A remark on the L1-lower semicontinuity for integral functionals in BV

We study the L1-lower semicontinuity in BV of an integral functional of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\displaystyle{{ \int_{{\Omega}}f(x,u,\nabla u)dx+ \int_{{\Omega}} {{-\kern-,375cm\int}}_{{u^-(x)}}^{{u^+(x)}}f^\infty(x,t,\frac{{D^su}}{{|D^su|}})dt d|D^su|}}}}$\end{document}. Our assumptions on f extend previous results recently obtained by Gori, Maggi and Marcellini in the case where the above functional is restricted to W1,1.