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.

}, doi = {10.1515/geofl-2015-0005}, url = {http://urania.sissa.it/xmlui/handle/1963/34481}, author = {Stefano Amato and Lucia Tealdi and Giovanni Bellettini} } @article {2014, title = {Constrained BV functions on double coverings for Plateau{\textquoteright}s type problems}, journal = {Adv. Calc. Var.}, year = {2015}, abstract = {We link Brakke{\textquoteright}s "soap films" covering construction with the theory of finite perimeter sets, in order to study Plateau{\textquoteright}s problem without fixing a priori the topology of the solution. The minimization is set up in the class of $BV$ functions defined on a double covering space of the complement of an $(n - 2)$-dimensional smooth compact manifold $S$ without boundary. The main novelty of our approach stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. The model allows to avoid all issues concerning the presence of the boundary $S$. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on $Γ$-convergence theory.

}, author = {Stefano Amato and Giovanni Bellettini and Maurizio Paolini} } @mastersthesis {2015, title = {Some results on anisotropic mean curvature and other phase-transition problems}, year = {2015}, school = {SISSA}, abstract = {The present thesis is divided into three parts. In the first part, we analyze a suitable regularization {\textemdash} which we call nonlinear multidomain model {\textemdash} of the motion of a hypersurface under smooth anisotropic mean curvature flow. The second part of the thesis deals with crystalline mean curvature of facets of a solid set of R^3 . Finally, in the third part we study a phase-transition model for Plateau{\textquoteright}s type problems based on the theory of coverings and of BV functions.}, keywords = {Anisotropic mean curvature}, author = {Stefano Amato} } @article {2013, title = {The nonlinear multidomain model: a new formal asymptotic analysis.}, journal = {Geometry Partial Differential Equations {\textendash} proceedings, CRM Series (15), 2013.}, number = {SISSA preprint;SISSA 54/2013/MATE}, year = {2013}, abstract = {We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m- equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.

}, keywords = {bidomain model, anisotropic mean curvature, star-shaped combination}, isbn = {8876424724}, author = {Stefano Amato and Giovanni Bellettini and Maurizio Paolini} }