偏微分方程（第一卷 英文版）
作者：（德）索维尼 著
出版时间：2011年版
内容简介
《偏微分方程（第1卷）》是一部两卷集的偏微分方程教材。多变量椭圆，抛物和双曲方程是研究的主要对象，解决了PDE和多变量方法之间的关系。第一卷中集中研究了流形上的积分和微分，泛函解析基础，映射的Brouwer度，广义解析函数和圆周同调这些议题，在这一卷中通过积分表示论解决偏微分方程问题，第二卷中讲述函数解析解法。书中各章的独立性较强，有一定偏微分方程基本知识的读者可以独立阅读各章。
目录
i differentiation and integration on manifolds
§1 the weierstraβ approximation theorem
§2 parameter-invariant integrals and differentialforms
§3 the exterior derivative of differential forms
§4 the stokes integral theorem for manifolds
§5 the integral theorems of gaub and stokes
§6 curvilinear integrals
§7 the lemma of poineare
§8 co-derivatives and the laplace-beltrami operator
§9 some historical notices to chapter i
ii foundations of functional analysis
§1 daniell's integral with examples
§2 extension of daniell's integral to lebesgue'sintegral
§3 measurable sets
§4 measurable functions
§5 riemann's and lebesgue's integral on rectangles
§6 banach and hilbert spaces
§7 the lebesgue spaces lp(x)
§8 bounded linear funetionals on lp(x) and weakconvergence
§9 some historical notices to chapter ii
iii brouwer's degree of mapping with geometric applications
§1 the winding number
§2 the degree of mapping in rn
§3 geometric existence theorems
§4 the index of a mapping
§5 the product theorem
§6 theorems of jordan-brouwer
iv generalized analytic functions
§1 the cauchy-riemann differential equation
§2 holomorphic functions in cn
§3 geometric behavior of holomorphic functions in c
§4 isolated singularities and the general residuetheorem
§5 the inhomogeneous cauchy-riemann differentialequation
§6 pseudoholomorphic functions
§7 conformal mappings
§8 boundary behavior of conformal mappings
§9 some historical notices to chapter iv
v potential theory and spherical harmonics
§1 poisson's differential equation in rn
§2 poisson's integral formula with applications
§3 dirichlet's problem for the laplace equation in rn
§4 theory of spherical harmonics: fourier series
§5 theory of spherical harmonics in n variables
vi linear partial differential equations in rn
§1 the maximum principle for elliptic differentialequations
§2 quasilinear elliptic differential equations
§3 the heat equation
§4 characteristic surfaces
§5 the wave equation in rn for n = 1, 3, 2
§6 the wave equation in rn for n _2
§7 the inhomogeneous wave equation and aninitial-boundary-value problem
§8 classification, transformation and reduction ofpartialdifferential equations
§9 some historical notices to the chapters v and vi
references
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