Global calibrations for the non-homogeneous Mumford-Shah functional
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 1 (2002) no. 3, p. 603-648
Using a calibration method we prove that, if ΓΩ is a closed regular hypersurface and if the function g is discontinuous along Γ and regular outside, then the function u β which solves Δu β =β(u β -g)inΩΓ ν u β =0onΩΓ is in turn discontinuous along Γ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional ΩS u |u| 2 dx+ n-1 (S u )+β ΩS u (u-g) 2 dx, over SBV(Ω), for β large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.
@article{ASNSP_2002_5_1_3_603_0,
     author = {Morini, Massimiliano},
     title = {Global calibrations for the non-homogeneous Mumford-Shah functional},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {3},
     year = {2002},
     pages = {603-648},
     zbl = {1170.49308},
     zbl = {pre02217017},
     mrnumber = {1990674},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_3_603_0}
}
Morini, Massimiliano. Global calibrations for the non-homogeneous Mumford-Shah functional. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 1 (2002) no. 3, pp. 603-648. http://www.numdam.org/item/ASNSP_2002_5_1_3_603_0/

[1] G. Alberti - G. Bouchitté - G. Dal Maso, The calibration method for the Mumford-Shah functional, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 249-254. | MR 1711069 | Zbl 0948.49005

[2] G. Alberti - G. Bouchitté - G. Dal Maso G., The calibration method for the Mumford-Shah functional and free discontinuity problems, Preprint SISSA, Trieste, 2001. | MR 2001706 | Zbl 1015.49008

[3] L. Ambrosio, Movimenti minimizzanti, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur. 113 (1995), 191-246. | MR 1387558 | Zbl 0957.49029

[4] L. Ambrosio, A compactness theorem for a new class of variational problems, Boll. Un. Mat. It. 3-B (1989), 857-881. | MR 1032614 | Zbl 0767.49001

[5] L. Ambrosio - N. Fusco - D. Pallara, “Functions of Bounded Variation and Free-Discontinuity Problems”, Oxford University Press, Oxford, 2000. | MR 1857292 | Zbl 0957.49001

[6] A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré, Anal. Non Linéaire. 13 (1996), 485-528. | Numdam | MR 1404319 | Zbl 0883.49004

[7] A. Chambolle - F. Doveri, Minimizing movements of the Mumford-Shah energy, Discrete Contin. Dynam. Systems 3 (1997), 153-174. | MR 1432071 | Zbl 0948.35073

[8] G. Dal Maso - M. G. Mora - M. Morini, Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity set, J. Math. Pures Appl. 79 (2000), 141-162. | MR 1749156 | Zbl 0962.49013

[9] E. De Giorgi - L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), 199-210. | MR 1152641 | Zbl 0715.49014

[10] M. C. Delfour - J. P. Zolésio, Shape Analysis via oriented distance functions, J. Funct. Anal. 123 (1994), 129-201. | MR 1279299 | Zbl 0814.49032

[11] T. De Pauw - D. Smets, On explicit solutions for the problem of Mumford and Shah, Comm. Contemp. Math. 1 (1999), 201-212. | MR 1696099 | Zbl 0953.49022

[12] M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 145-193. | Numdam | MR 1658873 | Zbl 0931.49010

[13] P. Grisvard, “Majorations en norme du maximum de la résolvante du laplacien dans un polygone. Nonlinear partial differential equations and their applications”, Collège de France Seminar, Vol. XII (Paris, 1991-1993), 87-96, Pitman Res. | MR 1291845 | Zbl 0811.35022

[14] P. Grisvard, “Elliptic Problems in Nonsmooth Domains”, Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. | MR 775683 | Zbl 0695.35060

[15] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. | MR 1329547 | Zbl 1261.35001 | Zbl 0816.35001

[16] M. G. Mora, Local calibrations for minimizers of the Mumford-Shah functional with a triple junction, Preprint SISSA, Trieste, 2001. | MR 1901148 | Zbl 1017.49018

[17] M. G. Mora - M. Morini, Functional depending on curvatures with constraints, Rend. Sem. Mat. Univ. Padova 104 (2000), 173-199. | Numdam | MR 1809356 | Zbl 1017.49019

[18] M. G. Mora - M. Morini, Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set, To appear on Ann. Inst. H. Poincaré, Anal. non linéaire. | Numdam | MR 1841127 | Zbl 1052.49018

[19] D. Mumford - J. Shah, Boundary detection by minimizing functionals, I, Proc. IEEE Conf. on Computer Vision and Pattern Recognition (San Francisco, 1985).

[20] D. Mumford - J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577-685. | MR 997568 | Zbl 0691.49036

[21] T. J. Richardson, Limit theorems for a variational problem arising in computer vision, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 1-49. | Numdam | MR 1183756 | Zbl 0757.49027