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

Abstract : 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.
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Contributor : Jean-Jacques Alibert <>
Submitted on : Tuesday, May 20, 2014 - 4:53:57 PM
Last modification on : Tuesday, June 19, 2018 - 3:50:01 PM

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

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