On Singularities of Mappings with a Majorant Integrable by Spheres

The paper is devoted to the study of the boundary behavior of mappings with finite distortion, more precisely, open discrete mappings with moduli conditions similar to the Poletsky inequality in the inverse direction. We study the case when some majorant in the above inequality is integrable over spheres centered at each finite point. It is established that the indicated mappings have a continuous extension to an isolated boundary point of the boundary of a domain whenever the mapping \(f\) omits at least one point.

The proof of the main results is step-by-step and is based on the following logic: we prove that (1) no two different boundary points of the corresponding mapped domain can belong to the limit set of the mapping \(f\) at the point \(x_0\), and (2) the cluster set of a mapping at a given point is, in principle, always a singleton, provided that this mapping omits at least one finite point. The proofs of these statements are made by contradiction. This contradiction, in turn, is ensured by the property of approaching continua in the preimage under the mapping on the one hand, and by the upper bound on the mapped families of paths (taking into account the definition of the mapping through this upper bound) on the other hand.

In order to construct the above approaching continua, we use some geometric constructions that take into account the nature (definition) of the class of mappings under study. In particular, to prove that the cluster set under the mapping does not contain more than one boundary point, we use segments joining elements of some approximate sequences with its limit points, and the corresponding converging continua which are the pre-images of these segments under the mapping.

Note that the result obtained in the article was previously obtained by us first for homeomorphisms, then for open discrete closed mappings, and then for open discrete mappings but with an integrable majorant in the inverse Poletsky inequality. We also previously established a similar result, but in the case where the mapping omits at least two points.