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
Semi-Linear 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 57
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 Equations 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 193
5．The Dirichlet Problem for the Equation of Constant Mean Curvature 203
Small Solutions 204
The Volume Functional 206
Wente's Uniqueness Result 208
Local Compactness 209
Large Solutions 212
6．Harmonic Maps of Riemannian Surfaces 214
The Euler-Lagrange Equations for Harmonic Maps 215
Bochner identity 217
The Homotopy Problem and its Functional Analytic Setting 217
Existence and Non-Existence Results 220
The Evolution of Harmonic Maps 221
Appendix A 237
Sobolev Spaces 237
H？lder Spaces 238
Imbedding Theorems 238
Density Theorem 239
Trace and Extension Theorems239—Poincaré Inequality 240
Appendix B 242
Schauder Estimates 242
LP-Theory 242
Weak Solutions 243
A Reg-ularity Result 243
Maximum Principle 245
Weak Maximum Principle 246
Application 247
Appendix C 248
Fréchet Differentiability 248
Natural Growth Conditions 250
References 251
Index 271