De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions

TL;DR

This paper proves convergence of a multiphase mean curvature flow scheme with arbitrary surface tensions and mobilities, extending previous results to more general and realistic models of grain growth.

Contribution

It introduces a new convergence proof for the Merriman-Bence-Osher scheme accommodating general surface tensions and mobilities, using gradient flow theory.

Findings

The scheme converges to a sharp energy-dissipation relation.

The proof applies to models with arbitrary mobilities and surface tensions.

The approach generalizes previous convergence results.

Abstract

We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedo\u{g}lu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.