2013/10/09 by Giovanni Bellettini, Bellettini, Giovanni, Maurizio Paolini +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1310.2443
openalex publication_date 2013/10/09 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
In this paper we provide an estimate from above for the value of the relaxed\narea functional for a map defined on a bounded domain of the plane with values\nin the plane and discontinuous on a regular simple curve with two endpoints. We\nshow that, under suitable assumptions, the relaxed area does not exceed the\narea of the regular part of the map, with the addition of a singular term\nmeasuring the area of a disk type solution of the Plateau's problem spanning\nthe two traces of the map on the jump. The result is valid also when the area\nminimizing surface has self intersections. A key element in our argument is to\nshow the existence of what we call a semicartesian parametrization of this\nsurface, namely a conformal parametrization defined on a suitable parameter\nspace, which is the identity in the first component. To prove our result,\nvarious tools of parametric minimal surface theory are used, as well as some\nresult from Morse theory.\n