Calculus of Variations and Geometric Measure Theory

S. Conti - C. De Lellis - S. Mueller - M. Romeo

Polyconvexity equals rank-one convexity for connected isotropic sets in $M^{2\times 2}$

created on 14 Nov 2002
modified by delellis on 03 May 2011


Published Paper

Inserted: 14 nov 2002
Last Updated: 3 may 2011

Journal: C. R. Math. Acad. Sci. Paris
Volume: 337
Number: 4
Pages: 233-238
Year: 2003


We give a short, self-contained argument showing that, for compact connected sets in $M^{2\times 2}$ 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.

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