On Caratheodory prime ends extension for unclosed Orlicz-Sobolev classes

TL;DR

This paper investigates boundary extension problems for open, discrete Orlicz-Sobolev mappings using prime ends, extending classical Caratheodory results beyond conformal mappings.

Contribution

It generalizes Caratheodory's boundary extension results to a broader class of mappings in Orlicz-Sobolev spaces that are open and discrete but not necessarily closed.

Findings

Established boundary extension criteria for Orlicz-Sobolev mappings

Extended classical conformal mapping results to more general settings

Provided new insights into prime ends in the context of Sobolev classes

Abstract

We study problems related to continuous boundary extension of mappings of Orlicz-Sobolev classes in terms of prime ends. The results we obtain concern the case when the mappings are open, discrete, but not closed (not preserving the boundary of a domain). These results generalize the well-known results of Caratheodory on boundary extension of conformal mappings.