Regularity of Minimal Surfaces [electronic resource] /by Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba.
by Dierkes, Ulrich [author.]; Hildebrandt, Stefan [author.]; Tromba, Anthony J [author.]; SpringerLink (Online service).
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BookSeries: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics: 340Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2010.Description: XVII, 623 p. online resource.ISBN: 9783642117008.Subject(s): Mathematics | Functions of complex variables | Differential equations, partial | Global differential geometry | Mathematics | Calculus of Variations and Optimal Control, Optimization | Differential Geometry | Partial Differential Equations | Functions of a Complex Variable | Theoretical, Mathematical and Computational PhysicsDDC classification: 515.64 Online resources: Click here to access online | Item type | Current location | Call number | Status | Date due | Barcode |
|---|---|---|---|---|---|
| QA402.3 (Browse shelf) | Available | ||||
| QA402.5-QA402.6 (Browse shelf) | Available | ||||
| Long Loan | MAIN LIBRARY | QA315-316 (Browse shelf) | Available |
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| QA402.3 Constructive Aspects of Functional Analysis | QA402.3 Calculus of Variations, Classical and Modern | QA402.3 Minimal Surfaces | QA402.3 Regularity of Minimal Surfaces | QA402.3 Handbook of Power Systems II | QA402.3 Handbook of Networks in Power Systems I | QA402.3 Handbook of Networks in Power Systems II |
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.
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.
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