On conditions for polyconvexity

We give an example of a smooth function f : ℝ2×2 → ℝ, which is not polyconvex and which has the property that its restriction to any ball B ⊂ ℝ2×2 of radius one can be extended to a smooth polyconvex function fB : ℝ2×2 → ℝ. In particular, it implies that there exists no 'local condition' which is necessary and sufficient for polyconvexity of functions g : ℝn×m → ℝ, where n, m ≥ 2. We also briefly discuss connections with quasiconvexity. © 2000 American Mathematical Society