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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.
