A rank-one convex, nonpolyconvex isotropic function on $\textrm {GL}^{\!+}(2)$ with compact connected sublevel sets. Issue 2 (April 2022)

• Record Type: Journal Article
• Authors: Voss, Jendrik; Ghiba, Ionel-Dumitrel; Martin, Robert J.; Neff, Patrizio
• Is Part Of: Proceedings. Volume 152:Issue 2(2022)
• Page Start: 356
• Page End: 381
• Publication Date: 2022-04
• Subjects: quasiconvexity -- rank-one convexity -- polyconvexity -- nonlinear elasticity -- hyperelasticity -- weak lower semi-continuity -- isotropic sets -- calculus of variations; 26B25 -- 26A51 -- 74B20; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2021.9
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.