Variational Methods in Nonlinear Analysis: With Applications in Optimization and Partial Differential Equations / Edition 1 available in Paperback
Variational Methods in Nonlinear Analysis: With Applications in Optimization and Partial Differential Equations / Edition 1
- ISBN-10:
- 3110647362
- ISBN-13:
- 9783110647365
- Pub. Date:
- 04/06/2020
- Publisher:
- De Gruyter
- ISBN-10:
- 3110647362
- ISBN-13:
- 9783110647365
- Pub. Date:
- 04/06/2020
- Publisher:
- De Gruyter
Variational Methods in Nonlinear Analysis: With Applications in Optimization and Partial Differential Equations / Edition 1
Paperback
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Product Details
| ISBN-13: | 9783110647365 |
|---|---|
| Publisher: | De Gruyter |
| Publication date: | 04/06/2020 |
| Series: | De Gruyter Textbook |
| Pages: | 499 |
| Product dimensions: | 6.69(w) x 9.45(h) x (d) |
| Age Range: | 18 Years |
About the Author
Table of Contents
Preface IX
Notation XXI
1 Preliminaries 1
1.1 Fundamentals in the theory of Banach spaces 1
1.1.1 Linear operators and functional 1
1.1.2 The dual X* 5
1.1.3 The bidual X** 6
1.1.4 Different choices of topology on a Banach space: strong and weak topologies 8
1.1.5 The strong topology on X 9
1.1.6 The weak* topology on X* 10
1.1.7 The weak topology on X 14
1.2 Convex subsets of Banach spaces and their properties 22
1.2.1 Definitions and elementary properties 22
1.2.2 Separation of convex sets and its consequences 24
1.3 Compact operators and completely continuous operators 28
1.4 Lebesgue spaces 33
1.5 Sobolev spaces 39
1.6 Lebesgue-Bochner and Sobolev-Bochner spaces 48
1.7 Multivalued maps 52
1.7.1 Michael's selection theorem 55
1.8 Appendix 57
1.8.1 The finite intersection property 57
1.8.2 Spaces of continuous functions and the Arzelà-Ascoli theorem 57
1.8.3 Partitions of unity 58
1.8.4 Hölder continuous functions 58
1.8.5 Lebesgue points 59
2 Differentiability, convexity and optimization in Banach spaces 61
2.1 Differentiability in Banach spaces 61
2.1.1 The directional derivative 61
2.1.2 The Gâteaux derivative 61
2.1.3 The Fréchet derivative 64
2.1.4 C1 functionals 66
2.1.5 Connections between Gâteaux, Fréchet differentiability and continuity 67
2.1.6 Vector valued maps and higher order derivatives 69
2.2 General results on optimization problems 71
2.3 Convex functions 75
2.3.1 Basic definitions, properties and examples 76
2.3.2 Three important examples: the indicator, Minkowski and support Functions 79
2.3.3 Convexity and semicontinuity 80
2.3.4 Convexity and continuity 83
2.3.5 Convexity and differentiability 83
2.4 Optimization and convexity 87
2.5 Projections in Hilbert spaces 89
2.6 Geometric properties of Banach spaces related to convexity 92
2.6.1 Strictly, uniformly and locally uniformly convex Banach spaces 92
2.6.2 Convexity and the duality map 97
2.7 Appendix 101
2.7.1 Proof of Proposition 2.3.20 101
2.7.2 Proof of Proposition 2.3.21 102
3 Fixed-point theorems and their applications 105
3.1 Banach fixed-point theorem 105
3.1.1 The Banach fixed-point theorem and generalizations 105
3.1.2 Solvability of differential equations 107
3.1.3 Nonlinear integral equations 108
3.1.4 The inverse and the implicit function theorems 109
3.1.5 Iterative schemes for the solution of operator equations 114
3.2 The Brouwer fixed-point theorem and its consequences 116
3.2.1 Some topological notions 116
3.2.2 Various forms of the Brouwer fixed-point theorem 117
3.2.3 Brouwer's theorem and surjectivity of coercive maps 119
3.2.4 Application of Brouwer's theorem in mathematical economics 120
3.2.5 Failure of Brouwer's theorem in infinite dimensions 122
3.3 Schauder fixed-point theorem and Leray-Schauder alternative 123
3.3.1 Schauder fixed-point theorem 124
3.3.2 Application of Schauder fixed-point theorem to the solvability of nonlinear integral equations 125
3.3.3 The Leray-Schauder principle 127
3.3.4 Application of the Leray-Schauder alternative to nonlinear integral equations 128
3.4 Fixed-point theorems for nonexpansive maps 130
3.4.1 The Browder fixed-point theorem 131
3.4.2 The Krasnoselskii-Mann algorithm 134
3.5 A fixed-point theorem for multivalued maps 138
3.6 The Ekeland variational principle and Caristi's fixed-point theorem 140
3.6.1 The Ekeland variational principle 141
3.6.2 Caristi's fixed-point theorem 143
3.6.3 Applications: approximation of critical points 144
3.7 Appendix 145
3.7.1 The Gronwall inequality 145
3.7.2 Composition of averaged operators 145
4 Nonsmooth analysis; the subdifferential 147
4.1 The subdifferential: definition and examples 147
4.2 The subdifferential for convex functions 151
4.2.1 Existence and fundamental properties 151
4.2.2 The subdifferential and the right-hand side directional derivative 154
4.3 Subdifferential calculus 157
4.4 The subdifferential and the duality map 162
4.5 Approximation of the subdifferential and density of its domain 166
4.6 The subdifferential and optimization 168
4.7 The Moreau proximity operator in Hilbert spaces 170
4.7.1 Definition and fundamental properties 170
4.7.2 The Moreau-Yosida approximation 174
4.8 The proximity operator and numerical optimization algorithms 177
4.8.1 The standard proximal method 178
4.8.2 The forward-backward and the Douglas-Rachford scheme 184
5 Minimax theorems and duality 191
5.1 A minimax theorem 191
5.2 Conjugate functions 196
5.2.1 The Legendre-Fenchel conjugate 196
5.2.2 The biconjugate function 200
5.2.3 The subdifferential and the Legendre-Fenchel transform 204
5.3 The inf-convolution 207
5.4 Duality and optimization: Fenchel duality 212
5.5 Minimax and convex duality methods 222
5.5.1 A general framework 223
5.5.2 Applications and examples 231
5.6 Primal dual algorithms 235
5.7 Appendix 243
5.7.1 Proof of Proposition 5.1.2 243
5.7.2 Proof of Lemma 5.5.1 245
5.7.3 Proof of relation (5.61) 247
6 The calculus of variations 249
6.1 Motivation 249
6.2 Warm up: variational theory of the Laplacian 251
6.2.1 The Dirichlet functional and the Poisson equation 252
6.2.2 Regularity properties for the solutions of Poisson-type equations 257
6.2.3 Laplacian eigenvalue problems 261
6.3 Semicontinuity of integral functional 268
6.3.1 Semicontinuity in Lebesgue spaces 269
6.3.2 Semicontinuity in Sobolev spaces 273
6.4 A general problem from the calculus of variations 274
6.5 Differentiate functionals and connection with nonlinear PDEs: the Euler-Lagrange equation 276
6.6 Regularity results in the calculus of variations 279
6.6.1 The De Giorgi class 281
6.6.2 Hölder continuity of minimizers 284
6.6.3 Further regularity 288
6.7 A semilinear elliptic problem and its variational formulation 294
6.7.1 The case where sub and supersolutions exist 295
6.7.2 Growth conditions on the nonlinearity 300
6.7.3 Regularity for semilinear problems 306
6.8 A variational formulation of the p-Laplacian 307
6.8.1 The p-Laplacian Poisson equation 307
6.8.2 A quasilinear nonlinear elliptic equation involving the p-Laplacian 309
6.9 Appendix 314
6.9.1 A version of the Riemann-Lebesgue lemma 314
6.9.2 Proof of Theorem 6.6.3 315
6.9.3 Proof of Theorem 6.6.5 327
6.9.4 Proof of generalized Caccioppoli estimates 332
7 Variational inequalities 335
7.1 Motivation 335
7.2 Warm up: free boundary value problems for the Laplacian 336
7.3 The Lax-Milgram-Stampacchia theory 341
7.4 Variational inequalities of the second kind 348
7.5 Approximation methods and numerical techniques 351
7.5.1 The penalization method 351
7.5.2 Internal approximation schemes 356
7.6 Application: boundary and free boundary value problems 359
7.6.1 An important class of bilinear forms 359
7.6.2 Boundary value problems 362
7.6.3 Free boundary value problems 369
7.6.4 Semilinear variational inequalities 376
7.7 Appendix 378
7.7.1 An elementary lemma 378
8 Critical point theory 379
8.1 Motivation 379
8.2 The mountain pass and the saddle point theorems 380
8.2.1 The mountain pass theorem 380
8.2.2 Generalizations of the mountain pass theorem 381
8.2.3 The saddle point theorem 384
8.3 Applications in semilinear elliptic problems 385
8.3.1 Superlinear growth at infinity 386
8.3.2 Nonresonant semilinear problems with asymptotic linear growth at infinity and the saddle point theorem 390
8.3.3 Resonant semilinear problems and the saddle point theorem 394
8.4 Applications in quasilinear elliptic problems 399
8.4.1 The p-Laplacian and the mountain pass theorem 399
8.4.2 Resonant problems for the p-Laplacian and the saddle point theorem 404
9 Monotone-type operators 409
9.1 Motivation 409
9.2 Monotone operators 410
9.2.1 Monotone operators, definitions and examples 410
9.2.2 Local boundedness of monotone operators 411
9.2.3 Hemicontinuity and demicontinuity 413
9.2.4 Surjectivity of monotone operators and the Minty-Browder theory 414
9.3 Maximal monotone operators 417
9.3.1 Maximal monotone operators definitions and examples 417
9.3.2 Properties of maximal monotone operators 418
9.3.3 Criteria for maximal monotonicity 421
9.3.4 Surjectivity results 422
9.3.5 Maximal monotonicity of the subdifferential and the duality map 427
9.3.6 Yosida approximation and applications 429
9.3.7 Sum of maximal monotone operators 439
9.4 Pseudomonotone operators 442
9.4.1 Pseudomonotone operators, definitions and examples 442
9.4.2 Surjectivity results for pseudomonotone operators 444
9.5 Applications of monotone-type operators 446
9.5.1 Quasilinear elliptic equations 446
9.5.2 Semillnear elliptic inclusions 448
9.5.3 Variational inequalities with monotone-type operators 450
9.5.4 Gradient flows in Hubert spaces 451
9.5.5 The Cauchy problem in evolution triples 455
Bibliography 463
Index 469