  For any $M, n \geq 2$ and any open set $\Omega \subset \mathbb{R}^n$ we find
a smooth, strongly polyconvex function $F\colon \mathbb{R}^{M\times n}\to
\mathbb{R}$ and a Lipschitz map $u\colon \mathbb{R}^n \to \mathbb{R}^M$ that is
a weak local minimizer of the energy \[
  \int_{\Omega} F(Du). \] but with nowhere continuous partial derivatives. This
extends celebrated results by M\"uller-Sver\'ak and Sz\'ekelyhidi to higher
dimensions.
