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An example of a quasiconvex function that is not polyconvex in two dimensions. (English) Zbl 0761.26009
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 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 $$|\gamma|\leq 2/\sqrt 3$$.

##### MSC:
 26B25 Convexity of real functions of several variables, generalizations
##### Keywords:
quasiconvex functions; polyconvex; rank-one convex
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##### References:
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