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Uniform rectifiability and singular sets. (English) Zbl 0908.49030

The paper deals with a quantitative notion of rectifiability, called uniform rectifiability [a general reference for this concept is G. David and 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 \(L^{1,d}\).
Reviewer: L.Ambrosio (Pavia)

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
28A75 Length, area, volume, other geometric measure theory
49J10 Existence theories for free problems in two or more independent variables

Citations:

Zbl 0832.42008

References:

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