偏微分方程（第二卷 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2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。本书是第二卷主要讲述了banach空间算子方程的可解性，hilbert空间线性算子和谱理论；线性椭圆微分方程的schauder理论；微分方程弱解；非线性偏微分方程；非线性椭圆系统和微分几何应用。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。目次：banach空间中的算子；hilbert空间线性算子；线性椭圆微分方程；非线性偏微分方程；非线性椭圆系统。读者对象：数学专业的本科生，研究生和相关的科研人员。
目录
vii operators in banach spaces
1 fixed point theorems
2 the leray-schauder degree of mapping
3 fundamental properties for the degree of mapping
4 linear operators in banach spaces
5 some historical notices to the chapters iii and vii
viii linear operators in hilbert spaces
1 various eigenvalue problems
2 singular integral equations
3 the abstract hilbert space
4 bounded linear operators in hilbert spaces
5 unitary operators
6 completely continuous operators in hilbert spaces
7 spectral theory for completely continuous hermitianoperators
8 the sturm-liouville eigenvalue problem
9 weyl's eigenvalue problem for the laplace operator
9 some historical notices to chapter viii
ix linear elliptic differential equations
1 the differential equation △φ+p(x, y)φx+q(x, y)φy=r(x, y)
2 the schwarzian integral formula
3 the riemann-hilbert boundary value problem
4 potential-theoretic estimates.
5 schauder's continuity method
6 existence and regularity theorems
7 the schauder estimates
8 some historical notices to chapter ix
x weak solutions of elliptic differential equations
1 sobolev spaces
2 embedding and compactness
3 existence of weak solutions
4 boundedness of weak solutions
5 hslder continuity of weak solutions
6 weak potential-theoretic estimates
7 boundary behavior of weak solutions
8 equations in divergence form
9 green's function for elliptic operators
10 spectral theory of the laplace-beltrami operator
11 some historical notices to chapter x
xi nonlinear partial differential equations
1 the fundamental forms and curvatures of a surface
2 two-dimensional parametric integrals
3 quasilinear hyperbolic differential equations and systems ofsecond order (characteristic parameters)
4 cauchy's initial value problem for quasilinear hyperbolicdifferential equations and systems of second order
5 riemann's integration method
6 bernstein's analyticity theorem
7 some historical notices to chapter xi
xii nonlinear elliptic systems
1 maximum principles for the h-surface system
2 gradient estimates for nonlinear elliptic systems
3 global estimates for nonlinear systems
4 the dirichlet problem for nonlinear elliptic systems
5 distortion estimates for plane elliptic systems
6 a curvature estimate for minimal surfaces
7 global estimates for conformal mappings with respect toriemannian metrics
8 introduction of conformal parameters into a riemannianmetric
9 the uniformization method for quasilinear elliptic differentialequations and the dirichlet problem
10 an outlook on plateau's problem
11 some historical notices to chapter xii
references
index
