Convergence of perturbed Allen-Cahn equations to forced mean curvature flow

TL;DR

This paper proves that certain perturbed Allen-Cahn equations converge to a forced mean curvature flow in the sharp interface limit, extending previous results to include square-integrable perturbations and discussing applications.

Contribution

It introduces a generalized formulation for forced mean curvature flow and demonstrates convergence of perturbed Allen-Cahn equations under new conditions.

Findings

Convergence of perturbed Allen-Cahn equations to forced mean curvature flow

Extension of convergence results to square-integrable perturbations

Application of previous Allen-Cahn action functional results

Abstract

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally we discuss some applications.