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Chapter Ⅰ．The Direct Methods in the Calculus of Variations 1

1．Lower Semi-continuity 2

Degenerate Elliptic Equations 4

Minimal Partitioning Hypersurfaces 6

Minimal Hypersurfaces in Riemannian Manifolds 7

A General Lower Semi-continuity Result 8

2．Constraints 13

Semilinear Elliptic Boundary Value Problems 14

Perron's Method in a Variational Guise 16

The Classical Plateau Problem 19

3．Compensated Compactness 25

Applications in Elasticity 29

Convergence Results for Nonlinear Elliptic Equations 32

Hardy Space Methods 35

4．The Concentration-Compactness Principle 36

Existence of Extremal Functions for Sobolev Embeddings 42

5．Ekeland's Variational Principle 51

Existence of Minimizers for Quasi-convex Functionals 54

6．Duality 58

Hamiltonian Systems 60

Periodic Solutions of Nonlinear Wave Equations 65

7．Minimization Problems Depending on Parameters 69

Harmonic Maps with Singularities 71

Chapter Ⅱ．Minimax Methods 74

1．The Finite Dimensional Case 74

2．The Palais-Smale Condition 77

3．A General Deformation Lemma 81

Pseudo-gradient Flows on Banach Spaces 81

Pseudo-gradient Flows on Manifolds 85

4．The Minimax Principle 87

Closed Geodesics on Spheres 89

5．Index Theory 94

Krasnoselskii Genus 94

Minimax Principles for Even Functionals 96

Applications to Semilinear Elliptic Problems 98

General Index Theories 99

Ljusternik-Schnirelman Category 100

A Geometrical S1-Index 101

Multiple Periodic Orbits of Hamiltonian Systems 103

6．The Mountain Pass Lemma and its Variants 108

Applications to Semilinear Elliptic Boundary Value Problems 110

The Symmetric Mountain Pass Lemma 112

Application to Semilinear Equa-tions with Symmetry 116

7．Perturbation Theory 118

Applications to Semilinear Elliptic Equations 120

8．Linking 125

Applications to Semilinear Elliptic Equations 128

Applications to Hamil-tonian Systems 130

9．Parameter Dependence 137

10．Critical Points of Mountain Pass Type 143

Multiple Solutions of Coercive Elliptic Problems 147

11．Non-differentiable Functionals 150

12．Ljusternik-Schnirelman Theory on Convex Sets 162

Applications to Semilinear Elliptic Boundary Value Problems 166

Chapter Ⅲ．Limit Cases of the Palais-Smale Condition 169

1．Poho？aev's Non-existence Result 170

2．The Brezis-Nirenberg Result 173

Constrained Minimization 174

The Unconstrained Case:Local Compact-ness 175

Multiple Solutions 180

3．The Effect of Topology 183

A Global Compactness Result 184

Positive Solutions on Annular-Shaped Regions 190

4．The Yamabe Problem 194

The Variational Approach 195

The Locally Conformally Flat Case 197

The Yamabe Flow 198

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

Convergence of the Yamabe Flow in the General Case 204

The Compact Case u∞＞0 211

Bubbling:The Case u∞？0 216

5．The Dirichlet Problem for the Equation of Constant Mean Curvature 220

Small Solutions 221

The Volume Functional 223

Wente's Uniqueness Result 225

Local Compactness 226

Large Solutions 229

6．Harmonic Maps of Riemannian Surfaces 231

The Euler-Lagrange Equations for Harmonic Maps 232

Bochner identity 234

The Homotopy Problem and its Functional Analytic Setting 234

Existence and Non-existence Results 237

The Heat Flow for Harmonic Maps 238

The Global Existence Result 239

The Proof of Theorem 6.6 242

Finite-Time Blow-Up 253

Reverse Bubbling and Nonuniqueness 257

Appendix A 263

Sobolev Spaces 263

H？lder Spaces 264

Imbedding Theorems 264

Density Theorem 265

Trace and Extension Theorems 265

Poincaré Inequality 266

Appendix B 268

Schauder Estimates 268

Lp-Theory 268

Weak Solutions 269

A Reg-ularity Result 269

Maximum Principle 271

Weak Maximum Principle 272

Application 273

Appendix C 274

Fréchet Differentiability 274

Natural Growth Conditions 276

References 277

Index 301