| Preface | p. ix |
| Introduction and Preliminaries | |
| Introduction | p. 1 |
| Notation | p. 10 |
| Tangency and Comparison Theorems for Elliptic Inequalities | |
| The contributions of Eberhard Hopf | p. 13 |
| Tangency and comparison principles for quasilinear inequalities | p. 21 |
| Maximum and sweeping principles for quasilinear inequalities | p. 25 |
| Comparison theorems for divergence structure inequalities | p. 30 |
| Tangency theorems via Harnack's inequality | p. 34 |
| Uniqueness of the Dirichlet problem | p. 37 |
| The boundary point lemma | p. 39 |
| Appendix: Proof of Eberhard Hopf's maximum principle | p. 42 |
| Notes | p. 46 |
| Problems | p. 46 |
| Maximum Principles for Divergence Structure Elliptic Differential Inequalities | |
| Distribution solutions | p. 51 |
| Maximum principles for homogeneous inequalities | p. 54 |
| A maximum principle for thin sets | p. 59 |
| A comparison theorem in W[superscript 1, p] ([Omega]) | p. 61 |
| Comparison theorems for singular elliptic inequalities | p. 62 |
| Strongly degenerate operators | p. 68 |
| Maximum principles for non-homogeneous elliptic inequalities | p. 72 |
| Uniqueness of the singular Dirichlet problem | p. 78 |
| Appendix: Sobolev's inequality | p. 79 |
| Notes | p. 81 |
| Problems | p. 81 |
| Boundary Value Problems for Nonlinear Ordinary Differential Equations | |
| Preliminary lemmas | p. 83 |
| Existence theorems | p. 89 |
| Existence theorems on a half-line | p. 92 |
| The end point lemma | p. 96 |
| Appendix: Proof of Proposition 4.2.1 | p. 97 |
| Problems | p. 101 |
| The Strong Maximum Principle and the Compact Support Principle | |
| The strong maximum principle | p. 103 |
| The compact support principle | p. 105 |
| A special case | p. 107 |
| Strong maximum principle: Generalized version | p. 110 |
| A boundary point lemma | p. 119 |
| Compact support principle: Generalized version | p. 120 |
| Notes | p. 125 |
| Problems | p. 126 |
| Non-homogeneous Divergence Structure Inequalities | |
| Maximum principles for structured inequalities | p. 127 |
| Proof of Theorems 6.1.1 and 6.1.2 | p. 131 |
| Proof of Theorem 6.1.3 and the first part of Theorem 6.1.5 | p. 139 |
| Proof of Theorem 6.1.4 and the second part of Theorem 6.1.5 | p. 142 |
| The case p = 1 and the mean curvature equation | p. 146 |
| Notes | p. 150 |
| Problems | p. 150 |
| The Harnack Inequality | |
| Local boundedness and the weak Harnack inequality | p. 153 |
| The Harnack inequality | p. 163 |
| Holder continuity | p. 166 |
| The case p [Characters not reproducible] n | p. 171 |
| Appendix. The John-Nirenberg theorem | p. 173 |
| Notes | p. 179 |
| Problems | p. 180 |
| Applications | |
| Cauchy-Liouville Theorems | p. 181 |
| Radial symmetry | p. 186 |
| Symmetry for overdetermined boundary value problems | p. 195 |
| The phenomenon of dead cores | p. 203 |
| The strong maximum principle for Riemannian manifolds | p. 218 |
| Problems | p. 220 |
| Bibliography | p. 223 |
| Subject Index | p. 233 |
| Author Index | p. 235 |
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