Leseprobe
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Plateau's Problem and the Calculus of Variations. (MN-35)
Michael Struwe
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
This book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("H-surfaces") and its analytical framework. A comprehensive overview of the classical existence and regularity theory for disc-type minimal and H-surfaces is given and recent advances toward general structure theorems concerning the existence of multiple solutions are explored in full detail.
The book focuses on the author's derivation of the Morse-inequalities and in particular the mountain-pass-lemma of Morse-Tompkins and Shiffman for minimal surfaces and the proof of the existence of large (unstable) H-surfaces (Rellich's conjecture) due to Brezis-Coron, Steffen, and the author. Many related results are covered as well. More than the geometric aspects of Plateau's problem (which have been exhaustively covered elsewhere), the author stresses the analytic side. The emphasis lies on the variational method.
Originally published in 1989.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
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Parametrization, Closed geodesic, Modulus of continuity, Dirichlet integral, Differentiable function, Surface area, Complex number, Riemann mapping theorem, Branch point, Contradiction, Plateau's problem, Estimation, Dirichlet problem, W0, Compact space, Direct method in the calculus of variations, Existential quantification, Parameter space, Theorem, Mathematician, Complex analysis, Embedding, Minimal surface, Parallel projection, Bernhard Riemann, C0, Conjecture, Quantity, Metric space, Second derivative, Variational principle, Convex set, Linear differential equation, Jordan curve theorem, Quadratic growth, Banach space, Calculus of variations, Mathematical problem, Big O notation, Nonparametric statistics, Mean curvature, Mathematical analysis, Total curvature, Uniform convergence, Maximum principle, Euler–Lagrange equation, Global analysis, Variational method (quantum mechanics), Convex curve, Special case, Boundary value problem, Geometric measure theory, Sign (mathematics), Weak solution, Morse theory, Conformal map, Normal (geometry), Tangent space, Partial differential equation