On boundary extension of one class of mappings in terms of prime ends

Sevost’yanov, Evgeny; Skvortsov, S. A.; Sverchevska, I.A.

doi:10.30970/ms.53.1.29-40

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…where C := max For some other studies of mappings with conditions on the distortion of the modulus of families of paths, see, for example, in [20][21][22][23][24][25][26].…”

Section: IImentioning

confidence: 99%

On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

Dovhopiatyi,

Ilkevych,

Sevost'yanov

et al. 2024

Mat. Stud.

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This article is devoted to the study of mappings with bounded andfinite distortion defined in some domain of the Euclidean space. Weconsider mappings that satisfy some upper estimates for thedistortion of the modulus of families of paths, where the order ofthe modulus equals to $p,$ $n-1<p\leqslant n.$ The main problemstudied in the manuscript is the investigation of the boundarybehavior of such mappings, more precisely, the distortion of thedistance under mappings near boundary points. The publication isprimarily devoted to definition domains with ``bad boundaries'', inwhich the mappings not even have a continuous extension to theboundary in the Euclidean sense. However, we introduce the conceptof a quasiconformal regular domain in which the specified continuousextension is valid and the corresponding distance distortionestimates are satisfied; however, both must be understood in thesense of the so-called prime ends. More precisely, such estimateshold in the case when the mapping acts from a quasiconformal regulardomain to an Ahlfors regular domain with the Poincar\'e inequality.The consideration of domains that are Ahlfors regular and satisfythe Poincar\'e inequality is due to the fact that, lower estimates forthe modulus of families of paths through the diameter of thecorresponding sets hold in these domains. (There are the so-calledLoewner-type estimates). We consider homeomorphisms and mappingswith branching separately. The main analytical condition under whichthe results of the paper were obtained is the finiteness of theintegral averages of some majorant involved in the defining modulusinequality under infinitesimal balls. This condition includes thesituation of quasiconformal and quasiregular mappings, because forthem the specified majorant is itself bounded in a definitiondomain. Also, the results of the article are valid for more generalclasses for which Poletsky-type upper moduli inequalities aresatisfied, for example, for mappings with finite length distortion.

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“…where C := max For some other studies of mappings with conditions on the distortion of the modulus of families of paths, see, for example, in [20][21][22][23][24][25][26].…”

Section: IImentioning

confidence: 99%

On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

Dovhopiatyi,

Ilkevych,

Sevost'yanov

et al. 2024

Mat. Stud.

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“…Let us prove that this lifting is whole, for which we use the general scheme from [Sev 1 , Proof of Lemma 2.1], cf. [SeSkSv,Lemma 1]. Suppose the opposite, namely that c = b.…”

Section: Continuous Extension Of Mappings To the Boundarymentioning

confidence: 99%

On the inverse Poletskii inequality in metric spaces and prime ends

Sevost'yanov

2021

Preprint

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We study mappings defined in the domain of a metric space that distort the modulus of families of paths by the type of the inverse Poletskii inequality. Under certain conditions, it is proved that such mappings have a continuous extension to the boundary of the domain in terms of prime ends. Under some additional conditions, the families of such mappings are equicontinuous in the closure of the domain with respect to the space of prime ends.

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On the Inverse Poletsky Inequality in Metric Spaces and Prime Ends

Sevost’yanov

2023

Anal Math

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On boundary extension of one class of mappings in terms of prime ends

Cited by 3 publications

References 18 publications

On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

On the inverse Poletskii inequality in metric spaces and prime ends

On the Inverse Poletsky Inequality in Metric Spaces and Prime Ends

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