A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension

For any $M,n\ge 2$ and any open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ we find a smooth, strongly polyconvex function $F\text{\hspace{0.17em}⁣}:{\mathbb{R}}^{M\times n}\to \mathbb{R}$ and a Lipschitz map $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to {\mathbb{R}}^M$ that is a weak local minimizer of the energy $${\int }_{\mathrm{\Omega }}F(Du)\mathrm{.}$$ but with nowhere continuous partial derivatives. This extends celebrated results by Müller-Sverák and Székelyhidi to higher dimensions.