Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, p. 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 $B\left(0,1\right)\subset {ℝ}^{N}$ that contains the origin, with ${d}_{H}\left(K,P\right)\le \epsilon$, and if $u$ is an energy minimizing function in $B\left(0,1\right)\setminus K$ with restricted values on $\partial B\left(0,1\right)\setminus K$, then the normalized energy of $u$ in $B\left(0,r\right)\setminus K$ is bounded by the normalized energy of $u$ in $B\left(0,1\right)\setminus K$. 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 $\left(\epsilon ,{\epsilon }_{0}\right)$-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.

Classification:  49Q20,  49Q05
@article{ASNSP_2010_5_9_2_351_0,
author = {Lemenant, Antoine},
title = {Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
pages = {351-384},
zbl = {1197.49050},
mrnumber = {2731160},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_351_0}
}

Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, pp. 351-384. http://www.numdam.org/item/ASNSP_2010_5_9_2_351_0/

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