*Submitted Paper*

**Inserted:** 18 may 2015

**Last Updated:** 18 may 2015

**Year:** 2015

**Abstract:**

Given an anisotropy $φ$ on $R^3$, we discuss the relations between the $φ$-calibrability of a facet $F ⊂ ∂E$ of a solid crystal $E$, and the capillary problem on a capillary tube with base $F$. When $F$ is parallel to a facet of $Bφ$ (the unit ball of $φ$), $φ$-calibrability is equivalent to show the existence of a $φ$-subunitary vector field in $F$ with suitable normal trace on $∂F$, and with constant divergence equal to the $φ$-mean curvature of $F$. Assuming $E$ convex at $F$, $Bφ$ a disk, and $F$ (strictly) $φ$-calibrable, such a vector field is obtained by solving the capillary problem on $F$ in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict $φ$-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class $C^{1,1}$, the solution of the total variation flow starting at $1_F$.

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