On a Crystalline Variational Problem, Part I: First Variation and Global L-infinity Regularity

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