×

Monotonicity and separation for the Mumford-Shah problem. (English) Zbl 1038.49022

This paper is an important contribution to the understanding of the regularity properties of minimizers of the Mumford-Shah functional \[ J(u,K):=\int_{\Omega\setminus K}| \nabla u| ^2+\alpha (u-g)^2\,dx+ \beta{\mathcal H}^1(K). \] We know after Bonnet’s paper that the (partial) regularity of the jump set \(K\) could be achieved if we had a classification of the so-called global Mumford-Shah minimizers. We know also, after previous work by Bonnet-David and Alberti-Bouchitté-Dal Maso, that the list of global minimizers contains the empty set, the half line, the full line, and the propeller (three half lines meeting with equal angles). One of the main results of the paper is that the only minimizers \(K\) for which the complement of \(K\) is not connected are the line and the propeller. This reduces the classification problem to the sets \(K\) such that their complement is connected. The proof is based on new monotonicity estimates for the normalized energy \[ {2\over r}\int_{B_r(x)\setminus K}| \nabla u| ^2\,dx+ {1\over r}{\mathcal H}^1(K\cap B_r(x)). \] Another result is that the classification can indeed be achieved for global minimizers contained in a sector. As a byproduct, by a standard blow-up argument, the authors obtain the boundary regularity of the jump set of minimizers. This last result has also been obtained independently by Maddalena and Solimini.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Ambrosio, L., Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111, 291-322 (1990) · Zbl 0711.49064
[2] Ambrosio, L.; Pallara, D., Partial regularity of free discontinuity sets I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24, 1-38 (1997) · Zbl 0896.49023
[3] Ambrosio, L.; Fusco, N.; Pallara, D., Partial regularity of free discontinuity sets II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24, 39-62 (1997) · Zbl 0896.49024
[4] Bonnet, A., On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 13, 4, 485-528 (1996) · Zbl 0883.49004
[5] Bonnet, A.; David, G., Cracktip is a global Mumford-Shah minimizer. Cracktip is a global Mumford-Shah minimizer, Astérisque, 274 (2001), SMF · Zbl 1014.49009
[6] Dal Maso, G.; Morel, J.-M.; Solimini, S., A variational method in image segmentation: Existence and approximation results, Acta Math., 168, 89-151 (1992) · Zbl 0772.49006
[7] David, G., \(C^1\) arcs for minimizers of the Mumford-Shah functional, SIAM. J. Appl. Math., 56, 3, 783-888 (1996) · Zbl 0870.49020
[8] David, G.; Semmes, S., Analysis of and on Uniformly Rectifiable Sets. Analysis of and on Uniformly Rectifiable Sets, AMS Series of Mathematical Surveys and Monographs, 38 (1993) · Zbl 0832.42008
[9] David, G.; Semmes, S., On the singular sets of minimizers of the Mumford-Shah functional, J. Math. Pures Appl., 75, 299-342 (1996) · Zbl 0853.49010
[10] De Giorgi, E., Problemi con discontinuità libera, Int. Symp. Renato Caccioppoli, Napoli, Sept. 20-22, 1989, Ricerche Mat. (suppl.), 40, 203-214 (1991)
[11] De Giorgi, E.; Carriero, M.; Leaci, A., Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal., 108, 195-218 (1989) · Zbl 0682.49002
[12] Falconer, K., The Geometry of Fractal Sets (1984), Cambridge University Press · Zbl 0587.28004
[13] Federer, H., Geometric Measure Theory. Geometric Measure Theory, Grundlehren der Mathematischen Wissenschaften, 153 (1969), Springer-Verlag · Zbl 0176.00801
[14] Hardy, G.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge University Press · Zbl 0047.05302
[15] Léger, J.-C., Flatness and finiteness in the Mumford-Shah problem, J. Math. Pures Appl. (9), 78, 4, 431-459 (1999) · Zbl 0942.49030
[16] Lops, F. A.; Maddalena, F.; Solimini, S., Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 18, 639-673 (2001) · Zbl 1001.49018
[17] Maddalena, F.; Solimini, S., Blow-up techniques and regularity near the boundary for free discontinuity problems, Advanced Nonlinear Studies, 1, 2 (2001) · Zbl 1044.49026
[18] Mattila, P., Geometry of Sets and Measures in Euclidean Space. Geometry of Sets and Measures in Euclidean Space, Cambridge Studies in Advanced Mathematics, 44 (1995), Cambridge University Press · Zbl 0819.28004
[19] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42, 577-685 (1989) · Zbl 0691.49036
[20] Newman, M. H.A., Elements of the Topology of Plane Sets of Points (1961), Cambridge University Press: Cambridge University Press New York · Zbl 0098.00209
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.