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

TL;DR

This paper investigates the lower semicontinuity of certain integral functionals in the space $W^{1,1} imes L^p$, providing conditions and integral representations for their lower semicontinuous envelopes under weak-star topology.

Contribution

It offers new insights into the lower semicontinuity properties of functionals in $W^{1,1} imes L^p$ and characterizes their envelopes when not inherently lower semicontinuous.

Findings

Characterization of lower semicontinuity conditions

Integral representation of the lower semicontinuous envelope

Extension of results to functionals with gradient dependence

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.