TY  - BOOK
AU  - 
AU  - 
AU  - 
ED  - SpringerLink (Online service)
TI  - Regularity of Minimal Surfaces
T2  - Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,
SN  - 9783642117008
AV  - QA315-316 
U1  - 515.64 23
PY  - 2010///
CY  - Berlin, Heidelberg
PB  - Springer Berlin Heidelberg
KW  - Mathematics
KW  - Functions of complex variables
KW  - Differential equations, partial
KW  - Global differential geometry
KW  - Calculus of Variations and Optimal Control, Optimization
KW  - Differential Geometry
KW  - Partial Differential Equations
KW  - Functions of a Complex Variable
KW  - Theoretical, Mathematical and Computational Physics
N1  - 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; ZDB-2-SMA
N2  - 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 Plateauós 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 Plateauós problem have no interior branch points
UR  - http://dx.doi.org/10.1007/978-3-642-11700-8
ER  -