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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
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