# 偏微分方程导论（第2版）

2011-01

ISBN: 9787510029714

• 《偏微分方程导论（第2版）》是一部数学专业研究生的偏微分方程教程。其旨在让读者更好地了解偏微分方程的经典基础结果，为读者更深层次学习这方面的专著和教程提供现代理论观点。这是第二版，较第一版增加了不少练习，专门增加了一章讲述拟微分算子，增加了不少材料，内容更加丰富。书中的前五章讲述经典理论，如一阶方程，局部存在性定理，数学物理基础偏微分方程，适时地运用现代物理技巧解释长期研究的话题。最后三章专注于现代理论，索伯列夫空间，椭?边界值问题和拟微分算子。 chapter0
preliminaries
a.notationsanddefinitions
c.convolutions
d.thefouriertransform
e.distributions
f.compactoperators
chapter1
localexistencetheory
a.basicconcepts
b.realfirstorderequations
c.thegeneralcauchyproblem
d.thecauchy-kowalevskitheorem
e.localsolvability：thelewyexample
f.constant-coeffcientoperators：fundamentalsolutions

chapter2
thelaplaceoperator
a.symmetrypropertiesofthelaplacian
b.basicpropertiesofharmonicfunctions
c.thefundamentalsolution
d.thedirichletandneumannproblems
e.thegreensfunction
f.dirichletsprinciple
g.thedirichletprobleminahalf-space
h.thedirichletprobleminaball

chapter3
layerpotentials
a.thesetup
b.integraloperators
c.doublelayerpotentials
d.singlelayerpotentials
e.solutionoftheproblems
f.furtherremarks

chapter4
theheatoperator
a.thegaussiankernel
b.functionsofthelaplacian
c.theheatequationinboundeddomains

chapter5
thewaveoperator
a.thecauchyproblem
b.solutionofthecauchyproblem
c.theinhomogeneousequation
d.fourieranalysisofthewaveoperator
e.thewaveequationinboundeddomains

chapter6
thel2theoryofderivatives
a.sobolevspacesonr
b.furtherresultsonsobolevspaces
c.localregularityofellipticoperators
d.constant-coefficienthypoellipticoperators
e.sobolevspacesonboundeddomains

chapter7
ellipticboundaryvalueproblems
a.strongellipticity
b.onintegrationbyparts
c.dirichletformsandboundaryconditions
d.thecoerciveestimate
e.existence，uniqueness，andeigenvalues
f.regularityattheboundary：thesecondordercase
g.furtherresultsandtechniques
h.epilogue：thereturnofthegreensfunction

chapter8
pseudodifferentialoperators
a.basicdefinitionsandproperties
b.kernelsofpseudodifferentialoperators
c.asymptoticexpansionsofsymbols
e.sobolevestimates
f.ellipticoperators
g.introductiontomicrolocalanalysis
h.changeofcoordinates
bibliography
indexofsymbols
index
• ##### 内容简介:
《偏微分方程导论（第2版）》是一部数学专业研究生的偏微分方程教程。其旨在让读者更好地了解偏微分方程的经典基础结果，为读者更深层次学习这方面的专著和教程提供现代理论观点。这是第二版，较第一版增加了不少练习，专门增加了一章讲述拟微分算子，增加了不少材料，内容更加丰富。书中的前五章讲述经典理论，如一阶方程，局部存在性定理，数学物理基础偏微分方程，适时地运用现代物理技巧解释长期研究的话题。最后三章专注于现代理论，索伯列夫空间，椭?边界值问题和拟微分算子。
• ##### 目录:
chapter0
preliminaries
a.notationsanddefinitions
c.convolutions
d.thefouriertransform
e.distributions
f.compactoperators
chapter1
localexistencetheory
a.basicconcepts
b.realfirstorderequations
c.thegeneralcauchyproblem
d.thecauchy-kowalevskitheorem
e.localsolvability：thelewyexample
f.constant-coeffcientoperators：fundamentalsolutions

chapter2
thelaplaceoperator
a.symmetrypropertiesofthelaplacian
b.basicpropertiesofharmonicfunctions
c.thefundamentalsolution
d.thedirichletandneumannproblems
e.thegreensfunction
f.dirichletsprinciple
g.thedirichletprobleminahalf-space
h.thedirichletprobleminaball

chapter3
layerpotentials
a.thesetup
b.integraloperators
c.doublelayerpotentials
d.singlelayerpotentials
e.solutionoftheproblems
f.furtherremarks

chapter4
theheatoperator
a.thegaussiankernel
b.functionsofthelaplacian
c.theheatequationinboundeddomains

chapter5
thewaveoperator
a.thecauchyproblem
b.solutionofthecauchyproblem
c.theinhomogeneousequation
d.fourieranalysisofthewaveoperator
e.thewaveequationinboundeddomains

chapter6
thel2theoryofderivatives
a.sobolevspacesonr
b.furtherresultsonsobolevspaces
c.localregularityofellipticoperators
d.constant-coefficienthypoellipticoperators
e.sobolevspacesonboundeddomains

chapter7
ellipticboundaryvalueproblems
a.strongellipticity
b.onintegrationbyparts
c.dirichletformsandboundaryconditions
d.thecoerciveestimate
e.existence，uniqueness，andeigenvalues
f.regularityattheboundary：thesecondordercase
g.furtherresultsandtechniques
h.epilogue：thereturnofthegreensfunction

chapter8
pseudodifferentialoperators
a.basicdefinitionsandproperties
b.kernelsofpseudodifferentialoperators
c.asymptoticexpansionsofsymbols
e.sobolevestimates
f.ellipticoperators
g.introductiontomicrolocalanalysis
h.changeofcoordinates
bibliography
indexofsymbols
index

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