Mathematical development, the author of this text observes, comes about through specific, easily understood problems that require difficult solutions and demand the use of new methods. Richard Courant employs this instructive approach in a text that balances the individuality of mathematical objects with the generality of mathematical methods.
Beginning with a discussion of Dirichlet's principle and the boundary-value problem of potential theory, the text proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Succeeding chapters explore the general problem of Douglas and conformal mapping of multiply connected domains, concluding with a survey of minimal surfaces with free boundaries and unstable 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 Appendix Index
