The calibration method for the Mumford-Shah functional and free-discontinuity problems

Calculus of Variations and Partial Differential Equations

Volumn 16, Issue 3, 2003, Pages 299-333

  Alberti, Giovanni ^a   Bouchitté, Guy ^b   Dal Maso, Gianni ^c  

^a UNIVERSITY OF PISA   (Italy)

^b UNIVERSITÉ DE TOULON   (France)

^c SISSA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037934282     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s005260100152     Document Type: Article

References (35)

1. G. Alberti, L. Ambrosio, X. Cabré: On a long-standing conjecture of E. De Giorgi: symmetry in 3d for general nonlinearities and a local minimality property. Acta Appl. Math. 65 (2001), 9-33.
2. 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.
3. L. Ambrosio: Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990), 291-322.
4. L. Ambrosio: Movimenti minimizzanti. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 19 (1995), 191-246.
5. L. Ambrosio: Corso introduttivo alia teoria geometrica della misura ed alle superficiminime. Appunti dei Corsi Tenuti da Docenti della Scuola (Notes of courses given by teachers at the School). Scuola Normale Superiore, Pisa, 1997.
6. L. Ambrosio, A. Braides: Functionals defined on partitions in sets of finite perimeter II: semicontinuity, relaxation, homogenization. J. Math. Pures Appl. 69 (1990), 307-333.
7. L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford Science Publications, Oxford, 1999.
8. A. Bonnet, G. David: Cracktip is a global Mumford-Shah minimizer. Astérisque, No. 274 (2001).
9. G. Bouchitté, A. Chambolle: Paper in preparation.
10. K.A. Brakke: Minimal cones on hypercubes. J. Geom. Anal. 1 (1991), 329-338.
11. K.A. Brakke: Numerical solution of soap film dual problems. Experiment. Math. 4 (1995), 269-287.
12. K.A. Brakke: Soap films and covering spaces. J. Geom. Anal. 5 (1995), 445-514.
13. 1. J. Convex Anal., 8 (2001), 149-170.
14. A. Chambolle, F. Doveri: Minimizing movements of the Mumford and Shah energy. Discrete Contin. Dynam. Systems 3 (1997), 153-174.
15. G. Dal Maso: The calibration method for free discontinuity problems. European Congress of Mathematics. Vol. I. (Barcelona, July 10-14, 2000). Edited by C. Casacuberta et al. Progress in Mathematics, 201. Birkhäuser Verlag, Basel, 2001.
16. G. Dal Maso, M.G. Mora, M. Morini: Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets. J. Math. Pures Appl. (9) 79 (2000), 141-162.
17. G. Dal Maso, J.-M. Morel, S. Solimini: A variational method in image segmentation: existence and approximation results. Acta Math. 168 (1992), 89-151.
18. E. De Giorgi: New problems on minimizing movements. Boundary value problems for partial differential equations and applications (dedicated to E. Magenes), 81-98, edited by J.-L. Lions and C. Baiocchi. Research Notes in Applied Mathematics, 29. Masson, Paris. 1993.
19. E. De Giorgi, M. Carriero, A. Leaci: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989), 195-218.
20. L.C. Evans, R.F. Gariepy: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992.
21. H. Federer: Geometric measure theory. Grundlehren der mathematischen Wissenschaften, 153. Springer-Verlag, Berlin-New York, 1969. Reprinted in the series Classics in Mathematics. Springer-Verlag, Berlin-Heidelberg, 1996.
22. H. Federer: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24 (1974-75), 351-407.
23. E. Giusti: Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser, Boston, 1984.
24. M. Gobbino: Gradient flow for the one-dimensional Mumford-Shah functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 145-193.
25. M. Gobbino: Minimizing movements and evolution problems in Euclidean spaces. Ann. Mat. Pura Appl. (4) 176 (1999), 29-48.
26. A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.
27. M. Miranda: Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 515-542.
28. M.G. Mora, M. Morini: Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity sets. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 18 (2001), 403-436.
29. J.-M. Morel, S. Solimini: Variational methods in image segmentation. Progress in Nonlinear Differential Equations and their Applications, 14. Birkhäuser Boston, Boston, 1995.
30. F. Morgan: Calibrations and new singularities in area-minimizing surfaces: a survey. Variational methods (Paris, 1988), 329-342. edited by H. Berestycki et al. Progr. Nonlinear Differential Equations Appl., 4. Birkhäuser Boston, Boston, 1990.
31. F. Morgan, G. Lawlor: Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms. Pacific J. Math. 166 (1994), 55-83.
32. M. Morini: Global calibrations for the non-homogeneous Mumford-Shah functional. Preprint SISSA, May 2001 (http://www.sissa.it/fa/publications/pub.html).
33. D. Mumford, J. Shah: Boundary detection by minimizing functionals. IEEE Conference on Computer Vision an Pattern Recognition. San Francisco, 1985.
34. D. Mumford, J. Shah: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989), 577-685.
35. G.M. Troianiello: Elliptic differential equations and obstacle problems. The University Series in Mathematics. Plenum Press, New York, 1987.