On the lower semicontinuity of supremal functional under differential constraint

We study the weak* lower semicontinuity of supremal functionals under a differential constraint that is described by a constant-rank partial differential operator A. The notion of A-Young quasiconvexity, which is introduced here, provides a sufficient condition when the supremand function 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.