  We study integral functionals constrained to divergence-free vector fields in
$L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions,
$1<p<\infty$. We prove that as the thickness of the domain goes to zero, the
Gamma-limit with respect to weak convergence in $L^p$ is always given by the
associated functional with convexified energy density wherever it is finite.
Remarkably, this happens despite the fact that relaxation of nonconvex
functionals subject to the limiting constraint can give rise to a nonlocal
functional as illustrated in an example.
