Calculus of Variations and Geometric Measure Theory
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M. Amar - V. De Cicco - N. Fusco

Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

created by amar on 13 Jul 2006
modified on 25 Sep 2008


Published Paper

Inserted: 13 jul 2006
Last Updated: 25 sep 2008

Volume: 14
Pages: 456-477
Year: 2008


New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in $\BV(\Om)$ are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further $\BV$ dependence, a representation formula for the relaxed functional is also obtained.


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