  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.
