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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
Zbl 0046.10803
Full Text:
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