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

For any M, n ≥ 2 and any open set Ω ⊂ Rⁿ we find a smooth, strongly polyconvex function F RM× n→ R and a Lipschitz map u Rⁿ → RM that is a weak local minimizer of the energy \[ ∫Ω F(Du). \] but with nowhere continuous partial derivatives. This extends celebrated results by M\"uller-Sver\'ak and Sz\'ekelyhidi to higher dimensions.