ON THE MEAN CURVATURE FLOW OF GRAIN BOUNDARIES

Suppose that Gamma(0) subset of Rn+1 is a closed countably n-rectifiable set whose complement Rn+1\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.