Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. Novaga - M. Paolini

On a crystalline variational problem, part II: $BV$-regularity and structure of minimizers on facets

created on 22 Dec 2001


Published Paper

Inserted: 22 dec 2001

Journal: Arch. Rational Mech. Anal.
Volume: 157
Number: 3
Pages: 193-217
Year: 2001


For a nonsmooth positively one homogeneous convex function $\phi : \ensuremath{\mathbb R^n} \to [0,+\infty[$, it is possible to introduce the class ${\mathcal R}_\phi(\ensuremath{\mathbb R^n})$ of smooth boundaries with respect to $\phi$, to define their $\phi$-mean curvature $\kappa_\phi$, and to prove that, for $E \in {\mathcal R}_\phi(\ensuremath{\mathbb R^n})$, there holds $\kappa_\phi \in L^\infty(\dE)$ \cite{BeNoPa1:00}. Based on these results, we continue the analysis on the structure of $\partial E$ and on the regularity properties of $\kappa_\phi$. We prove that a facet $F$ of $\partial E$ is Lipschitz (up to negligible sets) and that $\kappa_\phi$ has bounded variation on $F$. Further properties of the jump set of $\kappa_\phi$ are inspected: in particular, in three space dimensions, we relate the sublevel sets of $\kappa_\phi$ on $F$ with the geometry of the Wulff shape $\mathcal{W}_\phi}:= \{\p \leq 1\}$.