Calculus of Variations and Geometric Measure Theory

S. Stuvard - Y. Tonegawa

An existence theorem for Brakke flow with fixed boundary conditions

created by stuvard on 06 Dec 2019
modified on 28 Jan 2021


Published Paper

Inserted: 6 dec 2019
Last Updated: 28 jan 2021

Journal: Calc. Var. Partial Differential Equations
Volume: 60
Year: 2021

ArXiv: 1912.02404 PDF
Links: arXiv link


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.

Keywords: minimal surfaces, varifolds, mean curvature flow, Plateau's problem