A note on relaxation with constraints on the determinant

We consider multiple integrals of the Calculus of Variations of the form $E(u)=int W(x,u(x),Du(x)), dx$ where $W$ is a Carath'eodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, $det Du>0$ or $det Du=1$, respectively. Under suitable growth and lower semicontinuity assumptions in the $u$ variable we prove that the functional $int W^{qc}(x,u(x),Du(x)), dx$ is an upper bound for the relaxation of $E$ and coincides with the relaxation if the quasiconvex envelope $W^{qc}$ of $W$ is polyconvex and satisfies $p$ growth from below for $p$ bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal.217 (2015) 413-437] relative to the case where $W$ depends only on the gradient variable.