變分法(第4版)（簡體書）

本書是《變分法》第四版，主要講述在非線性偏微分方程和哈密頓系統中的應用，繼第一版出版十八年再次全新呈現。整本書都做了大量的修改，僅500多條參考書目就將其價值大大提升。第四版中主要講述變分微積分，增加了該領域的最新進展。這也是一部變分法學習的教程，特別講述了Yamabe流的收斂和脹開現象以及最新研究發現的調和映射和曲面中熱流的向後小泡形成。目次：本書分為三部分，第一部分主要闡述變分法中的經典直接法及其擴張，第二部分討論極小極大化方法，第三部分介紹變分法中的新理論。
讀者對象：適合研究微分幾何和偏微分方程的學生、老師和科研工作者。

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

Almost twenty years after conception of the first edition， it was a challenge to prepare an updated version of this text on the Calculus of Variations. The field has truely advanced dramatically since that time， to an extent that I find it impossible to give a comprehensive account of all the many important developments that have occurred since the last edition appeared. Fortunately， an excellent overview of the most significant results， with a focus on functional analytic and Morse theoretical aspects of the Calculus of Variations， can be found in the recent survey paper by Ekeland-Ghoussoub [1]. I therefore haveonly added new material directly related to the themes originally covered.
Even with this restriction， a selection had to be made. In view of the fact that flow methods are emerging as the natural tool for studying variational problems in the field of Geometric Analysis， an emphasis was placed on advances in this domain. In particular， the present edition includes the proof for the convergence of the Yamabe flow on an arbitrary closed manifold of dimension 3 m 5 for initial data allowing at most single-point blow-up.Moreover， we give a detailed treatment of the phenomenon of blow-up and discuss the newly discovered results for backward bubbling in the heat flow for harmonic maps of surfaces.
Aside from these more significant additions， a number of smaller changes have been made throughout the text， thereby taking care not to spoil the freshness of the original presentation. References have been updated， whenever possible， and several mistakes that had survived the past revisions have now been eliminated. I would like to thank Silvia Cingolani， Irene Fouseca， Emmanuel Hebey， and Maximilian Schultz for helpful comments in this regard. Moreover，I am indebted to Gilles Angelsberg， Ruben Jakob， Reto Miiller， and Melanie Rupfiin， for carefully proof-reading the new material.
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