On the mean curvature flow of grain boundaries

TL;DR

This paper proves the global existence of mean curvature flow starting from certain initial sets with finite or controlled measure, using Brakke's formulation, and describes the evolution of grain boundaries as moving open sets.

Contribution

It establishes the global-in-time existence of mean curvature flow for complex initial sets with finite or exponential measure growth, extending previous results to more general initial geometries.

Findings

Existence of a finite family of open sets evolving continuously.

Boundaries of these sets match the space-time support of the flow.

Flow exists globally in time under specified measure conditions.

Abstract

Suppose that $\Gamma_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus \Gamma_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $\Gamma_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $\Gamma_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow.