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

Abstract

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.