On conditions for polyconvexity
Jan Kristensen · Proceedings of the American Mathematical Society · 1999
We give an example of a smooth function f : R 2 × 2 → R 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 ⊂ R 2 × 2 B \subset \mathbb R^{2\times 2} of radius one can be extended to a smooth polyconvex function f B : R 2 × 2 → R 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 : R n × m → R g: {\mathbb R}^{n \times m} \to \mathbb R , where n n , m ≥ 2 m \geq 2 . We also briefly discuss connections with quasiconvexity.