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Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets. (English) Zbl 1197.49050
Summary: Let \(P\) be a hyperplane in \(\mathbb R^N\), and denote by \(d_H\) the Hausdorff distance. We show that for all positive radius \(r < 1\) there is an \(\varepsilon > 0\), such that if \(K\) is a Reifenberg-flat set in \(B(0,1) \subset \mathbb R^N\) that contains the origin, with \(d_H (K ,P) \leq \varepsilon \), and if \(u\) is an energy minimizing function in \(B(0, 1)\setminus K\) with restricted values on \(\partial B(0, 1)\setminus K\), then the normalized energy of \(u\) in \(B(0, r)\setminus K\) is bounded by the normalized energy of \(u\) in \(B(0, 1)\setminus K\). We also prove the same result in \(\mathbb R^3\) when \(K\) is an \(\epsilon \)-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type \(\mathbb Y\) and \(\mathbb T\). Moreover, the result is still true for a further generalization of sets called (\(\varepsilon , \varepsilon _0\))-minimal. This article is a preliminary study for a forthcoming paper where a regularity result for the singular set of the Mumford-Shah functional close to minimal cones in \(\mathbb R^3\) is proved by the same author.

49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
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