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

On the uniqueness of minimizers for a class of variational problems with Polyconvex integrand