Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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.


References [Enhancements On Off] (What's this?)

  • 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)

Similar Articles

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.

American Mathematical Society