Ukrainian Mathematical Bulletin

Volume 4, Issue 1, 2007 pp. 79-115.

The Beltrami Equation and Ring Homeomorphisms

Authors: Vladimir Ryazanov, Uri Srebro, and Eduard Yakubov
Author institution: Vladimir Ryazanov, Institute of Applied Mathematics and Mechanics, NAS of Ukraine, Roze Luxemburg str. 74, 83114, Donetsk, Ukraine, ryazanov@iamm.ac.donetsk.ua, ryazanov@www.math.helsinki.fi, vlryazanov1@rambler.ru Uri Srebro, Technion Israel Institute of Technology, Haifa 32000, Israel, srebro@math.technion.ac.il Eduard Yakubov, Holon Academic Institute of Technology, 52 Golomb St., P.O.Box 305, Holon 58102, Israel, yakubov@hit.ac.il

Summary: With the aid of results by Gehring, we introduce and study plane ring $Q$-homeomorphisms. This study is then applied to deriving general principles on the existence and uniqueness of homeomorphic ACL solutions to the Beltrami equation extending earlier results. In particular, we obtain new existence criteria which are expressed in terms of finite mean oscillation majorants for tangential dilatations. Moreover, we give a new proof of our generalization of the well-known Lehto existence theorem that has, in turn, a number of other consequences.

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