An examination of approaches to easy-to-understand but difficult-to-solve mathematical problems, this classic text begins with a discussion of Dirichlet's principle and the boundary value problem of potential theory, then proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Also explores minimal surfaces with free boundaries and unstable minimal surfaces. 1950 edition.
Republication of the New York, 1950 edition.

Table of Contents for Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces

Introduction
I. Dirichlet's Principle and the Boundary Value Problem of Potential Theory
1. Dirichlet's Principle
2. Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk
3. Dirichlet's integral and quadratic functionals
4. Further preparation
5. Proof of Dirichlet's Principle for general domains
6. Alternative Proof of Dirichlet's Principle
7. Conformal mapping of simply and doubly connected domains
8. Dirichlet's Principle for free boundary values. Natural boundary conditions
II. Conformal Mapping on Parallel-Slit Domains
1. Introduction
2. Solution of variational problem II
3. Conformal mapping of plane domains on slit domains
4. Riemann domains
5. General Riemann domains. Uniformization
6. Riemann domains defined by non-overlapping cells
7. Conformal mapping of domains not of genus zero
III. Plateau's Problem
1. Introduction
2. Formulation and solution of basic variational problems
3. Proof by conformal mapping that solution is a minimal surface
4. First variation of Dirichlet's integral
5. Additional remarks
6. Unsolved problems
7. First variation and method of descent
8. Dependence of area on boundary
IV. The General Problem of Douglas
1. Introduction
2. Solution of variational problem for k-fold connected domains
3. Further discussion of solution
4. Generalization to higher topological structure
V. Conformal Mapping of Multiply Connected Domains
1. Introduction
2. Conformal mapping on circular domains
3. Mapping theorems for a general class of normal domains
4. Conformal mapping on Riemann surfaces bounded by unit circles
5. Uniqueness theorems
6. Supplementary remarks
7. Existence of solution for variational problem in two dimensions
VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces
1. Introduction
2. Free boundaries. Preparations
3. Minimal surfaces with partly free boundaries
4. Minimal surfaces spanning closed manifolds
5. Properties of the free boundary. Transversality
6. Unstable minimal surfaces with prescribed polygonal boundaries
7. Unstable minimal surfaces in rectifiable contours
8. Continuity of Dirichlet's integral under transformation of r-space
Bibliography, Chapters I to VI
Appendix. Some Recent Developments in the Theory of Conformal Mapping. by M. Schiffer
1. Green's function and boundary value problems
2. Dirichlet integrals for harmonic functions
3. Variation of the Green's formula
Bibliography to App
Index

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