On equicontinuity of Orlicz--Sobolev class in s closure of a domain

TL;DR

This paper investigates the equicontinuity of Orlicz--Sobolev class homeomorphisms in domain closures, establishing conditions involving prime ends and boundary regularity that ensure uniform behavior.

Contribution

It provides new theorems linking equicontinuity of Orlicz--Sobolev homeomorphisms to boundary properties and dilatation conditions in regular domains.

Findings

Classes are equicontinuous under boundary restrictions.

Equicontinuity depends on the inner dilatation having finite mean oscillation.

Results apply to domains with specific boundary regularity.

Abstract

A behavior of homeomorphisms of Orlicz--Sobolev classes in a closure of a domain is investigated. There are obtained theorems about equicontinuity of classes mentioned above in terms of prime ends of regular domains. In particular, it is proved that above classes are equicontinuous in domains with some restrictions on it's boundaries provided that the corresponding inner dilatation of order $p$ has a majorant of finite mean oscillation at every point.