Polyconvexity equals rank-one convexity for connected isotropic sets in M^2×2

Sergio Conti, Camillo De Lellis, Stefan Müller, and Mario Romeo

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Submission date: 06. Nov. 2002
Pages: 8
published in: Comptes rendus mathematique, 337 (2003) 4, p. 233-238 
DOI number (of the published article): 10.1016/S1631-073X(03)00333-9
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Abstract:
We give a short, self-contained argument showing that, for compact connected sets in 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.