Regularity of the singular set for Mumford-Shah minimizers in 3 near a minimal cone
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 561-609.

We prove that if (u,K) is a minimizer of the Mumford-Shah functional in an open set Ω of 3 , and if xK and r>0 are such that K is close enough to a minimal cone of type (a plane), 𝕐 (three half planes meeting at x with 120 angles) or 𝕋 (cone over the 6 edges of a regular tetrahedron centered at x) in terms of Hausdorff distance in B(x,r), then K is C 1,α equivalent to the minimal cone in B(x,cr) where c<1 is a universal constant.

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Classification: 49Q20, 49Q05
Lemenant, Antoine 1

1 Université Denis Diderot - Paris 7 U.F.R de Mathématiques Site Chevaleret Case 7012 175, rue du Chevaleret 75205 Paris Cedex 13 (France) lemenant@ann.jussieu.fr
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Lemenant, Antoine. Regularity of the singular set for Mumford-Shah minimizers in $\protect \mathbb{R}^3$ near a minimal cone. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 561-609. http://www.numdam.org/item/ASNSP_2011_5_10_3_561_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, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. | MR | Zbl

[3] A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré Anal. Non Linéaire (4) 13 (1996), 485–528. | EuDML | Numdam | MR | Zbl

[4] G. David, C 1+α -regularity for two-dimensional almost-minimal sets in n , J. Geom. Anal. 20 (2010), 837–954. | MR | Zbl

[5] G. David, C 1 -arcs for minimizers of the Mumford-Shah functional, SIAM J. Appl. Math. (3) 56 (1996), 783–888. | MR | Zbl

[6] G. David, Singular sets of minimizers for the Mumford-Shah functional, Vol. 233 of “Progress in Mathematics”, Birkhäuser Verlag, Basel, 2005. | MR | Zbl

[7] G. David, Hölder regularity of two-dimensional almost-minimal sets in n , Ann. Fac. Sci. Toulouse Math. (1) 18 (2009), 65–246. | EuDML | Numdam | MR | Zbl

[8] G. David, T. De Pauw and T. Toro, A generalization of Reifenberg’s theorem in 3 , Geom. Funct. Anal. (4) 18 (2008), 1168–1235. | MR | Zbl

[9] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set., Arch. Ration. Mech. Anal. 108 (1989), 195–218. | MR | Zbl

[10] H. Federer, “Geometric Measure Theory”, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. | MR | Zbl

[11] H. Koch, G. Leoni and M. Morini, On optimal regularity of free boundary problems and a conjecture of De Giorgi, Comm. Pure Appl. Math. (8) 58 (2005), 1051–1076. | MR | Zbl

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

[13] A. Lemenant, On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone, Rend. Sem. Mat. Univ. Padova 122 (2009). | EuDML | Numdam | MR | Zbl

[14] A. Lemenant, Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 9 (2010), 1–34. | Numdam | MR | Zbl

[15] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577–685. | MR | Zbl

[16] 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