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AuthorCourant, Richard. author
TitleDirichlet's Principle, Conformal Mapping, and Minimal Surfaces [electronic resource] / by Richard Courant
ImprintNew York, NY : Springer New York, 1950
Connect tohttp://dx.doi.org/10.1007/978-1-4612-9917-2
Descript XI, 332 p. online resource

CONTENT

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. 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 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)


Mathematics Differential geometry Mathematics Differential Geometry



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