Consistency result for a non monotone scheme for anisotropic mean curvature flow

Abstract

In this paper, we propose a new scheme for anisotropic motion by mean curvature in ${\mathbb{R}}^d$. The scheme consists of a phase-field approximation of the motion, where the nonlinear diffusive terms in the corresponding anisotropic Allen-Cahn equation are linearized in the Fourier space. In real space, this corresponds to the convolution with a specific kernel of the form

We analyse the resulting scheme, following the work of Ishii-Pires-Souganidis on the convergence of the Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main difficulty here, is that the kernel $K_{\varphi ,t}$ is not positive and that its moments of order 2 are not in $L^1({\mathbb{R}}^d)$. Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow.