Optimal regularity for codimension one minimal surfaces with a free boundary
type: Item
instance of: scholarly article
title: Optimal regularity for codimension one minimal surfaces with a free boundary
zbMATH Open document ID: 0609.35012
DOI: 10.1007/BF01165891
author: Michael Grüter
published in: Manuscripta Mathematica
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155224
review text: We consider the problem to minimize n-dimensional area among currents T whose boundary (or part of it) is supposed to lie in a given hypersurface S of \({\mathbb{R}}^{n+1}\). It is shown that the set of singular points of T - i.e. the closure of those points where T is not an embedded submanifold with boundary - has Hausdorff-dimension at most n-7. As in the case of interior regularity this result turns out to be optimal. The methods used are those of Geometric Measure Theory. The proof uses reflection arguments combined with regularity results (by Grüter and Jost) for varifolds with free boundaries. The existence of singular points is reduced to the existence of singular minimizing cones by Federer's dimension reduction argument.
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 3985732
zbMATH Keywords: minimal surfaces
zbMATH Keywords: free boundary
zbMATH Keywords: set of singular points
zbMATH Keywords: Hausdorff- dimension
zbMATH Keywords: interior regularity
zbMATH Keywords: geometric measure theory
zbMATH Keywords: reflection arguments
zbMATH Keywords: varifolds with free boundaries
zbMATH Keywords: singular minimizing cones
MaRDI profile type: Publication
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OpenAlex ID: W156929693
description: scientific article; zbMATH DE number 3985732