Polyconvexity for functions of a system of closed differential forms

• Published in: Calculus of variations and partial differential equations 2018-02, Vol.57 (1), p.1-26, Article 26
• Main Author: Šilhavý, M.
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Convexity; Elasticity; Existence theorems; Functionals; Integers; Integrals; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-017-1298-2

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 I ( ω 1 , … , ω s ) = ∫ Ω f ( ω 1 , … , ω s ) d x where ω 1 , … , ω s are closed differential forms on a bounded open set Ω ⊂ 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 ω 1 , … , ω s . Thus, for example, a function f = f ( ω 1 , … , ω s ) is mult. ext. polyconvex if and only if f ( ω 1 , … , ω s ) = Φ ( … , ω 1 q 1 ∧ ⋯ ∧ ω s q s , … ) where q 1 , … , q s ranges a finite set of integers and Φ 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 (Šilhavý in Math Mech Solids, 2017 . http://journals.sagepub.com/doi/metrics/10.1177/1081286517696536 ).