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.

中文翻译：

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具有Dirichlet或动态边界条件的平均曲率流的混合公式

我们考虑了具有Dirichlet或动态边界条件的Allen-Cahn方程的尖锐边界极限，并给出了其极限的形变特征，该极限形式上是具有Dirichlet或动态边界条件的平均曲率流。为了显示极限的存在，我们在边界消失和差异测度的某些一致有界性的情况下应用了相场法。为此，我们通过在边界上的褶皱的第一个变化扩展了在这些边界条件下的通常Brakke流。