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

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 $\mathbb{R}^{4\times 4}_\text{sym}$ 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 $\mathbb{R}^{3\times 3}_\text{sym}$.