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

I. Dirichlets Principle and the Boundary Value Problem of Potential Theory.- 1. Dirichlets Principle.- Definitions.- Original statement of Dirichlets Principle.- General objection: A variational problem need not he solvable.- Minimizing sequences.- Explicit expression for Dirichlets integral over a circle. Specific objection to Dirichlets Principle.- Correct formulation of Dirichlets Principle.- 2. Semicontinuity of Dirichlets integral. Dirichlets Principle for circular disk.- 3. Dirichlets integral and quadratic functionals.- 4. Further preparation.- Convergence of a sequence of harmonic functions.- Oscillation of functions appraised by Dirichlets integral.- Invariance of Dirichlets integral under conformal mapping. Applications.- Dirichlets Principle for a circle with partly free boundary.- 5. Proof of Dirichlets Principle for general domains.- Direct methods in the calculus of variations.- Construction of the harmonic function u by a smoothing process.- Proof that D[ul = d.- Proof that u attains prescribed boundary values.- Generalizations.- 6. Alternative proof of Dirichlets Principle.- Fundamental integral inequality.- Solution of variational problem I.- 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.- Classes of normal domains. Parallel-slit domains.- Variational problem: Motivation and formulation.- 2. Solution of variational problem II.- Construction of the function u.- Continuous dependence of the solution on the domain.- 3. Conformal mapping of plane domains on slit domains.- Mapping of k-fold connected domains.- Mapping on slit domains for domains G of infinite connectivity.- Half-plane slit domains. Moduli.- Boundary mapping.- 4. Riemann domains.- The sewing theorem.- 5. General Riemann domains. Uniformisation.- 6. Riemann domains defined by non-overlapping cells.- 7. Conformal lS²