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

Abstract

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