Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 332 pages
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 Jesse Douglas' 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."
 

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Contents

Introduction
1
Semicontinuity of Dirichlets integral Dirichlets Principle for cir
11
Proof of Dirichlets Principle for general domains
23
Alternative proof of Dirichlets Principle
31
Conformal mapping of simply and doubly connected domains
38
Plateaus Problem
95
The General Problem of Douglas
141
W Conformal Mapping of Multiply Connected Domains
167
Minimal surfaces with partly free boundaries
206
Minimal surfaces spanning closed manifolds
213
Unstable minimal surfaces with prescribed polygonal boundaries
223
Unstable minimal surfaces in rectifiable contours
236
Bibliography Chapters I to WI
245
Dirichlet integrals for harmonic functions
266
Variation of the Greens function
292
Bibliography to Appendix
319

W Conformal Mapping of Multiply Connected DomainsContinued
184
WI Minimal Surfaces with Free Boundaries and Unstable Minimal Sur
199
Supplementary Notes 1977
331
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