Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

Tim Bastian Laux and Thilo Martin Simon

Contact the author: Please use for correspondence this email.
Submission date: 23. Jun. 2016
Pages: 25
published in: Communications on pure and applied mathematics, 71 (2018) 8, p. 1597-1647 
DOI number (of the published article): 10.1002/cpa.21747
Bibtex
with the following different title: Convergence of the Allen-Cahn equation to multiphase mean curvature flow
MSC-Numbers: 35A15, 35K57, 35K93
Keywords and phrases: mean curvature flow, Allen-Cahn equation
Download full preprint: PDF (480 kB)

Abstract:
We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit.