*Published Paper*

**Inserted:** 22 dec 2001

**Journal:** Arch. Rational Mech. Anal.

**Volume:** 157

**Number:** 3

**Pages:** 165-191

**Year:** 2001

**Abstract:**

Let $\phi : \ensuremath{\mathbb R^n} \to [0,+\infty[$ be a given positively one-homogeneous convex function, and let $\mathcal{W}_\phi := \{\phi \leq 1\}$. Motivated by motion by crystalline mean curvature in three dimensions, we introduce and study the class ${\mathcal R}_\phi(\ensuremath{\mathbb R^n} )$ of ``smooth'' boundaries in the relative geometry induced by the ambient Banach space $(\ensuremath{\mathbb R^n} , \phi)$. One can realize that, even when $\mathcal{W}_\phi$ is a polytope, ${\mathcal R}_\phi({\ensuremath{\mathbb R^n}})$ 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 source of several technical difficulties, related to the geometry of Lipschitz manifolds. Given a boundary $\partial E$ in the class ${\mathcal R}_\phi({\ensuremath{\mathbb R^n}})$, we rigorously compute the first variation of the corresponding anisotropic perimet 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.