Higher integrability for the gradient of Mumford-Shah almost-minimizers

Piontek, Sebastian; Schmidt, Thomas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 6.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We extend a recent higher-integrability result for the gradient of minimizers of the Mumford-Shah functional to a suitable class of almost-minimizers. The extension crucially depends on an $L^{\infty }$ gradient estimate up to regular portions of the discontinuity set of an almost-minimizer.

[1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation. Boll. Unione Mat. Ital., VII. Ser., B 3 (1989) 857–881 | MR | Zbl

[2] L. Ambrosio, Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291–322 | DOI | MR | Zbl

[3] L. Ambrosio, N. Fusco and J. E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional. Calc. Var. Partial Differ. Equ. 16 (2003) 187–215 | DOI | MR | Zbl

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

[5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications. Clarendon Press (2000) | DOI | MR | Zbl

[6] L. Ambrosio and D. Pallara, Partial regularity of free discontinuity sets, I. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24 (1997) 1–38 | Numdam | MR | Zbl

[7] G. Anzellotti, On the $C^{1,\alpha }$-regularity of $\omega$-minima of quadratic functionals. Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 2 (1983) 195–212 | MR | Zbl

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

[9] G. David, C¹-arcs for minimizers of the Mumford-Shah functional. SIAM J. Appl. Math. 56 (1996) 783–888 | DOI | MR | Zbl

[10] G. David, Singular Sets of Minimizers for the Mumford-Shah Functional. Progress in Mathematics. Birkhäuser (2005) | MR | Zbl

[11] E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82 (1988) 199–210 | Zbl

[12] E. De Giorgi and L. Ambrosio, New functionals in calculus of variations. In Nonsmooth Optimization and Related Topics, edited by F.H. Clarke, V.F. Dem’Yanov and F. Giannessi. Springer Boston (1989) 49–59 | DOI | MR | Zbl

[13] 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 | DOI | MR | Zbl

[14] F. Duzaar, A. Gastel and J. F. Grotowski, Partial regularity for almost minimizers of quasi-convex integrals. SIAM J. Math. Anal. 32 (2000) 665–687 | DOI | MR | Zbl

[15] C. De Lellis and M. Focardi, Higher integrability of the gradient for minimizers of the 2d Mumford-Shah energy. J. Math. Pures Appl. (9) 100 (2013) 391–409 | DOI | MR | Zbl

[16] C. De Lellis, M. Focardi and B. Ruffini, A note on the Hausdorff dimension of the singular set for minimizers of the Mumford-Shah energy. Adv. Calc. Var. 7 (2014) 539–545 | DOI | MR | Zbl

[17] G. De Philippis and A. Figalli, Higher integrability for minimizers of the Mumford-Shah functional. Arch. Ration. Mech. Anal. 213 (2014) 491–502 | DOI | MR | Zbl

[18] M. Focardi, Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture. In Free discontinuity problems, edited by N. Fusco and A. Pratelli. Edizioni della Normale Scuola Normale Superiore, Pisa (2016) 1–68 | MR | Zbl

[19] N. Fusco, An overview of the Mumford-Shah problem. Milan J. Math. 71 (2003) 95–119 | DOI | MR | Zbl

[20] J.C. Leger, Flatness and finiteness in the Mumford-Shah problem. J. Math. Pures Appl. (9), 78 (1999) 431–459 | DOI | MR | Zbl

[21] F. Maddalena and S. Solimini, Concentration and flatness properties of the singular set of bisected balls. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30 (2001) 623–659 | Numdam | MR | Zbl

[22] F. Maddalena and S. Solimini, Regularity properties of free discontinuity sets. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18 (2001) 675–685 | DOI | Numdam | MR | Zbl

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

[24] S. Piontek, Höhere Integrierbarkeit für Fast-Minimierer des Mumford-Shah-Funktionals. Master Thesis. University Erlangen-Nuremberg (2016)

[25] S. Rigot, Big pieces of $C^{1,\alpha }$-graphs for minimizers of the Mumford-Shah functional. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29 (2000) 329–349 | Numdam | MR | Zbl