Sobolev Homeomorphism that Cannot be Approximated by Diffeomorphisms in W1,1

Abstract

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).