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Rank-one convexity does not imply quasiconvexity. (English) Zbl 0777.49015
Let \(\Omega\subset\mathbb{R}^n\) be an open ball. Consider Lipschitz functions \(u: \Omega\to\mathbb{R}^m\). Denote by \(Du\) the gradient of \(u\). Let \(f\) be a (smooth) function on \(m\times n\) matrices which satisfies the Legendre-Hadamard condition: \((\partial^2 f(X)/\partial X_{ij}\partial X_{kl})\bar u^i\xi^j\bar u^k\xi^l\geq 0\) for each \(\bar u\in\mathbb{R}^m\), each \(\xi\in\mathbb{R}^n\) and each matrix \(X\). C. B. Morrey [Pac. J. Math. 2, 25–53 (1952; Zbl 0046.10803)] posed the following question: under the above assumptions, is the integral \(\int_\Omega f(Du)\) lower-semicontinuous with respect to the uniform convergence of uniformly Lipschitz functions? This paper shows that for \(m\ge 3\) and \(n\ge 2\) the answer is no.
Reviewer: V. Šverák (Bonn)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
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References:
[1] Ball, J. Math, pures et appl. 69 pp 241– (1990)
[2] Ball, J. of Func. Anal. 41 pp 2– (1981) · Zbl 0459.35020
[3] DOI: 10.1007/BF00275731 · Zbl 0565.49010
[4] DOI: 10.1007/BF01597353 · Zbl 0019.35203
[5] Tartar, Nonlinear analysis and mechanics: Heriot-Watt Symposium IV pp 136– (1979)
[6] DOI: 10.1098/rspa.1991.0073 · Zbl 0741.49016
[7] Šverák, Proc. Roy. Soc. Edinburgh 114A pp 237– (1990) · Zbl 0714.49024
[8] Sivaloganathan, Ann. Inst. H. Poincaré, Analise Non linéaire 5 pp 99– (1988) · Zbl 0664.73006
[9] Serre, J. Math. pures et appl. 62 pp 177– (1983)
[10] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803
[11] Morrey, Multiple integrals in the calculus of variations (1966) · Zbl 0142.38701
[12] DOI: 10.1007/BF01135336 · Zbl 0825.73029
[13] Dacorogna, Material instabilities in continuum mechanics pp 77– (1988)
[14] Dacorogna, Direct methods in the calculus of variations (1988) · Zbl 0703.49001
[15] DOI: 10.1007/BF00279992 · Zbl 0368.73040
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