01509nas a2200133 4500008004100000245011100041210006900152520096400221653002001185100002501205700001801230700002201248856010501270 2013 en d00aOn the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity0 aarea of the graph of a piecewise smooth map from the plane to th3 aIn 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.10aArea functional1 aBellettini, Giovanni1 aTealdi, Lucia1 aPaolini, Maurizio uhttp://www.math.sissa.it/publication/area-graph-piecewise-smooth-map-plane-plane-curve-discontinuity