偏微分方程（第1卷）

　　Introduction 1 Basic Theory of ODE and Vector Fields Introduction 1 The derivative 2 Fundamental local existence theorem for ODE 3 Inverse function and implicit function theorems 4 Constant-coefficient linear systems; exponentiation of matrices 5 Variable-coefficient linear systems of ODE： Duhamel＇s principle 6 Dependence of solutions on initial data and on other parameters 7 Flows and vector fields 8 Lie brackets 9 Commuting flows; Frobenius＇s theorem 10 Hamiltonian systems 11 Geodesics 12 Variational problems and the stationary action principle 13 Differential forms 14 The symplectic form and canonical transformations 15 First-order， scalar， nonlinear PDE

暂缺《偏微分方程（第1卷）》作者简介

ContentsofVolumesIIandIII
Introduction
1BasicTheoryofODEandVectorFields
Introduction
1Thederivative
2FundamentallocalexistencetheoremforODE
3Inversefunctionandimplicitfunctiontheorems
4Constant-coefficientlinearsystems;exponentiationofmatrices
5Variable-coefficientlinearsystemsofODE:Duhamel'sprinciple
6Dependenceofsolutionsoninitialdataandonotherparameters
7Flowsandvectorfields
8Liebrackets
9Commutingflows;Frobenius'stheorem
10Hamiltoniansystems
11Geodesics
12Variationalproblemsandthestationaryactionprinciple
13Differentialforms
14Thesymplecticformandcanonicaltransformations
15First-order,scalar,nonlinearPDE
16CompletelyintegrableHamiltoniansystems
17Examplesofintegrablesystems;centralforceproblems
18Relativisticmotion
19Topologicalapplicationsofdifferentialforms
20Criticalpointsandindexofavectorfield
ANonsmoothvectorfields
References
2TheLaplaceEquationandWaveEquation
Introduction
1Vibratingstringsandmembranes
2Thedivergenceofavectorfield
3Thecovariantderivativeanddivergenceoftensorfields
4TheLaplaceoperatoronaRiemannianmanifold
5Thewaveequationonaproductmanifoldandenergy
conservation
6Uniquenessandfinitepropagationspeed
7Lorentzmanifoldsandstress-energytensors
8Moregeneralhyperbolicequations;energyestimates
9Thesymbolofadifferentialoperatorandageneral
Green-Stokesformula
10TheHodgeLaplacianonk-forms
11Maxwell'sequations
References
FourierAnalysis,Distributions,andConstant-Coefficient
LinearPDE
Introduction
1Fourierseries
2Harmonicfunctionsandholomorphicfunctionsintheplane
3TheFouriertransform
4Distributionsandtempereddistributions
5Theclassicalevolutionequations
6Radialdistributions,polarcoordinates,andBesselfunctions
7ThemethodofimagesandPoisson'ssummationformula
8Homogeneousdistributionsandprincipalvaluedistributions
9Ellipticoperators
10Localsolvabilityofconstant-coefficientPDE
11ThediscreteFouriertransform
12ThefastFouriertransform
AThemightyGaussianandthesublimegammafunction
References
SobolevSpaces
Introduction
1SobolevspacesonR"
2Thecomplexinterpolationmethod
3Sobolevspacesoncompactmanifolds
4Sobolevspacesonboundeddomains
5TheSobolevspacesHs0()
6TheSchwartzkerneltheorem
References
5LinearEllipticEquations
Introduction
1ExistenceandregularityofsolutionstotheDirichletproblem
2Theweakandstrongmaximumprinciples
3TheDirichletproblemontheballinR"
4TheRiemannmappingtheorem(smoothboundary)
5TheDirichletproblemonadomainwitharoughboundary
6TheRiemannmappingtheorem(roughboundary)
7TheNeumannboundaryproblem
8TheHodgedecompositionandharmonicforms
9NaturalboundaryproblemsfortheHodgeLaplacian
10Isothermalcoordinatesandconformalstructuresonsurfaces
11Generalellipticboundaryproblems
12Operatorpropertiesofregularboundaryproblems
ASpacesofgeneralizedfunctionsonmanifoldswithboundary
BTheMayer-VietorissequenceindeRhamcohomology
References
6LinearEvolutionEquations
Introduction
1Theheatequationandthewaveequationonboundeddomains
2Theheatequationandwaveequationonunboundeddomains
3Maxwell'sequations
4TheCauchy-Kowalewskytheorem
5Hyperbolicsystems
6Geometricaloptics
7Theformationofcaustics
ASomeBanachspacesofharmonicfunctions
BThestationaryphasemethod
References
AOutlineofFunctionalAnalysis
Introduction
1Banachspaces
2Hilbertspaces
3Frechetspaces;locallyconvexspaces
4Duality
5Linearoperators
6Compactoperators
7Fredholmoperators
8Unboundedoperators
9Semigroups
References
Manifolds,VectorBundles,andLieGroups
Introduction
1Metricspacesandtopologicalspaces
2Manifolds
3Vectorbundles
4Sard'stheorem
5Liegroups
6TheCampbell-Hausdorffformula
7RepresentationsofLiegroupsandLiealgebras
8RepresentationsofcompactLiegroups
9RepresentationsofSU(2)andrelatedgroups
References
Index