Preface ........................................................ ix
1. Introduction and Preliminaries
1.1. Introduction ............................................ 1
1.2. Notation ............................................... 10
2. Tangency and Comparison Theorems for Elliptic Inequalities
2.1. The contributions of Eberhard Hopf ..................... 13
2.2. Tangency and comparison principles for quasilinear
inequalities ........................................... 21
2.3. Maximum and sweeping principles for quasilinear
inequalities ........................................... 25
2.4. Comparison theorems for divergence structure
inequalities ........................................... 30
2.5. Tangency theorems via Harnack's inequality ............. 34
2.6. Uniqueness of the Dirichlet problem .................... 37
2.7. The boundary point lemma ............................... 39
2.8. Appendix: Proof of Eberhard Hopf's
maximum principle ...................................... 42
Notes .......................................................... 46
Problems ....................................................... 46
3. Maximum Principles for Divergence Structure Elliptic
Differential Inequalities
3.1. Distribution solutions ................................. 51
3.2. Maximum principles for homogeneous inequalities ........ 54
3.3. A maximum principle for thin sets ...................... 59
3.4. A comparison theorem in W1,P(Ω) ......................... 61
3.5. Comparison theorems for singular elliptic
inequalities ........................................... 62
3.6. Strongly degenerate operators 68
3.7. Maximum principles for non-homogeneous elliptic
inequalities ........................................... 72
3.8. Uniqueness of the singular Dirichlet problem ........... 78
3.9. Appendix: Sobolev's inequality ......................... 79
Notes .......................................................... 81
Problems ....................................................... 81
4. Boundary Value Problems for Nonlinear Ordinary
Differential Equations
4.1. Preliminary lemmas ..................................... 83
4.2. Existence theorems ..................................... 89
4.3. Existence theorems on a half-line ...................... 92
4.4. The end point lemma .................................... 96
4.5. Appendix: Proof of Proposition 4.2.1 ................... 97
Problems ...................................................... 101
5. The Strong Maximum Principle and the Compact
Support Principle
5.1. The strong maximum principle .......................... 103
5.2. The compact support principle ......................... 105
5.3. A special case ........................................ 107
5.4. Strong maximum principle: Generalized version ......... 110
5.5. A boundary point lemma ................................ 119
5.6. Compact support principle: Generalized version ........ 120
Notes ......................................................... 125
Problems ...................................................... 126
6. Non-homogeneous Divergence Structure Inequalities
6.1. Maximum principles for structured inequalities ........ 127
6.2. Proof of Theorems 6.1.1 and 6.1.2 ..................... 131
6.3. Proof of Theorem 6.1.3 and the first part
of Theorem 6.1.5 ...................................... 139
6.4. Proof of Theorem 6.1.4 and the second part
of Theorem 6.1.5 ...................................... 142
6.5. The case p = 1 and the mean curvature equation ........ 146
Notes ......................................................... 150
Problems ...................................................... 150
7. The Harnack Inequality
7.1. Local boundedness and the weak Harnack inequality ..... 153
7.2. The Harnack inequality ................................ 163
7.3. Holder continuity ..................................... 166
7.4. The case p ≥ n ........................................ 171
7.5. Appendix. The John-Nirenberg theorem .................. 173
Notes ......................................................... 179
Problems ...................................................... 180
8. Applications
8.1. Cauchy-Liouville Theorems ............................. 181
8.2. Radial symmetry ....................................... 186
8.3. Symmetry for overdetermined boundary value
problems .............................................. 195
8.4. The phenomenon of dead cores .......................... 203
8.5. The strong maximum principle for Riemannian
manifolds ............................................. 218
Problems ...................................................... 220
Bibliography .................................................. 223
Subject Index ................................................. 233
Author Index .................................................. 235
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