Existence and uniqueness for anisotropic and crystalline mean curvature flows

Abstract

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) “flat flow” in the case of general, including crystalline, anisotropies, solving a long-standing open question.