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On convergence rates for solutions of approximate mean curvature equations. (English) Zbl 1241.53057
Author’s abstract: L.C. Evans and J. Spruck [J. Differ. Geom. 33, No. 3, 635–681 (1991; Zbl 0726.53029)] considered an approximate equation for the level-set equation of the mean curvature flow and proved the convergence of solutions. K. Deckelnick [Interfaces Free Bound. 2, No. 2, 117–142 (2000; Zbl 0997.65112)] 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.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35D40 Viscosity solutions to PDEs
Full Text: DOI
[1] 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. · Zbl 0788.68153 · doi:10.1007/BF00375127 · doi.org
[2] 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. · Zbl 0735.35082
[3] 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. · Zbl 0696.35087
[4] 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. · Zbl 0755.35015
[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. · Zbl 0874.65066 · doi:10.1007/s002110050228 · doi.org
[6] Klaus Deckelnick, Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow, Interfaces Free Bound. 2 (2000), no. 2, 117 – 142. · Zbl 0997.65112 · doi:10.4171/IFB/15 · doi.org
[7] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635 – 681. · Zbl 0726.53029
[8] Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. · Zbl 1096.53039
[9] Yoshikazu Giga and Shun’ichi Goto, Motion of hypersurfaces and geometric equations, J. Math. Soc. Japan 44 (1992), no. 1, 99 – 111. · Zbl 0739.53005 · doi:10.2969/jmsj/04410099 · doi.org
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