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In this paper we provide an estimate from above for the value of the relaxed area functional A(u,Ω) for an R²-valued map u defined on a bounded domain Ω of the plane and discontinuous on a C² simple curve Ju ⊂ Ω, with two endpoints. We show that, under certain assumptions on u, A(u,Ω) does not exceed the area of the regular part of u, with the addition of a singular term measuring the area of a disk-type solution Σmin of the Plateau's problem spanning the two traces of u on Ju. The result is valid also when Σmin has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of Σmin, namely a conformal parametrization of Σmin 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 results from Morse theory. [ABSTRACT FROM AUTHOR]

We show that the completion problem of reconstructing the hidden arcs of the contours of an image, given only the visible ones, has a solution. More precisely we prove that, given an oriented plane graph K having as vertices only T-junctions and nonexterior terminal points, there exists an apparent contour G such that K is the visible part of G. This result is sharp, since the converse statement is easily seen to be satisfied. As a consequence, from K we can reconstruct a solid shape E in three-dimensional space such that K coincides with the visible part of the apparent contour of E. The main tools used to prove our result are a Morse description of K and the Huffman labelling for apparent contours. [ABSTRACT FROM AUTHOR]

We discuss some examples of nonuniqueness for the crystalline curvature flow, when the Wulff shape is a square not centered at the origin. [ABSTRACT FROM AUTHOR]