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

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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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Authors and Affiliations

  1. Laboratoire d'Analyse Numérique, Université Paul Sabatier, 31062, Toulouse, France

    Jean -Jacques Alibert & Bernard Dacorogna

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Jean -Jacques Alibert & Bernard Dacorogna

Authors
  1. Jean -Jacques Alibert
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  2. Bernard Dacorogna
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Communicated by J. Ball

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Alibert, J.J., Dacorogna, B. An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal. 117, 155–166 (1992). https://doi.org/10.1007/BF00387763

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  • DOI: https://doi.org/10.1007/BF00387763

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