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Anisotropic mean curvature on facets and relations with capillarity

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Given an anisotropy ɸ on R3, we discuss the relations between the ɸ-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 ̃︀ BFɸ of the unit ball of ɸ, ɸ-calibrability is equivalent to show the existence of a ɸ-subunitary vector field in F, with suitable normal trace on @F, and with constant divergence equal to the ɸ-mean curvature of F. Assuming E convex at F, ̃︀ BFɸ a disk, and F (strictly) ɸ-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 ɸ-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C1,1, the solution of the total variation flow starting at 1F.
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  • S.I.S.S.A., via Bonomea 265, 34136, Trieste, Italy
  • Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca
    Scientifica 1, 00133 Roma, Italy
  • INFN Laboratori Nazionali di Frascati (LNF), via E. Fermi 40, Frascati 00044 Roma, Italy
  • S.I.S.S.A., via Bonomea 265, 34136, Trieste, Italy
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