  This paper relates the lower semi-continuity of an integral functional in the
compensated compactness setting of vector fields satisfying a constant-rank
first-order differential constraint, to closed $\mathcal{A}$-$p$ quasiconvexity
of the integrand. The lower semi-continuous envelope of relaxation is
identified for continuous, but potentially extended real-valued integrands. We
discuss the continuity assumption and show that when it is dropped our notion
of quasiconvexity is still equivalent to lower semi-continuity of the integrand
under an additional assumption on the characteristic cone of $\mathcal{A}$.
