Lower Semi-Continuity for $\mathcal{A}$-Quasiconvex Functionals under Convex Restrictions

We show weak lower semi-continuity of functionals assuming the new notion of a “convexly constrained” $\mathcal{A}$-quasiconvex integrand. We assume $\mathcal{A}$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent “convexly constrained” $\mathcal{A}$-free sequences. In a motivating example, the integrand is $-{\det }^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.