变分法 原书第4版 英文版【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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Chapter Ⅰ．The Direct Methods in the Calculus of Variations1

1．Lower Semi-continuity2

Degenerate Elliptic Equations4

Minimal Partitioning Hypersurfaces6

Minimal Hypersurfaces in Riemannian Manifolds7

A General Lower Semi-continuity Result8

2．Constraints13

Semilinear Elliptic Boundary Value Problems14

Perron's Method in a Variational Guise16

The Classical Plateau Problem19

3．Compensated Compactness25

Applications in Elasticity29

Convergence Results for Nonlinear Elliptic Equations32

Hardy Space Methods35

4．The Concentration-Compactness Principle36

Existence of Extremal Functions for Sobolev Embeddings42

5．Ekeland's Variational Principle51

Existence of Minimizers for Quasi-convex Functionals54

6．Duality58

Hamiltonian Systems60

Periodic Solutions of Nonlinear Wave Equations65

7．Minimization Problems Depending on Parameters69

Harmonic Maps with Singularities71

Chapter Ⅱ．Minimax Methods74

1．The Finite Dimensional Case74

2．The Palais-Smale Condition77

3．A General Deformation Lemma81

Pseudo-gradient Flows on Banach Spaces81

Pseudo-gradient Flows on Manifolds85

4．The Minimax Principle87

Closed Geodesics on Spheres89

5．Index Theory94

Krasnoselskii Genus94

Minimax Principles for Even Functionals96

Applications to Semilinear Elliptic Problems98

General Index Theories99

Ljusternik-Schnirelman Category100

A Geometrical S1-Index101

Multiple Periodic Orbits of Hamiltonian Systems103

6．The Mountain Pass Lemma and its Variants108

Applications to Semilinear Elliptic Boundary Value Problems110

The Symmetric Mountain Pass Lemma112

Application to Semilinear Equa-tions with Symmetry116

7．Perturbation Theory118

Applications to Semilinear Elliptic Equations120

8．Linking125

Applications to Semilinear Elliptic Equations128

Applications to Hamil-tonian Systems130

9．Parameter Dependence137

10．Critical Points of Mountain Pass Type143

Multiple Solutions of Coercive Elliptic Problems147

11．Non-differentiable Functionals150

12．Ljusternik-Schnirelman Theory on Convex Sets162

Applications to Semilinear Elliptic Boundary Value Problems166

Chapter Ⅲ．Limit Cases of the Palais-Smale Condition169

1．Poho？aev's Non-existence Result170

2．The Brezis-Nirenberg Result173

Constrained Minimization174

The Unconstrained Case:Local Compact-ness175

Multiple Solutions180

3．The Effect of Topology183

A Global Compactness Result184

Positive Solutions on Annular-Shaped Regions190

4．The Yamabe Problem194

The Variational Approach195

The Locally Conformally Flat Case197

The Yamabe Flow198

The Proof of Theorem 4.9(following Ye[1])200

Convergence of the Yamabe Flow in the General Case204

The Compact Case u∞＞0211

Bubbling:The Case u∞？0216

5．The Dirichlet Problem for the Equation of Constant Mean Curvature220

Small Solutions221

The Volume Functional223

Wente's Uniqueness Result225

Local Compactness226

Large Solutions229

6．Harmonic Maps of Riemannian Surfaces231

The Euler-Lagrange Equations for Harmonic Maps232

Bochner identity234

The Homotopy Problem and its Functional Analytic Setting234

Existence and Non-existence Results237

The Heat Flow for Harmonic Maps238

The Global Existence Result239

The Proof of Theorem 6.6242

Finite-Time Blow-Up253

Reverse Bubbling and Nonuniqueness257

Appendix A263

Sobolev Spaces263

H？lder Spaces264

Imbedding Theorems264

Density Theorem265

Trace and Extension Theorems265

Poincaré Inequality266

Appendix B268

Schauder Estimates268

Lp-Theory268

Weak Solutions269

A Reg-ularity Result269

Maximum Principle271

Weak Maximum Principle272

Application273

Appendix C274

Fréchet Differentiability274

Natural Growth Conditions276

References277

Index301