Abstract

We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions methods.

MSC codes

  1. 65N12
  2. 35K65
  3. 53C21
  4. 76T05

Keywords

  1. motion by mean curvature
  2. approximation scheme
  3. viscosity solutions

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
G. Barles, B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 557–579
2.
G. Barles, H. M. Soner, P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439–469
3.
G. Barles, P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271–283
4.
J. Bence, B. Merriman, S. Osher, Diffusion generated motion by mean curvature, preprint
5.
Yun Gang Chen, Yoshikazu Giga, Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749–786
6.
M. Crandall, L. C. Evans, P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502
7.
M. 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), 1–67
8.
M. Crandall, P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1–42
9.
Lawrence C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533–557
10.
L. C. Evans, J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635–681
L. C. Evans, J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc., 330 (1992), 321–332
L. C. Evans, J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal., 2 (1992), 121–150
Lawrence C. Evans, Joel Spruck, Motion of level sets by mean curvature. IV, J. Geom. Anal., 5 (1995), 77–114
11.
Hitoshi Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33–77
12.
P. Mascarenhas, Diffusion generated motion by mean curvature, Cam report, Math Dept., University of California, Los Angeles, 1992
13.
Stanley Osher, James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49
14.
Halil Mete Soner, Motion of a set by the curvature of its boundary, J. Differential Equations, 101 (1993), 313–372

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 484 - 500
ISSN (online): 1095-7170

History

Submitted: 31 December 1992
Accepted: 27 September 1993
Published online: 14 July 2006

MSC codes

  1. 65N12
  2. 35K65
  3. 53C21
  4. 76T05

Keywords

  1. motion by mean curvature
  2. approximation scheme
  3. viscosity solutions

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media