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

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 some uniform boundedness property 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.