An existence theorem for Brakke flow with fixed boundary conditions

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<jats:title>Abstract</jats:title><jats:p>Consider an arbitrary closed, countably <jats:italic>n</jats:italic>-rectifiable set in a strictly convex <jats:inline-formula><jats:alternatives><jats:tex-math>$$(n+1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-dimensional domain, and suppose that the set has finite <jats:italic>n</jats:italic>-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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$t \uparrow \infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>↑</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.</jats:p>

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