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MiS Preprint
100/2007

On the relationship between rank-(n-1) convexity and S-quasiconvexity

Mariapia Palombaro

Abstract

We prove that rank-$(n-1)$ convexity does not imply ${\mathcal S}$-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in the space of $m\times n$ matrices for $m>n$, by adapting the well-known Šverák's counterexample to the solenoidal setting. On the other hand, we also remark that rank-$(n-1)$ convexity and ${\mathcal S}$-quasiconvexity turn out to be equivalent in the space of $n\times n$ diagonal matrices. This follows by a generalization of Müller's work.

Received:
Nov 7, 2007
Published:
Nov 7, 2007
MSC Codes:
49J45
Keywords:
a-quasiconvexity, quasiconvexity, lower semicontinuity

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Preprint
2007 Repository Open Access
Mariapia Palombaro

On the relationship between rank-\((n-1)\) convexity and \({\mathcal S}\)-quasiconvexity