On convergence rates for solutions of approximate mean curvature equations

Mitake, Hiroyoshi

doi:10.1090/S0002-9939-2011-11002-5

On convergence rates for solutions of approximate mean curvature equations
HTML articles powered by AMS MathViewer

by Hiroyoshi Mitake PDF

Proc. Amer. Math. Soc. 139 (2011), 3691-3696 Request permission

Abstract:

Evans and Spruck (1991) considered an approximate equation for the level-set equation of the mean curvature flow and proved the convergence of solutions. Deckelnick (2000) established a rate for the convergence. In this paper, we will provide a simple proof for the same result as that of Deckelnick. Moreover, we consider generalized mean curvature equations and introduce approximate equations for them and then establish a rate for the convergence.

References

Luis Alvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean-Michel Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal. 123 (1993), no. 3, 199–257. MR 1225209, DOI 10.1007/BF00375127
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 207–210. MR 1030181
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Michael G. Crandall and Pierre-Louis Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math. 75 (1996), no. 1, 17–41. MR 1417861, DOI 10.1007/s002110050228
Klaus Deckelnick, Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow, Interfaces Free Bound. 2 (2000), no. 2, 117–142. MR 1760409, DOI 10.4171/IFB/15
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. MR 2238463
Yoshikazu Giga and Shun’ichi Goto, Motion of hypersurfaces and geometric equations, J. Math. Soc. Japan 44 (1992), no. 1, 99–111. MR 1139660, DOI 10.2969/jmsj/04410099