Linear Second Order Elliptic Operators

by Julian Lopez-gomez

Overview

The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions.Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators.Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein-Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.

Product Details

ISBN-13:	9789814440240
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/10/2013
Pages:	356
Product dimensions:	5.90(w) x 9.10(h) x 1.00(d)

Table of Contents

Preface vii

1 The minimum principle 1

1.1 Concept of ellipticity. First consequences 2

1.2 Minimum principle of E. Hopf 5

1.3 Interior sphere properties 12

1.4 Boundary lemma of E. Hopf 19

1.5 Positivity properties of super-harmonic functions 23

1.6 Uniform decay property of E. Hopf 25

1.7 The generalized minimum principle of M. H. Protter and H. F. Weinberger 30

1.8 Appendix: Smooth domains 32

1.9 Comments on Chapter 1 38

2 Classifying supersolutions 41

2.1 First classification theorem 42

2.2 Existence of positive strict supersolutions 47

2.3 Positivity of the resolvent operator 52

2.4 Behavior of the positive supersolutions on γ⁰ 52

2.5 Second classification theorem 53

2.6 Appendix: Partitions of the unity 58

2.7 Comments on Chapter 2 60

3 Representation theorems 63

3.1 The projection on a closed convex set 65

3.2 The orthogonal projection on a closed subspace 69

3.3 The representation theorem of F. Riesz 71

3.4 Continuity and coercivity of bilinear forms 75

3.5 The theorem of G. Stampacchia 76

3.6 The theorem of P. D. Lax and A. N. Milgram 78

3.7 Projecting on a closed convex set of a u.c. B-space 78

3.7.1 Basic concepts and preliminaries 79

3.7.2 The projection theorem 82

3.7.3 The projection on a closed linear subspace 85

3.7.4 The projection on a closed hyperplane 87

3.8 Comments on Chapter 3 88

4 Existence of weak solutions 91

4.1 Preliminaries. Sobolev spaces 93

4.1.1 Test functions 93

4.1.2 Weak derivatives. Sobolev spaces 94

4.1.3 Holder spaces of continuous functions 97

4.1.4 Sobolev's imbeddings 98

4.1.5 Compact imbeddings 101

4.2 Trace operators 102

4.3 Weak solutions 114

4.4 Continuity of the associated bilinear form 117

4.5 Invertibility of (4.4) when β ≥ 0 118

4.5.1 Coercivity of the associated bilinear form 118

4.5.2 Existence of weak solutions. The resolvent operator 120

4.6 Invertibility of (4.4) for arbitrary β 122

4.7 Comments on Chapter 4 126

5 Regularity of weak solutions 129

5.1 L^p(R^N)-estimates for the Laplacian 131

5.2 L^p(Ω)-estimates for the Laplacian 135

5.3 General elliptic L^p(Ω)-estimates when γ₁ = 0 138

5.4 The method of continuity 139

5.5 Regularity of weak solutions when Γ₁ = 0 141

5.6 A first glance to the general case when Γ₁ ≠ 0 147

5.7 Comments on Chapter 5 152

6 The Krein-Rutman theorem 155

6.1 Orderings. Ordered Banach spaces 155

6.2 Spectral theory of linear compact operators 161

6.3 The Krein-Rutman theorem 164

6.4 Preliminaries of the proof of Theorem 6.3 166

6.5 Proof of Theorem 6.3 171

6.6 Comments on Chapter 6 184

7 The strong maximum principle 187

7.1 Minimum principle of J. M. Bony 189

7.2 The existence of the principal eigenvalue 195

7.3 Two equivalent weak eigenvalue problems 206

7.4 Simplicity and dominance of σ[L, B, Ω] 208

7.4.1 Proof of the strict dominance in case Γ₀ = 0 210

7.4.2 Proof of the strict dominance in case ω¹ = 0 212

7.4.3 Proof of the strict dominance in the general case 215

7.5 The strong maximum principle 215

7.6 The classical minimum principles revisited 217

7.7 Comments on Chapter 7 220

8 Properties of the principal eigenvalue 225

8.1 Monotonicity properties 226

8.2 Point-wise min-max characterizations 229

8.3 Concavity with respect to the potential 232

8.4 Stability of Ω along the Dirichlet components of ∂Ω 234

8.4.1 Proof of Proposition 8.5 236

8.4.2 Proof of Theorem 8.4 240

8.5 Continuous dependence with respect to Ω 240

8.6 Continuous dependence with respect to β(x) 254

8.7 Asymptotic behavior of σ(Beta;) as min β ↑ ∞ 260

8.8 Lower estimates of σ[L, D, Ω] in terms of |Ω 264

8.9 Comments on Chapter 8 267

9 Principal eigenvalues of linear weighted boundary value problems 273

9.1 General properties of the map Σ(λ) 274

9.2 Characterizing the existence of a principal eigenvalue 278

9.3 Ascertaining lim_λ→∞ σ[L + λV, B, Ω] when V ≥ 0 284

9.3.1 The simplest case 285

9.3.2 The admissible V's satisfying the main theorem 287

9.3.3 The main theorem 290

9.4 Characterizing the existence of principal eigenvalues for admissible potentials 307

9.5 Comments on Chapter 9 311

Bibliography 319

Index 331