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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.

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