Boundary regularity for solutions of a partitioning problem
type: Item
instance of: scholarly article
title: Boundary regularity for solutions of a partitioning problem
zbMATH Open document ID: 0613.49029
DOI: 10.1007/BF00250810
author: Michael Grüter
published in: Archive for Rational Mechanics and Analysis
publication date: 1987-01-01T00:00:00Z
review text: For a bounded smooth domain \(\Omega \subset {\mathbb{R}}^{n+1}\) and \(0<\sigma <1\) the author uses standard arguments coming from the theory of sets of finite perimeter to produce a set \(E\subset \Omega\) satisfying meas E\(=\sigma \cdot \text{meas}\Omega\) such that \(\partial E\cap \Omega\) has minimal area among all such sets. While interior regularity was proved by \textit{E. Gonzalez}, \textit{Z. U. Massari} and \textit{I. Tamanini} [Indiana Univ. Math. J. 32, 25-37 (1983; Zbl 0486.49024)] the author extends their result up to the boundary by showing \({\mathbb{H}}-\dim (\sin g T)\leq n-7\) for the set of singularities in \({\bar \Omega}\) of the associated rectifiable current \(T:=\partialEL\Omega\). Moreover, near a regular point \(x\in \delta \Omega\) spt T and \(\partial \Omega\) intersect orthogonally, and the varifold associated to T has constant generalized mean curvature. The proofs are based on the author's earlier work on the regularity for minimal surfaces with a free boundary.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 3992293
zbMATH Keywords: partitioning problem
zbMATH Keywords: regularity of minimizing varifolds
zbMATH Keywords: sets of finite perimeter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3992293