The thresholding scheme for mean curvature flow and de Giorgi's ideas for minimizing movements

TL;DR

This paper investigates the thresholding scheme for mean curvature flow, connecting it to De Giorgi's gradient flow ideas, and derives a weak formulation through energy dissipation principles.

Contribution

It establishes a link between the thresholding scheme and De Giorgi's minimizing movements framework for mean curvature flow.

Findings

Thresholding can be viewed as a minimizing movements scheme.

A dissipation-based weak formulation of mean curvature flow is derived.

The analysis is restricted to a single interface for clarity.

Abstract

We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flow as the steepest descent of the interfacial area. The basis of our analysis is the observation by Esedoglu and the second author that thresholding can be interpreted as a minimizing movements scheme for an energy that approximates the interfacial area. De Giorgi's framework provides an optimal energy dissipation relation for the scheme in which we pass to the limit to derive a dissipation-based weak formulation of mean curvature flow. Although applicable in the general setting of arbitrary networks, here we restrict ourselves to the case of a single interface, which allows for a compact, self-contained presentation.