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On isotropic rank 1 convex functions

Published online by Cambridge University Press:  14 November 2011

Miroslav Šilhavý
Miroslav Šilhavý
Affiliation:
Mathematical Institute of the AV ČR, Žitná 25, 115 67 Prague 1. Czech Republic
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Abstract

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 1081 - 1105
Copyright
Copyright © Royal Society of Edinburgh 1999

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