Calculus of Variations and Geometric Measure Theory
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Y. Giga - F. Onoue - K. Takasao

A varifold formulation of mean curvature ow with Dirichlet or dynamic boundary conditions

created by onoue on 07 Oct 2019
modified on 09 Oct 2020


Submitted Paper

Inserted: 7 oct 2019
Last Updated: 9 oct 2020

Year: 2018

ArXiv: 1810.09107 PDF


We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the assumption that the discrepancy measure vanishes on the boundary. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

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