Sala Seminari (Dip. Matematica).
The thresholding scheme, a time discretization for mean-curvature flow was introduced by Meriman, Bence and Osher in 1992. In the talk we present new convergence results for modern variants of this scheme, in particular in the multi-phase case with arbitrary surface tensions. The first 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 and Otto in 2014. 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 that $\Gamma$-converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren, Taylor and Wang in 1993 and Luckhaus and Sturzenhecker in 1995, 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 is however not ensured by the compactness coming from the basic estimates. We will also discuss new convergence results for volume-preserving mean-curvature flow and forced mean-curvature flow. — Based on joint works with Felix Otto (MPI MIS Leipzig) and Drew Swartz (Purdue University)