In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to ``fill the hole'' in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

%B Annali di Matematica Pura ed Applicata (1923 -) %8 Jul %G eng %U https://doi.org/10.1007/s10231-019-00887-0 %R 10.1007/s10231-019-00887-0 %0 Report %D 2019 %T On the square distance function from a manifold with boundary %A Giovanni Bellettini %A Alaa Elshorbagy %XWe characterize arbitrary codimensional smooth manifolds $\mathcal{M}$ with boundary embedded in $\mathbb{R}^n$ using the square distance function and the signed distance function from $\mathcal{M}$ and from its boundary. The results are localized in an open set.

%G eng %U http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf %0 Journal Article %J Journal de Mathématiques Pures et Appliquées %D 2018 %T Minimizing movements for mean curvature flow of droplets with prescribed contact angle %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K Capillary functional %K Mean curvature flow with prescribed contact angle %K Minimizing movements %K Sets of finite perimeter %XWe study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren–Taylor–Wang and Luckhaus–Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. Résumé Nous étudions le mouvement par courbure moyenne d'une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit éventuellement non constant. En utilisant les solutions construites comme limites d'un algorithme d'approximation dû à Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l'existence d'une évolution faible, et sa compatibilité avec une solution au sens des distributions. Nous démontrons également plusieurs résultats de comparaison.

%B Journal de Mathématiques Pures et Appliquées %V 117 %P 1 - 58 %G eng %U http://www.sciencedirect.com/science/article/pii/S0021782418300825 %R https://doi.org/10.1016/j.matpur.2018.06.003 %0 Journal Article %J SIAM Journal on Mathematical Analysis %D 2018 %T Minimizing Movements for Mean Curvature Flow of Partitions %A Giovanni Bellettini %A Shokhrukh Kholmatov %XWe prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

%B SIAM Journal on Mathematical Analysis %V 50 %P 4117-4148 %G eng %U https://doi.org/10.1137/17M1159294 %R 10.1137/17M1159294 %0 Journal Article %J Communications on Pure & Applied Analysis %D 2017 %T Minimizers of anisotropic perimeters with cylindrical norms %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K anisotropic Bernstein problem; %K minimal cones %K Non parametric minimal surfaces %K Sets of finite perimeter %XWe study various regularity properties of minimizers of the Φ–perimeter, where Φ is a norm. Under suitable assumptions on Φ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

%B Communications on Pure & Applied Analysis %V 16 %P 1427 %G eng %U http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d %R 10.3934/cpaa.2017068 %0 Journal Article %J ESAIM: COCV %D 2016 %T On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity %A Giovanni Bellettini %A Lucia Tealdi %A Maurizio Paolini %K Area functional %XIn 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.

%B ESAIM: COCV %V 22 %P 29-63 %G en %U https://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html %N 1 %1 7257 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Lucia Tealdi (ltealdi@sissa.it) on 2013-11-29T13:51:49Z No. of bitstreams: 1 bellettini_paolini_tealdi_SISSAprprint.pdf: 705021 bytes, checksum: 98a550aeb5925de05f6b419569ccd283 (MD5) %R 10.1051/cocv/2014065 %0 Journal Article %D 2015 %T Anisotropic mean curvature on facets and relations with capillarity %A Stefano Amato %A Lucia Tealdi %A Giovanni Bellettini %XWe 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.

%I de Gruyter %G en_US %U http://urania.sissa.it/xmlui/handle/1963/34481 %1 34663 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by Stefano Amato (samato@sissa.it) on 2015-08-06T08:33:21Z No. of bitstreams: 1 amato_bellettini_tealdi.pdf: 561517 bytes, checksum: ee2370b8b8ae04078e1e8eae84d37bdd (MD5) %R 10.1515/geofl-2015-0005 %0 Journal Article %J Adv. Calc. Var. %D 2015 %T Constrained BV functions on double coverings for Plateau's type problems %A Stefano Amato %A Giovanni Bellettini %A Maurizio Paolini %XWe 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.

%B Adv. Calc. Var. %G en_US %1 7597 %$ Submitted by Stefano Amato (samato@sissa.it) on 2014-12-12T07:20:56Z No. of bitstreams: 1 bvcovering_gen.pdf: 520748 bytes, checksum: 7b3d78994ff2d2ae5cbbe87eaa5f0777 (MD5) %0 Report %D 2015 %T Results on the minimization of the Dirichlet functional among semicartesian parametrizations %A Lucia Tealdi %A Giovanni Bellettini %A Maurizio Paolini %XWe 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.

%G en %U http://urania.sissa.it/xmlui/handle/1963/34488 %1 34671 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by Lucia Tealdi (ltealdi@sissa.it) on 2015-08-07T16:12:37Z No. of bitstreams: 1 bellettini_paolini_tealdi_semicart.pdf: 338138 bytes, checksum: c689a5b64d30dc600fe0429b72996c7d (MD5) %0 Report %D 2015 %T Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity %A Lucia Tealdi %A Giovanni Bellettini %A Maurizio Paolini %XWe 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.

%G en %U http://urania.sissa.it/xmlui/handle/1963/34483 %1 34670 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by Lucia Tealdi (ltealdi@sissa.it) on 2015-08-07T16:06:48Z No. of bitstreams: 1 bellettini_paolini_tealdi_semicartesian.pdf: 623410 bytes, checksum: 871f66c4686a39ad7d1bc996f0e2b0aa (MD5) %0 Journal Article %J Geometry Partial Differential Equations – proceedings, CRM Series (15), 2013. %D 2013 %T The nonlinear multidomain model: a new formal asymptotic analysis. %A Stefano Amato %A Giovanni Bellettini %A Maurizio Paolini %K bidomain model, anisotropic mean curvature, star-shaped combination %XWe 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.

%B Geometry Partial Differential Equations – proceedings, CRM Series (15), 2013. %@ 8876424724 %G en %1 7259 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Stefano Amato (samato@sissa.it) on 2013-11-29T15:38:34Z No. of bitstreams: 1 bidomain_gen.pdf: 320909 bytes, checksum: a47cfa6c6dec3318d5d75f8a6a4a425b (MD5) %0 Journal Article %D 1995 %T Special functions of bounded deformation %A Giovanni Bellettini %A Alessandra Coscia %A Gianni Dal Maso %I SISSA Library %G en %U http://hdl.handle.net/1963/978 %1 3476 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:41:13Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995