Gradient flow for the one-dimensional Mumford-Shah functional

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1. [ 1 ] L. Ambrosio, A Compactness Theorem for a New Class of Functions of Bounded Variation, Boll. Un. Mat. Ital.3-B (1989), 857-881. Zbl0767.49001MR1032614
2. [2] L. Ambrosio, Existence Theory for a New Class of Variational Problems, Arch. Rational Mech. Anal.111 (1990), 291-322. Zbl0711.49064MR1068374
3. [3] L. Ambrosio, Free Discontinuity Problems and Special Functions with Bounded Variation, Proceedings ECM2Budapest1996, Progress in Mathematics168 (1998), 15-35. Zbl0909.49002MR1645795
4. [4] L. Ambrosio - A. Braides, Energies in SB V and Variational Models in Fracture Mechanics, in Homogenization and Applications to Material Sciences, (D. Cioranescu, A. Damlamian, P. Donato eds.), Gakuto, Gakkotosho, Tokio, Japan, 1997, p. 1-22. Zbl0904.73045MR1473974
5. [5] L. Ambrosio - V.M. Tortorelli, Approximation of Functionals Depending on Jumps by Elliptic Functionals via r-Convergence, Comm. Pure Appl. Math.43 (1990), 999-1036. Zbl0722.49020MR1075076
6. [6] L. Ambrosio-. M. Tortorelli, On the Approximation of Free Discontinuity Problems, Boll. Un. Mat. Ital.6-B (1992), 105-123. Zbl0776.49029MR1164940
7. [7] A. Braides - G. Dal Maso, Nonlocal Approximation of the Mumford-Shah Functional, Calc. Var. Partial Differential Equations5 (1997), 293-322. Zbl0873.49009MR1450713
8. [8] H. Brezis, "Opérateures Maximaux Monotones et Semigroups de Contraction dans les Espaces de Hilbert", North-Holland Mathematics Studies, n. 5 (1973). Zbl0252.47055MR348562
9. [9] H. Brezis, Analyse Fonctionnelle: Theorie et Applications, Masson, 1987. Zbl0511.46001MR697382
10. [10] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math.55 (1995), 827-863. Zbl0830.49015MR1331589
11. [11] A. Chambolle - G. Dal Maso, Discrete Approximation of the Mumford-Shah Functional in Dimension two, RAIRO Modèl. Math. Anal. Numèr., to appear Zbl0943.49011MR1726478
12. [12] A. Chambolle - F. Doveri, Minimizing Movements of the Mumford and Shah Energy, Discrete and Continuous Dynamical Systems, vol. 3, n. 2 (1997), 153-174. Zbl0948.35073MR1432071
13. [13] G. Dal Maso, "An Introduction to Γ-convergence", Birkhäuser, Boston, 1993. Zbl0816.49001
14. [14] E. De Giorgi - G. Carriero - A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal.108 (1989), 195-218. Zbl0682.49002MR1012174
15. [15] E. De Giorgi - T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.58 (1975), 842-850. Zbl0339.49005MR448194
16. [16] E. De Giorgi - A. Marino - M. Tosques, Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.73 (1982), 6-14. Zbl0521.49011MR726279
17. [17] L.C. Evans - R.F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
18. [18] M. Gobbino, Finite Difference Approximation of the Mumford-Shah Functional, Comm. Pure Appl. Math.51 (1998), 197-228. Zbl0888.49013MR1488299
19. [19] A.A. Griffiths, The phenomenon of rupture and flow in solids, Phil. Trans. Royal Soc. London Ser.A221 (1920), 163-198. 
20. [20] T. Ilmanen, Convergence of the Allen-Cahn Equation to Brakke's Motion by Mean Curvature, J. Differential Geom. 38 (1993), 417-461. Zbl0784.53035MR1237490
21. [21] D. Mumford - J. Shah, Optimal Approximation by Piecewise Smooth Functions and Associated Variational Problem, Comm. Pure Appl. Math.17 (1989), 577-685. Zbl0691.49036MR997568
22. [22] W.P. Ziemer, "Weakly Differentiable Functions", Springer, Berlin, 1989. Zbl0692.46022MR1014685

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