Regularity of Minimal Surfaces - Ulrich Dierkes, Stefan Hildebrandt, Anthony Tromba

Regularity of Minimal Surfaces

Buch | Softcover
XVII, 623 Seiten
2012 | 2. Softcover reprint of hardcover 2nd ed. 2010
Springer Berlin (Verlag)
978-3-642-26521-1 (ISBN)
160,49 inkl. MwSt
This is the second of a three-volume treatise on minimal surfaces. It deals with basic regularity results for minimal surfaces concerning their boundary behavior at Plateau boundaries and free boundaries.
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 5.11.2012
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 973 g
Themenwelt Mathematik / Informatik Mathematik Analysis
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 3-642-26521-9 / 3642265219
ISBN-13 978-3-642-26521-1 / 9783642265211
Zustand Neuware
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