Uniform rectifiability and singular sets
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 \(L^{1,d}\).
scientific article; zbMATH DE number 936268
scientific article