  We prove the existence, uniqueness, and regularity of minimizers of a
polyconvex functional in two and three dimensions, which corresponds to the
$H^1$ projection of measure-preserving maps. Our result introduces a new
criteria on the uniqueness of the minimizer, based on the smallness of the
lagrange multiplier. No estimate on the second derivatives of the pressure is
needed to get a unique global minimizer. As an application, we construct a
minimizing movement scheme to construct $L^r$ solutions of the Navier-Stokes
equation for a short time interval.
