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

TL;DR

This paper develops a varifold-based framework to describe the limit of the Allen-Cahn equation with boundary conditions, capturing mean curvature flow with Dirichlet or dynamic boundaries.

Contribution

It introduces a novel varifold formulation for mean curvature flow with boundary conditions, extending Brakke flow to boundary scenarios.

Findings

Established a varifold characterization of the sharp interface limit.

Extended Brakke flow to include boundary conditions.

Proved existence of the limit using phase field methods.

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 vanishing on the boundary and the boundedness of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.