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A counterexample in the vectorial calculus of variations. (English) Zbl 0641.49007
Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 77-83 (1988).
[For the entire collection see Zbl 0627.00023.]
The authors consider integrals of the calculus of variations such as \(\int_{\Omega}f(Du)dx\), where u is a vector-valued function. They give and example showing that rank one convexity of f does not imply polyconvexity also if f is not represented as a function g(\(\lambda\),\(\mu)\) of the eigenvalues of (Du \(T,Du)^{1/2}\).
Reviewer: R.Schianchi

49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
49J20 Existence theories for optimal control problems involving partial differential equations