The Maximum Principle (eBook)

Contents 6
Preface 9
Introduction and Preliminaries 11
1.1 Introduction 11
1.2 Notation 20
Tangency and Comparison Theorems for Elliptic Inequalities 22
2.1 The contributions of Eberhard Hopf 22
2.2 Tangency and comparison principles for quasilinear inequalities 30
2.3 Maximum and sweeping principles for quasilinear inequalities 34
2.4 Comparison theorems for divergence structure inequalities 39
2.5 Tangency theorems via Harnack’s inequality 43
2.6 Uniqueness of the Dirichlet problem 46
2.7 The boundary point lemma 48
2.8 Appendix: Proof of Eberhard Hopf’s maximum principle 51
Notes 55
Problems 55
Maximum Principles for Divergence Structure Elliptic Differential Inequalities 59
3.1 Distribution solutions 59
3.2 Maximum principles for homogeneous inequalities 62
3.3 A maximum principle for thin sets 67
3.4 A comparison theorem in 69
3.5 Comparison theorems for singular elliptic inequalities 70
3.6 Strongly degenerate operators 76
3.7 Maximum principles for non-homogeneous elliptic inequalities 80
3.8 Uniqueness of the singular Dirichlet problem 86
3.9 Appendix: Sobolev’s inequality 87
Notes 89
Problems 89
Boundary Value Problems for Nonlinear Ordinary Differential Equations 91
4.1 Preliminary lemmas 91
4.2 Existence theorems 97
4.3 Existence theorems on a half-line 100
4.4 The end point lemma 104
4.5 Appendix: Proof of Proposition 4.2.1 105
Problems 109
The Strong Maximum Principle and the Compact Support Principle 110
5.1 The strong maximum principle 110
5.2 The compact support principle 112
5.3 A special case 114
5.4 Strong maximum principle: Generalized version 117
5.5 A boundary point lemma 126
5.6 Compact support principle: Generalized version 127
Notes 132
Problems 133
Non-homogeneous Divergence Structure Inequalities 134
6.1 Maximum principles for structured inequalities 134
6.2 Proof of Theorems 6.1.1 and 6.1.2 138
6.3 Proof of Theorem 6.1.3 and the .rst part of Theorem 6.1.5 146
6.4 Proof of Theorem 6.1.4 and the second part of Theorem 6.1.5 149
6.5 The case p = 1 and the mean curvature equation 153
Notes 157
Problems 157
The Harnack Inequality 159
7.1 Local boundedness and the weak Harnack inequality 159
7.2 The Harnack inequality 169
7.3 Hölder continuity 172
7.4 The case 177
7.5 Appendix. The John–Nirenberg theorem 179
Notes 185
Problems 186
Applications 187
8.1 Cauchy–Liouville Theorems 187
8.2 Radial symmetry 192
8.3 Symmetry for overdetermined boundary value problems 201
8.4 The phenomenon of dead cores 209
8.5 The strong maximum principle for Riemannian manifolds 224
Problems 226
Bibliography 228
Subject Index 238
Author Index 240

Chapter 1 Introduction and Preliminaries (p. 1-2)

1.1 Introduction

The maximum principles of Eberhard Hopf are classical and bedrock results of the theory of second order elliptic partial differential equations. They go back to the maximum principle for harmonic functions, already known to Gauss in 1839 on the basis of the mean value theorem. On the other hand, they carry forward to the maximum principles of Gilbarg, Trudinger and Serrin, and the maximum principles for singular quasilinear elliptic differential inequalities, a theory initiated particularly by V´azquez and Diaz in the 1980s, but with earlier intimations in the work of Benilan, Brezis and Crandall. The purpose of the present work is to provide a clear explanation of the various maximum principles available for second-order elliptic equations, from their beginnings in linear theory to recent work on nonlinear equations, operators and inequalities. While simple in essence, these results lend themselves to a quite remarkable number of subtle uses when combined appropriately with other notions.

The first chapter concerns tangency and comparison theorems, based to begin with on the pioneering results of Eberhard Hopf. Section 2.1 includes in particular a discussion of Hopf’s nonlinear contributions, which are in fact not nearly as well known as his classical linear principle. We continue with a treatment of quasilinear equations and inequalities, with linear equations of course being an important special case. We consider both non-singular and singular cases, that is, in the latter case, equations which lose ellipticity at special values of the gradient of solutions, particularly at critical points. The concern here with singular equations arises from their growing importance in variational theory and applied mathematics, as well as their from specific theoretical interest, e.g., the celebrated p-Laplace operator .p.

The results of Hopf apply specifically to C2 solutions of elliptic differential inequalities. In many cases, however, especially when the equations and inequalities in question are expressed in divergence form, as in the calculus of variations, one can expect solutions to be no more than of class C1 or even only weakly differentiable in some Sobolev space. The solutions then must naturally be taken in a distribution sense. Correspondingly, in such cases, the study of maximum principles requires new techniques as alternatives to Hopf’s approach. These methods, necessarily integral in nature, originally arose from the work of a number of mathematicians, going back as far as Tonelli, Leray and Morrey in the years 1928–1935. Sections 2.4 and 2.5 are devoted specifically to C1 solutions of divergence structure inequalities, allowing both singular and non-singular operators. Theorem 2.4.1 and its attendant corollaries are of special interest for their simplicity and elegance, see also the corresponding uniqueness result for the singular Dirichlet problem (2.6.2). We note also the Tangency Theorem 2.5.2 obtained from the weak Harnack inequality (Section 7.1). Chapter 3 continues the study of divergence structure inequalities, but for more general operators for which the methods of Chapter 2 are inadequate.