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Some examples of rank one convex functions in dimension two. (English) Zbl 0722.49018
Summary: We study the rank one convexity of some functions f($$\xi$$) where $$\xi$$ is a $$2\times 2$$ matrix. Examples such as $$| \xi |^{2\alpha}+h(\det \xi)$$ and $$| \xi |^{2\alpha}(| \xi |^ 2-\gamma \det \xi)$$ are investigated. Numerical computations are done on the example of the first author and P. Marcellini [in: Material instabilities in continuous mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 77-83 (1988; Zbl 0641.49007)], $$| \xi |^ 4-\frac{4}{\sqrt{3}}| \xi |^ 2 \det \xi$$, indicating that this function is quasiconvex.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 26A51 Convexity of real functions in one variable, generalizations
##### Keywords:
Sobolev space; quasiconvexity; rank one convexity
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##### References:
 [1] Hove, Proc. Koninkl. Ned. Akad. 50 pp 18– (1947) [2] DOI: 10.1007/BF00126996 · Zbl 0323.73010 [3] Dacorogna, Material Instabilities in Continuum Mechanics, Proceedings pp 77– (1988) [4] Dacorogna, Direct methods in the calculus of variations (1989) · Zbl 0703.49001 [5] DOI: 10.1016/0022-1236(84)90041-7 · Zbl 0549.46019 [6] DOI: 10.1007/978-1-4613-8704-6_2 [7] Ball, Arch. Rational Mech. Anal. 64 pp 337– (1977) [8] Aubert, C. R. Acad. Sci. Paris 290 pp 537– (1980) [9] Aubert, Proc. Roy. Soc. Edin. 106 pp 237– (1987) [10] DOI: 10.1090/S0002-9904-1938-06730-4 · Zbl 0018.24201 [11] DOI: 10.1007/BF01597353 · Zbl 0019.35203 [12] Serre, J. Math. Pures Appl. 62 pp 117– (1983) [13] DOI: 10.1090/S0002-9904-1938-06778-X · Zbl 0018.38802 [14] DOI: 10.1098/rspa.1972.0026 · Zbl 0257.73034 [15] Morrey, Multiple integrals in the calculus of variations (1966) · Zbl 0142.38701 [16] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803 [17] DOI: 10.1007/BF01442177 · Zbl 0567.49007 [18] MacShane, Bull. Amer. Math. Soc. 45 (1939) [19] DOI: 10.1002/cpa.3160390107 · Zbl 0609.49008 [20] DOI: 10.1090/S0002-9947-1940-0002839-X · JFM 66.0047.03
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