On the approximation of front propagation problems with nonlocal terms

 Access to full text

 Full (PDF)

1. [1] L. Alvarez, F. Guichard, P.L. Lions and J-.M. Morel, Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123 (1993) 199–257. Zbl0788.68153
2. [2] L. Ambrosio, Geometric evolution problems, distance function and viscosity solutions, in Calculus of variations and partial differential equations. Topics on geometrical evolution problems and degree theory, G. Buttazzo et al. Eds., Based on a summer school, Pisa, Italy, September 1996. Springer, Berlin (2000) 5–93; 327–337 . Zbl0956.35002
3. [3] G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484–500. Zbl0831.65138
4. [4] G. Barles and P.M. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4 (1991) 271–283. Zbl0729.65077
5. [5] G. Barles, H.M. Soner and P.M. Souganidis, Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439–469. Zbl0785.35049
6. [6] G. Barles and P.M. Souganidis, A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141 (1998) 237–296. Zbl0904.35034
7. [7] H. Blum, Biological shape and visual science. J. Theor. Biology 38 (1973) 205–287. 
8. [8] J. Bence, B. Merriman and S. Osher, Diffusion motion generated by mean curvature. CAM Report 92-18. Dept of Mathematics. University of California Los Angeles (1992). 
9. [9] P. Cardaliaguet, On front propagation problems with nonlocal terms. Adv. Differential Equation 5 (1999) 213–268. Zbl1029.53081
10. [10] F. Cao, Partial differential equations and mathematical morphology. J. Math. Pures Appl. 77 (1998) 909–941. Zbl0920.35040
11. [11] F. Catte, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion. SIAM J. Numer. Anal. 32 (1995) 1895–1909. Zbl0841.68124
12. [12] Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991) 749–786. Zbl0696.35087
13. [13] X. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term. Nonlinear Anal. T.M.A. 28 (1997) 1283–1298. Zbl0883.35013
14. [14] M. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solution of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. Zbl0755.35015
15. [15] M. Crandall and P.-L. Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer. Math. 75 (1996) 17–41. Zbl0874.65066
16. [16] J. Escher and G. Simonett, Moving surfaces and abstract parabolic evolution equations. Topics in nonlinear analysis, Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser, Basel (1999) 183–212. Zbl0920.35066
17. [17] L.C. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991) 635–681. Zbl0726.53029
18. [18] Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40 (1990) 443–470. Zbl0836.35009
19. [19] F. Guichard and J.M. Morel, Partial differential equation and image iterative filtering. Tutorial of ICIP 95, Washington D.C., (1995). 
20. [20] H. Ishii, A generalization of the Bence-Merriman and Osher algorithm for motion by mean curvature, in Proceedings of the international conference on curvature flows and related topics, Levico, Italy, June 27 – July 2nd 1994, A. Damlamian et al. Eds. GAKUTO Int. Ser., Math. Sci. Appl. 5, Gakkotosho, Tokyo (1995) 111–127 . Zbl0844.35043
21. [21] H. Ishii, Gauss curvature flow and its approximation, in Proceedings of the international conference on free boundary problems: theory and applications, Chiba, Japan, November 7-13 1999, N. Kenmochi Ed. GAKUTO Int. Ser., Math. Sci. Appl. 14, Gakkotosho, Tokyo (2000) 198–206. Zbl0987.53027
22. [22] S. Osher and J.A. Sethian, Front propagation with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 79 (1998) 12–49. Zbl0659.65132
23. [23] D. Pasquignon, Computation of skeleton by PDE. IEEE-ICIP, Washington (1995). 
24. [24] D. Pasquignon, Approximation of viscosity solution by morphological filters. ESAIM: COCV 4 (1999) 335–359. Zbl0929.65063
25. [25] J.A. Sethian, Level set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge Monographs Appl. Comput. Math. 3, Cambridge University Press, Cambridge (1996). Zbl0859.76004MR1700751
26. [26] H.M. Soner, Front propagation, in Boundaries, interfaces and transitions, (Banff, AB, 1995) CRM Proc. Lect. Notes 13, Amer. Math. Soc., Providence RI (1998) 185–206. Zbl0914.35065
27. [27] L. Vincent, Files d’attentes et algorithmes morphologiques. Thèse mines de Paris (1992).