Implicit time discretization for the mean curvature flow of mean convex sets

TL;DR

This paper analyzes an implicit time discretization scheme for mean curvature flow, demonstrating that it preserves mean convexity and converges strongly in BV, leading to unconditional convergence results in this setting.

Contribution

It proves that the Almgren-Taylor-Wang scheme maintains mean convexity and achieves strict BV convergence for mean convex sets, strengthening previous conditional results.

Findings

Scheme preserves strict mean convexity.

Arrival time functions converge strictly in BV.

Conditional convergence results become unconditional for mean convex sets.

Abstract

In this note we analyze the Almgren-Taylor-Wang scheme for mean curvature flow in the case of mean convex initial conditions. We show that the scheme preserves strict mean convexity and, by compensated compactness techniques, that the arrival time functions converge strictly in \(BV\). In particular, this establishes the convergence of the time-integrated perimeters of the approximations. As a corollary, the conditional convergence result of Luckhaus-Sturzenhecker becomes unconditonal in the mean convex case.