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EnglishIn 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.
| Autori: | |
| Titolo: | Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s10231-016-0556-9 |
| URL: | http://springerlink.metapress.com/app/home/journal.asp?wasp=cmw755wvtg0qvm8kjj1q&referrer=parent&backto=linkingpublicationresults,1:108198,1 |
| Data di pubblicazione: | 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. |
| Lingua: | Inglese |
| Rivista: | |
| Citazione: | 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] |
| Appare nelle tipologie: | Articolo in rivista, Nota a sentenza |
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