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Isometries between Sobolev spaces
Author(s) -
Biegert Markus,
Nittka Robin
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201190020
Subject(s) - mathematics , isometry (riemannian geometry) , sobolev space , bounded function , boundary (topology) , combinatorics , omega , space (punctuation) , mathematical analysis , physics , quantum mechanics , linguistics , philosophy
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.