On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

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 Ω of the plane taking values in the real plane and discontinuous on a simple curve, with two endpoints. We show that, under certain assumptions, the relaxed area does not exceed the area of the regular part of the function, 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 function across the jump set. The result is valid also when the minimal surface has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of the minimal 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 results from Morse theory.