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} } @mastersthesis {2015, title = {The relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces.}, year = {2015}, school = {SISSA}, abstract = {In this thesis we study the relaxation of the area functional w.r.t. the L^1 topology of a map from a bounded planar domain with values in the plane and jumping on a segment. We estimate from above the singular contribution of this functional due to the presence of the jump in terms of the infimum of the area among a suitable family of surfaces that we call semicartesian surfaces. In our analysis, we also introduce a different notion of area, namely the relaxation of the area w.r.t. a convergence stronger than the L^1 convergence, whose singular contribution is completely characterized in terms of suitable semicartesian area minimizing problems. We propose also some examples of maps for which the two notions of relaxation are different: these examples underline the highly non-local behaviour of the L^1-relaxation, and justify the introduction of the other functional. Some result about the existence of a semicartesian area-minimizing surface is also provided.}, keywords = {Area functional}, author = {Lucia Tealdi} } @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} }