Nonlinear analysis - theory and methods

Contents......Page 6
Introduction......Page 8
Keywords......Page 12
 1.1 Definitions, Density, and Approximation Results......Page 13
 1.2 The One-Dimensional Case......Page 25
 1.3 Duals of Sobolev Spaces......Page 27
 1.4 Absolute Continuity on Lines, the Chain Rule  and Consequences......Page 31
 1.5 Trace Theory......Page 38
 1.6 The Extension Operator......Page 47
 1.7 The Rellich–Kondrachov Theorem......Page 50
 1.8 The Poincaré and Poincaré–Wirtinger Inequalities......Page 55
 1.9 The Sobolev Embedding Theorem......Page 57
 1.10 Capacities. Miscellaneous Results......Page 70
 1.11 Remarks......Page 78
 2.1 Compact and Completely Continuous Maps......Page 82
 2.2 Proper Maps and Gradient Maps......Page 88
 2.3 Linear Compact Operators......Page 94
 2.4 Spectral Theory of Compact Linear Operators......Page 100
 2.5 Multifunctions......Page 111
 2.6 Monotone Maps: Definition and Basic Results......Page 124
 2.7 The Subdifferential and Duality Maps......Page 129
 2.8 Surjectivity and Characterizations of Maximal Monotonicity......Page 141
 2.9 Regularizations and Linear Monotone Operators......Page 150
 2.10 Operators of Monotone Type......Page 160
 2.11 Remarks......Page 171
3 Degree Theories......Page 177
 3.1 Brouwer Degree......Page 178
 3.2 The Leray–Schauder Degree......Page 198
 3.3 Degree for Multifunctions......Page 210
 3.4 Degree for (S)+-Maps......Page 213
 3.5 Degree for Maximal Monotone Perturbation of (S)+-Maps......Page 224
 3.6 Degree for Subdifferential Operators......Page 231
 3.7 Some Generalizations......Page 245
 3.8 Index of a ξ-Point......Page 263
 3.9 Remarks......Page 267
4 Partial Order, Fixed Point Theory, Variational Principles......Page 273
 4.1 Cones and Partial Order......Page 274
 4.2 Metric Fixed Points......Page 285
 4.3 Topological Fixed Points......Page 299
 4.4 Order Fixed Points and the Fixed Point Index......Page 313
 4.5 Fixed Points for Multifunctions......Page 324
 4.6 Abstract Variational Principles......Page 329
 4.7 Young Measures......Page 350
 4.8 Remarks......Page 366
5 Critical Point Theory......Page 371
 5.1 Pseudogradients and Compactness Conditions......Page 373
 5.2 Critical Points via Minimization—The Direct Method......Page 383
 5.3 Deformation Theorems......Page 389
 5.4 Minimax Theorems......Page 407
 5.5 Critical Points Under Constraints......Page 430
 5.6 Critical Points Under Symmetries......Page 437
 5.7 The Structure of the Critical Set......Page 454
 5.8 Remarks......Page 461
6 Morse Theory and Critical Groups......Page 467
 6.1 Elements of Algebraic Topology......Page 468
 6.2 Critical Groups, Morse Relations......Page 487
 6.3 Continuity and Homotopy Invariance of Critical Groups......Page 511
 6.4 Extended Gromoll–Meyer Theory......Page 516
 6.5 Local Extrema and Critical Points of Mountain  Pass Type......Page 536
 6.6 Computation of Critical Groups......Page 540
 6.7 Existence and Multiplicity of Critical Points......Page 555
 6.8 Remarks......Page 563
 BookmarkTitle:......Page 566
Index......Page 582