On boundary behavior of mappings of Sobolev and Orlicz--Sobolev class

TL;DR

This paper investigates the boundary behavior of Sobolev and Orlicz--Sobolev mappings in higher dimensions, establishing conditions under which these mappings extend continuously to boundary points based on dilatation properties.

Contribution

It provides new criteria involving finite mean oscillation and integral divergence for the continuous extension of Sobolev and Orlicz--Sobolev mappings to boundary points.

Findings

Mappings extend continuously if inner dilatation has FMO at the boundary

Continuous extension is also guaranteed by divergence of specific integrals

Results apply to mappings in ${R}^n$, $n \\ge 3$

Abstract

A boundary behavior of closed open discrete mappings of Sobolev and Orlicz--Sobolev classes in ${\Bbb R}^n,$ $n\ge 3,$ is studied. It is proved that, mappings mentioned above have a continuous extension to boundary point $x_0$ of a domain $D$ whenever its inner dilatation of order $p$ has a majorant $FMO$ (finite mean oscillation) at the point. Another sufficient condition of possibility of continuous extension is a divergence of some integral.