2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228599We 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].Analysis of PDEs49J45text