Elliptic differential equations and obstacle problems

Cover......Page 1
THE UNIVERSITY SERIES IN MATHEMATICS......Page 2
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Preface......Page 7
Contents......Page 11
Glossary of Basic Notations......Page 15
1. Function Spaces......Page 17
1.1.1. Banach and Hilbert Spaces......Page 18
1.1.2. Fixed Points and Compact Operators......Page 22
1.2.1. $C^k$ and $C^{k,\delta}$ Spaces......Page 24
1.2.2. Extensions......Page 27
1.2.3. Traces......Page 31
1.3. Lebesgue Spaces......Page 32
1.3.1. $L^p$ Spaces over $\Omega$......Page 33
1.3.2. Approximation by Convolution in $C^0$, $C^{0,\delta}$, $L^p$......Page 35
1.3.3. $L^p$ Spaces over $\Gamma$......Page 40
1.4.1. Definition and Basic Properties......Page 44
1.4.2. Equivalent Norms and Multipliers......Page 47
1.5. Sobolev Spaces......Page 54
1.5.1. Distributional Derivatives......Page 55
1.5.2. Difference Quotients......Page 58
1.5.3. $H^{k,p}$ Spaces: Definitions and First Properties......Page 60
1.5.4. Density Results......Page 63
1.5.5. Changes of Variables and Extensions......Page 66
1.6.1. Sobolev Inequalities I......Page 70
1.6.2. Rellich's Theorem with Some Applications......Page 74
1.6.3. Sobolev Inequalities II......Page 77
1.7.1. $H_0^{k,p}(\Omega)$ Spaces......Page 80
1.7.2. $H_1^{k,p}(\Omega\cup\Gamma)$ Spaces......Page 83
1.7.3. Boundary Values and $H{1/p',p}(\Gamma)$ Spaces......Page 85
1.7.4. Supplementary Results......Page 90
1.8.1. Some Notions from the Abstract Theory of Ordered Linear Spaces......Page 92
1.8.2. Inequalities and Lattice Properties in Function Spaces over $\Omega$......Page 95
1.8.3. Boundary Inequalities......Page 98
Problems......Page 101
2. The Variational Theory of Elliptic Boundary Value Problems......Page 105
2.1. Abstract Existence and Uniqueness Results......Page 107
2.2.1. Bilinear Forms......Page 110
2.2.2. The Weak Maximum Principle......Page 112
2.2.3. Interpretation of Solutions......Page 115
2.3. $L^s$ Regularity of Solutions......Page 118
2.4.1. Pointwise Bounds on Subsolutions......Page 126
2.4.2. Hoelder Continuity of Solutions......Page 129
2.4.3. $L^{2,\mu}$ Regularity of First Derivatives......Page 133
2.5.1. Regularity in the Interior......Page 141
2.5.2. Boundary and Global Regularity......Page 145
2.6. Interior Regularity for Nonlinear Equations......Page 149
2.6.1. Local Boundedness......Page 150
2.6.2. $H^2$ Regularity......Page 154
2.6.3. $H^{1,\infty}$ and $C^{k,\delta}$ Regularity......Page 156
Problems......Page 159
3. $H^{k,p}$ and $C^{k,\delta}$ Theory......Page 161
3.1.1. Homogeneous Equations with Constant Coefficients......Page 162
3.1.2. Nonhomogeneous Equations with Variable Coefficients......Page 166
3.2.1. Regularity of First Derivatives......Page 168
3.2.2. Regularity of Second Derivatives......Page 173
3.3. Interior $L^p$ Regularity of Derivatives......Page 176
3.4.1. Homogeneous Equations with Constant Coefficients......Page 181
3.4.2. Nonhomogeneous Equations with Variable Coefficients......Page 186
3.5.1. $L^{2,\mu}$ Regularity near the Boundary......Page 188
3.5.2. $L^p$ Regularity near the Boundary......Page 193
3.5.3. Global Regularity......Page 195
3.6.1. The Case of Smooth Coefficients......Page 196
3.6.2. The General Case......Page 199
3.7.1. Regularity of Solutions......Page 203
3.7.2. Maximum Principles......Page 205
3.7.3. Existence and Uniqueness......Page 210
3.8. The Marcinkiewicz Theorem and the John-Nirenberg Lemma......Page 212
Problems......Page 217
4. Variational Inequalities......Page 221
4.1.1. A Class of Minimum Problems......Page 222
4.1.2. Variational Inequalities......Page 225
4.2. Variational Inequalities for Nonlinear Operators......Page 231
4.2.1. Monotone and Pseudomonotone Operators......Page 232
4.2.2. Existence and Approximation of Solutions......Page 238
4.3.1. Convex Sets......Page 243
4.3.2. Bilinear Forms and Nonlinear Operators......Page 244
4.3.3. Interpretation of Solutions......Page 253
4.4. Existence and Uniqueness Results for a Class of Noncoercive Bilinear Forms......Page 257
4.4.1. Unilateral Variational Inequalities......Page 258
4.4.2. Bilateral Variational Inequalities......Page 263
4.5. Lewy-Stampacchia Inequalities and Applications to Regularity......Page 265
4.6.1. $H^{2,\infty}$ Regularity......Page 271
4.6.2. $H^2$ Regularity up to $\Gamma$ under General Conditions......Page 278
4.7.1. The Case of Continuous Leading Coefficients......Page 282
4.7.2. The Case of Holderian Leading Coefficients......Page 286
4.7.3. The Case of Discontinuous Leading Coefficients......Page 288
4.8. Lipschitz Regularity by the Penalty Method......Page 291
4.9. Problems Involving Natural Growth of Nonlinear Terms......Page 295
Problems......Page 304
5. Nonvariational Obstacle Problems......Page 307
5.1.1. Bilateral Problems......Page 308
5.1.2. Unilateral Problems......Page 310
5.1.3. An Approximation Result......Page 315
5.1.4. Systems of Unilateral Problems......Page 317
5.2. Differential Inequalities......Page 321
5.2.1. Interpolation Results......Page 322
5.2.2. A Global Bound......Page 324
5.2.3. A Local Bound......Page 328
5.3.1. Existence......Page 332
5.3.2. Uniqueness......Page 337
5.4.1. Generalized Solutions......Page 339
5.4.2. Implicit Unilateral Problems......Page 342
5.4.3. The Implicit Unilateral Problem of Stochastic Impulse Control......Page 345
Problems......Page 349
Bibliographical Notes......Page 351
References......Page 357
Index of Special Symbols and Abbreviations......Page 365
Index......Page 367