On the boundary behavior of quasiconformal mappings

2019 ◽  
Vol 16 (2) ◽  
pp. 289-300
Author(s):  
Vladimir Zorich
Keyword(s):  
Boundary Behavior ◽  
Quasiconformal Mappings ◽  
Open Questions

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

To the theory of quasiconformal mappings

2019 ◽  
Vol 16 (1) ◽  
pp. 141-147
Author(s):  
Vladimir Zorich
Keyword(s):  
Quasiconformal Mappings ◽  
Open Questions

The open questions of the theory of quasiconformal mappings that are adjacent to the field of studies of Professor Bogdan Bojarski are discussed.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

2021 ◽  
Vol 18 (2) ◽  
pp. 145-159
Author(s):  
Elena Afanas'eva ◽  
Viktoriia Bilet
Keyword(s):  
Riemannian Manifolds ◽  
Boundary Behavior ◽  
Quasiconformal Mappings ◽  
Uniform Domains

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

On behavior of inverse homeomorphisms in terms of prime ends

2019 ◽  
Vol 33 ◽  
pp. 188-203
Author(s):  
Evgeny Sevost'yanov ◽  
Sergei Skvortsov ◽  
N. Ilkevych
Keyword(s):  
Boundary Behavior ◽  
Type Inequality ◽  
Continuous Extension ◽  
Conformal Mappings ◽  
Quasiconformal Mappings ◽  
Inverse Mapping ◽  
Boundary Points ◽  
Prime Ends ◽  
Simply Connected ◽  
Simply Connected Domains

As is known, even conformal mappings of plane simply connected domains do not, generally speaking, have continuous boundary extension in the Euclidean sense. One of the minimum requirements necessary for such an extension is the local connectedness of the definition domain of the corresponding map on its boundary. Of course, quite a lot of simply connected domains do not have this property. For example, the unit disk with a cut along the positive part of the real axis is not locally connected at the boundary. In the same way, the mapped domain must also satisfy certain conditions necessary for the continuous extension of a mapping. The situation changes significantly if we are not talking about the Euclidean boundary behavior of mappings, but about extension in terms of the so-called prime ends. In this case, the domain of definition of mappings should be only regular, that is, this domain should be the image of a domain with a locally quasiconformal boundary under some quasiconformal mapping. A similar requirement also applies to the mapped domain. In this article, we study the equicontinuous families of maps at inner and boundary points in the case where the prime ends of the domain serve as boundary points. Relatively speaking, the paper consists of two parts, one of which contains a number of auxiliary statements, and the second, the final part of the work, contains the formulation of the main theorem and its proof. We consider a class of homeomorphisms of Euclidean space, inverse of which distort moduli of families of paths by the Poletsky type inequality. Note that these classes include most well-known mappings, such as conformal mappings, quasiconformal mappings, mappings with finite length and area distortion, and so on. It should be noted that under conformal mappings, distortion of the modulus of families of paths does not occur, therefore, when passing to inverse mappings, we remain in the class under study. A similar situation is in the case of quasiconformal mappings, since, as is known, the inverse mapping to a quasiconformal is also quasiconformal. In more general situations, the studied configurations can turn out to be much more complicated, in particular, the transition to inverse mappings can significantly change their properties (this is confirmed by specific examples of mappings, which are rather easy to construct in this case). This article is actually devoted to the study of this particular case, that is, when we are dealing with a certain family of homeomorphisms with an unbounded characteristic, in addition, mappings inverse to them are studied. In more detail, we consider mappings whose inverse satisfy the upper distortion estimate of the modulus of families of paths with integrable majorant. In the article, we proved that the families of the indicated mappings are equicontinuous both at the inner and boundary points of the domain, provided that the majorant responsible for the distortion of the modulus of the families of paths is integrable, besides that, the definition and mapped domains are regular, and the boundary points are prime ends of the definition domain. The results obtained in the paper are applicable to well-known classes of mappings, such as mappings with bounded and finite distortion, as well as to the Sobolev and Orlicz-Sobolev classes.

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