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

We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman, Bence and Osher. 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 movement scheme by Esedoglu et. al.. 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 movement 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, Taylor and Wang and Luckhaus and Sturzenhecker, which establish convergence of a more academic minimizing movement 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.