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

Abstract

The convergence of solutions of anisotropic Allen–Cahn equations is studied when the interface thickness parameter (denoted by $\epsilon$) 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.