Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets

Lemenant, Antoine

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 351-384.

Let $P$ be a hyperplane in ${ℝ}^N$, and denote by $d_H$ the Hausdorff distance. We show that for all positive radius $r<1$ there is an $\epsilon >0$, such that if $K$ is a Reifenberg-flat set in $B(0,1)\subset {ℝ}^N$ that contains the origin, with $d_H(K,P)\le \epsilon$, 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 ${ℝ}^3$ when $K$ is an $\epsilon$-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type $𝕐$ and $𝕋$. Moreover, the result is still true for a further generalization of sets called $(\epsilon ,{\epsilon }_0)$-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 ${ℝ}^3$ is proved by the same author.

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