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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the degenerate Beltrami equation


Authors: V. Gutlyanskii, O. Martio, T. Sugawa and M. Vuorinen
Journal: Trans. Amer. Math. Soc. 357 (2005), 875-900
MSC (2000): Primary 30C62
DOI: https://doi.org/10.1090/S0002-9947-04-03708-0
Published electronically: October 19, 2004
MathSciNet review: 2110425
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Abstract: We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient $\mu(z)$ has the norm $\Vert\mu\Vert _\infty=1.$Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of $\mu.$A uniqueness theorem is also proved when the singular set $\operatorname{Sing} (\mu)$ of $\mu$is contained in a totally disconnected compact set with an additional thinness condition on $\operatorname{Sing}(\mu).$


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Additional Information

V. Gutlyanskii
Affiliation: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 83114, Donetsk, Ukraine
Email: gut@iamm.ac.donetsk.ua

O. Martio
Affiliation: Department of Mathematics, P.O. Box 68 (Gustaf Hällströmin katu 2b), FIN–00014 University of Helsinki, Finland
Email: martio@cc.helsinki.fi

T. Sugawa
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, 739 – 8526 Higashi-Hiroshima, Japan
Email: sugawa@math.sci.hiroshima-u.ac.jp

M. Vuorinen
Affiliation: Department of Mathematics, FIN–20014 University of Turku, Finland
Email: vuorinen@csc.fi

DOI: https://doi.org/10.1090/S0002-9947-04-03708-0
Received by editor(s): February 11, 2002
Published electronically: October 19, 2004
Additional Notes: The third author was partially supported by the Academy of Finland and the JSPS while carrying out the present paper.
Article copyright: © Copyright 2004 American Mathematical Society