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

It is studied the lower semicontinuity of functionals of the type $\int_Ωf(x,u,v, \nabla u)dx$ 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.

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