Anisotropic Minkowski flows: well-posedness and asymptotic behaviour

We study anisotropic Minkowski-type curvature flows associated with a perimeter defined through oscillations on Wulff neighbourhoods. The anisotropy may be non-smooth, and in particular crystalline, while the mobility may differ from the anisotropy, provided a natural compatibility condition is satisfied. We compute the first variation of the perimeter and introduce a weak formulation in terms of distance functions and measure-valued nonlocal Cahn-Hoffman fields. By means of a nonlocal minimising movement scheme and suitable estimates for the evolution of Wulff shapes, we prove existence of weak superflows and subflows starting from arbitrary closed sets. Under the additional assumption that the initial level sets have compact boundary, a comparison principle yields existence and uniqueness of the level set flow. Finally, we prove that variational weak flows converge, up to fattening, as $r\searrow 0$ to the weak flow of the local anisotropic, possibly crystalline, curvature motion. The main step in the proof consists in averaging the nonlocal Cahn-Hoffman measures to obtain vector fields converging to a local Cahn-Hoffman field.