Convergence of the thresholding scheme for multi-phase mean-curvature flowReport as inadecuate

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Calculus of Variations and Partial Differential Equations

, 55:129

First Online: 11 October 2016Received: 27 April 2016Accepted: 23 August 2016


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 Esedoğlu et al. Commun Pure Appl Math 685:808–864, 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 312:387–438, 1993 and Luckhaus and Sturzenhecker Calculus Var Partial Differ Equ 32:253–271, 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.

Mathematics Subject Classification35A15 65M12 Communicated by M. Struwe.

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Author: Tim Laux - Felix Otto


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