# On the approximation of front propagation problems with nonlocal terms

Pierre Cardaliaguet; Denis Pasquignon

- Volume: 35, Issue: 3, page 437-462
- ISSN: 0764-583X

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topCardaliaguet, Pierre, and Pasquignon, Denis. "On the approximation of front propagation problems with nonlocal terms." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.3 (2001): 437-462. <http://eudml.org/doc/194057>.

@article{Cardaliaguet2001,

abstract = {We investigate the approximation of the evolution of compact hypersurfaces of $\mathbb \{R\}^N$ 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.},

author = {Cardaliaguet, Pierre, Pasquignon, Denis},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {front propagation; thinning; evolution; compact hypersurfaces; surface curvature},

language = {eng},

number = {3},

pages = {437-462},

publisher = {EDP-Sciences},

title = {On the approximation of front propagation problems with nonlocal terms},

url = {http://eudml.org/doc/194057},

volume = {35},

year = {2001},

}

TY - JOUR

AU - Cardaliaguet, Pierre

AU - Pasquignon, Denis

TI - On the approximation of front propagation problems with nonlocal terms

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2001

PB - EDP-Sciences

VL - 35

IS - 3

SP - 437

EP - 462

AB - We investigate the approximation of the evolution of compact hypersurfaces of $\mathbb {R}^N$ 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.

LA - eng

KW - front propagation; thinning; evolution; compact hypersurfaces; surface curvature

UR - http://eudml.org/doc/194057

ER -

## References

top- [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] 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] 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] 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] 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] 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] 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. Zbl1029.53081
- [10] F. Cao, Partial differential equations and mathematical morphology. J. Math. Pures Appl. 77 (1998) 909–941. Zbl0920.35040
- [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] 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] 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] 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] 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] 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] L.C. Evans and J. Spruck, Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991) 635–681. Zbl0726.53029
- [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. Zbl0836.35009
- [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 . Zbl0844.35043
- [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. Zbl0987.53027
- [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. Zbl0659.65132
- [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. Zbl0929.65063
- [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). Zbl0859.76004MR1700751
- [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] L. Vincent, Files d’attentes et algorithmes morphologiques. Thèse mines de Paris (1992).

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