On the mean curvature flow of grain boundaries

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.

64 pages, 6 figures