A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets

Jendrik Voss; Ionel-Dumitrel Ghiba; Robert J. Martin; Patrizio Neff

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arXiv · 2009.09690

A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets

Jendrik Voss·Ionel-Dumitrel Ghiba·Robert J. Martin·Patrizio Neff

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Abstract

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\operatorname{GL}^+(2)$ of invertible $2\times2-\,$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 $\operatorname{GL}^+(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W:\operatorname{GL}^+\to\mathbb{R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

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Jendrik Voss, Ionel-Dumitrel Ghiba, Robert J. Martin, Patrizio Neff. 2020-09-21. A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets. https://arxiv.org/abs/2009.09690

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