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

Tim Bastian Laux and Felix Otto

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Submission date: 06. May. 2015
Pages: 80
published in: Calculus of variations and partial differential equations, 55 (2016) 5, art-no. 129 
DOI number (of the published article): 10.1007/s00526-016-1053-0
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Abstract:
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 Γ-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.