Threshold dynamics type approximation schemes for propagating fronts. (English) Zbl 0935.53006

Ishii, Hitoshi; Pires, Gabriel E.; Souganidis, Panagiotis E.

J. Math. Soc. Japan 51, No. 2, 267-308 (1999).

The convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor is studied. These schemes are generalizations and extensions of the threshold dynamics models introduced by J. Gravner and D. Griffeath [Trans. Am. Math. Soc. 340, 837-870 (1993; Zbl 0791.58053)] to study cellular automata modelling of excitable media and by J. Bence, B. Merriman and S. Osher [“Diffusion generated motion by mean curvature”, preprint 1992] to study the mean curvature evolution. Two different scaling limits are studied, where one of them leads to curvature-independent motions, and the other leads to curvature dependent motions. The convergence of approximation schemes, which are combinations of different threshold dynamics, are studied as well. Some results on the asymptotic shape, which is described as the Wulff crystal, are presented, for fronts either moving with normal velocity depending on the normal direction and the curvature tensor, or propagating by threshold dynamics. The level set approach is used.

Reviewer: H.Ishii (Tokyo)

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
45L05	Theoretical approximation of solutions to integral equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

motion of hypersurfaces; approximation schemes; threshold dynamics; level set approach

Citations:

Zbl 0791.58053