01524nas a2200157   4500008004100000245011100041210006900152300001000221490000700231520095900238653002001197100002501217700001801242700002201260856008401282       2016                        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 th  a29-630 v223 a<p>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.</p>10aArea functional1 aBellettini, Giovanni1 aTealdi, Lucia1 aPaolini, Maurizio  uhttps://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html01509nas a2200121   4500008004100000245012400041210006900165260001000234520098200244653002001226100001801246856012301264       2015                        en  d00aThe relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces.0 arelaxed area of maps from the plane to the plane with a line dis  bSISSA3 aIn this thesis we study the relaxation of the area functional w.r.t. the L^1 topology of a map from a bounded planar domain with values in the plane and jumping on a segment.  
We estimate from above the singular contribution of this functional due to the presence of the jump in terms of the infimum of the area among a suitable family of surfaces that we call semicartesian surfaces. In our analysis, we also  introduce a different notion of area, namely the relaxation of the area w.r.t. a convergence stronger than the L^1 convergence, whose singular contribution is completely characterized in terms of suitable semicartesian area minimizing problems.  We propose also some examples of maps for which the two notions of relaxation are different: these examples underline the highly non-local behaviour of the L^1-relaxation, and justify the introduction of the other functional. Some result about the existence of a semicartesian area-minimizing surface is also provided.10aArea functional1 aTealdi, Lucia  uhttps://www.math.sissa.it/publication/relaxed-area-maps-plane-plane-line-discontinuity-and-role-semicartesian-surfaces