Lower semicontinuity of supremal functional under differential constraint

We study the weak* lower semicontinuity of functionals of the form $$ F(V)=supess_{x in Om} f(x,V (x)),, $$ where $Omsubset R^N$ is a bounded open set, $Vin L^{infty}(Omega;MM)cap Ker A$ and $A$ is a constant-rank partial differential operator. The notion of $A$-Young quasiconvexity, which is introduced here, provides a sufficient condition when $f(x,cdot)$ is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity.