On Caratheodory theorem for open discrete unclosed mappings

TL;DR

This paper investigates the boundary behavior and equicontinuity of open, discrete mappings satisfying inverse Poletsky inequalities, extending Caratheodory theorem concepts to non-locally connected boundaries and Orlicz-Sobolev classes.

Contribution

It extends Caratheodory theorem to mappings with non-locally connected boundaries, providing new equicontinuity results in the context of inverse Poletsky inequalities.

Findings

Established equicontinuity of mappings in the closure of the domain.

Extended Caratheodory theorem to non-locally connected boundary cases.

Derived results for Orlicz-Sobolev classes.

Abstract

We study mappings satisfying the inverse Poletsky-type inequality in a domain of the Euclidean space. Such inequalities are well known and play an important role in the study of quasiconformal and quasiregular mappings. We consider the case when the mapped domain, generally speaking, is not locally connected on its boundary. At the same time, we consider the situation when the mapping is open and discrete, but may not preserve the boundary of the domain. In terms of prime ends, we obtain results on the equicontinuity of families of such mappings in the closure of the definition domain. As a consequence, we also obtain the corresponding statement for Orlicz-Sobolev classes.