Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow

TL;DR

This paper proves that the constrained Allen-Cahn equation's solutions converge to Brakke's mean curvature flow, extending previous results to include constraints and connecting phase field models with geometric flows.

Contribution

It establishes the convergence of the constrained Allen-Cahn equation to Brakke's mean curvature flow, generalizing earlier unconstrained results.

Findings

Convergence of constrained Allen-Cahn solutions to Brakke's flow

Extension of previous convergence results to constrained equations

Connection between phase field models and geometric measure theory

Abstract

In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.