Abstract:
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.
