On the Hilbert problem for analytic functions in quasihyperbolic domains

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

doi:10.15407/dopovidi2019.02.023

Cited by 3 publications

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“…Note that D ∂ is almost smooth. Thus, there exist almost smooth Jordan curves (see (5) in [3]) with the quasihyperbolic boundary condition that are not quasi conformal curves.…”

Section: On Boundary-value Problems In Domains Without (A)-conditionmentioning

confidence: 99%

“…We substantially weaken the regularity conditions both on the functions λ and ϕ in the boundary condition (2) and on the boundary C of the domain D . On the one hand, we will deal with the coefficients λ of a countable bounded variation and the boundary data ϕ , which are measurable with respect to the logarithmic capacity, see the corresponding definitions in our previous paper [3]. On the other hand, we study here the Hilbert boundary-value problem in domains D with a more general boundary condition, see discussions in the next section.…”

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confidence: 99%

“…Гутлянский 1 , В.И. Рязанов 1,2 , Э. Якубов 3 , А.С. Ефимушкин 1 1 Институт прикладной математики и механики НАН Украины, Славянск…”

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On boundaryvalue problems in domains without (A)-condition

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 equations 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 its solutions. As consequences, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary-value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.

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“…Note that D ∂ is almost smooth. Thus, there exist almost smooth Jordan curves (see (5) in [3]) with the quasihyperbolic boundary condition that are not quasi conformal curves.…”

Section: On Boundary-value Problems In Domains Without (A)-conditionmentioning

confidence: 99%

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confidence: 99%

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On boundaryvalue problems in domains without (A)-condition

Gutlyanskiî¹,

Ryazanov²,

Yakubov³

et al. 2019

Dopov. Nac. akad. nauk Ukr.

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“…Then this result was extended to arbitrary smooth ( 1 C ) domains. Moreover, the following result was proved in [7] (see the next section for definitions):…”

Section: Introductionmentioning

confidence: 99%

“…The limit along all nontangential paths at ζ is called angular at the point. Following [7], we say that a Jordan curve Γ in  is almost smooth, if Γ has a tangent q.e. In particular, Γ is almost smooth, if Γ has a tangent at all its points except a countable collection.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions

Gutlyanskiî¹,

Nesmelova²,

Ryazanov³

et al. 2020

Dopov. Nac. akad. nauk Ukr.

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The study of the Dirichlet problem with arbitrary measurable boundary data for harmonic functions in the unit disk is due to the famous Luzin dissertation. Later on, the known monograph of Vekua was devoted to boundary-value problems for generalized analytic functions, but only with Hölder continuous boundary data. The present paper contains theorems on the existence of nonclassical solutions of Riemann and Hilbert problems for generalized analy tic functions with sources whose boundary data are measurable with respect to the logarithmic capacity. Our ap proach is based on the geometric interpretation of boundary values in comparison with the classical operator approach in PDE. On this basis, one can derive the corresponding existence theorems for the Poincaré problem on directional derivatives to the Poisson equations and, in particular, for the Neumann problem with arbitrary boundary data that are measurable with respect to the logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic inhomogeneous media.

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Boundary value problems for the generalized analytic and harmonic functions

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The study of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. Later on, the known monograph of Vekua has been devoted to boundary value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ bh\, =\, c\, ,$ where $\partial_{\bar z}\ :=\ \frac{1}{2}\left(\ \frac{\partial}{\partial x}\ +\ i\cdot\frac{\partial}{\partial y}\ \right),$ and it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in the corresponding domains $D\subset \mathbb C$. The present paper is a natural continuation of our articles on the Riemann, Hilbert, Dirichlet, Poincare and, in particular, Neumann boundary value problems for quasiconformal, analytic, harmonic and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here we extend the correspon\-ding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. It was also given relevant definitions and necessary references to the mentioned articles and comments on previous results. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

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On the Hilbert problem for analytic functions in quasihyperbolic domains

Cited by 3 publications

References 31 publications

On boundaryvalue problems in domains without (A)-condition

On boundaryvalue problems in domains without (A)-condition

Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions

Boundary value problems for the generalized analytic and harmonic functions

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