Let be a hyperplane in , and denote by the Hausdorff distance. We show that for all positive radius there is an , such that if is a Reifenberg-flat set in that contains the origin, with , and if is an energy minimizing function in with restricted values on , then the normalized energy of in is bounded by the normalized energy of in . We also prove the same result in when is an -minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type and . Moreover, the result is still true for a further generalization of sets called -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 is proved by the same author.
@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}, pages = {351--384}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {2}, year = {2010}, mrnumber = {2731160}, zbl = {1197.49050}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2010_5_9_2_351_0/} }
TY - JOUR AU - Lemenant, Antoine TI - Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 351 EP - 384 VL - 9 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2010_5_9_2_351_0/ LA - en ID - ASNSP_2010_5_9_2_351_0 ER -
%0 Journal Article %A Lemenant, Antoine %T Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 351-384 %V 9 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2010_5_9_2_351_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2010_5_9_2_351_0/
[1] Partial regularity of free discontinuity sets II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 39–62. | EuDML | Numdam | MR | Zbl
, and ,[2] “Fonctions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. | MR
, and ,[3] On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 485–528. | EuDML | Numdam | MR | Zbl
,[4] “Elliptic Boundary Value Problems on Corner Domains”, Vol. 1341, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. | MR
,[5] Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory 15 (1992), 227–261. | MR | Zbl
,[6] -regularity for two dimensional almost-minimal sets in , J. Geom. Anal., to appear. | MR
,[7] “Singular Sets of Minimizers for the Mumford-Shah Functional”, Birkhäuser Verlag, 2005. | MR | Zbl
,[8] A generalisation of Reifenberg’s theorem in , Geom. Funct. Anal. 18 (2008), 1168–1235. | MR | Zbl
, and ,[9] Regularity of the singular set for Mumford-Shah minimizers in near a minimal cone, preprint, 2008.
,[10] “Sur la régularité des minimiseurs de Mumford-Shah en dimension et supérieure”, Thesis Université Paris Sud XI, Orsay, 2008.
,[11] Solution of the plateau problem for -dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. | MR | Zbl
,[12] The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103 (1976), 489–539. | MR | Zbl
,