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On conditions for polyconvexity

Author: Jan Kristensen
Journal: Proc. Amer. Math. Soc. 128 (2000), 1793-1797
MSC (1991): Primary 49J10, 49J45
Published electronically: October 29, 1999
MathSciNet review: 1670399
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Abstract: We give an example of a smooth function $f: \mathbb R^{2\times 2} \to \mathbb R$, which is not polyconvex and which has the property that its restriction to any ball $B \subset \mathbb R^{2\times 2}$ of radius one can be extended to a smooth polyconvex function $f_{B}: \mathbb R^{2\times 2} \to \mathbb R$. In particular, it implies that there exists no `local condition' which is necessary and sufficient for polyconvexity of functions $g: {\mathbb R}^{n \times m} \to \mathbb R$, where $n$, $m \geq 2$. We also briefly discuss connections with quasiconvexity.

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Additional Information

Jan Kristensen
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom

Keywords: Polyconvexity, quasiconvexity, rank-$1$ convexity
Received by editor(s): July 29, 1998
Published electronically: October 29, 1999
Additional Notes: Supported by the Danish Natural Science Research Council through grant no. 9501304.
Communicated by: Steven R. Bell
Article copyright: © Copyright 2000 American Mathematical Society