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.

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

ONOUE, Fumihiko;
2018

Abstract

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.
2018
Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, by Ministry of Education, Culture, Sports, Science, and Technology, Japan
Professor Yoshikazu Giga, The University of Tokyo
Mathematics - Analysis of PDEs; Mathematics - Analysis of PDEs; Mathematics - Differential Geometry; 53C44, 49Q20
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/78997
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