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Prime ends and Orlicz-Sobolev classes


Authors: D. A. Kovtonyuk and V. I. Ryazanov
Translated by: the authors
Original publication: Algebra i Analiz, tom 27 (2015), nomer 5.
Journal: St. Petersburg Math. J. 27 (2016), 765-788
MSC (2010): Primary 30C65
DOI: https://doi.org/10.1090/spmj/1416
Published electronically: July 26, 2016
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Abstract: A canonical representation of prime ends is obtained in the case of regular spatial domains, and the boundary behavior is studied for the so-called lower $ Q$-homeomorphisms, which generalize the quasiconformal mappings in a natural way. In particular, a series of efficient conditions on a function $ Q$ are found for continuous and homeomorphic extendibility to the boundary along prime ends. On that basis, a theory is developed that describes the boundary behavior of mappings in the Sobolev and Orlicz-Sobolev classes and also of finitely bi-Lipschitz mappings, which are a far-reaching generalization of the well-known classes of isometries and quasiisometries.


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Additional Information

D. A. Kovtonyuk
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg Str. 74, Donetsk 83114, Ukraine
Email: denis_kovtonyuk@bk.ru

V. I. Ryazanov
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg Str. 74, Donetsk 83114, Ukraine
Email: vl_ryazanov1@mail.ru

DOI: https://doi.org/10.1090/spmj/1416
Keywords: Prime ends, regular domains, boundary behavior, mappings with finite distortion, lower $Q$-homeomorphisms, ring $Q$-homeomorphisms, Orlicz--Sobolev classes, finitely bi-Lipschitz mappings
Received by editor(s): December 8, 2014
Published electronically: July 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society