A solution to Morrey's problem in $\mathbb{R}^{2\times m}$

Abstract

We construct, for any $p\in(1,\infty)$, $p$-homogeneous rank-one convex integrands $F\colon\mathbb R^{2\times m}\to \mathbb R$ that are nowhere quasiconvex when $m$ is large. When $p$ is sufficiently close to $4$, such examples can be constructed on $\mathbb R^{2\times 4}$. Related constructions give, for every $p$, conjugation- and transposition-invariant examples on $\mathbb R^{d\times d}$, and examples on $\mathbb R^{4\times 2}$ for every $p\neq 2$.