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

Abstract

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