Carathéodory boundary extensions for generalized quasiregular mappings

Abstract

This paper is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality with an integrable majorant, are open, and discrete, and not necessarily preserve the boundary of a domain. Under some conditions on the geometry of these domains, it is proved that the specified mappings have a continuous boundary extension. The result is valid even in a more general form, when the majorant mentioned above is integrable over almost all concentric spheres centered at each point. We also have proved some results on equicontinuity of the family of above mappings in the closure of a domain.