On a crystalline variational problem, Part I : First variation and global L∞ regularity

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