Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

TL;DR

This paper proves that solutions to the Allen-Cahn equation with arbitrary potentials converge to De Giorgi's multiphase mean curvature flow, establishing a link between phase field models and geometric evolution equations.

Contribution

It demonstrates the convergence of Allen-Cahn solutions to De Giorgi's BV-solutions for multiphase mean curvature flow and shows their equivalence to varifold solutions, ensuring uniqueness.

Findings

Convergence of Allen-Cahn solutions to BV-solutions

Equivalence of BV-solutions and varifold solutions

Uniqueness of solutions in a weak-strong sense

Abstract

This paper presents a conditional convergence result of solutions to the Allen--Cahn equation with arbitrary potentials to a De Giorgi type $ \mathrm{BV} $-solution to multiphase mean curvature flow. Moreover we show that De Giorgi type $\mathrm{BV} $-solutions are De Giorgi type varifold solutions, and thus our solution is unique in a weak-strong sense.