On equicontinuity of homeomorphisms of the Orlicz and Orlicz – Sobolev classes in the closure of a domain

Abstract

We investigate the behavior of homeomorphisms of the Orlicz – Sobolev classes in the closure of a domain. The theorems on equicontinuity of the indicated classes are obtained in terms of the prime ends of regular domains. In particular, it is shown that indicated classes are equicontinuous in domains with certain restrictions imposed on their boundaries provided that the corresponding inner dilatation of order p has a majorant of finite mean oscillation at every point. We also prove theorems on the (pointwise) equicontinuity of the analyzed classes in the case of locally connected boundaries.