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

Abstract

We construct, for any exponent $p\in(1,\infty)$, $p$-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 $\mathbb{R}^{2\times 3}$: for $p=4$, our example is an explicit quartic polynomial. 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 $p\neq 2$.