Morrey's problem in ${\mathbb{R}}^{2\times 4}$ and ${\mathbb{R}}_{\mathrm{sym}}^{3\times 3}$

Abstract

We find an explicit rank-one convex non-quasiconvex integrand in ${\mathbb{R}}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map ${\mathbb{T}}^4\to {\mathbb{R}}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in ${\mathbb{R}}_{\text{sym}}^{4\times 4}$ which is not quasiconvex. These examples are obtained by transpositions and restrictions of Grabovsky's example of a rank-one convex, non-quasiconvex integrand in ${\mathbb{R}}^{8\times 2}.$ We also modify Šverák's example to construct a rank-one convex non-quasiconvex integrand in ${\mathbb{R}}_{\text{sym}}^{3\times 3}$.