Isometries between Sobolev spaces

Let Ω 1 and Ω 2 be bounded, connected open sets in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^N$\end{document} with continuous boundary, and let p > 2. We show that every positive linear isometry T from W 1, p (Ω 1 ) to W 1, p (Ω 2 ) that satisfies \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{1,p}_0(\Omega _2) \subset TW^{1,p}_0(\Omega _1)$\end{document} corresponds to a rigid motion of the space, i.e., Tu = u ○ξ for an isometry ξ of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^N$\end{document} , and more precisely ξ(Ω 2 ) = Ω 1 . We also prove similar results for less regular domains, and we obtain partial results also for p = 2.