Strong Convergence of the Thresholding Scheme for the Mean Curvature Flow of Mean Convex Sets

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