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 N , and denote by d H the Hausdorff distance. We show that for all positive radius r<1 there is an ε>0, such that if K is a Reifenberg-flat set in B(0,1) N that contains the origin, with d H (K,P)ε, and if u is an energy minimizing function in B(0,1)K with restricted values on B(0,1)K, then the normalized energy of u in B(0,r)K is bounded by the normalized energy of u in B(0,1)K. We also prove the same result in 3 when K is an ε-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 (ε,ε 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 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}
}
Lemenant, Antoine. 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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