Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets
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 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
Lemenant, Antoine 1

1 Université Paris XI, Bureau 15 Bâtiment 430, 91400 Orsay, France
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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, Série 5, Tome 9 (2010) no. 2, pp. 351-384. http://archive.numdam.org/item/ASNSP_2010_5_9_2_351_0/

[1] L. Ambrosio, N. Fusco and D. Pallara, Partial regularity of free discontinuity sets II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 39–62. | EuDML | Numdam | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, “Fonctions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. | MR

[3] A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 485–528. | EuDML | Numdam | MR | Zbl

[4] M. Dauge, “Elliptic Boundary Value Problems on Corner Domains”, Vol. 1341, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. | MR

[5] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory 15 (1992), 227–261. | MR | Zbl

[6] G. David, C 1+α -regularity for two dimensional almost-minimal sets in n , J. Geom. Anal., to appear. | MR

[7] G. David, “Singular Sets of Minimizers for the Mumford-Shah Functional”, Birkhäuser Verlag, 2005. | MR | Zbl

[8] G. David, T. De Pauw and T. Toro, A generalisation of Reifenberg’s theorem in 3 , Geom. Funct. Anal. 18 (2008), 1168–1235. | MR | Zbl

[9] A. Lemenant, Regularity of the singular set for Mumford-Shah minimizers in 3 near a minimal cone, preprint, 2008.

[10] A. Lemenant, “Sur la régularité des minimiseurs de Mumford-Shah en dimension 3 et supérieure”, Thesis Université Paris Sud XI, Orsay, 2008.

[11] E. R. Reifenberg, Solution of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. | MR | Zbl

[12] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103 (1976), 489–539. | MR | Zbl