Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

TL;DR

This paper analyzes how a heterogeneous Allen-Cahn equation models phase separation and converges to weighted mean curvature flow, providing new insights into surface tension effects and solution uniqueness.

Contribution

It establishes the asymptotic behavior of phase field energies and proves convergence to weighted mean curvature flow with a weak-strong uniqueness principle.

Findings

Convergence of phase field energies to weighted mean curvature flow.

Derivation of a Gibbs-Thomson relation for heterogeneous surface tensions.

Weak-strong uniqueness principle for solutions of weighted mean curvature flow.

Abstract

We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of the diffuse interface vanishes. This convergence result allows us to deduce a Gibbs-Thomson relation for heterogeneous surface tensions. Proceeding from this information, we prove that (weak) solutions of the Allen-Cahn equation with space dependent potential converge to a BV solution of weighted mean curvature flow, under an energy convergence hypothesis. Additionally, relying on the relative energy technique, we establish a weak-strong uniqueness principle for solutions of weighted mean curvature flow.