On the mean curvature flow of grain boundaries

Abstract

Suppose that Λ0 ⊂ ℝn+1 is a closed countably n-rectifiable set whose complement ℝn+1\ Λ0 consists of more than one connected component. Assume that the n-dimensional Hausdorff measure of Λ0 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 Λ0. 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.