1 results
A rank-one convex, nonpolyconvex isotropic function on
$\textrm {GL}^{\!+}(2)$ with compact connected sublevel sets
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 2 / April 2022
- Published online by Cambridge University Press:
- 02 February 2022, pp. 356-381
- Print publication:
- April 2022
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- Article
- Export citation
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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.
