On singularities of mappings with a finite length distortion

TL;DR

This paper investigates conditions under which certain mappings with finite length distortion can be continuously extended to boundary points, focusing on integrability conditions near cluster points.

Contribution

It establishes that integrability of a mapping's characteristic on spheres near a boundary point guarantees its continuous extension to that point.

Findings

Integrability of the mapping's characteristic ensures boundary extension.

Continuous extension is possible if the characteristic is Lebesgue integrable.

Results apply to mappings with finite length distortion near boundary points.

Abstract

We study the possibility of a continuous extension of a class of mappings to an isolated point on the boundary of a domain. We show that if some characteristic of this mapping is integrable on almost all spheres in the neighborhood of at least one point of the corresponding cluster set, then this mapping has a continuous extension to the specified point. In particular, this assertion is true if the specified characteristic is simply Lebesgue integrable in the neighborhood of at least one limit point.