An example of a quasiconvex function that is not polyconvex in two dimensions (Q1192276)

Let \(\mathbb{R}^{2\times 2}\) be the set of all \(2\times 2\) real matrices endowed with the Euclidean norm, let \(\gamma\) be a real number, and let \(f_ \gamma:\mathbb{R}^{2\times 2}\to\mathbb{R}\) be defined by \(f_ \gamma(\xi)=|\xi|^ 2(|\xi|^ 2-2\gamma \text{det} \xi)\). The authors continue joint investigations by the second author and \textit{P. Marcellini} [see Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 77-83 (1988; Zbl 0641.49007)] and prove the following assertions: (i) \(f_ \gamma\) is convex if and only if \(|\gamma|\leq 2\sqrt 2/3\); (ii) \(f_ \gamma\) is polyconvex if and only if \(|\gamma|\leq 1\); (iii) there exists an \(\varepsilon>0\) such that \(f_ \gamma\) is quasiconvex if and only if \(|\gamma|\leq 1+\varepsilon\); (iv) \(f_ \gamma\) is rank-one convex if and only if \(|\gamma|\leq 2/\sqrt 3\).