We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $\mathbb{R}^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface.We establish longtime-existence of the flow and investigate the projection of the flowing surface onto $\mathbb{R}^n$, the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in $\mathbb{R}^n$ with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.

中文翻译：

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图形平均曲率流的阴影

我们考虑初始表面的平均曲率流，它是$ \ mathbb {R} ^ n $中定义的某个域上的函数图。如果该图不完整，则在曲面的边界处施加恒定的Dirichlet边界条件。建立流的长时间存在状态，并研究流向投影在阴影\\ mathbb {R} ^ n $上的投影流。该运动阴影可以看作是Dirichlet边界条件下$ \ mathbb {R} ^ n $中超表面平均曲率流的一个弱解。此外，我们提供了一个独立兴趣的引理，以尊重曲率条件的方式局部平移两个平滑开放集的交集的边界。