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

Fusco, Nicola; Cicco, Virginia De; Amar, Micol

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 456-477.

New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega$) 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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