偏微分方程

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

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.