Transformations that preserve Sobolev spaces
The following theorem is proved: If the mapping \(\phi\) : \(G\to G'\) between the domains G and G' in \({\mathbb{R}}^ n\) induces an isomorphism \(\phi^*: W^ 1_ p(G')\to W^ 1_ p(G)\) between the Sobolev-spaces with index p where \(p\in (n-1,n)\) then \(\phi\) coincides a.e. with a quasi-isometry.
scientific article; zbMATH DE number 3944222