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

TL;DR

This paper constructs examples of weak local minimizers for polyconvex functionals with nowhere continuous derivatives, extending previous results to higher dimensions and challenging regularity assumptions.

Contribution

It introduces smooth, strongly polyconvex functions with Lipschitz minimizers that have nowhere continuous derivatives in arbitrary dimensions.

Findings

Existence of weak local minimizers with nowhere continuous derivatives.

Extension of previous lower-dimensional results to higher dimensions.

Challenges to regularity expectations in polyconvex variational problems.

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.