On equicontinuity of mappings with branching in the closure of a domain

TL;DR

This paper investigates the local behavior and equicontinuity of certain classes of mappings with branching in the closure of a domain, under conditions involving a measurable function and boundary properties.

Contribution

It establishes conditions under which a family of open discrete mappings with quasiconformal characteristics is equicontinuous in the closure of the domain.

Findings

Family of mappings is equicontinuous under specified conditions.

Conditions involve measurable function Q(x) and boundary properties.

Results extend understanding of boundary behavior of mappings with branching.

Abstract

In the present paper, questions about a local behavior of mappings $f:D\rightarrow \overline{{\Bbb R}^n},$ $n\ge 2,$ in $\overline{D}$ are studied. Under some conditions on a measurable function $Q(x),$ $Q:D\rightarrow [0, \infty],$ and boundaries of $D$ and $D^{\,\prime}=f(D),$ it is showed that a family of open discrete map\-ping $f:D\rightarrow \overline{{\Bbb R}^n},$ $n\ge 2,$ with characteristic of quasiconformality $Q(x),$ is equicontinuous in $\overline{D}.$