偏微分方程

內(nèi)容概要

　　《偏微分方程(第1卷)》是一部兩卷集的偏微分方程教材。多變量橢圓，拋物和雙曲方程是研究的主要對象，解決了PDE和多變量方法之間的關(guān)系。第一卷中集中研究了流形上的積分和微分，泛函解析基礎(chǔ)，映射的Brouwer度，廣義解析函數(shù)和圓周同調(diào)這些議題，在這一卷中通過積分表示論解決偏微分方程問題，第二卷中講述函數(shù)解析解法。書中各章的獨立性較強，有一定偏微分方程基本知識的讀者可以獨立閱讀各章。
　　

作者簡介

作者：(德國)索維尼 (Friedrich Sauvigny)

書籍目錄

i differentiation and integration on manifolds
 §1 the weierstraβ approximation theorem
 §2 parameter-invariant integrals and differential
forms
 §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's
integral
 §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 weak
convergence
 §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 residue
theorem
 §5 the inhomogeneous cauchy-riemann differential
equation
 §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 differential
equations
 §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 an
initial-boundary-value problem
 §8 classification, transformation and reduction of
partialdifferential equations
 §9 some historical notices to the chapters v and vi
references
index

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