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 {\textquoteleft}{\textquoteleft}fill the hole{\textquoteright}{\textquoteright} 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.

}, issn = {1618-1891}, doi = {10.1007/s10231-019-00887-0}, url = {https://doi.org/10.1007/s10231-019-00887-0}, author = {Giovanni Bellettini and Alaa Elshorbagy and Maurizio Paolini and Riccardo Scala} } @article {2019, title = {On the square distance function from a manifold with boundary}, year = {2019}, abstract = {We 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.

}, url = {http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf}, author = {Giovanni Bellettini and Alaa Elshorbagy} } @article {BELLETTINI20181, title = {Minimizing movements for mean curvature flow of droplets with prescribed contact angle}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {117}, year = {2018}, pages = {1 - 58}, abstract = {We 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{\textendash}Taylor{\textendash}Wang and Luckhaus{\textendash}Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. R{\'e}sum{\'e} Nous {\'e}tudions le mouvement par courbure moyenne d{\textquoteright}une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit {\'e}ventuellement non constant. En utilisant les solutions construites comme limites d{\textquoteright}un algorithme d{\textquoteright}approximation d{\^u} {\`a} Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l{\textquoteright}existence d{\textquoteright}une {\'e}volution faible, et sa compatibilit{\'e} avec une solution au sens des distributions. Nous d{\'e}montrons {\'e}galement plusieurs r{\'e}sultats de comparaison.

}, keywords = {Capillary functional, Mean curvature flow with prescribed contact angle, Minimizing movements, Sets of finite perimeter}, issn = {0021-7824}, doi = {https://doi.org/10.1016/j.matpur.2018.06.003}, url = {http://www.sciencedirect.com/science/article/pii/S0021782418300825}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @article {doi:10.1137/17M1159294, title = {Minimizing Movements for Mean Curvature Flow of Partitions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {50}, number = {4}, year = {2018}, pages = {4117-4148}, abstract = {We 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.

}, doi = {10.1137/17M1159294}, url = {https://doi.org/10.1137/17M1159294}, author = {Giovanni Bellettini and Shokhrukh Kholmatov} } @article {1534-0392_2017_4_1427, title = {Minimizers of anisotropic perimeters with cylindrical norms}, journal = {Communications on Pure \& Applied Analysis}, volume = {16}, number = {1534-0392_2017_4_142}, year = {2017}, pages = {1427}, abstract = {We study various regularity properties of minimizers of the Φ{\textendash}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.

}, keywords = {anisotropic Bernstein problem;, minimal cones, Non parametric minimal surfaces, Sets of finite perimeter}, issn = {1534-0392}, doi = {10.3934/cpaa.2017068}, url = {http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @article {2013, title = {On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity}, journal = {ESAIM: COCV}, volume = {22}, number = {arXiv:1310.2443;}, year = {2016}, pages = {29-63}, 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}, doi = {10.1051/cocv/2014065}, url = {https://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html}, author = {Giovanni Bellettini and Lucia Tealdi and Maurizio Paolini} } @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 {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} } @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 {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} }