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內容簡介

In this second volume, FUNCTIONAL ANALYTIC METHODS, we continue our textbook PARTIAL DIFFERENTIAL EQUATIONS OF GEOMETRY AND PHYSICS.From both areas we shall answer central questions such as curvature estimates or eigenvalue problems, for instance. With the title of our textbook we also want to emphasize the pure and applied aspects of partial differential equa-tions. It turns out that the concepts of solutions are permanently extended ia the theory of partial differential equations. Here the classical methods do not lose their significance. Besides the n-dimensional theory we equally want to present the two-dimensional theory - so important to our geometric intuition.We shall solve the differential equations by the continuity method, the vari-ational method or the topological method. The continuity method may be preferred from a geometric point of view, since the stability of the solution is investigated there. The variational method is very attractive from the physi-cal point of view； however, difficult regularity questions for the weak solution appear with this method. The topological method controls the whole set of solutions during the deformation of the problem, and does not depend onuniqueness as does the variational method.

內頁插圖

目錄

7 Operators in Banach Spaces
Fixed point theorems
The Leray-Schauder degree of mapping
Fundamental properties for the degree of mapping
Linear operators in Banach spaces
Some historical notices to the chapters III and VII

8 Linear Operators in Hilbert Spaces
Various eigenvalue problems
Singular integral equations
The abstract Hilbert space
Bounded linear operators in Hilbert spaces
Unitary operators
Completely continuous operators in Hilbert spaces
Spectral theory for completely continuous Hermitian operators
The Sturm-Liouville eigenvalue problem
Weyl's eigenvalue problem for the Laplace operator
Some historical notices to chapter VIII

9 Linear Elliptic Differential Equations
The differential equation
The Schwarzian integral formula
The Riemann-Hilbert boundary value problem
Potential-theoretic estimates
Schauder's continuity method
Existence and regularity theorems
The Schauder estimates
Some historical notices to chapter IX

10 Weak Solutions of Elliptic Differential Equations
Sobolev spaces
Embedding and compactness
Existence of weak solutions
Boundedness of weak solutions
HSlder continuity of weak solutions
Weak potential-theoretic estimates
Boundary behavior of weak solutions
Equations in divergence form
Green's function for elliptic operators
Spectral theory of the Laplace-Beltrami operator
Some historical notices to chapter X

11 Nonlinear Partial Differential Equations
The fundamental forms and curvatures of a surface
Two-dimensional parametric integrals
Quasilinear hyperbolic differential equations and systems of second order （Characteristic parameters）
Cauchy's initial value problem for quasilinear hyperbolic
differential equations and systems of second order
Riemann's integration method
Bernstein's analyticity theorem
Some historical notices to chapter XI

12 Nonlinear Elliptic Systems
Maximum principles for the H-surface system
Gradient estimates for nonlinear elliptic systems
Global estimates for nonlinear systems
The Dirichlet problem for nonlinear elliptic systems
Distortion estimates for plane elliptic systems
A curvature estimate for minimal surfaces
Global estimates for conformal mappings with respect to Riemannian metrics
Introduction of conformal parameters into a Riemannian metric
The uniformization method for quasilinear elliptic differential equations and the Dirichlet problem
An outlook on Plateau's problem
Some historical notices to chapter XII
References
Index

前言/序言

偏微分方程（第2捲） [Partial Differential Equations 2] 下載 mobi epub pdf txt 電子書

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