In this paper, we estimate the area of the graph of a map u: Ω⊂ R2→ R2 discontinuous on a segment Ju, with Ju either compactly contained in the bounded open set Ω , or starting and ending on ∂Ω. We characterize A¯ ∞(u, Ω) , the relaxed area functional in a sort of uniform convergence, in terms of the infimum of the area of those surfaces in R3 spanning the graphs of the traces of u on the two sides of Ju and having what we have called a semicartesian structure. We exhibit examples showing that A¯ (u, Ω) , the relaxed area in L1(Ω; R2) , may depend on the values of u far from Ju and also on the relative position of Ju with respect to ∂Ω. These examples confirm the highly non-local behavior of A¯ (u, ·) and justify the interest in the study of A¯ ∞. Finally we prove that A¯ (u, ·) is not subadditive for a rather large class of discontinuous maps u.

Bellettini, G., Paolini, M., Tealdi, L., Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity, <<ANNALI DI MATEMATICA PURA ED APPLICATA>>, 2016; 195 (6): 2131-2170. [doi:10.1007/s10231-016-0556-9] [http://hdl.handle.net/10807/99225]

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

Paolini, Maurizio
Primo
;
2016

Abstract

In this paper, we estimate the area of the graph of a map u: Ω⊂ R2→ R2 discontinuous on a segment Ju, with Ju either compactly contained in the bounded open set Ω , or starting and ending on ∂Ω. We characterize A¯ ∞(u, Ω) , the relaxed area functional in a sort of uniform convergence, in terms of the infimum of the area of those surfaces in R3 spanning the graphs of the traces of u on the two sides of Ju and having what we have called a semicartesian structure. We exhibit examples showing that A¯ (u, Ω) , the relaxed area in L1(Ω; R2) , may depend on the values of u far from Ju and also on the relative position of Ju with respect to ∂Ω. These examples confirm the highly non-local behavior of A¯ (u, ·) and justify the interest in the study of A¯ ∞. Finally we prove that A¯ (u, ·) is not subadditive for a rather large class of discontinuous maps u.
Inglese
Bellettini, G., Paolini, M., Tealdi, L., Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity, <<ANNALI DI MATEMATICA PURA ED APPLICATA>>, 2016; 195 (6): 2131-2170. [doi:10.1007/s10231-016-0556-9] [http://hdl.handle.net/10807/99225]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/99225
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