Equicontinuity of homeomorphisms with unbounded characteristic

The article is devoted to the study of the boundary properties of homeomorphisms $f\colon D\to D'$, $D,D'\subset\mathbb R^n$, satisfying some geometric conditions responsible for the control of the measure of distortion of families of curves in $D$. Under additional requirements on the boundaries $\partial D$ and $\partial D'$ of the domains, we prove that the family of all such homeomorphisms is equicontinuous in $\overline D$.