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

TL;DR

This paper investigates the relationship between rank-$(n-1)$ convexity and ${ m extbf{S}}$-quasiconvexity, showing they are not equivalent in general but are in the space of $n imes n$ diagonal matrices, using counterexamples and generalizations.

Contribution

It demonstrates that rank-$(n-1)$ convexity does not imply ${ m extbf{S}}$-quasiconvexity in ${ m M}^{m imes n}$ for $m>n$, and establishes their equivalence in $n imes n$ diagonal matrices.

Findings

Rank-$(n-1)$ convexity does not imply ${ m extbf{S}}$-quasiconvexity in ${ m M}^{m imes n}$ for $m>n$.

In $n imes n$ diagonal matrices, rank-$(n-1)$ convexity and ${ m extbf{S}}$-quasiconvexity are equivalent.

The results adapt Sverak's counterexample and Mueller's work to different settings.

Abstract

We prove that rank-$(n-1)$ convexity does not imply ${\mathcal S}$-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in ${\mathbb M}^{m\times n}$ for $m>n$, by adapting the well-known Sverak's counterexample [5] 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 Mueller's work [4].