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

Maurizio Paolini, Giovanni Bellettini, Lucia Tealdi

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1 Citazioni (Scopus)

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.
Lingua originaleEnglish
pagine (da-a)2131-2170
Numero di pagine40
RivistaAnnali di Matematica Pura ed Applicata
Volume195
DOI
Stato di pubblicazionePubblicato - 2016

Keywords

  • Applied Mathematics
  • Area of graphs
  • Relaxed area functional
  • Semicartesian surfaces

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