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

TitleSemicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity
Publication TypePreprint
2015
AuthorsTealdi, L, Bellettini, G, Paolini, M

We address the problem of estimating the area of the graph of a map u, defined on a bounded planar domain O and taking values in the plane, jumping on a segment J, either compactly contained in O or having both the end points on the boundary of O.
We define the relaxation of the area functional w.r.t. a sort of uniform
convergence, and we characterize it in terms of the infimum of the area among those surfaces in the space spanning the graphs of the traces of u on the two side of J and having what we have called a semicartesian structure.
We exhibit examples showing that the relaxed area functional w.r.t the L^1 convergence may depend also on the values of u far from J, and on the relative position of J w.r.t. the boundary of O; these examples confirm the non-local behaviour of the L^1 relaxed area functional, and justify the interest in studying the relaxation w.r.t. a stronger convergence. We prove also that the L^1 relaxed area functional in non-subadditive for a rather class of maps.

http://urania.sissa.it/xmlui/handle/1963/34483

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