An example of a quasiconvex function that is not polyconvex in two dimensions
Résumé
We study the different notions of convexity for the function f γ(ξ) = |ξ|2 (|ξ|2 − 2γ det ξ) where ξ ε ℝ2×2, introduced by Dacorogna & Marcellini. We show that f γ is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦γ¦≦ 2/3 √2, 1, 1+ɛ (for some ɛ>0), 2/√3, respectively.