# On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations

### Yoshikazu Giga

University of Tokyo, Japan### Takeshi Ohtsuka

University of Tokyo, Japan### Reiner Schätzle

Universität Tübingen, Germany

## Abstract

The convergence of solutions of anisotropic Allen–Cahn equations is studied when the interface thickness parameter (denoted by $ε$) tends to zero. It is shown that the convergence to a level set solution of the corresponding anisotropic interface equations is uniform with respect to the derivatives of a surface energy density function. As an application the crystalline motion of interfaces is shown to be approximated by the anisotropic Allen–Cahn equations.

## Cite this article

Yoshikazu Giga, Takeshi Ohtsuka, Reiner Schätzle, On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations. Interfaces Free Bound. 8 (2006), no. 3, pp. 317–348

DOI 10.4171/IFB/146