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Anisotropic mean curvature on facets and relations with capillarity

TitleAnisotropic mean curvature on facets and relations with capillarity
Publication TypeJournal Article
Year of Publication2015
AuthorsAmato, S, Tealdi, L, Bellettini, G

We discuss the relations between the anisotropic calibrability of a facet F of a solid crystal E, and the capillary problem on a capillary tube with base F.
When F is parallel to a facet of the Wulff shape,
calibrability is equivalent to show the existence of
an anisotropic subunitary vector field in $F, with suitable normal trace
on the boundary of the facet, and with constant divergence
equal to the anisotropic mean curvature of F.
When the Wulff shape is a cylynder, assuming E convex at F,
and F (strictly) 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 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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