TY - RPRT
T1 - On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity
Y1 - 2013
A1 - Giovanni Bellettini
A1 - Lucia Tealdi
A1 - Maurizio Paolini
KW - Area functional
AB - 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.
U1 - 7257
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -