Diffeomorphic approximation of $W^{1,1}$ planar Sobolev homeomorphisms

Abstract

Let $\Omega \subseteq {\mathbb{R}}^2$ be a domain and let $f\in W^{1,1}(\Omega ,{\mathbb{R}}^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega )$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(\Omega ,{\mathbb{R}}^2)$ and uniformly.