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 $F:{\mathbb{R}}^{2\times m}\to \mathbb{R}$ 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 $p\ne 2$.

arXiv comment30 pages. The new version includes new examples in dimension 2x3 and 3x2, and some proofs have been streamlined. Comments are more than welcome