Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

TL;DR

This paper proves the strong convergence of the thresholding scheme for mean curvature flow in the two-phase mean convex setting, showing that the discrete approximation's energy converges to the continuous limit.

Contribution

It establishes the unconditional convergence of the scheme in the mean convex case and extends the analysis to anisotropic flows with non-negative kernels.

Findings

Time-integrated energy of the approximation converges to that of the limit.

Conditional convergence results become unconditional in the mean convex case.

Extension of the scheme to anisotropic flows is possible.

Abstract

In this work, we analyze Merriman, Bence and Osher's thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoglu and Otto we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.