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We show weak lower semi-continuity of functionals assuming the new notion ofa "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 weshow that the functional is still (sequentially) weakly lower semi-continuousalong weakly convergent "convexly constrained" $\mathcal A$-free sequences. Ina motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convexconstraint is positive semi-definiteness of a matrix field.

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