The free-boundary Brakke flow

TL;DR

This paper introduces a new concept of free-boundary Brakke flow, establishing foundational results like compactness, regularity, and existence, which extend classical mean curvature flow theories to include boundary interactions.

Contribution

It develops the theory of free-boundary Brakke flows, including a compactness theorem, a Gaussian monotonicity formula, and existence results using elliptic regularization.

Findings

Established a compactness theorem for free-boundary Brakke flows.

Proved a Gaussian monotonicity formula applicable at all points.

Demonstrated existence of free-boundary Brakke flows via elliptic regularization.

Abstract

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.