Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow
Tim Bastian Laux and Thilo Martin Simon
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Submission date: 23. Jun. 2016
published in: Communications on pure and applied mathematics, 71 (2018) 8, p. 1597-1647
DOI number (of the published article): 10.1002/cpa.21747
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
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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.