An example of a quasiconvex function that is not polyconvex in two dimensions

Jean-Jacques Alibert (1) , Bernard Dacorogna (2)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 Section de Mathematiques [Lausanne]

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.

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Jacques Alibert, Bernard Dacorogna. An example of a quasiconvex function that is not polyconvex in two dimensions. Archive for Rational Mechanics and Analysis, 1992, 117 (2), pp.155-166. ⟨10.1007/BF00387763⟩. ⟨hal-00993881⟩