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 limdet ξ→0+ W (ξ) = ∞. We initiate a study which we hope would lead to a theory for the existence and uniqueness of minimizers of functionals of the form E(u) = ∫ Ω (W (∇u) − F · u)dx, 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 W (ξ) = f(ξ) + h(det ξ) – such that limt→0+ h(t) =∞ – 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.