On the approximation of front propagation problems with nonlocal terms
Cardaliaguet, Pierre ; Pasquignon, Denis
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001), p. 437-462 / Harvested from Numdam

We investigate the approximation of the evolution of compact hypersurfaces of N depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.

Publié le : 2001-01-01
Classification:  65M12,  35K22
@article{M2AN_2001__35_3_437_0,
     author = {Cardaliaguet, Pierre and Pasquignon, Denis},
     title = {On the approximation of front propagation problems with nonlocal terms},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {35},
     year = {2001},
     pages = {437-462},
     mrnumber = {1837079},
     zbl = {0992.65097},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2001__35_3_437_0}
}
Cardaliaguet, Pierre; Pasquignon, Denis. On the approximation of front propagation problems with nonlocal terms. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 437-462. http://gdmltest.u-ga.fr/item/M2AN_2001__35_3_437_0/

[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. | Zbl 0788.68153

[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 . | Zbl 0956.35002

[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. | Zbl 0831.65138

[4] G. Barles and P.M. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4 (1991) 271-283. | Zbl 0729.65077

[5] G. Barles, H.M. Soner and P.M. Souganidis, Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439-469. | Zbl 0785.35049

[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. | Zbl 0904.35034

[7] H. Blum, Biological shape and visual science. J. Theor. Biology 38 (1973) 205-287.

[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] P. Cardaliaguet, On front propagation problems with nonlocal terms. Adv. Differential Equation 5 (1999) 213-268. | Zbl 1029.53081

[10] F. Cao, Partial differential equations and mathematical morphology. J. Math. Pures Appl. 77 (1998) 909-941. | Zbl 0920.35040

[11] F. Catte, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion. SIAM J. Numer. Anal. 32 (1995) 1895-1909. | Zbl 0841.68124

[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. | Zbl 0696.35087

[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. | Zbl 0883.35013

[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. | Zbl 0755.35015

[15] M. Crandall and P.-L. Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer. Math. 75 (1996) 17-41. | Zbl 0874.65066

[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. | Zbl 0920.35066

[17] L.C. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991) 635-681. | Zbl 0726.53029

[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. | Zbl 0836.35009

[19] F. Guichard and J.M. Morel, Partial differential equation and image iterative filtering. Tutorial of ICIP 95, Washington D.C., (1995).

[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 . | Zbl 0844.35043

[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. | Zbl 0987.53027

[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. | Zbl 0659.65132

[23] D. Pasquignon, Computation of skeleton by PDE. IEEE-ICIP, Washington (1995).

[24] D. Pasquignon, Approximation of viscosity solution by morphological filters. ESAIM: COCV 4 (1999) 335-359. | Numdam | Zbl 0929.65063

[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). | MR 1700751 | Zbl 0859.76004

[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. | Zbl 0914.35065

[27] L. Vincent, Files d'attentes et algorithmes morphologiques. Thèse mines de Paris (1992).