On conditions for polyconvexity

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.