Uniform rectifiability and singular sets
type: Item
instance of: scholarly article
title: Uniform rectifiability and singular sets
zbMATH Open document ID: 0908.49030
DOI: 10.1016/S0294-1449(16)30109-3
author: Guy David
author: Stephen Semmes
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1999-03-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78387
review text: The paper deals with a quantitative notion of rectifiability, called uniform rectifiability [a general reference for this concept is \textit{G. David} and \textit{S. Semmes}: ``Analysis of and on uniformly rectifiable sets'' (1993; Zbl 0832.42008)]. The main result of the paper is the introduction of a new sufficient condition for uniform rectifiability in codimension 1, called WNPC (weak no Poincaré in the complement condition). This condition, which roughly speaking means the failure of Poincaré inequality in the complement of the set at most scales and locations, is stated in terms of Carleson sets and is applied in Section 5 of the paper to prove the uniform rectifiability of a higher dimensional version of the Mumford--Shah functional. The last section contains counterexamples showing that uniform rectifiability cannot be expected in general for sets \(E\subset{\mathbb R}^{d+1}\) with prescribed mean curvature \(H\), if \(H\) is assumed to be only in the Morrey space \(L^{1,d}\).
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 936268
zbMATH Keywords: uniform rectifiability
zbMATH Keywords: Mumford-Shah functional
zbMATH Keywords: singular sets
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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OpenAlex ID: W2472996316
description: scientific article; zbMATH DE number 936268