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变分法【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

变分法
  • M.Struwe著 著
  • 出版社： 北京;西安：世界图书出版公司
  • ISBN：7506247100
  • 出版时间：2000
  • 标注页数：272页
  • 文件大小：39MB
  • 文件页数：290页
  • 主题词：

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图书目录

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

Semi-Linear 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．Duality57

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 Equations 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 Problem193

5．The Dirichlet Problem for the Equation of Constant Mean Curvature203

Small Solutions204

The Volume Functional206

Wente's Uniqueness Result208

Local Compactness209

Large Solutions212

6．Harmonic Maps of Riemannian Surfaces214

The Euler-Lagrange Equations for Harmonic Maps215

Bochner identity217

The Homotopy Problem and its Functional Analytic Setting217

Existence and Non-Existence Results220

The Evolution of Harmonic Maps221

Appendix A237

Sobolev Spaces237

H？lder Spaces238

Imbedding Theorems238

Density Theorem239

Trace and Extension Theorems239—Poincaré Inequality240

Appendix B242

Schauder Estimates242

LP-Theory242

Weak Solutions243

A Reg-ularity Result243

Maximum Principle245

Weak Maximum Principle246

Application247

Appendix C248

Fréchet Differentiability248

Natural Growth Conditions250

References251

Index271

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