A Polyconvex Integrand; Euler-Lagrange Equations and Uniqueness of Equilibrium

In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy (Formula presemted.). We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form (Formula presented.), as well as their Euler-Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form (Formula presented.) - such that (Formula presented.) - which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler-Lagrange equations.