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

Let p be a given positively one-homogeneous convex function, and let W be the corresponding unit ball. Motivated by motion by crystalline mean curvature in three dimensions, we introduce and study a class of smooth boundaries in the relative geometry induced by p. One can realize that, even when W is a polytope, such smooth class cannot be reduced to the class of polyhedral boundaries (locally resembling the boundary of W). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of W) is source of several technical difficulties, related to the geometry of Lipschitz manifolds. Given a boundary ∂E in our class, we study 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 p-mean curvature kp of ∂E; the function kp is expected to be the initial velocity of ∂E, whenever ∂E is considered as the initial datum for the corresponding (nonsmooth) anisotropic mean curvature flow. We prove that kp is bounded on ∂E, and that it is of bounded variation on facets F of ∂E corresponding to facets of the boundary of W (if any). Finally, further regularity properties of kp and of its jump set are inspected: in particular, in three space dimensions, we relate the sublevel sets of kp on F with the geometry of the boundary of W. PART 1.