非线性分析方法 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 Linearization1

1.1 Differential Calculus in Banach Spaces1

1.1.1 Frechet Derivatives and Gateaux Derivatives2

1.1.2 Nemytscki Operator7

1.1.3 High-Order Derivatives9

1.2 Implicit Function Theorem and Continuity Method12

1.2.1 Inverse Function Theorem12

1.2.2 Applications17

1.2.3 Continuity Method23

1.3 Lyapunov-Schmidt Reduction and Bifurcation30

1.3.1 Bifurcation30

1.3.2 Lyapunov-Schmidt Reduction33

1.3.3 A Perturbation Problem43

1.3.4 Gluing47

1.3.5 Transversality49

1.4 Hard Implicit Function Theorem54

1.4.1 The Small Divisor Problem55

1.4.2 Nash-Moser Iteration62

2 Fixed-Point Theorems71

2.1 Order Method72

2.2 Convex Function and Its Subdifferentials80

2.2.1 Convex Functions80

2.2.2 Subdifferentials84

2.3 Convexity and Compactness87

2.4 Nonexpansive Maps104

2.5 Monotone Mappings109

2.6 Maximal Monotone Mapping120

3 Degree Theory and Applications127

3.1 The Notion of Topological Degree128

3.2 Fundamental Properties and Calculations of Brouwer Degrees137

3.3 Applications of Brouwer Degree148

3.3.1 Brouwer Fixed-Point Theorem148

3.3.2 The Borsuk-Ulam Theorem and Its Consequences148

3.3.3 Degrees for S1Equivariant Mappings151

3.3.4 Intersection153

3.4 Leray-Schauder Degrees155

3.5 The Global Bifurcation164

3.6 Applications175

3.6.1 Degree Theory on Closed Convex Sets175

3.6.2 Positive Solutions and the Scaling Method180

3.6.3 Krein-Rutman Theory for Positive Linear Operators185

3.6.4 Multiple Solutions189

3.6.5 A Free Boundary Problem192

3.6.6 Bridging193

3.7 Extensions195

3.7.1 Set-Valued Mappings195

3.7.2 Strict Set Contraction Mappings and Condensing Mappings198

3.7.3 Fredholm Mappings200

4 Minimization Methods205

4.1 Variational Principles206

4.1.1 Constraint Problems206

4.1.2 Euler-Lagrange Equation209

4.1.3 Dual Variational Principle212

4.2 Direct Method216

4.2.1 Fundamental Principle216

4.2.2 Examples217

4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement223

4.3 Quasi-Convexity231

4.3.1 Weak Continuity and Quasi-Convexity232

4.3.2 Morrey Theorem237

4.3.3 Nonlinear Elasticity242

4.4 Relaxation and Young Measure244

4.4.1 Relaxations245

4.4.2 Young Measure251

4.5 Other Function Spaces260

4.5.1 BV Space260

4.5.2 Hardy Space and BMO Space266

4.5.3 Compensation Compactness271

4.5.4 Applications to the Calculus of Variations274

4.6 Free Discontinuous Problems279

4.6.1 Γ-convergence279

4.6.2 A Phase Transition Problem280

4.6.3 Segmentation and Mumford-Shah Problem284

4.7 Concentration Compactness289

4.7.1 Concentration Function289

4.7.2 The Critical Sobolev Exponent and the Best Constants295

4.8 Minimax Methods301

4.8.1 Ekeland Variational Principle301

4.8.2 Minimax Principle303

4.8.3 Applications306

5 Topological and Variational Methods315

5.1 Morse Theory317

5.1.1 Introduction317

5.1.2 Deformation Theorem319

5.1.3 Critical Groups327

5.1.4 Global Theory334

5.1.5 Applications343

5.2 Minimax Principles(Revisited)347

5.2.1 A Minimax Principle347

5.2.2 Category and Ljusternik-Schnirelmann Multiplicity Theorem349

5.2.3 Cap Product354

5.2.4 Index Theorem358

5.2.5 Applications363

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture371

5.3.1 Hamiltonian Operator373

5.3.2 Periodic Solutions374

5.3.3 Weinstein Conjecture376

5.4 Prescribing Gaussian Curvature Problem on S2380

5.4.1 The Conformal Group and the Best Constant380

5.4.2 The Palais-Smale Sequence387

5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2389

5.5 Conley Index Theory392

5.5.1 Isolated Invariant Set393

5.5.2 Index Pair and Conley Index397

5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension408

Notes419

References425