The free-boundary Brakke flow

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White to the free-boundary setting. We use Ilmanen's elliptic regularization procedure to prove existence of free-boundary Brakke flows.