  We study Lipschitz critical points of the energy $\int_\Omega g(\det D u) \,
d x$ in two dimensions, where $g$ is a strictly convex function. We prove that
the Jacobian of any Lipschitz critical point is constant, and that the
Jacobians of sequences of approximately critical points converge strongly. The
latter result answers in particular an open problem posed by Kirchheim,
M\"uller and \v{S}ver\'ak in 2003.
