On discrete boundary extension of mappings in terms of prime ends

TL;DR

This paper investigates conditions under which mappings satisfying the inverse Poletsky inequality can be continuously extended to the boundary of a domain using prime ends, with results on discreteness of the extension.

Contribution

It establishes boundary extension criteria for inverse Poletsky inequality mappings in Euclidean spaces, including conditions for continuity and discreteness of the extension.

Findings

Mappings have continuous boundary extension via prime ends under integrability conditions.

Under additional conditions, the boundary extension is discrete.

The results apply to mappings satisfying the inverse Poletsky inequality in Euclidean domains.

Abstract

We study mappings that satisfy the inverse Poletsky inequality in a domain of the Euclidean space. Under certain conditions on the definition and mapped domains, it is established that they have a continuous extension to the boundary in terms of prime ends if the majorant involved in the Poletsky inequality is integrable over spheres. Under some additional conditions, the extension mentioned above is discrete.