Quasiminimal surfaces of codimension 1 and John domains
type: Item
instance of: scholarly article
title: Quasiminimal surfaces of codimension 1 and John domains
zbMATH Open document ID: 0921.49031
author: Guy David
author: Stephen Semmes
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: We study subsets \(E\) of codimension 1 of an annulus (say) in \(\mathbb{R}^n\) that separate the inside of the annulus from the outside and quasiminimize \((n-1)\)-dimensional Hausdorff measure, and the analogue of these sets in the context of BV and sets of finite perimeter. We show that quasiminimizers are uniformly rectifiable and Ahlfors-regular, and that their complement in \(\mathbb{R}^n\) is composed of two (connected) John domains with common boundary \(E\). In fact, these conditions characterize quasiminimizers. As an application we show that sets with not too large \((n-1)\)-dimensional Hausdorff measure which separate points in a definite way must contain a substantial piece of Lipschitz graph [see also \textit{P. W. Jones, N. W. Katz} and \textit{A. Vargas}, Rev. Mat. Iberoam. 13, No. 1, 189-210 (1997; Zbl 0908.49029)]. In this way we use area quasiminimizers to solve a problem in geometric measure theory.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 1060830
zbMATH Keywords: quasiminimal surfaces
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: John domains
zbMATH Keywords: area quasiminimizers
MaRDI profile type: Publication
full work available at URL: pjm.1998.183.213
OpenAlex ID: W2087794551
DOI: 10.2140/PJM.1998.183.213
description: scientific article; zbMATH DE number 1060830