Title: Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow,
Abstract: We show that every bounded subset of an Euclidean space can be approximated
by a set that admits a certain vector field, the so-called Cahn-Hoffman vector
field, that is subordinate to a given anisotropic metric and has a
square-integrable divergence. More generally, we introduce a concept of facets
as a kind of directed sets, and show that they can be approximated in a similar
manner.
We use this approximation to construct test functions necessary to prove the
comparison principle for viscosity solutions of the level set formulation of
the crystalline mean curvature flow that were recently introduced by the
authors. As a consequence, we obtain the well-posedness of the viscosity
solutions in an arbitrary dimension, which extends the validity of the result
in the previous paper.