On the lower semicontinuity of supremal functional under differential constraints

We study the weak∗ lower semicontinuity of functionals of the form (Equation Presented), where Ω ⊂ ℝ^N is a bounded open set, V ∈ L^∞ (Ω;double-struck M^dxN)∩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, ·) 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.