On a crystalline variational problem, part I: first variation and global L^infty regularity

Let φ : ℝn → [0. +∞[ be a given positively one-homogeneous convex function, and let Wφ := {φ ≦ 1}. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ℛφ(ℝn) of "smooth" boundaries in the relative geometry induced by the ambient Banach space (ℝn, φ). It can be seen that, even when Wφ is a polytope, ℛφ(ℝn) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂Wφ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂Wφ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary ∂E in the class ℛφ(ℝn), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on ∂E. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on ∂E. We define such a divergence to be the φ-mean curvature Kφ of ∂E; the function Kφ is expected to be the initial velocity of ∂E, whenever ∂E is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that Kφ is bounded on ∂E and that its sublevel sets are characterized through a variational inequality.