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Bücher Englische Bücher Minimal Surfaces
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Minimal Surfaces
Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (Broschiert, Englisch)
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Beschreibung
Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins… with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem andTomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates. Dieses Produkt haben wir gerade leider nicht auf Lager.
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Technische Daten
Erscheinungsdatum
01.12.2012
Sprache
Englisch
EAN
9783642265273
Herausgeber
Springer Berlin
Serien- oder Bandtitel
Grundlehren der mathematischen Wissenschaften
Sonderedition
Nein
Autor
Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny
Seitenanzahl
692
Auflage
2
Einbandart
Broschiert
Schlagwörter
Boundary value problem, Mean curvature, Minimal surface, Plateau's problem, calculus of variations, conformal mappings, curvature, differential geometry, differential geometry of surfaces, minimal surfaces, minimum, regularity theory, partial differential equations
Thema-Inhalt
PBU - Optimierung
PBMP - Differentielle und Riemannsche Geometrie
PBKJ - Differentialrechnung und -gleichungen
PBKD - Komplexe Analysis, komplexe Variablen, Funktionentheorie
PHU - Mathematische Physik
Höhe
235 mm
Breite
15.5 cm
Hersteller: Springer Gabler, Europaplatz 3, Heidelberg, Deutschland, 69115, ProductSafety@springernature.com, Springer Nature Customer Service Center GmbH
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