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

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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Bibliographic Details
Published in:Archive for rational mechanics and analysis 1992, Vol.117 (2), p.155-166
Main Authors: ALIBERT, J.-J, DACOROGNA, B
Format: Article
Language:English
Subjects:
Analysis of PDEs
Calculus of variations and optimal control
Exact sciences and technology
Mathematical analysis
Mathematics
Sciences and techniques of general use
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Summary: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.
ISSN:0003-9527
1432-0673
DOI:10.1007/BF00387763