Polyconvexity for functions of a system of closed differential forms

Abstract

This paper deals with the weakened convexity properties, mult. ext. quasiconvexity, mult. ext. one convexity, and mult. ext. polyconvexity, for integral functionals of the form $$\begin{aligned} I(\omega _1,\ldots ,\omega _s) = \int _\Omega f(\omega _1,\ldots ,\omega _s) dx \end{aligned}$$ where \(\omega _1,\ldots ,\omega _s\) are closed differential forms on a bounded open set \(\Omega \subset {\mathbb {R}}^n\). The main results of the paper are explicit descriptions of mult. ext. quasiaffine and mult ext. polyconvex functions. It turns out that these two classes consist, respectively, of linear and convex combinations of the set of all wedge products of exterior powers of the forms \(\omega _1,\ldots ,\omega _s\). Thus, for example, a function \(f=f(\omega _1,\ldots ,\omega _s)\) is mult. ext. polyconvex if and only if $$\begin{aligned} f(\omega _1,\ldots ,\omega _s) = \Phi (\ldots ,\omega _1^{q_1}\wedge \cdots \wedge \omega _s^{q_s},\ldots ) \end{aligned}$$ where \(q_1,\ldots ,q_s\) ranges a finite set of integers and \(\Phi \) is a convex function. An existence theorem for the minimum energy state is proved for mult. ext. polyconvex integrals. The polyconvexity in the calculus of variations and nonlinear elasticity are particular cases of mult. ext. polyconvexity. Our main motivation comes from electro-magneto-elastic interactions in continuous bodies. There the mult. ext. polyconvexity takes the form determined by an involved direct calculation in an earlier paper of the author (Silhavý in Math Mech Solids, 2017. http://journals.sagepub.com/doi/metrics/10.1177/1081286517696536).