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

## TL;DR

This paper proves the existence and uniqueness of minimizers for a class of elastic energy functionals involving polyconvex integrands, including degenerate and incompressible cases, extending prior results in elasticity theory.

## Contribution

It extends previous work by establishing uniqueness for a broader class of polyconvex integrands and analyzes minimizers in degenerate and incompressible scenarios.

## Key findings

- Existence and uniqueness of minimizers for polyconvex energy functionals.
- Minimizers in the degenerate case are measure-preserving maps.
- Incompressible minimizers can be obtained as limits of general displacements.

## Abstract

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.