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↑∞, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.
An existence theorem for Brakke flow with fixed boundary conditions / S. Stuvard, Y. Tonegawa. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 60:1(2021), pp. 43.1-43.53.
An existence theorem for Brakke flow with fixed boundary conditions
S. Stuvard;
2021
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↑∞, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.
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