Ukrainian Mathematical Bulletin

Volume 3, Issue 2, 2006 pp. 189-201.

To the Theory of Boundary Behavior of Space Mappings

Authors: Andrei A. Ignatiev, Vladimir I. Ryazanov
Author institution: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, R. Luxemburg str., 74, 83114, Donetsk, Ukraine ryaz@iamm.ac.donetsk.ua, ryazanov@iamm.ac.donetsk.ua, ryazanov@www.math.helsinki.fi

Summary: In this paper, we formulate a number of theorems about continuous and homeomorphic extensions of $Q$-homeomorphisms to regular boundaries. In particular, if a majorant of $Q$ has a finite mean oscillation in points of the boundary, we prove a generalization of the known theorem of Gehring--Martio on extension of quasiconformal mappings to the boundary. The results can be applied to various classes of mappings with finite distortion; these classes have been extensively studied lately by leading specialists in the theory of mappings in a large number of papers. In particular, the results can be applied to Sobolev class mappings.

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