
Regularity of Minimal Surfaces
Springer (Publisher)
2nd Edition
Published on 30. September 2010
Book
Hardback
XVII, 623 pages
978-3-642-11699-5 (ISBN)
Description
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.
Reviews / Votes
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)
More details
Series
Edition
2nd, rev. and enlarged ed. 2010
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Edition type
Revised edition
Illustrations
6 farbige Abbildungen, 62 s/w Abbildungen
XVII, 623 p. 68 illus., 6 illus. in color.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 39 mm
Weight
1121 gr
ISBN-13
978-3-642-11699-5 (9783642116995)
DOI
10.1007/978-3-642-11700-8
Schweitzer Classification
Other editions
Additional editions

Ulrich Dierkes | Stefan Hildebrandt | Anthony Tromba
Regularity of Minimal Surfaces
Book
11/2012
2nd Edition
Springer
€160.49
Shipment within 7-9 days

Ulrich Dierkes | Stefan Hildebrandt | Anthony Tromba
Regularity of Minimal Surfaces
E-Book
08/2010
2nd Edition
Springer
€149.79
Available for download
Previous edition
Book
11/1992
Springer
€85.59
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Persons
Content
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.