# Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow

## TL;DR

This paper develops a method to approximate general facets by regular facets with respect to anisotropic total variation energies, enabling the proof of well-posedness for crystalline mean curvature flow in arbitrary dimensions.

## Contribution

It introduces a new approximation technique for facets using Cahn-Hoffman vector fields, facilitating the analysis of crystalline mean curvature flow.

## Key findings

- Established the approximation of facets by regular facets.
- Proved the comparison principle for viscosity solutions.
- Demonstrated well-posedness of the flow in any dimension.

## 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.