Lower semicontinuous envelopes in $w^{1,1}\times l^p$

Abstract

It is studied the lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $(W^{1,1}\times L^p)$-weak \ast topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope.