Diffeomorphic approximation of Planar Sobolev Homeomorphisms in rearrangement invariant spaces

Abstract

Let $\Omega \subseteq {\mathcal{R}}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega ,{\mathcal{R}}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega )$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(\Omega ,{\mathcal{R}}^2)$.