2017/02/16 by Yoshikazu Giga, Giga, Yoshikazu, Norbert Požár +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1702.05220
openalex publication_date 2017/02/16 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We show that every bounded subset of an Euclidean space can be approximated\nby a set that admits a certain vector field, the so-called Cahn-Hoffman vector\nfield, that is subordinate to a given anisotropic metric and has a\nsquare-integrable divergence. More generally, we introduce a concept of facets\nas a kind of directed sets, and show that they can be approximated in a similar\nmanner.\n We use this approximation to construct test functions necessary to prove the\ncomparison principle for viscosity solutions of the level set formulation of\nthe crystalline mean curvature flow that were recently introduced by the\nauthors. As a consequence, we obtain the well-posedness of the viscosity\nsolutions in an arbitrary dimension, which extends the validity of the result\nin the previous paper.\n