本书是《变分法》第4版，主要讲述在非线性偏微分方程和哈密顿系统中的应用，继第一版出版十八年再次全新呈现。整本书都做了大量的修改，仅500多条参考书目就将其价值大大提升。第四版中主要讲述变分微积分，增加了该领域的最新进展。这也是一部变分法学习的教程，特别讲述了yamabe流的收敛和胀开现象以及最新研究发现的调和映射和曲面中热流的向后小泡形成。本书由斯特沃著。

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

semilinear 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 functional

-applications to semilinear elliptic problems

-general index theories

-ljusternik-schnirelman category

-a geometrical si-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 equa- tions 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 fhnctionals

12.ljnsternik-schnirelman theory on convex sets

applications to semilinear elliptic boundary value problems

Chapter Ⅲ.Limit cases of the palais-smale condition

1.pohozaev'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, 184 -positive solutions on annular-shaped regions, 190

4.the yamabe problem

the variational approach

-the locally conformally flat case

-the yamabe flow

-the proof of theorem4.9 （following ye [1]）

-convergence of the yamabe flow in the general case

-the compact case ucc

-bubbling： the casu

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 heat flow for harmonic maps

-the global existence result

-the proof of theorem 6.6

-finite-time blow-up

-reverse bubbling and nonuniqueness

appendix a

sobolev spaces

-hslder spaces

-imbedding theorems

-density theorem

-trace and extension theorems

-poincar4 inequality

appendix b

schauder estimates

-lp-theory

-weak solutions

-areg-ularityresult

-maximum principle

-weak maximum principle

-application

appendix c

frechet differentiability

-natural growth conditions

references

index