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Quasiminimal surfaces of codimension 1 and John domains. (English) Zbl 0921.49031
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 P. W. Jones, N. W. Katz and 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.
Reviewer: G.David (Orsay)

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 49Q05 Minimal surfaces and optimization
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