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

Laux, T; Simon, T.M.

Research article (journal) | Peer reviewed

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 multiphase 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. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and volume constraints.

Details about the publication

JournalCommunications on Pure and Applied Mathematics (Comm. Pure Appl. Math.)
Volume71
Issue8
Page range1597-1647
StatusPublished
Release year2018
Language in which the publication is writtenEnglish
DOI10.1002/cpa.21747
KeywordsAllen-Cahn equation; pure mathematics; applied mathematics

Authors from the University of Münster

Simon, Theresa
Juniorprofessorship of Applied Mathematics (Prof. Simon)