Mathematical Analysis/Calculus of Variations
Polyconvexity equals rank-one convexity for connected isotropic sets in 𝕄 2×2
Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 233-238.

We give a short, self-contained argument showing that, for compact connected sets in 𝕄 2×2 which are invariant under the left and right action of SO(2), polyconvexity is equivalent to rank-one convexity (and even to lamination convexity). As a corollary, the same holds for O(2)-invariant compact sets. These results were first proved by Cardaliaguet and Tahraoui. We also give an example showing that the assumption of connectedness is necessary in the SO(2) case.

Nous donnons un argument simple montrant que pour les ensembles connexes et compacts dans M2×2 qui sont invariants sous les actions à gauche et à droite de SO(2) la polyconvexité est équivalente à la 1-rang convéxité et même à la lamination-convexité. Comme corollaire la même chose est vraie pour les ensembles compacts O(2)-invariants. Ces résultats ont été démontrés par Cardaliaguet et Tahraoui pour la première fois. Nous donnons aussi un exemple montrant que l'hypothèse de connectivité est nécessaire pour le cas SO(2).

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DOI: 10.1016/S1631-073X(03)00333-9
Conti, Sergio 1; De Lellis, Camillo 1; Müller, Stefan 1; Romeo, Mario 1

1 Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
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Conti, Sergio; De Lellis, Camillo; Müller, Stefan; Romeo, Mario. Polyconvexity equals rank-one convexity for connected isotropic sets in $ \mathbb{M}^{\mathrm{2\times 2}}$. Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 233-238. doi : 10.1016/S1631-073X(03)00333-9. http://www.numdam.org/articles/10.1016/S1631-073X(03)00333-9/

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