An existence theorem for Brakke flow with fixed boundary conditions

TL;DR

This paper proves the existence of non-trivial Brakke flows with fixed boundary conditions starting from arbitrary rectifiable sets, and shows their convergence to solutions of Plateau's problem in a convex domain.

Contribution

It establishes an existence theorem for Brakke flows with fixed boundary data for arbitrary rectifiable sets in convex domains, linking geometric measure theory and mean curvature flow.

Findings

Existence of non-trivial Brakke flows with fixed boundary conditions.

Flows converge to stationary solutions of Plateau's problem.

Results apply to arbitrary rectifiable sets in convex domains.

Abstract

Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $t \uparrow \infty$, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.