On the boundary-value problems for quasiconformal functions in the plane

Gutlyanskiî, Vladimir; Ryazanov, Vladimir; Yefimushkin, A.S.

doi:10.1007/s10958-016-2769-2

Cited by 36 publications

(27 citation statements)

References 18 publications

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“…It is necessary to mention also that the theory of the boundary behavior of Sobolev's mappings has significant applications to the boundary value problems for the Beltrami equations and for analogs of the Laplace equation in anisotropic and inhomogeneous media, see e.g. [2], [7]- [10], [12], [13], [19], [22], [24] and relevant references therein. For basic definitions and notations, discussions and historic comments in the mapping theory on the Riemann surfaces, see our previous papers [25]- [27].…”

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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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confidence: 99%

Prime ends in the Sobolev mapping theory on Riemann surfaces

Ryazanov¹,

Волков²

2017

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“…By the above remarks, a desired function u is a real part of a solution f in class W 1,1 loc for the Beltrami equation (1.2) with µ ∈ C α given by the formula (7.11). By Lemma 1 in [27] µ is extended to a Hölder continuous function µ * : C → C of the class C α . Set k = max |µ(z)| < 1 in D. Then, for every…”

Section: )mentioning

confidence: 99%

“…The reader can find a rather comprehensive treatment of the theory in the new excellent books [10,11,29,45]. We also recommend to make familiar with the historic surveys contained in the monographs [20,41,55] on the topic with an exhaustive bibliography and take a look at our recent papers [25,27,47].…”

Section: Introductionmentioning

confidence: 99%

On the Hilbert problem for analytic functions in quasihyperbolic domains

Gutlyanskiî¹,

Ryazanov²,

Yakubov³

et al. 2019

Dopov. Nac. akad. nauk Ukr.

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We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring-Martio, generally speaking, without the standard (A)−condition by Ladyzhenskaya-Ural'tseva. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincare boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.

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“…The Dirichlet, Hilbert (Riemann-Hilbert), Neumann, Poincare and Riemann boundary value problems are basic in the theory of analytic functions and they are closely interconnected, see e.g. the monographs [24,56] and [72] for the history, and also the recent papers [21,34,[61][62][63][64][65][66][67][68][69] and [74]. for a prescribed continuous function ϕ : ∂D → R. The request (1.2) is too strong and has no sense if the boundary function ϕ is only measurable.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, it was demonstrated in [34,[61][62][63][64][65][66][67][68][69] and [74] that all other boundary value problems mentioned above for harmonic and analytic functions as well as their generalizations in the extended sense are successively reduced to the first boundary value problem. In particular, it is well-known that the Neumann problem has no classical solutions generally speaking even for smooth boundary data, see e.g.…”

Section: Introductionmentioning

confidence: 99%

On recent advances in boundary-value problems in the plane

Gutlyanskiî

Ryazanov

2017

J Math Sci

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The survey is devoted to recent advances in nonclassical solutions of the main boundary value problems such as the well-known Dirichlet, Hilbert, Neumann, Poincare and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometrical function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so-called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics, too. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary value problems. The latter concerns to the Laplace equation as well as to its counterparts in the potential theory for inhomogeneous and anisotropic media.

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On the boundary-value problems for quasiconformal functions in the plane

Cited by 36 publications

References 18 publications

Prime ends in the Sobolev mapping theory on Riemann surfaces

Prime ends in the Sobolev mapping theory on Riemann surfaces

On the Hilbert problem for analytic functions in quasihyperbolic domains

On recent advances in boundary-value problems in the plane

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