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On the approximation of front propagation problems with nonlocalterms

Published online by Cambridge University Press:  15 April 2002

Pierre Cardaliaguet  and
Denis Pasquignon
Pierre Cardaliaguet
Affiliation:
Université de Bretagne Occidentale, Faculté des Sciences et Techniques, Département de Mathématiques, 6 Avenue Victor Le Gorgeu, BP 809 29285 Brest Cedex, France. (Pierre.Cardaliaguet@univ-brest.fr)
Denis Pasquignon
Affiliation:
Centre de recherche CEREMADE, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France. (pasquig@ceremade.dauphine.fr)
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Abstract

We investigate the approximationof the evolution of compact hypersurfaces of $\mathbb{R}^N$ depending, not only on terms of curvature of the surface, but alsoon non local terms such as the measure of the set enclosedby the surface.

Keywords

Front propagationthinning.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 3 , May 2001 , pp. 437 - 462
Copyright
© EDP Sciences, SMAI, 2001

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References

Alvarez, L., Guichard, F., Lions, P.L. and J-.M. Morel, Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123 (1993) 199-257. CrossRef
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 .
Barles, G. and Georgelin, C., A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484-500. CrossRef
Barles, G. and Souganidis, P.M., Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4 (1991) 271-283.
Barles, G., Soner, H.M. and Souganidis, P.M., Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439-469. CrossRef
Barles, G. and Souganidis, P.M., A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141 (1998) 237-296. CrossRef
Blum, H., Biological shape and visual science. J. Theor. Biology 38 (1973) 205-287. CrossRef
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).
Cardaliaguet, P., On front propagation problems with nonlocal terms. Adv. Differential Equation 5 (1999) 213-268.
Cao, F., Partial differential equations and mathematical morphology. J. Math. Pures Appl. 77 (1998) 909-941. CrossRef
Catte, F., Dibos, F. and Koepfler, G., A morphological scheme for mean curvature motion. SIAM J. Numer. Anal. 32 (1995) 1895-1909. CrossRef
Chen, Y., Giga, Y. and Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991) 749-786. CrossRef
Chen, X., Hilhorst, D. and Logak, E., Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term. Nonlinear Anal. T.M.A. 28 (1997) 1283-1298. CrossRef
Crandall, M., Ishii, H. and Lions, P.-L., User's guide to viscosity solution of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. CrossRef
Crandall, M. and Lions, P.-L., Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer. Math. 75 (1996) 17-41. CrossRef
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.
Evans, L.C. and Spruck, J., Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991) 635-681. CrossRef
Giga, Y., Goto, S., Ishii, H. and Sato, M.-H., Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40 (1990) 443-470. CrossRef
F. Guichard and J.M. Morel, Partial differential equation and image iterative filtering. Tutorial of ICIP 95, Washington D.C., (1995).
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 .
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.
Osher, S. and Sethian, J.A., Front propagation with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 79 (1998) 12-49. CrossRef
D. Pasquignon, Computation of skeleton by PDE. IEEE-ICIP, Washington (1995).
Pasquignon, D., Approximation of viscosity solution by morphological filters. ESAIM: COCV 4 (1999) 335-359. CrossRef
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).
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.
L. Vincent, Files d'attentes et algorithmes morphologiques. Thèse mines de Paris (1992).