秩- $(n-1)$凸性与${\mathcal S}$ -拟凸性的关系
On the relationship between rank-$(n-1)$ convexity and ${\mathcal S}$-quasiconvexity
摘要：
我们证明了秩- $(n-1)$凸性并不意味着${\mathcal S}$ -拟凸性（即，对于$m>n$，在${\mathbb M}^{m\times n}$中，通过将众所周知的Sverak反例[5]适应于螺线管设置，另一方面，我们也注意到在对角矩阵$n\times n$空间中秩- $(n-1)$凸性和${\mathcal S}$-拟凸性是等价的.这是对穆勒工作的概括[4]。
英文摘要：
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].
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