Convergence of the thresholding scheme for multi-phase mean-curvature flow

Abstract

We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman et al. (Diffusion generated motion by mean curvature. Department of Mathematics, University of California, Los Angeles 1992). We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu et al. (Commun Pure Appl Math 68(5):808–864, 2015). This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movements scheme for an energy functional that $$\Gamma $$ -converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren et al. (SIAM J Control Optim 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calculus Var Partial Differ Equ 3(2):253–271, 1995), which establish convergence of a more academic minimizing movements scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which however is not ensured by the compactness coming from the basic estimates.