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.