On Carathéodory theorem for one class of mappings

Abstract

This article is devoted to the study of mappings of Orlicz-Sobolev classes that are not closed. In other words, we study the case when the mappings are open, discrete, but do not preserve the boundary of a domain (i.e., the corresponding cluster set of the mapping may contain inner points of the mapped domain). We investigated the problem of the existence of a limit of the above mapping at a fixed boundary point. We have established that such mappings have a continuous boundary extension whenever the corresponding Orlicz function satisfies the Calderon condition and the inner dilatation of the mapping satisfies some constraints on its growth, written in terms of singular parameters. In this case, we also impose conditions on the definition domain of the mapping. In particular, the domain must be locally connected on the boundary and finitely connected with respect to the pre-image of the corresponding cluster set. This approach is primarily related to the fulfillment of modulus inequalities of the Poletsky type in Orlicz-Sobolev classes. The available facts about the fulfillment of such inequalities require that the mapping be a homeomorphism, or more generally, an open discrete and closed mapping. If we consider more general classes of mappings, we cannot directly use the apparatus of modulus. Nevertheless, the conditions on the cluster set guarantee that the study of a mapping in an arbitrary domain can be reduced to some subdomain in which the mapping is already closed. The work is structured as follows. In the first part, we formulate the relevant definitions necessary for the formulation of the main result and present this formulation. The second section is devoted to the modulus of families of paths and surfaces. Here we formulate, in particular, a statement about the connection of Orlicz-Sobolev classes with upper modulus inequalities of the Poletsky type. The third part of the work is the proof of the main result, Theorem 1. The proof is carried out by the method of contradiction. The main tool of the proof is the apparatus of paths. We construct paths in the pre-image under the mapping, the diameter of which tends to zero. The ends of the paths are elements of sequences of points converging to a given point of the boundary. If the mapping has no boundary, the diameter of the mapped paths does not tend to zero. We show that the latter contradicts both the geometry of the definition domain and the proposed estimates of the growth of the dilatations of the mappings.