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Regularity of Minimal Surfaces

Ulrich Dierkes, Stefan Hildebrandt, Anthony Tromba (Autoren)

Buch | Hardcover

XVII, 623 Seiten

2010 | 2nd, rev. and enlarged ed. 2010
Springer Berlin (Verlag)
978-3-642-11699-5 (ISBN)

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Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas.This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau´s problem for H-surfaces in a Riemannian manifold.A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed.The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau´s problem have no interior branch points.

Boundary Behaviour of Minimal Surfaces.- Minimal Surfaces with Free Boundaries.- The Boundary Behaviour of Minimal Surfaces.- Singular Boundary Points of Minimal Surfaces.- Geometric Properties of Minimal Surfaces.- Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities.- The Thread Problem.- Branch Points.

From the reviews of the second edition:

"The most complete and thorough record of the ongoing efforts to justify Lagrange's optimism. ... contain a wealth of new material in the form of newly written chapters and sections ... . a compilation of results and proofs from a vast subject. Here were true scholars in the best sense of the word at work, creating their literary lifetime achievements. They wrote with love for detail, clarity and history, which makes them a pleasure to read. ... will become instantaneous classics." (Matthias Weber, The Mathematical Association of America, June, 2011)

Erscheint lt. Verlag	30.9.2010
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Co-Autor	Albrecht Küster
Zusatzinfo	XVII, 623 p. 68 illus., 6 illus. in color.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1020 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
Schlagworte	49Q05,53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20 • Boundary value problem • Calculus of Variations • Conformal Mappings • Differential Geometry • manifold • minimal surface • minimal surfaces • Minimum • Partial differential equations • regularity theory • Riemannian manifold
ISBN-10	3-642-11699-X / 364211699X
ISBN-13	978-3-642-11699-5 / 9783642116995
Zustand	Neuware