Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

TL;DR

This paper investigates the limiting behavior of an inhomogeneous, anisotropic Allen-Cahn equation, showing that interfaces evolve according to anisotropic mean curvature and providing precise estimates of transition layer thickness.

Contribution

It introduces a geometric framework using Finsler metrics to analyze the singular limit and proves new results on interface motion and layer thickness in anisotropic settings.

Findings

Interfaces evolve by anisotropic mean curvature.

Established a weak comparison principle.

Derived optimal estimates for transition layer thickness.

Abstract

We study the singular limit of a spatially inhomogeneous and anisotropic reaction-diffusion equation. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. The limit problem involves motion by anisotropic mean curvature. We also prove an optimal estimate of the thickness of the transition layer.