@article {2015,
title = {Anisotropic mean curvature on facets and relations with capillarity},
number = {Geometric Flows;1},
year = {2015},
publisher = {de Gruyter},
abstract = {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}
}
@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 {2014,
title = {Constrained BV functions on double coverings for Plateau{\textquoteright}s type problems},
year = {2014},
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}
}
@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}
}