Brakke's inequality for the thresholding scheme

TL;DR

This paper analyzes the thresholding scheme for mean curvature flow, proving a conditional convergence to Brakke's flow using variational methods and energy-dissipation inequalities.

Contribution

It introduces a localized variational interpretation of the scheme and establishes a convergence result under energy assumptions, advancing the theoretical understanding.

Findings

Conditional convergence to Brakke's mean curvature flow

Application of De Giorgi's variational interpolation

Energy-dissipation inequality in the limit

Abstract

We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke's notion of mean curvature flow. Our proof is based on a localized version of the minimizing movements interpretation of Esedo\u{g}lu and the second author. We apply De Giorgi's variational interpolation to the thresholding scheme and pass to the limit in the resulting energy-dissipation inequality. The result is conditional in the sense that we assume the time-integrated energies of the approximations to converge to those of the limit.