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