On equicontinuity of Sobolev classes in domains with locally connected boundary

TL;DR

This paper investigates the equicontinuity of Sobolev classes of homeomorphisms in domains with locally connected boundaries, establishing conditions involving finite mean oscillation of the inner dilatation of order p.

Contribution

It proves that Sobolev classes are equicontinuous in such domains under specific boundary and dilatation conditions, extending understanding of boundary behavior of these mappings.

Findings

Sobolev classes are equicontinuous in domains with locally connected boundaries.

Finite mean oscillation of the inner dilatation ensures equicontinuity.

Results apply to homeomorphisms in Orlicz classes with boundary restrictions.

Abstract

A behavior of homeomorphisms of Orlicz classes in a closure of a domain is investigated. It is proved that above classes are equicontinuous in the closure of 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.