%0 Journal Article %J ESAIM: COCV %D 2016 %T On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity %A Giovanni Bellettini %A Lucia Tealdi %A Maurizio Paolini %K Area functional %X

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.

%B ESAIM: COCV %V 22 %P 29-63 %G en %U https://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html %N 1 %1 7257 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Lucia Tealdi (ltealdi@sissa.it) on 2013-11-29T13:51:49Z No. of bitstreams: 1 bellettini_paolini_tealdi_SISSAprprint.pdf: 705021 bytes, checksum: 98a550aeb5925de05f6b419569ccd283 (MD5) %R 10.1051/cocv/2014065 %0 Thesis %D 2015 %T The relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces. %A Lucia Tealdi %K Area functional %X In 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. %I SISSA %G en %1 34732 %4 1 %# MAT/05 %$ Submitted by Lucia Tealdi (ltealdi@sissa.it) on 2015-09-21T12:23:44Z No. of bitstreams: 1 Tealdi_PhD_thesis.pdf: 917244 bytes, checksum: a6bfc90b8457a030be4c7314ee1f426f (MD5)