Shadows of graphical mean curvature flow

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.

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Received 19 April 2016

Accepted 19 June 2018

Published 11 March 2021