Existence and uniqueness for anisotropic and crystalline mean curvature flows
Provides a rigorous foundation for crystalline mean curvature flows, a key model in materials science and image processing, by resolving existence and uniqueness for general anisotropies.
The paper proves existence and uniqueness (up to fattening) for crystalline mean curvature flows with forcing and convex mobilities, introducing a new level set solution that satisfies comparison and stability. The results hold in any dimension for arbitrary initial closed sets.
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 a solution satisfies the comparison principle and a stability property 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 by-product of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) "flat flow" in the case of general, possibly crystalline, anisotropies is settled.