Chapter Ⅰ．The Direct Methods in the Calculus of Variations

1．Lower Semi-Continuity

Degenerate Elliptic Equations

Minimal Partitioning Hypersurfaces

Minimal Hypersurfaces in Riemannian Manifolds

A General Lower Semi-Continuity Result

2．Constraints

Semi-Linear Elliptic Boundary Value Problems

Perron's Method in a Variational Guise

The Classical Plateau Problem

3．Compensated Compactness

Applications in Elasticity

Convergence Results for Nonlinear Elliptic Equations

Hardy space methods

4．The Concentration-Compactness Principle

Existence of Extremal Functions for Sobolev Embeddings

5．Ekeland's Variational Principle

Existence of Minimizers for Quasi-Convex Functionals

6．Duality

Hamiltonian Systems

Periodic Solutions of Nonlinear Wave-Equations

7．Minimization Problems Depending on Parameters

Harmonic maps with singularities

Chapter Ⅱ．Minimax Methods

1．The Finite Dimensional Case

2．The Palais-Smale Condition

3．A General Deformation Lemma

Pseudo-Gradient Flows on Banach Spaces

Pseudo-Gradient Flows on Manifolds

4．The Minimax Principle

Closed Geodesics on Spheres

5．Index Theory

Krasnoselskii Genus

Minimax Principles for Even Functionals

Applications to Semilinear Elliptic Problems

General Index Theories

Ljusternik-Schnirelman Category

A Geometrical S1-Index

Multiple Periodic Orbits of Hamiltonian Systems

6．The Mountain Pass Lemma and its Variants

Applications to Semilinear Elliptic Boundary Value Problems

The Symmetric Mountain Pass Lemma

Application to Semilinear Equations with Symmetry

7．Perturbation Theory

Applications to Semilinear Elliptic Equations

8．Linking

Applications to Semilinear Elliptic Equations

Applications to Hamil-tonian Systems

9．Parameter Dependence

10．Critical Points of Mountain Pass Type

Multiple Solutions of Coercive Elliptic Problems

11．Non-Differentiable Functionals

12．Ljusternik-Schnirelman Theory on Convex Sets

Applications to Semilinear Elliptic Boundary Value Problems

Chapter Ⅲ．Limit Cases of the Palais-Smale Condition

1．Poho？aev's Non-Existence Result

2．The Brezis-Nirenberg Result

Constrained Minimization

The Unconstrained Case:Local Compact-ness

Multiple Solutions

3．The Effect of Topology

A Global Compactness Result

Positive Solutions on Annular-Shaped Regions

4．The Yamabe Problem

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

Small Solutions

The Volume Functional

Wente's Uniqueness Result

Local Compactness

Large Solutions

6．Harmonic Maps of Riemannian Surfaces

The Euler-Lagrange Equations for Harmonic Maps

Bochner identity

The Homotopy Problem and its Functional Analytic Setting

Existence and Non-Existence Results

The Evolution of Harmonic Maps

Appendix A

Sobolev Spaces

H？lder Spaces

Imbedding Theorems

Density Theorem

Trace and Extension Theorems239—Poincaré Inequality

Appendix B

Schauder Estimates

LP-Theory

Weak Solutions

A Reg-ularity Result

Maximum Principle

Weak Maximum Principle

Application

Appendix C

Fréchet Differentiability

Natural Growth Conditions

References

Index