Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
33/2015

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

Tim Bastian Laux and Felix Otto

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.

Received:
May 6, 2015
Published:
May 6, 2015

Related publications

inJournal
2016 Journal Open Access
Tim Bastian Laux and Felix Otto

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

In: Calculus of variations and partial differential equations, 55 (2016) 5, p. 129