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1 Linearization 1

1.1 Differential Calculus in Banach Spaces 1

1.1.1 Frechet Derivatives and Gateaux Derivatives 2

1.1.2 Nemytscki Operator 7

1.1.3 High-Order Derivatives 9

1.2 Implicit Function Theorem and Continuity Method 12

1.2.1 Inverse Function Theorem 12

1.2.2 Applications 17

1.2.3 Continuity Method 23

1.3 Lyapunov-Schmidt Reduction and Bifurcation 30

1.3.1 Bifurcation 30

1.3.2 Lyapunov-Schmidt Reduction 33

1.3.3 A Perturbation Problem 43

1.3.4 Gluing 47

1.3.5 Transversality 49

1.4 Hard Implicit Function Theorem 54

1.4.1 The Small Divisor Problem 55

1.4.2 Nash-Moser Iteration 62

2 Fixed-Point Theorems 71

2.1 Order Method 72

2.2 Convex Function and Its Subdifferentials 80

2.2.1 Convex Functions 80

2.2.2 Subdifferentials 84

2.3 Convexity and Compactness 87

2.4 Nonexpansive Maps 104

2.5 Monotone Mappings 109

2.6 Maximal Monotone Mapping 120

3 Degree Theory and Applications 127

3.1 The Notion of Topological Degree 128

3.2 Fundamental Properties and Calculations of Brouwer Degrees 137

3.3 Applications of Brouwer Degree 148

3.3.1 Brouwer Fixed-Point Theorem 148

3.3.2 The Borsuk-Ulam Theorem and Its Consequences 148

3.3.3 Degrees for S1Equivariant Mappings 151

3.3.4 Intersection 153

3.4 Leray-Schauder Degrees 155

3.5 The Global Bifurcation 164

3.6 Applications 175

3.6.1 Degree Theory on Closed Convex Sets 175

3.6.2 Positive Solutions and the Scaling Method 180

3.6.3 Krein-Rutman Theory for Positive Linear Operators 185

3.6.4 Multiple Solutions 189

3.6.5 A Free Boundary Problem 192

3.6.6 Bridging 193

3.7 Extensions 195

3.7.1 Set-Valued Mappings 195

3.7.2 Strict Set Contraction Mappings and Condensing Mappings 198

3.7.3 Fredholm Mappings 200

4 Minimization Methods 205

4.1 Variational Principles 206

4.1.1 Constraint Problems 206

4.1.2 Euler-Lagrange Equation 209

4.1.3 Dual Variational Principle 212

4.2 Direct Method 216

4.2.1 Fundamental Principle 216

4.2.2 Examples 217

4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement 223

4.3 Quasi-Convexity 231

4.3.1 Weak Continuity and Quasi-Convexity 232

4.3.2 Morrey Theorem 237

4.3.3 Nonlinear Elasticity 242

4.4 Relaxation and Young Measure 244

4.4.1 Relaxations 245

4.4.2 Young Measure 251

4.5 Other Function Spaces 260

4.5.1 BV Space 260

4.5.2 Hardy Space and BMO Space 266

4.5.3 Compensation Compactness 271

4.5.4 Applications to the Calculus of Variations 274

4.6 Free Discontinuous Problems 279

4.6.1 Γ-convergence 279

4.6.2 A Phase Transition Problem 280

4.6.3 Segmentation and Mumford-Shah Problem 284

4.7 Concentration Compactness 289

4.7.1 Concentration Function 289

4.7.2 The Critical Sobolev Exponent and the Best Constants 295

4.8 Minimax Methods 301

4.8.1 Ekeland Variational Principle 301

4.8.2 Minimax Principle 303

4.8.3 Applications 306

5 Topological and Variational Methods 315

5.1 Morse Theory 317

5.1.1 Introduction 317

5.1.2 Deformation Theorem 319

5.1.3 Critical Groups 327

5.1.4 Global Theory 334

5.1.5 Applications 343

5.2 Minimax Principles(Revisited) 347

5.2.1 A Minimax Principle 347

5.2.2 Category and Ljusternik-Schnirelmann Multiplicity Theorem 349

5.2.3 Cap Product 354

5.2.4 Index Theorem 358

5.2.5 Applications 363

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture 371

5.3.1 Hamiltonian Operator 373

5.3.2 Periodic Solutions 374

5.3.3 Weinstein Conjecture 376

5.4 Prescribing Gaussian Curvature Problem on S2 380

5.4.1 The Conformal Group and the Best Constant 380

5.4.2 The Palais-Smale Sequence 387

5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 389

5.5 Conley Index Theory 392

5.5.1 Isolated Invariant Set 393

5.5.2 Index Pair and Conley Index 397

5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension 408

Notes 419

References 425