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
內容試閱：
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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