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

TL;DR

This paper provides a counterexample to a conjecture by demonstrating an isotropic, rank-one convex function on  with compact connected sublevel sets that is not polyconvex, challenging previous assumptions in the field.

Contribution

The authors construct an explicit example of a non-polyconvex, rank-one convex, isotropic function on  with compact connected sublevel sets, disproving a longstanding conjecture.

Findings

Counterexample to the conjecture relating rank-one convexity and polyconvexity.

Existence of an isotropic, rank-one convex function on  that is not polyconvex.

The sublevel sets of the constructed function are compact and connected.

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.