Existence and uniqueness for a crystalline mean curvature flow

TL;DR

This paper proves existence and uniqueness of crystalline mean curvature flows with natural mobility in any dimension, using weak formulations and minimizing movements, applicable to arbitrary initial closed sets.

Contribution

It establishes a general existence and uniqueness theory for crystalline mean curvature flows with natural mobility in any dimension.

Findings

Existence of global-in-time solutions for crystalline mean curvature flow.

Uniqueness of solutions up to fattening in arbitrary dimensions.

Development of a weak formulation and minimizing movements approach.

Abstract

An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach.