Dirichlets Principle, Conformal Mapping, and Minimal Surfaces [Paperback]

I. Dirichlets Principle and the Boundary Value Problem of Potential Theory.- 1. Dirichlets Principle.- 2. Semicontinuity of Dirichlets integral. Dirichlets Principle for circular disk.- 3. Dirichlets integral and quadratic functionals.- 4. Further preparation.- 5. Proof of Dirichlets Principle for general domains.- 6. Alternative proof of Dirichlets Principle.- 7. Conformal mapping of simply and doubly connected domains.- 8. Dirichlets 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. Uniformisation.- 6. Riemann domains defined by non-overlapping cells.- 7. Conformal mapping of domains not of genus zero.- III. Plateaus 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 Dirichlets 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. MinimalSurfaces 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 olc'