01389nas a2200133 4500008004300000245007200043210006900115260001500184520094300199100001901142700001801161700002501179856005101204 2015 en_Ud 00aAnisotropic mean curvature on facets and relations with capillarity0 aAnisotropic mean curvature on facets and relations with capillar bde Gruyter3 aWe 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.1 aAmato, Stefano1 aTealdi, Lucia1 aBellettini, Giovanni uhttp://urania.sissa.it/xmlui/handle/1963/3448100947nas a2200121 4500008004100000245009700041210006900138520050200207100001800709700002500727700002200752856005100774 2015 en d00aResults on the minimization of the Dirichlet functional among semicartesian parametrizations0 aResults on the minimization of the Dirichlet functional among se3 aWe 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'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.1 aTealdi, Lucia1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://urania.sissa.it/xmlui/handle/1963/3448801475nas a2200121 4500008004100000245011300041210006900154520101400223100001801237700002501255700002201280856005101302 2015 en d00aSemicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity0 aSemicartesian surfaces and the relaxed area of maps from the pla3 aWe 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.1 aTealdi, Lucia1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://urania.sissa.it/xmlui/handle/1963/3448301209nas a2200121 4500008004300000245007700043210006900120520072600189100001900915700002500934700002200959856010600981 2014 en_Ud 00aConstrained BV functions on double coverings for Plateau's type problems0 aConstrained BV functions on double coverings for Plateaus type p3 aWe link Brakke's "soap films" covering construction with the theory of finite perimeter
sets, in order to study Plateau'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.1 aAmato, Stefano1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://www.math.sissa.it/publication/constrained-bv-functions-double-coverings-plateaus-type-problems01509nas a2200133 4500008004100000245011100041210006900152520096400221653002001185100002501205700001801230700002201248856010501270 2013 en d00aOn the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity0 aarea of the graph of a piecewise smooth map from the plane to th3 aIn 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'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.10aArea functional1 aBellettini, Giovanni1 aTealdi, Lucia1 aPaolini, Maurizio uhttp://www.math.sissa.it/publication/area-graph-piecewise-smooth-map-plane-plane-curve-discontinuity01012nas a2200145 4500008004100000020001500041245007100056210006500127520043600192653007200628100001900700700002500719700002200744856010000766 2013 en d a887642472400aThe nonlinear multidomain model: a new formal asymptotic analysis.0 anonlinear multidomain model a new formal asymptotic analysis3 aWe 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.10abidomain model, anisotropic mean curvature, star-shaped combination1 aAmato, Stefano1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://www.math.sissa.it/publication/nonlinear-multidomain-model-new-formal-asymptotic-analysis00375nas a2200121 4500008004100000245004500041210004500086260001800131100002500149700002300174700002100197856003500218 1995 en d00aSpecial functions of bounded deformation0 aSpecial functions of bounded deformation bSISSA Library1 aBellettini, Giovanni1 aCoscia, Alessandra1 aDal Maso, Gianni uhttp://hdl.handle.net/1963/978