| Abstrakt: |
This article is devoted to the study of mappings with bounded and finite distortion defined in some domain of the Euclidean space. We consider mappings that satisfy some upper estimates for the distortion of the modulus of families of paths, where the order of the modulus equals to p, n - 1 < p ⩽ n. The main problem studied in the manuscript is the investigation of the boundary behavior of such mappings, more precisely, the distortion of the distance under mappings near boundary points. The publication is primarily devoted to definition domains with "bad boundaries", in which the mappings not even have a continuous extension to the boundary in the Euclidean sense. However, we introduce the concept of a quasiconformal regular domain in which the specified continuous extension is valid and the corresponding distance distortion estimates are satisfied; however, both must be understood in the sense of the socalled prime ends. More precisely, such estimates hold in the case when the mapping acts from a quasiconformal regular domain to an Ahlfors regular domain with the Poincar'e inequality. The consideration of domains that are Ahlfors regular and satisfy the Poincar'e inequality is due to the fact that, lower estimates for the modulus of families of paths through the diameter of the corresponding sets hold in these domains. (There are the so-called Loewner-type estimates). We consider homeomorphisms and mappings with branching separately. The main analytical condition under which the results of the paper were obtained is the finiteness of the integral averages of some majorant involved in the defining modulus inequality under infinitesimal balls. This condition includes the situation of quasiconformal and quasiregular mappings, because for them the specified majorant is itself bounded in a definition domain. Also, the results of the article are valid for more general classes for which Poletsky-type upper moduli inequalities are satisfied, for example, for mappings with finite length distortion. [ABSTRACT FROM AUTHOR] |
