On theory of multivalent solutions for Riemann Hilbert problem
It is proved the existence of multivalent solutions for the Riemann–Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and princ...
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nasplib_isofts_kiev_ua-123456789-1408412025-02-09T17:04:05Z On theory of multivalent solutions for Riemann Hilbert problem К теории многозначных решений задачи Римана–Гильберта До теорiї многозначних рiшень задачi Рiмана–Гiльберта Ryazanov, V. It is proved the existence of multivalent solutions for the Riemann–Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthemore, it is shown that the dimension of the spaces of these solutions is infinite. Доказано существование многозначных решений задачи Римана–Гильберта при общих предположениях конечносвязных областей, ограниченных взаимно непересекающимися жордановыми кривыми, измеримых коэффициентах и измеримых граничных данных. Теорема сформулирована в терминах гармонической меры и главных асимптотических значений. Также приведен соответствующий усиленный критерий для областей со спрямляемыми границами, сформулированный в терминах натурального параметра длины и некасательных пределов. Кроме того, показано, что размерность пространства найденных решений бесконечна. Доведено iснування багатозначних рiшень задачi Рiмана–Гiльберта при загальних припущеннях кiнцевозв’язних областей, обмежених взаємно неперетинаючими жордановими кривими, вимiрних коефiцiєнтах i вимiрних граничних даних. Теорема сформульована в термiнах гармонiйної мiри i головних асимптотичних значень. Також наведено вiдповiдний посилений критерiй для областей зi спрямлюваними межами, сформульований в термiнах натурального параметра довжини i недотичних границь. Крiм того, показано, що розмiрнiсть простору знайдених рiшень нескiнченна. 2015 Article On theory of multivalent solutions for Riemann Hilbert problem / V. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Слов’янськ: ІПММ НАН України, 2015. — Т. 29. — С. 87-93. — Бібліогр.: 19 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/140841 517.5 en Труды Института прикладной математики и механики application/pdf Інститут прикладної математики і механіки НАН України |
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It is proved the existence of multivalent solutions for the Riemann–Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthemore, it is shown that the dimension of the spaces of these solutions is infinite. |
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Ryazanov, V. |
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Ryazanov, V. On theory of multivalent solutions for Riemann Hilbert problem Труды Института прикладной математики и механики |
| author_facet |
Ryazanov, V. |
| author_sort |
Ryazanov, V. |
| title |
On theory of multivalent solutions for Riemann Hilbert problem |
| title_short |
On theory of multivalent solutions for Riemann Hilbert problem |
| title_full |
On theory of multivalent solutions for Riemann Hilbert problem |
| title_fullStr |
On theory of multivalent solutions for Riemann Hilbert problem |
| title_full_unstemmed |
On theory of multivalent solutions for Riemann Hilbert problem |
| title_sort |
on theory of multivalent solutions for riemann hilbert problem |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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https://nasplib.isofts.kiev.ua/handle/123456789/140841 |
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On theory of multivalent solutions for Riemann Hilbert problem / V. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Слов’янськ: ІПММ НАН України, 2015. — Т. 29. — С. 87-93. — Бібліогр.: 19 назв. — англ. |
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Труды Института прикладной математики и механики |
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ISSN 1683-4720 Труды ИПММ НАН Украины. 2015. Том 29
UDK 517.5
c©2015. V. Ryazanov
ON THEORY OF MULTIVALENT SOLUTIONS
FOR RIEMANN–HILBERT PROBLEM
It is proved the existence of multivalent solutions for the Riemann–Hilbert problem in the general
settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coeffi-
cients and measurable boundary data. The theorem is formulated in terms of harmonic measure and
principal asymptotic values. It is also given the corresponding reinforced criterion for domains with
rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthemore,
it is shown that the dimension of the spaces of these solutions is infinite.
Keywords: Riemann–Hilbert problem, Jordan curves, harmonic measures, principal asymptotic
values, nontangential limits.
1. Introduction.
This note is a continuation of the paper [16] where the Riemann–Hilbert problem
was resolved in these general settings for simply connected domains. At the present
paper, on the basis of [16] and a theorem due to Poincare, see e.g. Section VI.1 in [6],
it is given a resolution of the problem for finitely connected domains.
Recall that boundary value problems for analytic functions are due to the Riemann
dissertation (1851), also to works of Hilbert (1904, 1912, 1924) and Poincaré (1910).
The Riemann dissertation contained a general setting of a problem on finding analytic
functions with a connection between their real and imaginary parts on the boundary.
The first concrete problem of such a type has been proposed by Hilbert (1904) and
called by the Hilbert problem or the Riemann–Hilbert problem. That consists in finding
an analytic function f inside of a domain bounded by a rectifiable Jordan curve C with
the boundary condition
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) ∀ ζ ∈ C (1)
where it was assumed by him that the functions λ and ϕ are continuously differentiable
with respect to the natural parameter s on C and, moreover, |λ| 6= 0 everywhere on C.
Hence without loss of generality one can assume that |λ| ≡ 1 on C.
The first way for solving this problem based on the theory of singular integral
equations was proposed by Hilbert (1904), see [7]. This attempt was not quite successful
because of the theory of singular integral equations has been not yet enough developed
at that time. However, just that way became the main approach in this research
direction, see e.g. the monographs [3, 11] and [19]. In particular, the existence of
solutions to this problem was in that way proved for Hölder continuous λ and ϕ, see e.g.
[3]. But subsequent weakening conditions on λ and ϕ in this way led to strengthening
conditions on the contour C, say to the Lyapunov curves or to the Radon condition of
bounded rotation or even to smooth curves.
87
V. Ryazanov
However, Hilbert (1905) has resolved his problem with the above settings to (1) in
the second way based on the reduction it to solving the corresponding two Dirichlet
problems, see e.g. [8]. It was recently shown in [16] that the latter approach makes
possible to obtain perfectly general results in the problem for the arbitrary Jordan
domains with coefficients λ and boundary data ϕ that are only measurable with respect
to the harmonic measure.
The key was the following Gehring result on the Dirichlet problem for harmonic
functions: if ϕ : R → R is 2π-periodic, measurable and finite a.e. with respect to the
Lebesgue measure, then there is a harmonic function in the unit disk D = {z ∈ C :
|z| < 1} such that u(z)→ ϕ(ϑ) for a.e. ϑ as z → eiϑ along any nontangential path, see
[5], see also [17]. But the way of the reduction of the Riemann–Hilbert problem to the
corresponding 2 Dirichlet problems was original in [16].
2. The case of circular domains.
Let us start from the simplest kind of multiply connected domains. Recall that a
domain D in C = C ∪ {∞} is called circular if its boundary consists of finite number
of mutually disjoint circles and points. We call such a domain nondegenerate if its
boundary consists only of circles.
Theorem 2.1 Let D∗ be a bounded nondegenerate circular domain and let λ :
∂D∗ → C, |λ(ζ)| ≡ 1, and ϕ : ∂D∗ → R be measurable functions. Then there exist
multivalent analytic functions f : D∗ → C such that
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) (2)
along any nontangential path to a.e. ζ ∈ ∂D∗.
Proof. Indeed, by the Poincare theorem, see e.g. Theorem VI.1 in [6], there is a
locally conformal mapping g of the unit disc D = {z ∈ C : |z| < 1} onto D∗. Let
h : D∗ → D be the corresponding multivalent analytic function that is inverse to g. D∗
without a finite number of cuts is simply connected and hence h has there only single-
valued branches that are extended to the boundary by the Caratheodory theorem.
By Section VI.2 in [6], ∂D without a countable set of its points consists of a coun-
table collection of arcs every of which is a one-to-one image of a circle in ∂D∗ without
its one point under every extended branch of h. Note that by the reflection principle g
is conformally extended into a neighborhood of every such arc and, thus, nontangential
paths to its points go into nontangential paths to the corresponding points of circles in
∂D∗ and inversely.
Setting Λ = λ ◦ g and Φ = ϕ ◦ g with the extended g on the given arcs of ∂D we
obtain measurable functions on ∂D. Thus, by Theorem 2.1 in [16] there exist analytic
functions F : D→ C such that
lim
w→η
Re {Λ(η) · F (w)} = Φ(η) (3)
along any nontangential path to a.e. η ∈ ∂D. By the above arguments, we see that
f = F ◦ h are desired multivalent analytic solutions of (2). 2
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On theory of multivalent solutions for Riemann–Hilbert problem
In particular, choosing λ ≡ 1 in (2), we obtain the following statement.
Proposition 2.2. Let D∗ be a bounded nondegenerate circular domain and let
ϕ : ∂D∗ → R be a measurable function. Then there exist multivalent analytic functions
f : D∗ → C such that
lim
z→ζ
Re f(z) = ϕ(ζ) (4)
along any nontangential path to a.e. ζ ∈ ∂D∗.
3. The case of rectifiable Jordan curves.
To resolve the Riemann–Hilbert problem in the case of domains bounded by a finite
number of rectifiable Jordan curves we should extend to this case the known results of
Caratheodory (1912), Lindelöf (1917), F. and M. Riesz (1916) and Lavrentiev (1936)
for Jordan’s domains.
Lemma 3.1 Let D be a bounded domain in C whose boundary components are
Jordan curves, D∗ be a bounded nondegenerate circular domain in C and let ω : D → D∗
be a conformal mapping. Then
(i) ω can be extended to a homeomorphism of D onto D∗;
(ii) arg [ω(ζ) − ω(z)] − arg [ζ − z] → const as z → ζ whenever ∂D has a tangent
at ζ ∈ ∂D;
(iii) for rectifiable ∂D, length ω−1(E) = 0 whenever |E| = 0, E ⊂ ∂D∗;
(iv) for rectifiable ∂D, |ω(E)| = 0 whenever length E = 0, E ⊂ ∂D.
Proof. (i) Indeed, we are able to transform D∗ into a simply connected domain D∗
through a finite sequence of cuts. Thus, we come to the desired conclusion applying the
Caratheodory theorems to simply connected domains D∗ and D∗ := ω−1(D∗), see e.g.
Theorem 9.4 in [2] and Theorem II.C.1 in [9].
(ii) In the construction from the previous item, we may assume that the point
ζ is not the end of the cuts in D generated by the cuts in D∗ under the extended
mapping ω−1. Thus, we come to the desired conclusion twice applying the Caratheodory
theorems, the reflection principle for conformal mappings and the Lindelöf theorem for
the Jordan domains, see e.g. Theorem II.C.2 in [9].
Points (iii) and (iv) are proved similarly to the last item on the basis of the
corresponding results of F. and M. Riesz and Lavrentiev for Jordan domains with
rectifiable boundaries, see e.g. Theorem II.D.2 in [9], and [10], see also the point III.1.5
in [14]. 2
Theorem 3.2. Let D be a bounded domain in C whose boundary components are
rectifiable Jordan curves and λ : ∂D → C, |λ(ζ)| ≡ 1, and ϕ : ∂D → R be measurable
functions with respect to the natural parameter on ∂D. Then there exist multivalent
analytic functions f : D→ C such that along any nontangential path
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (5)
with respect to the natural parameters of the boundary components of D.
89
V. Ryazanov
Proof. This case is reduced to the case of a bounded nondegenerate circular domain
D∗ in the following way. First, there is a conformal mapping ω of D onto a circular
domain D∗, see e.g. Theorem V.6.2 in [6]. Note that D∗ is not degenerate because
isolated singularities of conformal mappings are removable that is due to the well-
known Weierstrass theorem, see e.g. Theorem 1.2 in [2]. Without loss of generality, we
may assume that D∗ is bounded.
By point (i) in Lemma 3.1 ω can be extended to a homeomorphisms of D onto
D∗. If ∂D is rectifiable, then by point (iii) in Lemma 3.1 length ω−1(E) = 0 whenever
E ⊂ ∂D∗ with |E| = 0, and by (iv) in Lemma 3.1, conversely, |ω(E)| = 0 whenever
E ⊂ ∂D with length E = 0.
In the last case ω and ω−1 transform measurable sets into measurable sets. Indeed,
every measurable set is the union of a sigma-compact set and a set of measure zero,
see e.g. Theorem III(6.6) in [18], and continuous mappings transform compact sets into
compact sets. Thus, a function ϕ : ∂D → R is measurable with respect to the natural
parameter on ∂D if and only if the function Φ = ϕ◦ω−1 : ∂D∗ → R is measurable with
respect to the natural parameter on ∂D∗.
By point (ii) in Lemma 3.1, if ∂D has a tangent at a point ζ ∈ ∂D, then arg [ω(ζ)−
ω(z)]− arg [ζ − z]→ const as z → ζ. In other words, the conformal images of sectors
in D with a vertex at ζ is asymptotically the same as sectors in D∗ with a vertex at
w = ω(ζ). Thus, nontangential paths in D are transformed under ω into nontangential
paths in D∗ and inversely. Finally, a rectifiable Jordan curve has a tangent a.e. with
respect to the natural parameter and, thus, Theorem 3.2 follows from Theorem 2.1. 2
In particular, choosing λ ≡ 1 in (5), we obtain the following statement.
Proposition 3.3 Let D be a bounded domain in C whose boundary components
are rectifiable Jordan curves and let ϕ : ∂D → R be measurable. Then there exist
multivalent analytic functions f : D → C such that
lim
z→ζ
Re f(z) = ϕ(ζ) for a.e. ζ ∈ ∂D (6)
along any nontangential path with respect to the natural parameters of the boundary
components of ∂D.
4. The case of arbitrary Jordan curves.
The conceptions of a harmonic measure introduced by R. Nevanlinna in [12] and a
principal asymptotic value based on one nice result of F. Bagemihl [1] make possible
with a great simplicity and generality to formulate the existence theorems for the
Dirichlet and Riemann–Hilbert problems.
First of all, given a measurable set E ⊆ ∂D and a point z ∈ D, a harmonic measure
of E at z relative to D is the value at z of the bounded harmonic function u in D
with the boundary values 1 a.e. on E and 0 a.e on ∂D \E. In particular, by the mean
value theorem for harmonic functions, the harmonic measure of E at 0 relative to D is
equal to |E|/2π. In general, the geometric sense of the harmonic measure of E at z0
relative to D is the angular measure of view of E from the point z0 in radians divided
90
On theory of multivalent solutions for Riemann–Hilbert problem
by 2π. Hence the harmonic measure on ∂D has also the corresponding probabilistic
interpretation. The harmonic measure in domains D bounded by finite collections of
Jordan curves is defined in a similar way.
Next, a Jordan curve generally speaking has no tangents. Hence we need a replacement
for the notion of a nontangential limit. In this connection, recall Theorem 2 in [1], see
also Theorem III.1.8 in [13], stating that, for any function Ω : D → C, for all pairs of
arcs γ1 and γ2 in D terminating at ζ ∈ ∂D, except a countable set of ζ ∈ ∂D,
C(Ω, γ1) ∩ C(Ω, γ2) 6= ∅ (7)
where C(Ω, γ) denotes the cluster set of Ω at ζ along γ, i.e.,
C(Ω, γ) = {w ∈ C : Ω(zn)→ w, zn → ζ, zn ∈ γ} .
Applying the Poincare mapping, branches of its inverse mapping and their boundary
behavior, see e.g. Theorem VI.1 and Section VI.2 in [6], we extend this result to
arbitrary domains D bounded by a finite number of Jordan curves, cf. the proof of
Theorem 2.1.
Now, given a function Ω : D → C and ζ ∈ ∂D, denote by P (Ω, ζ) the intersection
of all cluster sets C(Ω, γ) for arcs γ in D terminating at ζ. Later on, we call the points
of the set P (Ω, ζ) principal asymptotic values of Ω at ζ. Note that, if Ω has a limit
along at least one arc in D terminating at a point ζ ∈ ∂D with the property (7), then
the principal asymptotic value is unique.
Theorem 4.1. Let D be a bounded domain in C whose boundary components are
Jordan curves and let λ : ∂D → C, |λ(ζ)| ≡ 1, and ϕ : ∂D → R be measurable functions
with respect to harmonic measures in D. Then there exist multivalent analytic functions
f : D→ C such that
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (8)
with respect to harmonic measures in D in the sense of the unique principal asymptotic
value.
Proof. By the reasons of the first item in the proof of Theorem 3.2, there is a
conformal mapping ω of D onto a bounded nondegenerate circular domain D∗ in C.
Set Λ = λ◦Ω and Φ = ϕ◦Ω where Ω := ω−1 extended to ∂D∗ by point (i) in Lemma 3.1.
Note that harmonic measure zero is invariant under conformal mappings. Thus,
arguing as in the third item of the proof to Theorem 3.2, we conclude that the functions
Λ and Φ are measurable with respect to harmonic measures in D∗.
By Theorem 2.1 there exist multivalent analytic functions F : D∗ → C such that
lim
w→η
Re {Λ(η) · F (w)} = Φ(η)
along any nontangential path to a.e. η ∈ ∂D∗.
91
V. Ryazanov
By the construction the functions f := F ◦ ω are desired multivalent analytic
solutions of (8) in view of the Bagemihl result. 2
In particular, choosing λ ≡ 1 in (8), we obtain the following consequence.
Proposition 4.2. Let D be a bounded domain in C whose boundary components
are Jordan curves and let ϕ : ∂D → R be a measurable function with respect to harmonic
measures in D. Then there exist multivalent analytic functions f : D → C such that
lim
z→ζ
Re f(z) = ϕ(ζ) for a.e. ζ ∈ ∂D (9)
with respect to harmonic measures in D in the sense of the unique principal asymptotic
value.
5. On dimension of spaces of solutions.
By the Lindelöf maximum principle, see e.g. Lemma 1.1 in [4], it follows the
uniqueness theorem for the Dirichlet problem in the class of bounded harmonic functions
on the unit disk. Our multivalent analytic solutions are generally speaking not bounded
and we have the new phenomena.
Theorem 5.1 The spaces of solutions of the Riemann–Hilbert problem in Theo-
rems 2.1, 3.2 and 4.1 and in Propositions 2.2, 3.3 and 4.2 have the infinite dimension.
Proof. By Theorem 5.1 in [16] the space of solutions of the problem (3) has the
infinite dimension. Thus, the conclusion follows by the construction of these solutions
in the given theorems through the reduction to (3). 2
1. Bagemihl F. Curvilinear cluster sets of arbitrary functions // Proc. Nat. Acad. Sci. U.S.A. – 2015.
– V. 41. – P. 379-382.
2. Collingwood E.F., Lohwator A.J. The theory of cluster sets // Cambridge Tracts in Math. and
Math. Physics, No. 56, Cambridge Univ. Press, Cambridge, 1966.
3. Gakhov F.D. Boundary value problems. – Dover Publications Inc., New York, 1990.
4. Garnett J.B., Marshall D.E. Harmonic Measure. – Cambridge Univ. Press, Cambridge, 2005.
5. Gehring F.W. On the Dirichlet problem // Michigan Math. J. – 1955—1956. – V. 3. – P. 201.
6. Goluzin G.M. Geometric theory of functions of a complex variable. – Transl. of Math. Monographs,
Vol. 26, American Mathematical Society, Providence, R.I. 1969.
7. Hilbert D. Über eine Anwendung der Integralgleichungen auf eine Problem der Funktionentheorie.
– Verhandl. des III Int. Math. Kongr., Heidelberg, 1904.
8. Hilbert D. Grundzüge einer algemeinen Theorie der Integralgleichungen. – Leipzig, Berlin, 1912.
9. Koosis P. Introduction toHp spaces. – 2nd ed., Cambridge Tracts in Mathematics, 115, Cambridge
Univ. Press, Cambridge, 1998.
10. Lavrentiev M. On some boundary problems in the theory of univalent functions // Mat. Sbornik
N.S. – 1936. – V. 1, No.43. – P. 815–846 [in Russian].
11. Muskhelishvili N.I. Singular integral equations. Boundary problems of function theory and their
application to mathematical physics. – Dover Publications. Inc., New York, 1992.
12. Nevanlinna R. Eindeutige analytische Funktionen. – Ann Arbor, Michigan, 1944.
13. Noshiro K. Cluster sets. – Springer-Verlag, Berlin etc., 1960.
14. Priwalov I.I. Randeigenschaften analytischer Funktionen. – Hochschulbücher für Mathematik, Bd.
25, Deutscher Verlag der Wissenschaften, Berlin, 1956.
15. Riesz M. Sur les functions conjuguees // Math. Z. – 1927. – Vol. 27, No. 2. – P. 218-244.
92
On theory of multivalent solutions for Riemann–Hilbert problem
16. Ryazanov V. On the Riemann-Hilbert Problem without Index // Ann. Univ. Bucharest, Ser.
Math. – 2014. – Vol. 5 (LXIII), No. 1. – P. 169-178.
17. Ryazanov V. Infinite dimension of solutions of the Dirichlet problem // Open Math. (the former
Central European J. Math.). – 2015. – V. 13, no. 1. – P. 348-350.
18. Saks S. Theory of the integral. – Warsaw, 1937; Dover Publications Inc., New York, 1964.
19. Vekua I. N. Generalized analytic functions. – Pergamon Press, London etc., 1962.
В. И. Рязанов
К теории многозначных решений задачи Римана–Гильберта.
Доказано существование многозначных решений задачи Римана–Гильберта при общих предпо-
ложениях конечносвязных областей, ограниченных взаимно непересекающимися жордановыми
кривыми, измеримых коэффициентах и измеримых граничных данных. Теорема сформулиро-
вана в терминах гармонической меры и главных асимптотических значений. Также приведен
соответствующий усиленный критерий для областей со спрямляемыми границами, сформули-
рованный в терминах натурального параметра длины и некасательных пределов. Кроме того,
показано, что размерность пространства найденных решений бесконечна.
Ключевые слова: задача Римана–Гильберта, жордановы кривые, гармоническая мера, глав-
ные асимптотические значения, некасательные пределы.
В. I. Рязанов
До теорiї многозначних рiшень задачi Рiмана–Гiльберта.
Доведено iснування багатозначних рiшень задачi Рiмана–Гiльберта при загальних припущеннях
кiнцевозв’язних областей, обмежених взаємно неперетинаючими жордановими кривими, вимiр-
них коефiцiєнтах i вимiрних граничних даних. Теорема сформульована в термiнах гармонiйної
мiри i головних асимптотичних значень. Також наведено вiдповiдний посилений критерiй для
областей зi спрямлюваними межами, сформульований в термiнах натурального параметра до-
вжини i недотичних границь. Крiм того, показано, що розмiрнiсть простору знайдених рiшень
нескiнченна.
Ключовi слова: задача Рiмана–Гiльберта, жордановi кривi, гармонiйна мiра, головнi асимп-
тотичнi значення, недотичнi границi.
Ин-т прикл. математики и механики НАН Украины, Славянск
vl.ryazanov1@gmail.com
Received 11.10.15
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