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MiS Preprint
1/1999
On Conditions for Polyconvexity
Jan Kristensen
Abstract
We give an example of a smooth function $f:\mathbb{R}^{2 \times 2} \to \mathbb{R}$, which is not polyconvex and that 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.
