An existence theorem for Brakke flow with fixed boundary conditions

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.

51 pages. v2 clarifies some details in Section 7 and adds 2 figures. Final version, to appear on Calc. Var. PDE