@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}
}
@article {2015,
title = {Results on the minimization of the Dirichlet functional among semicartesian parametrizations},
year = {2015},
note = {The article is compsed of 18 pages and is recorded in PDF format},
abstract = {We start to investigate the existence of conformal minimizers for the Dirichlet functional in the setting of the so-called semicartesian parametrizations, adapting to this context some techniques used in solving the classical Plateau{\textquoteright}s problem.
The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in the study of the relaxed area functional for maps from the plane to the plane jumping on a line.},
url = {http://urania.sissa.it/xmlui/handle/1963/34488},
author = {Lucia Tealdi and Giovanni Bellettini and Maurizio Paolini}
}
@article {2015,
title = {Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity},
year = {2015},
note = {The preprint is compsed of 37 pages and is recorded in PDF format},
abstract = {We address the problem of estimating the area of the graph of a map u, defined on a bounded planar domain O and taking values in the plane, jumping on a segment J, either compactly contained in O or having both the end points on the boundary of O.
We define the relaxation of the area functional w.r.t. a sort of uniform
convergence, and we characterize it in terms of the infimum of the area among those surfaces in the space spanning the graphs of the traces of u on the two side of J and having what we have called a semicartesian structure.
We exhibit examples showing that the relaxed area functional w.r.t the L^1 convergence may depend also on the values of u far from J, and on the relative position of J w.r.t. the boundary of O; these examples confirm the non-local behaviour of the L^1 relaxed area functional, and justify the interest in studying the relaxation w.r.t. a stronger convergence. We prove also that the L^1 relaxed area functional in non-subadditive for a rather class of maps.},
url = {http://urania.sissa.it/xmlui/handle/1963/34483},
author = {Lucia Tealdi and Giovanni Bellettini and Maurizio Paolini}
}
@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 = {On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity},
number = {arXiv:1310.2443;},
year = {2013},
abstract = {In this paper we provide an estimate from above for the value of the relaxed
area functional for a map defined on a bounded domain of the plane with values
in the plane and discontinuous on a regular simple curve with two endpoints. We
show that, under suitable assumptions, the relaxed area does not exceed the
area of the regular part of the map, with the addition of a singular term
measuring the area of a disk type solution of the Plateau{\textquoteright}s problem spanning
the two traces of the map on the jump. The result is valid also when the area
minimizing surface has self intersections. A key element in our argument is to
show the existence of what we call a semicartesian parametrization of this
surface, namely a conformal parametrization defined on a suitable parameter
space, which is the identity in the first component. To prove our result,
various tools of parametric minimal surface theory are used, as well as some
result from Morse theory.},
keywords = {Area functional},
author = {Giovanni Bellettini and Lucia Tealdi 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}
}
@article {1995,
title = {Special functions of bounded deformation},
number = {SISSA;76/95/M},
year = {1995},
publisher = {SISSA Library},
url = {http://hdl.handle.net/1963/978},
author = {Giovanni Bellettini and Alessandra Coscia and Gianni Dal Maso}
}