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

TL;DR

This paper introduces a new phase-field scheme for anisotropic mean curvature flow, analyzing its convergence despite the non-positivity of the convolution kernel, extending previous isotropic results.

Contribution

It proposes a novel linearized Fourier-based scheme for anisotropic mean curvature flow and proves its consistency despite kernel non-positivity.

Findings

Scheme is consistent with anisotropic mean curvature flow.

Convolution kernel is non-positive and not integrable in second moments.

Analysis extends convergence results to anisotropic settings.

Abstract

In this paper, we propose a new scheme for anisotropic motion by mean curvature in $\R^d$. 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 kernel of the form \[ K_{\phi,t}(x) = \F^{-1}\left[ e^{-4\pi^2 t \phi^o(\xi)} \right](x). \] 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 $K_{\phi,t}$ is not positive and that its moments of order 2 are not in $L^1(\R^d)$. Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow.