Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity

created by tealdi on 07 Aug 2015

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Inserted: 7 aug 2015
Last Updated: 7 aug 2015

Year: 2015

Abstract:

In this paper we estimate the area of the graph of a map $u$ defined on a bounded open planar set $\Omega$ with values in the plane, discontinuous on a segment $J$, with $J$ either compactly contained in $\Omega$, or starting and ending on $\partial \Omega$. We characterize $\overline{A}^\infty(u,\Omega)$, the relaxed area functional in a sort of uniform convergence, in terms of the infimum of the area of those surfaces in the space spanning the graphs of the traces of $u$ on the two sides of $J$ and having what we have called a semicartesian structure. We exhibit examples showing that $\overline{A}(u,\Omega)$, the relaxed area in $L^1$, may depend on the values of $u$ far from $J$ and also on the relative position of $J$ with respect to $\partial\Omega$. These examples confirm the highly non-local behaviour of $\overline{A}(u,\cdot)$, and justify the interest in the study of the relaxation w.r.t. the stronger convergence. Finally we prove that $\rel(u,\cdot)$ is not subadditive for a rather large class of discontinuous maps $u$.