arXiv subjects
Publications and source records attributed to Zarina Kovba.
The paper is devoted to the study of the boundary behavior of mappings. We consider mappings that satisfy inverse moduli inequalities of Poletskii type, under which the images of the domain under the mappings may change. It is proved that a classes of such mappings are equicontinuous with respect to prime ends of some domain if the majorant in the indicated modulus inequality is integrable.
We study problems related to continuous boundary extension of mappings of Orlicz-Sobolev classes in terms of prime ends. The results we obtain concern the case when the mappings are open, discrete, but not closed (not preserving the boundary of a domain). These results generalize the well-known results of Caratheodory on boundary extension of conformal mappings.
We study mappings satisfying the inverse Poletsky-type inequality in a domain of the Euclidean space. Such inequalities are well known and play an important role in the study of quasiconformal and quasiregular mappings. We consider the case when the mapped domain, generally speaking, is not locally connected on its boundary. At the same time, we consider the situation when the mapping is open and discrete, but may not preserve the boundary of the domain. In terms of prime ends, we obtain results on the equicontinuity of families of such mappings in the closure of the definition domain. As a consequence, we also obtain the corresponding statement for Orlicz-Sobolev classes.
We investigate distortion estimates for mappings at boundary points of a domain. We consider mappings of the Sobolev and Orlicz-Sobolev classes and some other classes of mappings that do not preserve the boundary of a domain. For above mappings, we establish distortion estimates at boundary points. In particular, under certain conditions on the characteristics of the mappings, we show that they are Hölder continuous. In the manuscript we consider both the case of ``good'' boundaries and ``domains with prime ends''. We have obtained not only Hölder-type estimates, but also some more general ones under appropriate (more general) conditions on the characteristic.
We study the problem of the lower bounds of the modulus of families of paths of order $p,$ $p>n-1,$ and their connection with the geometry of domains containing the specified families. Among other things, we have proved an analogue of Näkki's theorem on the positivity of the $p$-module of families of paths joining a pair of continua in the given domain. The geometry of domains with a strongly accessible boundary in the sense of the $p$-modulus of families of paths was also studied. We show that domains with a $p$-strongly accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely connected at their boundary. The mentioned result generalizes Näkki's result, which was proved for uniform domains in the case of a conformal modulus.