| Preface Introduction Ⅰ. Dirichlet's Principle and the Boundary Value Problem of Potential Theory 1. Dirichlet's Principle Definitions Original statement of Dirichlet's Principle General objection: A variational problem need not be solvable Minimizing sequences Explicit expression for Dirichlet's integral over a circle. Spe-cific objection to Dirichlet's Principle Correct formulation of Dirichlet's Principle 2. Semicontinuity of Dirichlet's integral. Diriehlet's Principle for cir-cular disk 3. Dirichlet's integral and quadratic functionals 4. Further preparation Convergence of a sequence of harmonic functions Oscillation of functions appraised by Dirichlet's integral Invariance of Dirichlet's integral under eonformal mapping. Applications Dirichlet's Principle for a circle with partly free boundary 5. Proof of Dirichlet's Principle for general domains Direct methods in the calculus of variations Construction of the harmonic function u by a "smoothing proc ess" Proof that D[u] = d Proof that u attains prescribed boundary values Generalizations 6. Alternative proof of Diriehlet's Principle. Fundamental integral inequallty. Solution of variational problem I 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 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 con nectivity Half-plane slit domains. Moduli Boundary mapping 4. Riemann domains Introduction The "sewing theorem" 5. General Riemann domains. Uniformization 6. Riemann domains defined by non-overlapping cells 7. Conformal mapping of domains not of genus zero Introduction Description of slit domains not of genus zero The mapping theorem Remarks. Half-plane slit domains Ⅲ. Plateau's Problem 1. Introduction 2. Formulation and solution of basic variational problems. Notations Fundamental lemma. Solution of minimum problem Remarks. Semicontinuity 3. Proof by conformal mapping that solution is a minimal surface 4. First variation of Dirichlet's integral Variation in general space of admissible functions First variation in space of harmonic vectors Proof that stationary vectors represent minimal surfaces 5. Additional remarks. Biunique correspondence of boundary points Relative minima Proof that solution of variational problem solves problem of least area Role of conformal mapping in solution of Plateau's problem 6. Unsolved problems Analytic extension of minimal surfaces Uniqueness. Boundaries spanning infinitely many minimal surfaces Branch points of minimal surfaces Ⅳ.Plateau's Problem-Continued Ⅴ.The General Problem of Douglas Ⅵ.Conformal Mapping of Multiply Connected Domains Ⅶ.Conformal Mapping of Multiply Connnected Domains-Continued Ⅷ.Minimal Surfaces with free Boundaries and Unstable Minimal Surfaces Ⅸ.Minimal Surfaces with free Boundaries and Unstable Minimal Surfaces-Continued Ⅹ.Bibliography,Chapters I to VI Ⅺ.Appendix.Some Recent Developments in the Theory if Conformal Mapping Bibligraphy to Appendix Index |
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