On the Hilbert problem for analytic functions in quasihyperbolic domains

On the Hilbert problem for analytic functions in quasihyperbolic domains

We study the Hilbert boundary-value problem for analytic functions in the Jordan domains satisfying the quasi-hyperbolic boundary condition by Gehring—Martio. Assuming that the coefficients of the problem are functions of the countably bounded variation and the boundary data are measurable with r...

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Datum:2019
Hauptverfasser: Gutlyanskii, V.Ya., Ryazanov, V.I., Yakubov, E., Yefimushkin, A.S.
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Veröffentlicht: Видавничий дім "Академперіодика" НАН України 2019
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spelling irk-123456789-1505032019-04-09T01:25:34Z On the Hilbert problem for analytic functions in quasihyperbolic domains Gutlyanskii, V.Ya. Ryazanov, V.I. Yakubov, E. Yefimushkin, A.S. Математика We study the Hilbert boundary-value problem for analytic functions in the Jordan domains satisfying the quasi-hyperbolic boundary condition by Gehring—Martio. Assuming that the coefficients of the problem are functions of the countably bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of solutions of the problem in terms of angular limits. As consequences, we derive the corresponding results concerning the Dirichlet, Neumann, and Poincaré boundary-value problems for harmonic functions. Досліджено граничну задачу Гільберта для аналітичних функцій в жорданових областях, які задовольняють квазігіперболічну умову Герінга Мартіо. З припущенням, що коефіцієнти задачі є функціями зліченно-обмеженої варіації і граничні дані є вимірними відносно логарифмічної ємності, доведено існування розв'язків задачі в термінах кутових границь. Як наслідки отримано відповідні результати для крайових задач Діріхле, Неймана і Пуанкаре для гармонічних функцій. Исследована краевая задача Гильберта для аналитических функций в жордановых областях, удовлетворяющих квазигиперболическому условию Геринга Мартио. С предположением, что коэффициенты задачи являются функциями счетно-ограниченной вариации, а граничные данные измеримы относительно логарифмической емкости, доказано существование решений задачи в терминах угловых пределов. В качестве следствий получены соответствующие результаты для краевых задач Дирихле, Неймана и Пуанкаре для гармонических функций. 2019 Article On the Hilbert problem for analytic functions in quasihyperbolic domains / V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin // Доповіді Національної академії наук України. — 2019. — № 2. — С. 23-30. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2019.02.023 http://dspace.nbuv.gov.ua/handle/123456789/150503 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Математика
Математика
spellingShingle Математика
Математика
Gutlyanskii, V.Ya.
Ryazanov, V.I.
Yakubov, E.
Yefimushkin, A.S.
On the Hilbert problem for analytic functions in quasihyperbolic domains
Доповіді НАН України
description We study the Hilbert boundary-value problem for analytic functions in the Jordan domains satisfying the quasi-hyperbolic boundary condition by Gehring—Martio. Assuming that the coefficients of the problem are functions of the countably bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of solutions of the problem in terms of angular limits. As consequences, we derive the corresponding results concerning the Dirichlet, Neumann, and Poincaré boundary-value problems for harmonic functions.
format Article
author Gutlyanskii, V.Ya.
Ryazanov, V.I.
Yakubov, E.
Yefimushkin, A.S.
author_facet Gutlyanskii, V.Ya.
Ryazanov, V.I.
Yakubov, E.
Yefimushkin, A.S.
author_sort Gutlyanskii, V.Ya.
title On the Hilbert problem for analytic functions in quasihyperbolic domains
title_short On the Hilbert problem for analytic functions in quasihyperbolic domains
title_full On the Hilbert problem for analytic functions in quasihyperbolic domains
title_fullStr On the Hilbert problem for analytic functions in quasihyperbolic domains
title_full_unstemmed On the Hilbert problem for analytic functions in quasihyperbolic domains
title_sort on the hilbert problem for analytic functions in quasihyperbolic domains
publisher Видавничий дім "Академперіодика" НАН України
publishDate 2019
topic_facet Математика
url http://dspace.nbuv.gov.ua/handle/123456789/150503
citation_txt On the Hilbert problem for analytic functions in quasihyperbolic domains / V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin // Доповіді Національної академії наук України. — 2019. — № 2. — С. 23-30. — Бібліогр.: 15 назв. — англ.
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fulltext 23ISSN 1025­6415. Допов. Нац. акад. наук Укр. 2019. № 2 © V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin, 2019 1. Introduction. D. Hilbert studied the boundary­value problem formulated as follows: To find an analytic function ( )f z in a domain D bounded by a rectifiable Jordan contour C that satis­ fies the boundary condition lim Re{ ( ) ( )} ( ) z f z C →ς λ ζ = ϕ ζ ∀ζ ∈ , (1) where both the coefficient λ and the boundary data ϕ of the problem are continuously differen­ tiable with respect to the natural parameter s on C . Moreover, it was assumed by Hilbert that 0λ ≠ everywhere on C . The latter allows us, without loss of generality, to consider that | | 1λ ≡ on C . In this case, the quantity Re { }fλ from the left in (1) means a projection of f onto the direction λ interpreted as vectors in 2 ¡ . Historic surveys in the subject can be found in the recent papers [1­3]. Here, we substantially weaken the regularity conditions on the functions λ and ϕ in the boundary condition (1) and on the boundary C of the domain D . On the one hand, we will deal with the coefficients λ of coun­ tably bounded variation and the boundary data ϕ which are measurable with respect to the loga­ doi: https://doi.org/10.15407/dopovidi2019.02.023 UDC 517.5 V.Ya. Gutlyanskii 1, V.I. Ryazanov 1,2, E. Yakubov 3, A.S. Yefimushkin 1 1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk 2 Bogdan Khmelnytsky National University of Cherkasy 3 Holon Institute of Technology, Israel E­mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com, yakubov@hit.ac.il, eduardyakubov@gmail.com, a.yefimushkin@gmail.com On the Hilbert problem for analytic functions in quasihyperbolic domains Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskii We study the Hilbert boundary­value problem for analytic functions in the Jordan domains satisfying the quasi­ hyperbolic boundary condition by Gehring—Martio. Assuming that the coefficients of the problem are functions of the countably bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of solutions of the problem in terms of angular limits. As consequences, we derive the correspond­ ing results concerning the Dirichlet, Neumann, and Poincaré boundary­value problems for harmonic functions. Keywords: Hilbert, Dirichlet, Neumann, and Poincaré boundary­value problems, analytic and harmonic functions, quasihyperbolic boundary condition, logarithmic capacity, angular limits. 24 ISSN 1025­6415. Dopov. Nac. akad. nauk Ukr. 2019. № 2 V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin rithmic capacity. On the other hand, the fundamental Becker—Pommerenke result in [4] allows us to study the Hilbert boundary­value problem in domains D with the quasihyperbolic boundary condition introduced in [5]. Recall that the quasihyperbolic distance between points z and 0z in a domain D ⊂£ is the quantity 0( , ) : inf / ( , ) ,Dk z z ds d D γ γ = ζ ∂∫ where ( , )d Dζ ∂ denotes the Euclidean distance from the point Dζ ∈ to D∂ , and the infimum is taken over all rectifiable curves γ joining the points z and 0z in D , see [6]. Further, it is said that a domain D satisfies the quasihyperbolic boundary condition if, for constants a and b and a point 0z D∈ , 0 0 ( , ) ( , ) ln . ( , )D d z D k z z a b z D d z D ∂ + ∀ ∈ ∂ � (2) Let D be a Jordan domain in £ such that it has a tangent at a point Dζ ∈∂ A path in D terminating at ζ is called nontangential if its part in a neighborhood of ζ lies inside of an angle in D with the vertex at ζ . The limit along all nontangential paths at ζ is called angular at the point. The latter notion is a standard tool for the study of the boundary behavior of analytic and harmonic functions, see, e.g., [7­9]. Further, the Hilbert boundary condition (1) will be under­ stood precisely in the sense of angular limit. The notion of the logarithmic capacity is the important tool for our research. Dealing with measurable boundary data functions ( )ϕ ζ with respect to the logarithmic capacity, see defini­ tions in [3], we will use the abbreviation q.e. ( quasieverywhere) on a set ,E ⊂£ if a property holds for all Eζ ∈ except its subset of zero logarithmic capacity [10]. 2. Some definitions and preliminary remarks. Later on, D denotes the unit disk { :| | 1}z z∈ <£ . Given a Jordan domain D in £ , we call : Dλ ∂ →£ a function of bounded variation, write ( )Dλ ∈ ∂BV , if 1 1 ( ) : sup | ( ) ( ) | j k j j j V D = λ + = ∂ = λ ζ − λ ζ < ∞∑ (3) where the supremum is taken over all finite collections of points j Dζ ∈∂ , 1, ,j k= … , with the cyclic order meaning that jζ lies between 1j+ζ and 1j−ζ for every 1, ,j k= … . Here, we assume that 1 1 0k+ζ = ζ = ζ . The quantity ( )V Dλ ∂ is called the variation of the function λ . Now, we call : Dλ ∂ →£ a function of the countably bounded variation, write ( )Dλ ∈ ∂CBV , if there is a countable collection of mutually disjoint arcs nγ of D∂ , 1, 2,n = … on each of which the restriction of λ is of bounded variation nV , sup n n V < ∞ , and the set \ nD∂ ∪γ has loga­ rithmic capacity zero. In particular, the latter holds true if \ nD∂ ∪γ is countable. It is clear that such functions can be singular enough. The following statement was proved as Proposition 5.1 in paper [3], where the function λα has been called by a function of the argument of λ . Proposition 1. For every function :λ ∂ → ∂D D of the class ( )∂DBV , there is a function :λα ∂ →¡D of the class ( )∂DBV with 3 / 2V Vα λλ π� such that ( ) exp{ ( )}i λλ ζ = α ζ for all ζ ∈∂D . Given a Jordan curve Γ ⊂£ , ( )cL∞ Γ denotes the class of all functions :α Γ →¡ which are measurable with respect to the logarithmic capacity and q.e. bounded. 25ISSN 1025­6415. Допов. Нац. акад. наук Укр. 2019. № 2 On the Hilbert problem for analytic functions in quasihyperbolic domains Proposition 2. For every function :λ ∂ → ∂D D in the class ( )∂DCBV , there is a function :λα ∂ →¡D in the class ( ) ( )cL∞ ∂ ∩ ∂D DCBV such that ( ) exp{ ( )} . . i q e onλλ ζ = α ζ ∂D . (4) Proof. Denote, by nλ , the function on ∂D that is equal to λ on nγ and to 1 outside of nγ . Let nα correspond to nλ by Proposition 1. Then its variation * 3 /2n nV V π� . With no loss of generality, we may assume that 0nα ≡ outside of nγ . Set 1 n n ∞ = α = α∑ . Then ( )α∈ ∂DCBV and ( ) exp{ ( )}iλ ζ = α ζ q.e. on ∂D Applying the corresponding shifts (divisible 2π ), we may change nα on nγ through n ∗α with * nα  π� at the middle point of nγ . Then it is clear that the new function ( )∗α ∈ ∂DCBV and ( ) exp{ ( )}i ∗λ ζ = α ζ q.e. on ∂D and, moreover, 3 /2nV∗α  π + π� on every nγ , i.e ∗α  is bounded outside of the set \ n∂ ∪γD . In addition, by construction, the function ∗α is conti­ nuous q.e. on ∂D Hence, ( )cL∗ ∞α ∈ ∂D . We say that a Jordan curve Γ in £ is almost smooth if Γ has a tangent quasieverywhere. Here, we say that a straight line L in £ is tangent to Γ at a point 0z ∈Γ if 00 ( , ) 0lim sup | |,z z dist z L z zz→ = −∈Γ . (5) In particular, Γ is almost smooth if Γ has a tangent at all its points except a countable set. The nature of such Jordan curves Γ is complicated enough because the countable set can be eve­ rywhere dense in Γ . Remark 1. By Corollary of Theorem 1 in [4], a conformal mapping of a Jordan domain D in £ with the quasihyperbolic boundary condition onto the unit disk D as well as its inverse are Hölder continuous in the closure of D and D , respectively. Since the logarithmic capacity of a set coincides with its transfinite diameter, these mappings keep the sets of the logarithmic capa­ city zero on the boundaries of D and D . Consequently, by Remark 2.1 in [3], such mappings also keep boundary functions which are measurable with respect to the logarithmic capacity. These facts are key for the research of the boundary­value problems in the given domains. 3. Correlation of conjugate harmonic functions. The following statement was first proven for the case of bounded variation in [3] as Theorem 5.1. Here, we give an alternative proof of this significant fact and extend it to the case of countably bounded variation. Lemma 1. Let :α ∂ →¡D be in the class ( ) ( )cL∞ ∂ ∩ ∂D DCBV , let :u →¡D be a bounded har­ monic function such that lim ( ) ( ) z u z →ς = α ζ (6) at every point of continuity of α , and let ν be its conjugate harmonic function. Then ν has the angular limit lim ( ) ( ) . . z z q e on →ς ν = β ζ ∂D , (7) where :β ∂ →¡D is measurable with respect to the logarithmic capacity. 26 ISSN 1025­6415. Dopov. Nac. akad. nauk Ukr. 2019. № 2 V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin Proof. Let us start from the case ( )α∈ ∂DBV . In this case, α has at most a countable set S of points of discontinuity and, consequently, S is of zero logarithmic capacity. Hence, by the gen­ eralized maximum principle, see the point 115 in [11], such a function u is unique and, thus, u can be represented as the Poisson integral of the function α , see Theorem I.D.2.2 in [8], 2 2 1 1 ( ) ( ) 2 1 2 cos( ) i itr u re e dt r t r π ϑ −π −= α π − ϑ − +∫ . (8) Here, the Poisson kernel is a real part of the analytic function ( ) / ( )z zζ + ζ − , iteζ = , iz re ϑ= , and, by the Weierstrass theorem, see Theorem 1.1.1 in [12], the Schwartz integral 1 ( ) : ( ) 2 z d f z i z∂ ζ + ζ= α ζ π ζ − ζ∫ D (9) gives the analytic function f u iv= + in D with Reu f= , Imv f= , and 1 ( ) ( ) ( ) 2 1 it it it it e z z F t f z e dt C dt e z e z π π − −π −π += α = + π π− −∫ ∫ , (10) where ( ) ( )it itF t e e−= α and 1 ( ) 2 itC e dt π −π = α π ∫ . By Theorem 2(c) in [13], the function ( )f z has angular limits ( )f ζ as z → ζ q.e. on ∂D , because the function F is of bounded variation. It re­ mains to note that ( ) lim ( )n n f f →∞ ζ = ζ , where ( ) ( )n nf f rζ = ζ , for an arbitrary sequence 1 0nr → − as n → ∞ q.e. on ∂D . Thus, ( )f ζ is measurable with respect to the logarithmic capacity, be­ cause the functions ( )nf ζ are so as continuous functions on ∂D , see 2.3.10 in [14]. Now, let ( )α∈ ∂DCBV . Then its set of points of discontinuity is at most of zero logarithmic capacity. Hence, again by the generalized maximum principle, the bounded function u satisfying (6) is unique. Moreover, ( )cL∞α∈ ∂D and, consequently, u can be represented by the Poisson in­ tegral (8), and the Schwartz integral (9) gives the analytic function f u iv= + in D , where 2 1 2 sin( ) ( ) ( ) 2 1 2 cos( ) i itr t v re e dt r t r π ϑ −π ϑ −= α π − ϑ − +∫ . (11) Let us apply the linearity of the integral operator (11). Namely, denote, by χ , the character­ istic function of an arc ∗γ of ∂D , where α is of bounded variation from the definition of CBV . Setting ∗α = αχ and 0 ∗α = α − α , we have 0∗α = α + α . Then 0∗ν = ν + ν where ∗ν and 0ν corre­ spond to ∗α and 0α by formula (11). By the first item of the proof, there exists the angular limit * *( ) ( )lim z v z →ζ = β ζ q.e. on ∂D , where :∗β ∂ →¡D is a measurable function with respect to the logarithmic capacity. Moreover, it is evident from formula (11) that 0 0( ) ( )v z → β ζ as z → ζ for all ∗ζ ∈ γ , where 0 : ∗β γ →¡ is continuous on ∗γ . Thus, setting 0∗β = β +β on ∗γ , we obtain the conclusion of Lemma 1 because the collection of such arcs ∗γ is countable, and the completion of this collection on ∂D has zero logarithmic capacity. 27ISSN 1025­6415. Допов. Нац. акад. наук Укр. 2019. № 2 On the Hilbert problem for analytic functions in quasihyperbolic domains 4. The Hilbert problem for analytic functions. Theorem 1. Let :λ ∂ → ∂D D be in the class ( )∂DCBV and :ϕ ∂ →¡D be measurable with re­ spect to the logarithmic capacity. Then there is an analytic function :f →£D with the angular limit lim Re[ ( ) ( )] ( ) . . z f z q e on →ς λ ζ = ϕ ζ ∂D . (12) Proof. By Proposition 2, the function ( ) ( )cL∞ λα ∈ ∂ ∩ ∂D DCBV . Therefore, 1 ( ) : ( ) , 2 z d g z z i zλ ∂ + ζ ζ= α ζ ∈ π − ζ ζ∫ D D , is an analytic function with ( ) Re ( ) ( )u z g z λ= → α ζ as z → ζ for every ζ ∈∂D except a set of the discontinuity points for the function λα , which has zero logarithmic capacity, see Corolla­ ry IX.1.1 in [12] and Theorem I.D.2.2 in [8]. Note that the function ( ) exp{ ( )}A z ig z= is also ana lytic. By Lemma 1, there is a function :β ∂ →¡D that has the angular limit ( ) Im ( ) ( )z g zν = → β ζ ( ) Im ( ) ( )z g zν = → β ζ as z → ζ q.e. on ∂D and β is measurable with respect to the logarithmic capacity. Thus, by Corollary 4.1 in [3], there exists an analytic function :B →£D that has the angular limit ( ) Re ( ) ( )exp{ ( )}U z B z= → ϕ ζ β ζ as z → ζ q.e. on ∂D Finally, an elementary computation shows that the desired function has the form f AB= . Theorem 2. Let D be a Jordan domain with the quasihyperbolic boundary condition, let D∂ have a tangent q.e., let : , | ( ) | 1Dλ ∂ → λ ζ ≡£ , be in ( )D∂CBV , and let : Dϕ ∂ →¡ be measurable with respect to the logarithmic capacity. Then there is an analytic function :f D →£ with the angular limit lim Re[ ( ) ( )] ( ) . . z f z q e on D →ς λ ζ = ϕ ζ ∂ (13) Proof. Let g be a conformal mapping of D onto D that exists by the Riemann mapping theorem, see Theorem II.2.1 in [12], and by the Caratheodory theorem, see Theorem II.3.4 in [12], g be extended to a homeomorphism g% of D onto D . By Corollary of Theorem 1 in [4], * : | Dg g ∂= % and its inverse function are Hölder continuous. Then 1 *: ( )g−Λ = λ ∈ ∂o DCBV and 1 *: g−Φ = ϕo is measurable with respect to the logarithmic capacity by Remark 1. Thus, by Theorem 1, there is an analytic function :A →£D that has the angular limit lim Re{ ( ) ( )} ( ) . . A q e on ω→η Λ η ω = Φ η ∂D . (14) Let us consider the analytic function :f A g= o and show that f is desired. Indeed, by the Lindelöf theorem, see Theorem II.C.2 in [8], if D∂ has a tangent at a point ζ , then arg[ ( ) ( )] arg[ ] constg g z zζ − − ζ − → as z → ζ . In other words, the images under the conformal mapping g of sectors in D with a vertex at ζ is asymptotically the same as sectors in D with a vertex at ( )w g= ζ . Consequently, nontangential paths in D are transformed under g into non­ tangential paths in D and inversely q.e. on D∂ and ∂D respectively, because D is almost smooth and g∗ and 1g− ∗ keep sets of logarithmic capacity zero. Thus, (14) implies the existence of the angular limit (13) q.e. on D∂ . 28 ISSN 1025­6415. Dopov. Nac. akad. nauk Ukr. 2019. № 2 V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin 5. On Dirichlet, Neumann, and Poincaré problems. We reduce these boundary­value prob­ lems for harmonic functions to suitable Hilbert problems for analytic functions studied above. Corollary 1. Let D be a Jordan domain with the quasihyperbolic boundary condition and let D∂ have a tangent q.e. Suppose : Dϕ ∂ →¡ is measurable with respect to the logarithmic capacity. Then there exists a harmonic function :u D →£ that has the angular limit lim ( ) ( ) . . z u z q e on D →ς = ϕ ζ ∂ . (15) It is well known that the Neumann problem, in general, has no classical solution. The neces­ sary condition of solvability is that the integral of the function ϕ over ∂D is equal to zero [15]. Theorem 3. Let D be a Jordan domain with the quasihyperbolic boundary condition and let D∂ have a tangent q.e. Suppose that : , | ( ) | 1Dν ∂ → ν ζ ≡£ , is in the class CBV and : Dϕ ∂ →¡ is measurable with respect to the logarithmic capacity. Then there exists a harmonic function :u D →¡ with the angular limit lim ( ) . . z u q e on D →ς ∂ = ϕ ζ ∂ ∂ν . (16) Proof. Indeed, by Theorem 2, there exists an analytic function :f D →£ that has the an­ gular limit lim Re[ ( ) ( )] ( ) z f z →ς ν ζ = ϕ ζ (17) q.e. on D∂ . Note that an indefinite integral F of f in D is also an analytic function and, cor­ respondingly, the harmonic functions Re u F= and Im v F= satisfy the Cauchy—Riemann sys­ tem x yv u= − and y xv u= . Hence x x x x yf F F u iv u iu u′= = = + = − = ∇ where x yu u iu∇ = + is the gradient of the function u in the complex form. Thus, (16) follows from (17), i.e. u is the de sired harmonic function, because its directional derivative Re Re , u u u u ∂ = ν∇ = ν∇ = 〈ν ∇ 〉 ∂ν is the scalar product of ν and the gradient u∇ . Corollary 2. Let D be a Jordan domain in C with the quasihyperbolic boundary condition and let the unit interior normal ( )n ζ to the boundary D∂ be in the class CBV . Suppose that : Dϕ ∂ →¡ is measurable with respect to the logarithmic capacity. Then one can find a harmonic function :u D →£ such that q.e. on D∂ there exist: 1) the finite limit along the normal ( )n ζ ( ) : lim ( ) z u u z →ς ζ = , 2) the normal derivative 0 ( ) ( ) ( ) : lim ( ) t u u t n u n t→ ∂ ζ + − ζζ = = ϕ ζ ∂ , 3) the angular limit lim ( ) ( ) z u u z n n→ς ∂ ∂= = ζ ∂ ∂ 29ISSN 1025­6415. Допов. Нац. акад. наук Укр. 2019. № 2 On the Hilbert problem for analytic functions in quasihyperbolic domains REFERENCES 1. Gutlyanskii, V. Ya. & Ryazanov, V. I. (2017). On recent advances in boundary­value problems in the plane. J. Math. Sci., 221, No. 5, pp. 638­670. doi: https://doi.org/10.1007/s10958­017­3257­z 2. Gutlyanskii, V., Ryazanov, V. & Yefimushkin, A. (2016). On the boundary­value problems for quasiconformal functions in the plane. J. Math. Sci., 214, No. 2, pp. 200­219. doi: https://doi.org/10.1007/s10958­016­ 2769­2 3. Efimushkin, A. S. & Ryazanov, V. I. (2015). On the Riemann­Hilbert problem for the Beltrami equations in quasidisks. J. Math. Sci., 211, No. 5, pp. 646­659. doi: https://doi.org/10.1007/s10958­015­2621­0 4. Becker, J. & Pommerenke, Ch. (1982). Hölder continuity of conformal mappings and nonquasiconformal Jordan curves. Comment. Math. Helv., 57, No. 2, pp. 221­225. doi: https://doi.org/10.1007/BF02565858 5. Gehring, F. W. & Martio, O. (1985). Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10, pp. 203­219. 6. Gehring, F. W. & Palka, B. P. (1976). Quasiconformally homogeneous domains. J. Analyse Math., 30, pp. 172­ 199. doi: https://doi.org/10.1007/bf02786713 7. Duren, P. L. (1970). Theory of Hp spaces. Pure and Applied Mathematics, Vol. 38. New York: Academic Press. 8. Koosis, P. (1998). Introduction to Hp spaces. Cambridge Tracts in Mathematics, Vol.115. Cambridge: Cambridge Univ. Press. 9. Pommerenke, Ch. (1992). Boundary behaviour of conformal maps. Grundlehren der Mathematischen Wissenschaften, Vol. 299. Berlin: Springer. 10. Landkof, N. S. (1972). Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, Vol. 180. New York: Springer. 11. Nevanlinna, R. (1944). Eindeutige analytische Funktionen. Michigan: Ann Arbor. 12. Goluzin, G. M. (1969). Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, Vol. 26. Providence, R.I.: American Mathematical Society. 13. Twomey, J. B. (1988). Tangential boundary behaviour of the Cauchy integral. J. London Math. Soc. (2), 37, No. 3, pp. 447­454. doi: https://doi.org/10.1112/jlms/s2­37.3.447 14. Federer, H. (1969). Geometric Measure Theory. Berlin: Springer. 15. Mikhlin, S. G. (1978). Partielle differentialgleichungen in der mathematischen physik. Mathematische Lehrbücher und Monographien, Bd. 30. Berlin: Akademie. Received 10.12.2018 В.Я. Гутлянський 1, В.І. Рязанов 1,2, E. Якубов 3, А.С. Єфімушкін 1 1 Інститут прикладної математики і механіки НАН України, Слов’янськ 2 Черкаський національний університет ім. Богдана Хмельницького 3 Холонський інститут технологій, Ізраїль E­mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com, yakubov@hit.ac.il, eduardyakubov@gmail.com, a.yefimushkin@gmail.com ПРО ЗАДАЧУ ГІЛЬБЕРТА ДЛЯ АНАЛІТИЧНИХ ФУНКЦІЙ У КВАЗІГІПЕРБОЛІЧНИХ ОБЛАСТЯХ Досліджено граничну задачу Гільберта для аналітичних функцій в жорданових областях, які задоволь­ няють квазігіперболічну умову Герінга—Мартіо. З припущенням, що коефіцієнти задачі є функціями зліченно­обмеженої варіації і граничні дані є вимірними відносно логарифмічної ємності, доведено іс­ нування розв’язків задачі в термінах кутових границь. Як наслідки отримано відповідні результати для крайових задач Діріхле, Неймана і Пуанкаре для гармонічних функцій. Ключові слова: крайові задачі Гільберта, Діріхле, Неймана і Пуанкаре, аналітичні і гармонічні функції, квазігіперболічна гранична умова, логарифмічна ємність, кутова границя. 30 ISSN 1025­6415. Dopov. Nac. akad. nauk Ukr. 2019. № 2 V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov, A.S. Yefimushkin В.Я. Гутлянский 1, В.И. Рязанов 1,2, Э. Якубов 3, А.C. Ефимушкин 1 1 Институт прикладной математики и механики НАН Украины, Славянск 2 Черкасский национальный университет им. Богдана Хмельницкого 3 Холонский институт технологий, Израиль E­mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com, yakubov@hit.ac.il, eduardyakubov@gmail.com, a.yefimushkin@gmail.com О ЗАДАЧЕ ГИЛЬБЕРТА ДЛЯ АНАЛИТИЧЕСКИХ ФУНКЦИЙ В КВАЗИГИПЕРБОЛИЧЕСКИХ ОБЛАСТЯХ Исследована краевая задача Гильберта для аналитических функций в жордановых областях, удовле­ т воряющих квазигиперболическому условию Геринга—Мартио. С предположением, что коэффициенты задачи являются функциями счетно­ограниченной вариации, а граничные данные измеримы относитель­ но логарифмической емкости, доказано существование решений задачи в терминах угловых пределов. В качестве следствий получены соответствующие результаты для краевых задач Дирихле, Неймана и Пуанкаре для гармонических функций. Ключевые слова: краевые задачи Гильберта, Дирихле, Неймана и Пуанкаре, аналитические и гармони­ ческие функции, квазигиперболическое краевое условие, логарифмическая емкость, угловой предел.

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