The g-limit and the related gradient flow for singular perturbation functionals of perona-malik type

Transactions of the American Mathematical Society

Volumn 360, Issue 9, 2008, Pages 4929-4987

  Bellettini, G ^a,b   Fusco, G ^c  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

^b INFN LABORATORI NAZIONALI DI FRASCATI   (Italy)

^c UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

G convergence; Gradient flows; Minimizing movements; Singular perturbations

Indexed keywords

EID: 64849083904     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-08-04495-4     Document Type: Article

References (45)

1. G. Alberti, S. Müller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), 761-825. MR1823420 (2002f:49016) (Pubitemid 33282818)
2. R. Alicandro, A. Braides, M.S. Gelli: Free-discontinuity problems generated by singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1115-1129. MR1664085 (2000e:35012)
3. L. Ambrosio: Movimenti minimizzanti, Rend. Acc. Naz. Sc. XL, Mem. Mat. Appl. 113 (1995), 191-246. MR1387558 (97c:49044)
4. L. Ambrosio, C. De Lellis, C. Mantegazza: Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations, 9 (1999), 227-255. MR1731470 (2001g:74021)
5. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press (Oxford), 2000. MR1857292 (2003a:49002)
6. L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces and in the Wesserstein Space of Probability Measures, ETH Lectures in Mathematics, Birkhäuser, 2004.
7. G.I. Barenblatt, M. Bertsch, R. Dal Passo, M. Ughi: A degenerate pseudoparabolic regularisation of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal. 24 (1993), 1414-1439. MR1241152 (94h:35129)
8. G.I. Barenblatt, M. Bertsch, R. Dal Passo, V.M. Prostokishin, M. Ughi: A mathematical model of turbulent heat and mass transfer in stably stratified shear flow, J. Fluid Mech. 253 (1993), 341-358. MR1233902 (94g:80005)
9. G. Bellettini: On gradient flows of some non-convex functionals of Perona-Malik type, Oberwolfach Report 26/2004.
10. G. Bellettini, G. Fusco: A regularized Perona-Malik functional: Some aspects of the gradient dynamics, EQUADIFF 2003 Proceedings of the International Conference on Differential Equations, Hasselt, Belgium 22 - 26 July 2003, Freddy Dumortier, Henk Broer, Jean Mawhin, Andre Vanderbauwhede and Sjoerd Verduyn Lunel, eds., 639-644. MR2185104
11. x, Discrete Contin. Dyn. Systems, 16 (4), 2006, 783-842. MR2257160 (2007j:35080)
12. G. Bellettini, M. Novaga, E. Paolini: Global solutions to the gradient flow equation of a nonconvex functional, SIAM J. Math. Anal., 37 (5) (2006), 1657-1687. MR2215602 (2007a:35060)
13. G. Bouchitté, A. Braides, G. Buttazzo: Relaxation results for some free discontinuity problems, J. Reine Angew. Math. 458 (1995), 1-18. MR1310950 (95j:49014)
14. G. Bouchitté, C. Dubs, P. Seppecher: Regular approximation of free-discontinuity problems, Math. Mod. Meth. Appl. Sci. 10 (2000), 1073-1097. MR1780150 (2001k:49033)
15. A. Braides: Approximation of Free-Discontinuity Problems. Lecture Notes in Math., 1694, Springer 1998. MR1651773 (99j:49001)
16. A. Braides: G-Convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications, 2002. MR1968440 (2004e:49001)
17. R.J. Braun, J.W. Cahn, G.B. McFadden, A.A. Wheeler: Anisotropy of interfaces in an ordered alloy: a multiple order parameter model. Trans. Royal Soc. A 355 (1997), 1787-1832.
18. F. Catté, P.-L. Lions, J.-M. Morel, T. Coll: Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. MR1149092 (93a:94005)
19. G. Cortesani: Sequences of non-local functionals which approximate free-discontinuity problems, Arch. Rational Mech. Anal. 144 (1998), 357-402. MR1656480 (99j:49021)
20. G. Cortesani, R. Toader: A density result in SBV with respect to non-isotropic energies, Nonlinear Anal. 38 (1999), 1507-1519. MR1709990 (2000g:49018)
21. G. Dal Maso: An Introduction to G-convergence, Birkhäuser, Boston (1993). MR1201152 (94a:49001)
22. E. De Giorgi: New problems on minimizing movements, In Boundary Value Problems for Partial Differential Equations and Applications, C. Baiocchi and J.-L. Lions, eds., RMA: Res. Notes Appl. Math. 29, Masson, Paris (1993), 81-98. MR1260440 (95a:35057)
23. E. De Giorgi: Congetture riguardanti alcuni problemi di evoluzione, Duke Math. J. 81 (1995), 255-268
24. E. De Giorgi: Su alcuni problemi instabili legati alla teoria della visione, Atti del Convegno in onore di Carlo Ciliberto (Napoli, 1995), T. Bruno, P. Buonocore, L. Carbone, V. Esposito, eds., 91-98, La Città del Sole, Napoli, 1997.
25. 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., 68 (1980), 180-187. MR636814 (83m:49052)
26. S. DeMoulini: Young measure solutions for a nonlinear parabolic equation of forwardbackward type, SIAM J. Math. Anal. 27 (1996), 376-403. MR1377480 (97a:35096)
27. S. Esedoglu: An analysis of the Perona-Malik scheme, Comm. Pure Appl. Math. LIV (2001), 1442-1487. MR1852979 (2003i:94005)
28. F. Fierro, R. Goglione, M. Paolini: Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy, Math. Models Methods Appl. Sci. 8 (1998), 573-601. MR1634826 (99k:58040) (Pubitemid 128557617)
29. M. Gobbino: Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) XXVII (1998), 145-193. MR1658873 (99j:49003)
30. M. Gobbino: Minimizing movements and evolution problems in Euclidean spaces, Ann. Mat. Pura Appl. CLXXVI (1999), 29-48. MR1746533 (2001b:49011)
31. M. Gobbino: Entire solutions of the one-dimensional Perona-Malik equation, Comm. Partial Differential Equations 32 (2007), 719-743.
32. M.E. Gurtin, H.-M. Soner, P.E. Souganidis: Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations, 119 (1995), 54-108. MR1334488 (97a:73013)
33. K. Höllig, Existence of infinitely many solutions for a forward-backward heat equation, Trans. Amer. Math. Soc. 278 (1983), 299-316. MR697076 (84m:35062)
34. D. Horstmann, H.G. Othmer, K.J. Painter: Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math. 15 (2004), 546-576. MR2128611 (2006b:35166)
35. D. Horstmann, B. Schweizer: Comparison of two solution concepts for forward-backward diffusion, Preprint (2006).
36. B. Kawohl, N. Kutev: Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann. 311 (1998), 107-123. MR1624275 (99h:35110) (Pubitemid 128371246)
37. S. Kichenassamy: The Perona-Malik paradox, SIAM J. Appl. Math. 57 (2001), 1328-1342. MR1470926 (98i:35088)
38. A. Marino, C. Saccon, M. Tosques: Curves of maximal slope and parabolic variational inequalities on nonconvex constraints, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 281-330. MR1041899 (91d:58035)
39. M. Morini: Sequences of singularly perturbed functionals generating free-discontinuity problems, SIAM J. Math. Anal. 35, (2003), 759-805. MR2048406 (2005a:49023)
40. S. Müller: Variational models for microstructure and phase transitions, Calculus of variations and geometric evolution problems (Cetraro 1996), 85-210, Lecture Notes in Math., 1713, Springer, Berlin, 1999. MR1731640 (2001b:49019)
41. M. Nitzberg, D. Mumford, T. Shiota: Filtering, Segmentation and Depth, Lecture Notes in Computer Science, vol.662, Springer-Verlag, Berlin, 1993. MR1226232 (94h:92012)
42. K.J. Painter, D. Horstmann, H.G. Othmer: Localization in lattice and continuum models of reinforced random walks, Appl. Math. Lett. 16 (2003), 375-381. MR1961428
43. P. Perona, J. Malik: Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629-639.
44. P.I. Plotnikov: Passage to the limit with respect to viscosity in an equation with variable parabolicity direction, Differential Equations 30 (1994), 614-622. MR1299852 (95g:35085)
45. M. Slemrod: Dynamics of measure valued solutions to a backward-forward heat equation, J. Dynam. Differential Equations 3 (1991), 1-28. MR1094722 (92m:35124)