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.
Highlights
A time-parametrized family { (t)}t≥0 of n-dimensional surfaces in Rn+1 is called a mean curvature flow if the velocity of motion of (t) is equal to the mean curvature of (t) at each point and time
The aim of the present paper is to establish a global-in-time existence theorem for the MCF { (t)}t≥0 starting from a given surface 0 while keeping the boundary of (t) fixed for all times t ≥ 0
A global-in-time existence result for a Brakke flow without fixed boundary conditions was established by Kim and the second
Summary
A time-parametrized family { (t)}t≥0 of n-dimensional surfaces in Rn+1 (or in an open domain U ⊂ Rn+1) is called a mean curvature flow (abbreviated hereafter as MCF) if the velocity of motion of (t) is equal to the mean curvature of (t) at each point and time. The aim of the present paper is to establish a global-in-time existence theorem for the MCF { (t)}t≥0 starting from a given surface 0 while keeping the boundary of (t) fixed for all times t ≥ 0. Typical MCF under consideration in this setting may look like a moving network with multiple junctions for n = 1, or a moving cluster of bubbles for n = 2, and they may undergo various topological changes as they evolve. A global-in-time existence result for a Brakke flow without fixed boundary conditions was established by Kim and the second-
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