變分法

ChapterI.TheDirectMethodsintheCalculusofVariations
1.LowerSemi-Continuity
DegenerateEllipticEquations,4--MinimalPartitioningHypersurfaces,6
--MinimalHypersurfacesinRiemannianManifolds,7--AGeneralLower
Semi-ContinuityResult,8
2.Constraints
Semi-LinearEllipticBoundaryValueProblems,14--Perron'sMethodina
VariationalGuise,16--TheClassicalPlateauProblem,19
3.CompensatedCompactness
ApplicationsinElasticity,29--ConvergenceResultsforNonlinearElliptic
Equations,32--Hardyspacemethods,35
4.TheConcentration-CompactnessPrinciple
ExistenceofExtremalFunctionsforSobolevEmbeddings,42
5.Ekeland'sVariationalPrinciple
ExistenceofMinimizersforQuasi-ConvexFunctionals,54
6.Duality
HamiltonianSystems,60--PeriodicSolutionsofNonlinearWave-Equations,
65
7.MinimizationProblemsDependingonParameters
Harmonicmapswithsingularities,71
ChapterII.MinimaxMethods
1.TheFiniteDimensionalCase
2.ThePalais-SmaleCondition
3.AGeneralDeformationLemma
Pseudo-GradientFlowsonBanachSpaces,81--Pseudo-GradientFlowson
Manifolds,85
4.TheMinimaxPrinciple
ClosedGeodesicsonSpheres,89
5.IndexTheory
KrasnoselskiiGenus,94--MinimaxPrinciplesforEvenFunctionals,96--
ApplicationstoSemilinearEllipticProblems,98--GeneralIndexTheories,
99--Ljusternik-SchnirelmanCategory,100--AGeometricalS1-Index,101
--MultiplePeriodicOrbitsofHamiltonianSystems,103
6.TheMountainPassLemmaanditsVariants
ApplicationstoSemilinearEllipticBoundaryValueProblems,110--The
SymmetricMountainPassLemma,112--ApplicationtoSemilinearEquations
withSymmetry,116
7.PerturbationTheory
ApplicationstoSemilinearEllipticEquations,120
8.Linking
ApplicationstoSemilinearEllipticEquations.128--ApplicationstoHamil-
tonianSystems,130
9.ParameterDependence
10.CriticalPointsofMountainPassType
MultipleSolutionsofCoerciveEllipticProblems.147
11.Non-DifferentiableFunctionals
12.Ljusternik-SchnirelmanTheoryonConvexSets
ApplicationstoSemilinearEllipticBoundaryValueProblems.166
ChapterIII.LimitCasesofthePalais-SmaleCondition
1.Pohozaev'sNon-ExistenceResult
2.TheBrezis-NirenbergResult
ConstrainedMinimization,174--TheUnconstrainedCase:LocalCompact-
ness,175--MultipleSolutions,180
3.TheEffectofTopology
AGlobalCompactnessResult,184--PositiveSolutionsonAnnular-Shaped
Regions,190
4.TheYamabeProblem
5.TheDirichletProblemfortheEquationofConstantMeanCurvature
SmallSolutions,204--TheVolumeFunctional,206--Wente'sUniqueness
Result,208--LocalCompactness,209--LargeSolutions,212
6.HarmonicMapsofRiemannianSurfaces
TheEuler-LagrangeEquationsforHarmonicMaps,215--Bochneridentity,
217--TheHomotopyProblemanditsFunctionalAnalyticSetting,217--
ExistenceandNon-ExistenceResults,220--TheEvolutionofHarmonic
Maps.221
AppendixA
SobolevSpaces,237--H61derSpaces,238--ImbeddingTheorems,238--
DensityTheorem,239--TraceandExtensionTheorems,239--Poincare
Inequality,240
AppendixB
SchauderEstimates,242--LP-Theory,242--WeakSolutions.243--AReg-
ularityResult.243--MaximumPrinciple,245--WeakMaximumPrinciple,
246--Application.247
AppendixC
FrechetDifferentiability,248--NaturalGrowthConditions,250
References
Index