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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Boundary behavior and the Dirichlet problem for Beltrami equations


Authors: D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov and R. R. Salimov
Translated by: the authors
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 587-603
MSC (2010): Primary 30C65; Secondary 30C75, 35J46, 35J50, 35J56, 35J70, 35Q35
Published electronically: June 5, 2014
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Abstract: It is shown that a homeomorphic solution of the Beltrami equation $ \bar {\partial }f=\mu \,{\partial }f$ in the Sobolev class $ W^{1,1}_{\mathrm {loc}}$ is a so-called ring and, simultaneously, lower $ Q$-homeomorphism with $ Q(z)=K_{\mu }(z)$, where $ K_{\mu }(z)$ is the dilatation ratio of this equation. On this basis, the theory of the boundary behavior of such solutions is developed and, under certain conditions on $ K_{\mu }(z)$, the existence of regular solutions is established for the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. Also, the existence of pseudoregular as well as many-valued solutions is proved in the case of arbitrary finitely connected domains bounded by mutually disjoint Jordan curves.


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Additional Information

D. A. Kovtonyuk
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg str., Donetsk 83114, Ukraine
Email: denis{\textunderscore}kovtonyuk@bk.ru

I. V. Petkov
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg str., Donetsk 83114, Ukraine
Email: igorpetkov@list.ru

V. I. Ryazanov
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg str., Donetsk 83114, Ukraine
Email: vl.ryazanov1@gmail.com

R. R. Salimov
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg str., Donetsk 83114, Ukraine
Email: ruslan623@yandex.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2014-01308-8
PII: S 1061-0022(2014)01308-8
Keywords: Degenerate Beltrami equation, Dirichlet problem, boundary behavior, simply connected domains, regular solutions, multiply connected domains, pseudoregular solutions, many-valued solutions
Received by editor(s): June 6, 2012
Published electronically: June 5, 2014
Article copyright: © Copyright 2014 American Mathematical Society