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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 \(|\gamma|\leq 2/\sqrt 3\).

26B25 Convexity of real functions of several variables, generalizations
Full Text: DOI
[1] J. M. Ball [1], Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 64 (1977), 337-403. · Zbl 0368.73040
[2] J. M. Ball [2], Does rank one convexity imply quasiconvexity? In Metastability and incompletely posed problems, eds. S. Antman et al. Springer-Verlag (1987), 17-32.
[3] J. M. Ball [3], Sets of gradients with no rank one connections, J. Math. Pures et Appl. 69 (1990), 241-260. · Zbl 0644.49011
[4] B. Dacorogna [1], Direct methods in the calculus of variations, Springer-Verlag (1989). · Zbl 0703.49001
[5] B. Dacorogna, J. Douchet, W. Gangbo & J. Rappaz [1], Some examples of rank one convex functions in dimension two. Proc. of Royal Soc. of Edinburgh. 114 A (1990), 135-150. · Zbl 0722.49018
[6] B. Dacorogna & P. Marcellini [1], A counterexample in the vectorial calculus of variations. In Material instabilities in continuum mechanics, ed. by J. M. Ball, Oxford Univ. Press (1988), 77-83. · Zbl 0641.49007
[7] C. B. Morrey [1], Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25-53. · Zbl 0046.10803
[8] C. B. Morrey [2], Multiple integrals in the calculus of variations, Spinger-Verlag (1966). · Zbl 0142.38701
[9] C. G. Simader [1], On Dirichlet’s boundary value problem, Lecture Notes in Math., Vol. 268, Springer-Verlag (1972).
[10] V. ?veràk [1], Quasiconvex functions with subquadratic growth, to appear.
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