On convergence rates for solutions of approximate mean curvature equations

Author:
Hiroyoshi Mitake

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3691-3696

MSC (2010):
Primary 53C44, 65M15, 35D40

DOI:
https://doi.org/10.1090/S0002-9939-2011-11002-5

Published electronically:
March 30, 2011

MathSciNet review:
2813398

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**L. Alvarez, F. Guichard, P.-L. Lions, J.-M. Morel,*Axioms and fundamental equations of image processing*, Arch. Rational Mech. Anal. 123 (1993), no. 3, 199-257. MR**1225209 (94j:68306)****2.**Y. G. Chen, Y. Giga, S. Goto,*Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations*, Proc. Japan. Acad. 65 (1989), 207-210. MR**1030181 (91b:35049)****3.**Y. G. Chen, Y. Giga, S. Goto,*Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations*, J. Differential Geom. 33 (1991), no. 3, 749-786. MR**1100211 (93a:35093)****4.**M. G. Crandall, H. Ishii, P.-L. 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 (92j:35050)****5.**M. G. Crandall, P.-L. Lions,*Convergent difference schemes for nonlinear parabolic equations and mean curvature motion*, Numer. Math. 75 (1996), no. 1, 17-41. MR**1417861 (97j:65134)****6.**K. 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 (2001g:65098)****7.**L. C. Evans, J. Spruck,*Motion of level sets by mean curvature. I*, J. Differential Geom. 33 (1991), no. 3, 635-681. MR**1100206 (92h:35097)****8.**Y. Giga,*Surface evolution equations. A level set approach*, Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. MR**2238463 (2007j:53071)****9.**Y. Giga, S. Goto,*Motion of hypersurfaces and geometric equations*, J. Math. Soc. Japan 44 (1992), 99-111. MR**1139660 (93b:58025)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
53C44,
65M15,
35D40

Retrieve articles in all journals with MSC (2010): 53C44, 65M15, 35D40

Additional Information

**Hiroyoshi Mitake**

Affiliation:
Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Email:
mitake@hiroshima-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-2011-11002-5

Keywords:
Mean curvature equation,
affine curvature equation,
approximate equations,
convergence rate

Received by editor(s):
August 25, 2010

Published electronically:
March 30, 2011

Additional Notes:
This work was partially supported by the Research Fellowship (22-1725) for Young Researcher from JSPS

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.