内容简介

  《偏微分方程(第1卷)(第2版)》是一套3卷集经典名著,第一版曾影印出版,广受好评。第2版新增内容312页(3卷),这是第1卷。本卷在引入连续统力学、电磁学和复分析和实例的基础上,介绍了许多解决实际问题的方法,如傅里叶分析、分布理论和索伯列夫空间,这些方法可用于解决线性偏微分方程的基本问题。书中涉及的线性偏微分方程有拉普拉斯方程、热方程、波动方程、一般椭圆方程、双曲方程和抛物方程等。目次:偏微分方程和向量场基本理论;拉普拉斯方程和波动方程;傅里叶分析、分布函数和常系数线性偏微分方程;索伯列夫空间;线性椭圆方程;线性发展方程;泛函分析概述;流形、向量丛和李群。

目录

ContentsofVolumesIIandIII
Preface
1BasicTheoryofODEandVectorFields
1Thederivative
2FundamentallocalexistencetheoremforODE
3Inversefunctionandimplicitfunctiontheorems
4Constant-coefficientlinearsystems;exponentiationofmatrices
5Variable-coefficientlinearsystemsofODE:Duhamelsprinciple
6Dependenceofsolutionsoninitialdataandonotherparameters
7Flowsandvectorfields
8Liebrackets
9Commutingflows;Frobeniusstheorem
10Hamiltoniansystems
11Geodesics
12Variationalproblemsandthestationaryactionprinciple
13DifferentialformsN
14Thesymplecticformandcanonicaltransformations
15First-orderscalarnonlinearPDE
16Completelyintegrablehamiltoniansystems
17Examplesofintegrablesystems;centralforceproblems
18Relativisticmotion
19Topologicalapplicationsofdifferentialforms
20Criticalpointsandindexofavectorfield
ANonsmoothvectorfields
References

2TheLaplaceEquationandWaveEquation
1Vibratingstringsandmembranes
2Thedivergenceofavectorfield
3Thecovariantderivativeanddivergenceoftensorfields
4TheLaplaceoperatoronaRiemannianmanifold
5Thewaveequationonaproductmanifoldandenergyconservation
6Uniquenessandfinitepropagationspeed
7Lorentzmanifoldsandstress-energytensors
8Moregeneralhyperbolicequations;energyestimates
9ThesymbolofadifferentialoperatorandageneralGreen-Stokesformula
10TheHodgeLaplacianonk-forms
11Maxwellsequations
References

3FourierAnalysisDistributionsandConstant-CoefficientLinearPDE
1Fourierseries
2Harmonicfunctionsandholomorphicfunctionsintheplane
3TheFouriertransform
4Distributionsandtempereddistributions
5Theclassicalevolutionequations
6RadialdistributionspolarcoordinatesandBesselfunctions
7ThemethodofimagesandPoissonssummationformula
8Homogeneousdistributionsandprincipalvaluedistributions
9Ellipticoperators
10Localsolvabilityofconstant-coefficientPDE
11ThediscreteFouriertransform
12ThefastFouriertransform
AThemightyGaussianandthesublimegammafunction
References

4SobolevSpaces
1SobolevspacesonRn
2Thecomplexinterpolationmethod
3Sobolevspacesoncompactmanifolds
4Sobolevspacesonboundeddomains
5TheSobolevspacesH50(Ω)
6TheSchwartzkerneltheorem
7Sobolevspacesonroughdomains
References

5LinearEllipticEquations
1ExistenceandregularityofsolutionstotheDirichletproblem
2Theweakandstrongmaximumprinciples
3TheDirichletproblemontheba
4TheRiemannmappingtheorem(smoothboundary)
5TheDirichletproblemonadomainwitharoughboundary
6TheRiemannmappingtheorem(roughboundary)
7TheNeumannboundaryproblem
8TheHodgedecompositionandharmonicforms
9NaturalboundaryproblemsfortheHodgeLaplacian
10Isothermalcoordinatesandconformalstructuresonsurfaces
11Generalellipticboundaryproblems
12Operatorpropertiesofregularboundaryproblems
……
6LinearEvolutionEquations
AOutlineofFunctionaIAnalysis
BMarufoldsVectorBundlesandLieGroups
……

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