On the boundary behavior of mappings with finite distortion in the plane

Kovtonyuk, Denis; Petkov, Igor; Ryazanov, Vladimir

doi:10.1134/s1995080217020123

Cited by 13 publications

(12 citation statements)

References 65 publications

(70 reference statements)

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“…Introduction. The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [11] for the plane domains and in [14] for the spatial domains. The pointwise boundary behavior of the mappings with finite distortion in regular domains on Riemann surfaces was recently studied by us in [26].…”

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confidence: 99%

Prime ends in the Sobolev mapping theory on Riemann surfaces

Ryazanov¹,

Волков²

2017

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We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.1. Introduction. The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [11] for the plane domains and in [14] for the spatial domains. The pointwise boundary behavior of the mappings with finite distortion in regular domains on Riemann surfaces was recently studied by us in [26]. Moreover, the problem was investigated in regular domains on the Riemann manifolds for n ≥ 3 as well as in metric spaces, see e. ∈ (a, b). In what follows, |γ| denotes the locus of γ, i.e. the image γ ([a, b]).We act similarly to Caratheodory ([4]) under the definition of the prime ends of domains on a Riemann surface S, see Chapter 9 in [5]. First of all, recall that a continuous mapping σ : (ii) σ m splits D into 2 domains one of which contains σ m+1 and another one σ m−1 for every m > 1;2010 Mathematics Subject Classification: 30C62.

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Prime ends in the Sobolev mapping theory on Riemann surfaces

Ryazanov¹,

Волков²

2017

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“…On the basis of results of Section 8 in [8], we obtain the corresponding results on the boundary behavior of solutions of the Beltrami equations. Furthermore, it is sufficient in Theorem 2.1 to assume that K µ is integrable only in a neighborhood of ∂D or even more weak conditions which are due to Lemma 5.1 in [8].…”

Section: Boundary Behavior Of Solutions Of the Beltrami Equationsmentioning

confidence: 99%

“…Generalized homeomorphic solutions of the Beltrami equations are mappings with finite distortion whose boundary behavior with respect to prime ends in arbitrary finitely connected domains of C was studied in our last preprint [8] and we refer the reader to this text for historic comments, definitions and preliminary remarks.…”

Section: Introductionmentioning

confidence: 99%

The Dirichlet problem and prime ends

Kovtonyuk,

Petkov,

Ryazanov

2015

Preprint

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АннотацияIt is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ∂f = µ ∂f of the Sobolev class W 1,1 loc with respect to prime ends of domains. On this basis, under certain conditions on the complex coefficient µ, it is proved the existence of regular solutions of its Dirichlet problem in arbitrary simply connected domains and pseudoregular as well as multivalent solutions in arbitrary finitely connected domains with continuous boundary data in terms of prime ends.

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“…In the present paper, we will develop the theory of the boundary behavior of the so-called mappings with finite length distortion first introduced in paper [15] for R n , n 2, see also Chapter 8 in book [17]. As was shown in papers [8] and [9], such mappings, generally speaking, are not mappings with finite distortion by Iwaniec, because their first partial derivatives can be not locally integrable.…”

Section: Introductionmentioning

confidence: 99%

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Ryazanov

Волков

2020

J Math Sci

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The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries.We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Carathéodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Carathéodory prime ends are obtained.

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On the boundary behavior of mappings with finite distortion in the plane

Cited by 13 publications

References 65 publications

Prime ends in the Sobolev mapping theory on Riemann surfaces

Prime ends in the Sobolev mapping theory on Riemann surfaces

The Dirichlet problem and prime ends

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

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