Browsing by Author "Bellettini, Giovanni"

Now showing 1 - 4 of 4
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    Constrained BV functions on double coverings for Plateau's type problems
    (2014) Amato, Stefano; Bellettini, Giovanni; Paolini, Maurizio; Mathematics
    We 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.
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    On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity
    (2013-10-09) Bellettini, Giovanni; Tealdi, Lucia; Paolini, Maurizio; Mathematics
    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'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.
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    Results on the minimization of the Dirichlet functional among semicartesian parametrizations
    (2015-08-07) Tealdi, Lucia; Bellettini, Giovanni; Paolini, Maurizio; Mathematics
    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'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.
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    Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity
    (2015-08-07) Tealdi, Lucia; Bellettini, Giovanni; Paolini, Maurizio; Mathematics
    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.