It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this…mehr
It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this category. It is written in a style which, as the author hopes, expresses adequately the balance and tension between the individuality of mathematical objects and the generality of mathematical methods. The author has been interested in Dirichlet's Principle and its various applications since his days as a student under David Hilbert. Plans for writing a book on these topics were revived when JesseDouglas' work suggested to him a close connection between Dirichlet's Principle and basic problems concerning minimal sur faces. But war work and other duties intervened; even now, after much delay, the book appears in a much less polished and complete form than the author would have liked.
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. Uniformisation.- 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. 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 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 x-space.- Bibliography, Chapters I to VI.- 1. Green's function and boundary value problems.- Canonical conformal mappings.- Boundary value problems of second type and Neumann's function.- 2. Dirichlet integrals for harmonic functions.- Formal remarks..- Inequalities..- Conformal transformations.- An application to the theory of univalent functions.- Discontinuities of the kernels.- An eigenvalue problem.- Comparison theory.- An extremum problem in conformal mapping.- Mapping onto a circular domain.- Orthornormal systems.- 3. Variation of the Green's function.- Hadamard's variation formula.- Interior variations.- Application to the coefficient problem for univalent functions.- Boundary variations.- Lavrentieff's method.- Method of extremal length.- Concluding remarks.- Bibliography to Appendix.- Supplementary Notes (1977).
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. Uniformisation.- 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. 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 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 x-space.- Bibliography, Chapters I to VI.- 1. Green's function and boundary value problems.- Canonical conformal mappings.- Boundary value problems of second type and Neumann's function.- 2. Dirichlet integrals for harmonic functions.- Formal remarks..- Inequalities..- Conformal transformations.- An application to the theory of univalent functions.- Discontinuities of the kernels.- An eigenvalue problem.- Comparison theory.- An extremum problem in conformal mapping.- Mapping onto a circular domain.- Orthornormal systems.- 3. Variation of the Green's function.- Hadamard's variation formula.- Interior variations.- Application to the coefficient problem for univalent functions.- Boundary variations.- Lavrentieff's method.- Method of extremal length.- Concluding remarks.- Bibliography to Appendix.- Supplementary Notes (1977).
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