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Polyconvexity equals rank-one convexity for connected isotropic sets in M2×2


Conti, S; De Lellis, C; Müller, S; Romeo, M (2003). Polyconvexity equals rank-one convexity for connected isotropic sets in M2×2. Comptes Rendus Mathématique. Académie des Sciences. Paris, 337(4):233-238.

Abstract

We give a short, self-contained argument showing that, for compact connected sets in M2x2 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.

Abstract

We give a short, self-contained argument showing that, for compact connected sets in M2x2 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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:quasiconvexity; singular values; integral functionals
Language:English
Date:2003
Deposited On:17 Sep 2010 09:29
Last Modified:06 Dec 2017 20:49
Publisher:Elsevier
ISSN:1631-073X
Additional Information:English, French summary
Publisher DOI:https://doi.org/10.1016/S1631-073X(03)00333-9
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2009113
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1050.49010

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