On a crystalline variational problem. I: First variation and global \(L^\infty\) regularity.

*(English)*Zbl 0976.58016Summary: Let \(\phi: \mathbb{R}^n\to [0, +\infty)\) be a given positively one-homogeneous convex function, and let \({\mathcal W}_\phi := \{\phi \leq 1\}\). Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class \({\mathcal R}_\phi(\mathbb{R}^n)\) of “smooth” boundaries in the relative geometry induced by the ambient Banach space \((\mathbb{R}^n, \phi)\). It can be seen that, even when \({\mathcal W}_\phi\) is a polytope, \({\mathcal R}_\phi(\mathbb{R}^n)\) cannot be reduced to the class of polyhedral boundaries (locally resembling \(\partial {\mathcal W}_\phi)\). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of \(\partial{\mathcal W}_\phi)\) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary \(\partial E\) in the class \({\mathcal R}_\phi(\mathbb{R}^n)\), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on \(\partial E\). It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on \(\partial E\). We define such a divergence to be the \(\phi\)-mean curvature \(\kappa_\phi\) of \(\partial E\); the function \(\kappa_\phi\) is expected to be the initial velocity of \(\partial E\), whenever \(\partial E\) is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that \(\kappa_\phi\) is bounded on \(\partial E\) and that its sublevel sets are characterized through a variational inequality.