  We prove existence and uniqueness of minimizers for a family of energy
functionals that arises in Elasticity and involves polyconvex integrands over a
certain subset of displacement maps. This work extends previous results by Awi
and Gangbo to a larger class of integrands. First, we study these variational
problems over displacements for which the determinant is positive. Second, we
consider a limit case in which the functionals are degenerate. In that case,
the set of admissible displacements reduces to that of incompressible
displacements which are measure preserving maps. Finally, we establish that the
minimizer over the set of incompressible maps may be obtained as a limit of
minimizers corresponding to a sequence of minimization problems over general
displacements provided we have enough regularity on the dual problems. We point
out that these results defy the direct methods of the calculus of variations.
