ B q D p ≤ tB + C(t)D, p p  where C(t) := p−1 q and so  1 C(t)    −p q  t  −p q  ,   p−1 1 −t B+ C ≤ D, p pC(t)  for all t > 0. Picking t = q 2 gives the desired result. tp−p+1 and integrate over Ω (ii) Let t > p−1 p , multiply LA,p (E) = µ by E to obtain Z Z tp−p+1 E dµ = (tp − p + 1) |∇E|pA E tp−p dx, (7.90) Ω  which shows that E t ∈ Et  Ω  W01,p (Ω)  for p >  p−1 p .  If one uses as a minimizing  p−1 p  sequence ut := and sends t & they immediately see that (7.89) is optimal. p (iii) Again one uses ut := E t and sends t & p−1 . The result is immediate after using (7.90).  155  7.2. Main Results An important example is when A(x) is the identity matrix and E(x) = δ(x) := dist(x, ∂Ω) so |∇δ| = 1 a.e.. Then LA,p (δ) = −div(|∇δ|p−2 δ) = −∆δ =: µ which is non-negative if we further assume that Ω is convex. In this case we have the Lp analog of (7.2) (provided one relaxes the regularity assumption on the measure µ): Corollary 7.91. Suppose Ω is convex and δ(x) := dist(x, ∂Ω). Then for 1 < p < ∞ and u ∈ W01,p (Ω) we have   Z Z p−1 p |u|p p |∇u| dx ≥ dx, p p Ω Ω δ Z  p  |∇u| dx − Ω    p−1 p  p Z Ω  |u|p dx ≥ δp    p−1 p  p−1 Z Ω  |u|p dµ, δ p−1  where dµ := −∆δdx. Moreover all constants are optimal. The first inequality is due to [13].  156  Bibliography [1] Adimurthi, Anusha Sekar, Role of the fundamental solution in HardySobolev-type inequalities, Proceedings of the Royal Society of Edinburgh, 136A, 1111-1130, 2006 [2] F.G. Avkhadiev, K.J. Wirths, Unified Poincare and Hardy inequalities with sharp constants for convex domains, Math. Mech. 87, No. 8-9, 632-642 (2007) [3] B. Barbatis, S. Filippas, A. Tertikas, Series expansions for Lp Hardy inequalities, (August 20, 2007) preprint [4] H. Brezis, M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 217–237 (1998). [5] H. Brezis, M. Marcus, I.Shafrir, Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000), 177-191 [6] H. Brezis, J.L. Vazquez, Blowup Solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid 10 (1997), 443-469 [7] E. DiBenedetto. C 1 local regularity of weak solutions of degenerate elliptic equations, Nonlin. Analysis 7 (1983), 827850 [8] S. Filippas, A. Tertika, Optimizing improved Hardy inequalities, J.Funct. Anal. 192 (2002), n0.1, 186-233 [9] J.Fleckinger, E.M. Harrell II, F. Thelin, Boundary behaviour and estimates for solutions of equations containing the p-Laplacian, Electron. J. Differential Equations 38 (1999), 1-19 [10] 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. [11] B. Opic, A. Kufner, Hardy Type Inequalities, Pitman Research Notes in Mathematics, Vol. 219, Longman, New York, 1990 157  Chapter 7. Bibliography [12] V.G. Maz’ja Sobolev Spaces, Berlin, Springer-Verlag [13] M. Marcus, V.J. Mizel, Y. Pinchover, On the best constant for Hardy’s inequality in Rn , Transactions of the American Mathematical Society, Vol. 350, No. 8 (aug., 1998), pp 3237-3255 [14] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqns 51 (1984), 473484 [15] J.L. Vazuez, A Strong Maximum Principle for some Quasilinear Elliptic Equations, Appl. Math. Optim. 12:191-202 (1984) [16] J.L. Vazquez, E. Zuazua The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential, J. Funct. Anal. 173 (2000), 103-153  158  Chapter 8  Conclusion Chapters 2, 4, 5 and 6 all involve obtaining results concerning the regularity of the extremal solutions. In Chapter 2 the main issues is that the equation is nonvariational and hence the stability does not immediately allow the use of arbitrary test functions which can be used to obtain estimates. This is overcome through the use of a general Hardy inequality which allows the use of arbitrary test functions and then one proceeds as usual. In Chapter 3 we proved various curious results which were noticed numerically by various authors. In Chapter 4 we used the approach developed in [1] to show the extremal solution is bounded in low dimensions. This is not a complete straightforward extension of the case considered in [1] where f (u) = eu , since in their case f 0 (u) = f (u), which greatly streamlines the approach. To complete the picture one needs to show that the extremal solution is singular for N ≥ 9. Our approach showed this without needing to resort to a computer assisted proof. In Chapter 5 we obtained many results concerning the regularity of the extremal solution of fourth order problems on general domains. We mention that our results are vast improvements over the known results but, one expects they are not optimal after considering the known results on radial domains. The key ideas for our results were to test the stability on ψ := ∆u where u is a stable solution and then to obtain a pointwise lower bound on −∆u, which allows one to obtain estimates. Extending these results to optimal results, we suspect, will require a new idea. In Chapter 6 we obtained results concerning the regularity of the extremal solutions in the elliptic system given by −∆u = λev , −∆v = γeu . The main goal of this chapter was to show, at least in this special case, that the standard approach for the scalar case can be modified to handle the systems case. In Chapter 7 we examined very general Hardy inequalities and showed many known Hardy inequalities can be thought of a special cases of this general Hardy inequality. In addition improvements and weighted versions 159  8.1. Future directions were obtained. We believe this approach to Hardy inequalities is a somewhat unifying approach to the various Hardy inequalities.  8.1  Future directions  The results from Chapter 5 need to be improved since they are not optimal. As mentioned above we suspect this will require a new idea. The results from Chapter 6 needs to be extended to the case where the nonlinearities are not equal. Most of the equations examined in this thesis can be viewed as stationary solutions of various parabolic or hyperbolic models coming from certain physical models. These fourth order parabolic and hyperbolic models have not received much attention and this is possible future work. Many of the equations examined here also have various quasi linear versions and many of the results obtained here can possibly extended. This needs to be investigated.  160  Bibliography [1] 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.  161  

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