A non-local approximation of free discontinuity problems in SBV and SBD

Calculus of Variations and Partial Differential Equations

Volumn 25, Issue 1, 2006, Pages 33-62

  Negri, Matteo ^a  

^a UNIVERSITY OF PAVIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 30344452638     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-005-0356-3     Document Type: Article

References (35)

1. Alberti, G., Bouchitté, G., Seppecher, P.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144, 1-46 (1998)
2. Alicandro, R., Focardi, M., Gelli, M.S.: Finite-difference approximation of energies in fracture mechanics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4, 671-709 (2001)
3. Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111, 291-322 (1990)
4. Ambrosio, L., Coscia, A., Dal Maso. G.: Fine properties of Functions with Bounded Deformation. Arch. Rational Mech. Anal. 139, 201-238 (1997)
5. Ambrosio, L., Fusco, N., Pallara, D.: Special Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford (2000)
6. Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in SBD(Ω). Math. Z. 228, 337-351 (1998)
7. Benvenuti, E.: Damage integration in the strain space. Int J. Solids and Structures 41, 3167-3191 (2004)
8. Bouchitté, G., Braides, A., Buttazzo, G.: Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458, 1-18 (1995)
9. Bourdin, B., Francfort, G., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826 (2000)
10. Braides, A.: Approximation of free discontinuity problems, Springer, Berlin (1998)
11. Braides, A.: Γ-convergence for beginners, Oxford University Press, Oxford (2002)
12. Braides, A., Dal Maso, G.: Non-local approximation of the Mumford-Shah functional. Calc. Var. Partial Differ. Equ. 5, 293-322 (1997)
13. Braides, A., Garroni, A.: A remark on the non-local approximation of free discontinuity problems. Commun. Partial Differ. Equations 23, 817-829 (1998)
14. Catté, F., Lions, PL., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182-193 (1992)
15. Chambolle, A.: Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. Siam J. Appl. Math. 55, 827-863 (1995)
16. Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. 83, 929-954 (2004)
17. Chambolle, A.: Addendum to "An approximation result for special functions with bounded deformation": the N-dimensional case. J. Math. Pures Appl. 84, 137-145 (2005)
18. Comi, C.: A non-local model with tension and compression damage mechanisms. Bur. J. Mech. A Solids 20, 1-22 (2001)
19. Cortesani, G.: Sequences of non-local functionals which approximate free-discontinuity problems. Arch.Rational Mech. Anal. 144, 357-402 (1998)
20. Cortesani, G., Toader, R.: A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38, 585-604 (1999)
21. Cortesani, G., Toader, R.: Non-local approximation of non-isotropic free-discontinuity problems. SIAM J. Appl. Math. 59, 1507-1519 (1999)
22. Dal Maso, G.: An Introduction to Γ-convergence, Birkhäuser, Boston (1993)
23. Eringen, A.C.: A unified theory of thermomechanical materials. Int. J. Eng. Sci. 4, 179-202 (1966)
24. Eringen, A.C.: On nonlocal plasticity. Int. J. Eng. Sci. 19, 179-202 (1981)
25. Gobbino, M.: Finite difference approximation of the Mumford-Shah functional. Comm. Pure Appl. Math. 51, 197-228 (1998)
26. Gobbino, M., Mora, M.G.: Finite difference approximation of free discontinuity problems. Proc. Royal Soc. Edinburgh 131, 567-595 (2001)
27. Jirásek, M.: Nonlocal models for damage and fracture: comparison of approaches. Int. J. Solids Structures 35, 4133-4145 (1998)
28. Lussardi, L., Vitali, E.: Nonlocal approximation of free discontinuity problems with linear growth, (to appear)
29. Morini, M., Negri, M.: Mumford-Shah functional as Γ-limit of discrete Perona-Malik energies. Math. Models Methods Appl. Sci. 13, 785-805 (2003)
30. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated problems. Comm. Pure Appl. Math. 17, 577-685 (1989)
31. Negri, M.: A finite element approximation of the Griffith's model in fracture mechanics. Numer.Math. 95, 653-687 (2003)
32. Negri, M.: Convergence analysis for a smeared crack approach in brittle fracture. Preprint CVGMT
33. Nordström, N.: Biased anisotropic diffusion - a unified regularization and diffusion approach to edge detection. Image Vision Comput. 8, 318-327 (1990)
34. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans.Pattern Anal. Mach.Intell. 12, 629-639 (1990)
35. Pijaudier-Cabot, G., Bažant, Z.P.: Nonlocal damage theory. J. Eng. Mech. 113, 1512-1533 (1987)