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Hilbert boundary value problem for generalized
analytic functions with a singular line
Pavel Shabalin1[0000-0002-2791-3964], and Rafael Faizov1[0000-0003-4744-164X]
1Kazan State University of Architecture and Engineering, 420043 Kazan, Russia
Abstract. In this paper, we study an inhomogeneous Hilbert boundary
value problem with a finite index and a boundary condition on a circle for a
generalized Cauchy-Riemann equation with a singular coefficient. To solve
this problem, we conducted a complete study of the solvability of the Hilbert
boundary value problem of the theory of analytic functions with an infinite
index due to a finite number of points of a special type of vorticity. Based
on these results, we have derived a formula for the general solution and
studied the existence and number of solutions to the boundary value problem
of the theory of generalized analytic functions.
Keywords. Generalized analytical functions, Hilbert problem, infinite
index, refined zero-order whole functions.
1 Introduction
Analytical and generalized analytical functions are widely used in modeling various physical
and mechanical processes (for example [1, 2]). In particular, the theory of generalized analytic
functions has deep applications to mechanical problems of infinitesimal surface bends [3] and
problems of stress state of membrane shell theory [3, 4]. The Riemann boundary value problems
(in particular, the jump problem) [5, 6] and Hilbert [7, 8] play a significant role in this case. We
can interpret any infinitesimal bending of the surface as a certain state of stress equilibrium of
the shell. This state of the shell can be described by solving a homogeneous equation. We reduce
this equation to the Hilbert boundary value problem for analytic functions. The determination
of the hydrodynamic pressure on the side surfaces of the shell in the presence of a surface load
is also reduced to the Riemann-Hilbert problem of the theory of generalized analytical functions
[3], p. 491. So, the boundary value problems for generalized analytical functions are the
apparatus for solving mechanical problems, and the methods developed for the boundary
value problems of the theory of generalized analytical functions can also be used to solve
many nonlinear problems of the general bending problem.
Such well-known mathematicians as A.V. Bitsadze [9], M.I. Višik and O.A.
Ladyženskaya [10], V.A. Solonnikov [11], N.A. Zhura and A.P. Soldatov [12, 13], A.P.
Soldatov [14], A.B. Rasulov and A.P. Soldatov [15], and others were engaged in boundary
value problems for solving systems of differential equations. The theory of boundary value
problems for systems of elliptic differential equations turned out to be very promising for
applications in mechanics and physics.
Corresponding author: rafael.faizov2@gmail.com
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons
Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
In classical theory [3] generalized analytic functions U(z), z = x + iy, satisfy in some
domain a linear elliptic system of differential equations, which is usually given in the
following complex notation:
,
2
1
:=),(=)()(
y
i
x
zFUzBUzAU zz
(1)
where the coefficients and the right-hand side of the equation are summable functions with
some degree p > 2. Fundamental results in the theory of boundary value problems for
solutions of the system of the type (1) are obtained by I.N. Vekua [3, 4]. Notable contributions
to the development of this theory were made by J. Nitsche [14], B.V. Boyarsky [15], L.G.
MihaĭLov [16], A.P. Soldatov [17, 18].
The relaxation of restrictions on the coefficients of equation (1) is an important direction
in the development of the theory of generalized analytic functions. Z.D. Usmanov [19]
constructed a complete theory of generalized Cauchy-Riemann systems, whose coefficients
have a polar singularity of the 1st order at an isolated inner point of the domain. On the basis
of the developed analytical apparatus, he investigated local and global problems of the theory
of infinitesimal bends of surfaces with an isolated flattening point. The construction of the
solution of the generalized Cauchy-Riemann system with strong singularities of the
coefficients at an isolated point is also devoted to the work [20]. The study of solutions of
equation (1) in the case when the coefficients of this equation have singular lines is given in
the monograph [21], also [22, 23]. We emphasize that the presence of strong singularities in
the coefficients of equation (1) is not only a natural development of the classical theory of
generalized analytic functions, but is also in demand as models for problems of thin
momentless shells, axisymmetric field theory, and deformation problems [9, 3].
Boundary value problems for generalized Cauchy-Riemann equations with singular
coefficients were solved by N.R. Radjabov, A.B. Rasulov, A.P. Soldatov, U.S. Fedorov,
Bobodzhanova M.A., and others (for example, [24-27]). In these papers, the method for solving
the boundary value problem for equation (1) is based on a reduction to a similar problem for
analytic functions.In this case (for example, [28, 29]), it is possible to formulate boundary value
problems for generalized analytic functions with a singular line, when the coefficients of
equation (1) delegate their features to the boundary condition of the problem for analytic
functions and turn the latter into a problem with an infinite index. A.B. Rasulov investigated
some situations related to this effect, when the boundary value problem of the theory of
generalized analytic functions with a finite index has an infinite set of solutions [28, 29].
This article is written in line with these works of A.B. Rasulov. The paper contains a
detailed study of the solvability of the Hilbert boundary value problem with finite index for
a class of generalized analytic functions with a singular line. The results of the article
supplement the research of A.P. Soldatov and A.B. Rasulov [30], who, under certain
restrictions on singular coefficients, derived the formula for the general solution of equation
(1). This solution was used by the authors in reducing the boundary value problem for
equation (1) to a similar problem with a finite index for analytic functions.
By loosening the restrictions on the coefficients from [30], we reduce the solution of the
Hilbert problem for generalized analytic functions with a finite index to the problem for
analytic functions, but with an infinite index and two points of vorticity with a new type of
singularities. We construct a formula for the general solution of the problem, and conduct a
complete study of the solvability.
2 Materials and methods
In the unit circle D = {z = reiθ:0 < r < 1, 0 ≤ θ < 2π}, L = ∂D, the plane of a complex variable
z = x + iy = reiθ we consider the Hilbert boundary value problem about finding by the
boundary condition:
Re
[
e
iα(t)
U
(
t
)] =
f
(
t
),
t
∈
L
,
α
(
t
),
f
(
t
)
∈
H
(
L
),
(2)
solution U(z) of the generalized Cauchy-Riemann system with a singular line:
.
)(
=)(),(=)(
z
z
za
zAzFUzAU
z
(3)
We highlight that equation (3) is a special case of the studied by A.P. Soldatov and A.B.
Rasulov in [14] of the generalized Cauchy-Riemann equation:
∂z
̅U – A(z)U + B(z)U
̅ = F(z),
1,<<0 0,> ,
||
)(
=)( ,
||
)()(
=)( 1mn
zz
zb
zB
zz
zazz
zA mn
in a singly connected domain G with a smooth border, with singular (n=1) or super-singular
(n>1) line. For this equation, a formula for the general solution is derived, and a boundary
value problem with a combined boundary condition is set and solved. These conditions
combine the features of the linear conjugation problem and the Hilbert problem.
Following [30], we will assume that for A(z) there is such an analytic in D function α(z)
for that:
2.> ),(
)()(
=)( 0
0pDL
z
z
zaza
zA p
(4)
In [30] under additional conditions:
a
0
(
τ
0
) =
a
0
(
τ
1
) = 0,
a
0
(
t
)
∈
H
(
L
),
(5)
where τ0, τ1 – points of intersection of the contour L with a singular line (in this article τ0 = 1,
τ1 = –1), a formula for the general solution of equation (3) is derived, which for the domain
D takes the form:
U
(
z
) =
e
Ω(z)
[(
T
(
e
–Ω
F
))(
z
) +
ϕ
(
z
)],
(6)
function:
|,|ln)(
||ln)(
2
1
))((=)( 0
0
0zzzadt
zt
ttta
i
zTAz
L
(7)
Vekua integral operator:
,,
)(1
=))((20
0Dz
z
dA
zTA
D
acts [1] from Lp(D), p > 2, to the Helder class H(D
̅), the function ϕ(z) is analytic in the domain
G0 ⸦ D. We emphasize that conditions (5) also guarantee a finite index of the boundary value
problem for analytic functions, which is obtained when solving the problem considered in
[30] with a combined boundary condition.
In this paper, instead of condition (5), we assume that the following asymptotic formulas
are satisfied:
a
0
(
t
) =
a
0
(
τ
j
± 0) +
O
(|
t
–
τ
j
|),
t
→
τ
j
± 0,
j
= 0,
j
= 1,
a
0
(
τ
j
± 0) =
α
j
±
+
iβ
j
±
.
(8)
Accordingly, we will consider the boundary values of the function a0(z) to be Helder-
continuous on the upper L+ and lower L– arcs of the unit circle, including the ends. This leads
to the following changes in the proof of formula (6).
As in [14], we introduce Dε± = D ∩ {± Imz > ε}, ε – as a small positive number and denote
Dε = Dε+ ⋃ Dε–, the boundary of the region D/Dε is made up of the union lε of two segments
lε± and the union γε of two arcs of the unit circle γε,0, γε,1. These circles contain the points
τ0 = 1, τ1 = –1, respectively. We denote by the symbol TεA the integral Vekua operator on the
2
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
In classical theory [3] generalized analytic functions U(z), z = x + iy, satisfy in some
domain a linear elliptic system of differential equations, which is usually given in the
following complex notation:
,
2
1
:=),(=)()(
y
i
x
zFUzBUzAU zz
(1)
where the coefficients and the right-hand side of the equation are summable functions with
some degree p > 2. Fundamental results in the theory of boundary value problems for
solutions of the system of the type (1) are obtained by I.N. Vekua [3, 4]. Notable contributions
to the development of this theory were made by J. Nitsche [14], B.V. Boyarsky [15], L.G.
MihaĭLov [16], A.P. Soldatov [17, 18].
The relaxation of restrictions on the coefficients of equation (1) is an important direction
in the development of the theory of generalized analytic functions. Z.D. Usmanov [19]
constructed a complete theory of generalized Cauchy-Riemann systems, whose coefficients
have a polar singularity of the 1st order at an isolated inner point of the domain. On the basis
of the developed analytical apparatus, he investigated local and global problems of the theory
of infinitesimal bends of surfaces with an isolated flattening point. The construction of the
solution of the generalized Cauchy-Riemann system with strong singularities of the
coefficients at an isolated point is also devoted to the work [20]. The study of solutions of
equation (1) in the case when the coefficients of this equation have singular lines is given in
the monograph [21], also [22, 23]. We emphasize that the presence of strong singularities in
the coefficients of equation (1) is not only a natural development of the classical theory of
generalized analytic functions, but is also in demand as models for problems of thin
momentless shells, axisymmetric field theory, and deformation problems [9, 3].
Boundary value problems for generalized Cauchy-Riemann equations with singular
coefficients were solved by N.R. Radjabov, A.B. Rasulov, A.P. Soldatov, U.S. Fedorov,
Bobodzhanova M.A., and others (for example, [24-27]). In these papers, the method for solving
the boundary value problem for equation (1) is based on a reduction to a similar problem for
analytic functions.In this case (for example, [28, 29]), it is possible to formulate boundary value
problems for generalized analytic functions with a singular line, when the coefficients of
equation (1) delegate their features to the boundary condition of the problem for analytic
functions and turn the latter into a problem with an infinite index. A.B. Rasulov investigated
some situations related to this effect, when the boundary value problem of the theory of
generalized analytic functions with a finite index has an infinite set of solutions [28, 29].
This article is written in line with these works of A.B. Rasulov. The paper contains a
detailed study of the solvability of the Hilbert boundary value problem with finite index for
a class of generalized analytic functions with a singular line. The results of the article
supplement the research of A.P. Soldatov and A.B. Rasulov [30], who, under certain
restrictions on singular coefficients, derived the formula for the general solution of equation
(1). This solution was used by the authors in reducing the boundary value problem for
equation (1) to a similar problem with a finite index for analytic functions.
By loosening the restrictions on the coefficients from [30], we reduce the solution of the
Hilbert problem for generalized analytic functions with a finite index to the problem for
analytic functions, but with an infinite index and two points of vorticity with a new type of
singularities. We construct a formula for the general solution of the problem, and conduct a
complete study of the solvability.
2 Materials and methods
In the unit circle D = {z = reiθ:0 < r < 1, 0 ≤ θ < 2π}, L = ∂D, the plane of a complex variable
z = x + iy = reiθ we consider the Hilbert boundary value problem about finding by the
boundary condition:
Re
[
e
iα(t)
U
(
t
)] =
f
(
t
),
t
∈
L
,
α
(
t
),
f
(
t
)
∈
H
(
L
),
(2)
solution U(z) of the generalized Cauchy-Riemann system with a singular line:
.
)(
=)(),(=)(
z
z
za
zAzFUzAU
z
(3)
We highlight that equation (3) is a special case of the studied by A.P. Soldatov and A.B.
Rasulov in [14] of the generalized Cauchy-Riemann equation:
∂z
̅U – A(z)U + B(z)U
̅ = F(z),
1,<<0 0,> ,
||
)(
=)( ,
||
)()(
=)( 1mn
zz
zb
zB
zz
zazz
zA mn
in a singly connected domain G with a smooth border, with singular (n=1) or super-singular
(n>1) line. For this equation, a formula for the general solution is derived, and a boundary
value problem with a combined boundary condition is set and solved. These conditions
combine the features of the linear conjugation problem and the Hilbert problem.
Following [30], we will assume that for A(z) there is such an analytic in D function α(z)
for that:
2.> ),(
)()(
=)( 0
0pDL
z
z
zaza
zA p
(4)
In [30] under additional conditions:
a
0
(
τ
0
) =
a
0
(
τ
1
) = 0,
a
0
(
t
)
∈
H
(
L
),
(5)
where τ0, τ1 – points of intersection of the contour L with a singular line (in this article τ0 = 1,
τ1 = –1), a formula for the general solution of equation (3) is derived, which for the domain
D takes the form:
U
(
z
) =
e
Ω(z)
[(
T
(
e
–Ω
F
))(
z
) +
ϕ
(
z
)],
(6)
function:
|,|ln)(
||ln)(
2
1
))((=)( 0
0
0zzzadt
zt
ttta
i
zTAz
L
(7)
Vekua integral operator:
,,
)(1
=))((20
0Dz
z
dA
zTA
D
acts [1] from Lp(D), p > 2, to the Helder class H(D
̅), the function ϕ(z) is analytic in the domain
G0 ⸦ D. We emphasize that conditions (5) also guarantee a finite index of the boundary value
problem for analytic functions, which is obtained when solving the problem considered in
[30] with a combined boundary condition.
In this paper, instead of condition (5), we assume that the following asymptotic formulas
are satisfied:
a
0
(
t
) =
a
0
(
τ
j
± 0) +
O
(|
t
–
τ
j
|),
t
→
τ
j
± 0,
j
= 0,
j
= 1,
a
0
(
τ
j
± 0) =
α
j
±
+
iβ
j
±
.
(8)
Accordingly, we will consider the boundary values of the function a0(z) to be Helder-
continuous on the upper L+ and lower L– arcs of the unit circle, including the ends. This leads
to the following changes in the proof of formula (6).
As in [14], we introduce Dε± = D ∩ {± Imz > ε}, ε – as a small positive number and denote
Dε = Dε+ ⋃ Dε–, the boundary of the region D/Dε is made up of the union lε of two segments
lε± and the union γε of two arcs of the unit circle γε,0, γε,1. These circles contain the points
τ0 = 1, τ1 = –1, respectively. We denote by the symbol TεA the integral Vekua operator on the
3
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
union of domains Dε. We need to make sure that (TεA)(z), z ∈ K, converges uniformly at ε→0
to the limit of Ω(z) on any compact K, K ∈ D+ ⋃ D–, where D± = D ∩ {± Imz > 0}. Following
[30], we represent TεA in the form:
,
)(1
:=)(),())((=))((20
0z
d
zz
a
zIzIzATzAT
D
and the last integral is understood in the sense of:
}.|{|=,
)(
1
lim
=)( ,
2
0
,
0
zDD
z
d
zz
a
zI
D
We transform the right-hand side of the equality according to Green's formula:
,
||ln)(
2
1
||ln)(
2
1
=
)(
10
|=|
0
\
20
,
dt
zt
ttta
i
dt
zt
ttta
iz
d
zz
a
tzlLD
the segment lε– is oriented negatively. After moving to the limit in δ → 0 we get:
.||ln)(
||ln)(
2
1
=)( 0
0
\
zzzadt
zt
ttta
i
zI
lL
We will take into account that:
,
)(
2
2ln
=
||ln)(
2
100 dt
zt
ta
i
dt
zt
ttta
ill
now, for z ∈ K and sufficiently small values of ε by Cauchy's theorem, we have:
.
)(
2
2ln
=
)(
2
2ln 00 dt
zt
ta
i
dt
zt
ta
il
For z ∈ K and for ε → 0, using the Helder continuity on the arcs (including ends) L+ and
L– of the function a0(t), we derive:
0.
)(
2
2ln 0
dt
zt
ta
i
Thus,
,
||ln)(
2
1
||ln)(=
)(
1
lim 0
0
2
0
0
dt
zt
ttta
i
zzza
z
d
zz
a
LD
in this case, the Cauchy-type integral with two points of discontinuity of the density of the
logarithmic type has [31] near the point τj of the form:
),(
2
)(ln
0)](0)([
4
)(
ln
0)](0)([=
||ln)(
2
1
:=)(
00
2
00
0
z
z
aa
i
z
aadt
zt
ttta
i
z
j
jj
j
jj
L
(9)
where Ψ(z) – is an analytic in D function tending to a certain limit at z → τj, j =0, j = 1. In
accordance with equality (7), (9) the function Ω(z) has the form of a point τj the following
asymptotic representation:
(1).||ln)(
||ln
2
)(arg
2
||
ln
4
||ln
2
)(arg
2
||
ln
4
=)(
0
2
2
Ozzza
zzzi
zzzz
j
jj
j
jj
j
jj
j
jj
j
jj
j
jj
(10)
Here, by arg(z – τj) we mean a continuous branch in the domain D, whose boundary
values satisfy the equalities:
.2<)/2,(
,<0)/2,(3
=)(arg
jj
jj
j
i
e
Now we repeat the calculations from [14] and derive the formula for the general solution
of equation (3) in the form (6).
Using formula (6), we derive a formula for the general solution of the Hilbert boundary
value problem (2) for solutions U(z) of differential equation (3) in the class A of functions
U(z) with a product Ue–CΩ, bounded in D
̅ with some C ≥ 1.
3 Results
After we substitute function (6) into condition (2), we obtain the boundary condition for the
Hilbert problem for the analytic function ϕ(z) in the disk:
Re[eiα
*
(t)ϕ(t)] = f*(t), t
∈
L, t ≠ τ0, t ≠ τ1, α*(t) = α(t) + ImΩ(t),
f*
(
t
) =
f
(
t
)
e
–ReΩ(t)
–
Re
[
e
iα*(t)
(
T
(
e
–Ω
F
))(
t
)].
(11)
After we pass in formula (9) to the limit in z → t, t ∈ L, near singular points and introduce
the notation ,
4
7
4
5
= ,
4
1
4
13
= ,
4
9
4
3
=111000000
,
4
3
4
15
111
we get:
0,(1),||ln||
ln
4
=)(
0,(1),||ln||
ln
4
=)(
0000
2
00
0000
2
00
tOtttRe
tOtttRe
(12)
0.(1),|)|ln(||
ln
4
=)(
0,(1),|)|ln(||
ln
4
=)(
000
2
00
000
2
00
tOtOttIm
tOtOttIm
(13)
0,(1),||ln||
ln
4
=)(
0,(1),||ln||
ln
4
=)(
1111
2
11
1111
2
11
tOtttRe
tOtttRe
(14)
0.(1),|)|ln(||
ln
4
=)(
0,(1),|)|ln(||
ln
4
=)(
111
2
11
111
2
11
tOtOttIm
tOtOttIm
(15)
Thus, we reduced the Hilbert problem with a finite index for solving equation (1) to the
Hilbert problem (11) with an infinite index and two logarithmic vorticity points for the
analytic function. Since the Hilbert boundary value problem with such a vorticity character
has not yet been studied, we will carry out the solution of the problem in detail. The boundary
condition of the problem, taking into account the formulas (9), (12)-(15) and the equality ln|t
– t̅| = 2ln|t – t0| + 2ln|t – t1| – 2ln2, we will rewrite in the form:
Re
{
e
Φ(t)+2a0(t)ln((t – τ0)(t – τ1))
e
Γ+(t)
ϕ
(
t
)] =
f*
(
t
)
e
Γ0(t)
e
Re[Φ(t) + 2a0(t)ln((t – τ0)(t – τ1))]
,
(16)
where Γ+(t) – is the limit value on the contour L of the integral:
z
dt
z
L
)(1
=)(
4
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
union of domains Dε. We need to make sure that (TεA)(z), z ∈ K, converges uniformly at ε→0
to the limit of Ω(z) on any compact K, K ∈ D+ ⋃ D–, where D± = D ∩ {± Imz > 0}. Following
[30], we represent TεA in the form:
,
)(1
:=)(),())((=))((20
0z
d
zz
a
zIzIzATzAT
D
and the last integral is understood in the sense of:
}.|{|=,
)(
1
lim
=)( ,
2
0
,
0
zDD
z
d
zz
a
zI
D
We transform the right-hand side of the equality according to Green's formula:
,
||ln)(
2
1
||ln)(
2
1
=
)(
10
|=|
0
\
20
,
dt
zt
ttta
i
dt
zt
ttta
iz
d
zz
a
tzlLD
the segment lε– is oriented negatively. After moving to the limit in δ → 0 we get:
.||ln)(
||ln)(
2
1
=)( 0
0
\
zzzadt
zt
ttta
i
zI
lL
We will take into account that:
,
)(
2
2ln
=
||ln)(
2
100 dt
zt
ta
i
dt
zt
ttta
ill
now, for z ∈ K and sufficiently small values of ε by Cauchy's theorem, we have:
.
)(
2
2ln
=
)(
2
2ln 00 dt
zt
ta
i
dt
zt
ta
il
For z ∈ K and for ε → 0, using the Helder continuity on the arcs (including ends) L+ and
L– of the function a0(t), we derive:
0.
)(
2
2ln 0
dt
zt
ta
i
Thus,
,
||ln)(
2
1
||ln)(=
)(
1
lim 0
0
2
0
0
dt
zt
ttta
i
zzza
z
d
zz
a
LD
in this case, the Cauchy-type integral with two points of discontinuity of the density of the
logarithmic type has [31] near the point τj of the form:
),(
2
)(ln
0)](0)([
4
)(
ln
0)](0)([=
||ln)(
2
1
:=)(
00
2
00
0
z
z
aa
i
z
aadt
zt
ttta
i
z
j
jj
j
jj
L
(9)
where Ψ(z) – is an analytic in D function tending to a certain limit at z → τj, j =0, j = 1. In
accordance with equality (7), (9) the function Ω(z) has the form of a point τj the following
asymptotic representation:
(1).||ln)(
||ln
2
)(arg
2
||
ln
4
||ln
2
)(arg
2
||
ln
4
=)(
0
2
2
Ozzza
zzzi
zzzz
j
jj
j
jj
j
jj
j
jj
j
jj
j
jj
(10)
Here, by arg(z – τj) we mean a continuous branch in the domain D, whose boundary
values satisfy the equalities:
.2<)/2,(
,<0)/2,(3
=)(arg
jj
jj
j
i
e
Now we repeat the calculations from [14] and derive the formula for the general solution
of equation (3) in the form (6).
Using formula (6), we derive a formula for the general solution of the Hilbert boundary
value problem (2) for solutions U(z) of differential equation (3) in the class A of functions
U(z) with a product Ue–CΩ, bounded in D
̅ with some C ≥ 1.
3 Results
After we substitute function (6) into condition (2), we obtain the boundary condition for the
Hilbert problem for the analytic function ϕ(z) in the disk:
Re[eiα
*
(t)ϕ(t)] = f*(t), t
∈
L, t ≠ τ0, t ≠ τ1, α*(t) = α(t) + ImΩ(t),
f*
(
t
) =
f
(
t
)
e
–ReΩ(t)
–
Re
[
e
iα*(t)
(
T
(
e
–Ω
F
))(
t
)].
(11)
After we pass in formula (9) to the limit in z → t, t ∈ L, near singular points and introduce
the notation ,
4
7
4
5
= ,
4
1
4
13
= ,
4
9
4
3
=111000000
,
4
3
4
15
111
we get:
0,(1),||ln||
ln
4
=)(
0,(1),||ln||
ln
4
=)(
0000
2
00
0000
2
00
tOtttRe
tOtttRe
(12)
0.(1),|)|ln(||
ln
4
=)(
0,(1),|)|ln(||
ln
4
=)(
000
2
00
000
2
00
tOtOttIm
tOtOttIm
(13)
0,(1),||ln||
ln
4
=)(
0,(1),||ln||
ln
4
=)(
1111
2
11
1111
2
11
tOtttRe
tOtttRe
(14)
0.(1),|)|ln(||
ln
4
=)(
0,(1),|)|ln(||
ln
4
=)(
111
2
11
111
2
11
tOtOttIm
tOtOttIm
(15)
Thus, we reduced the Hilbert problem with a finite index for solving equation (1) to the
Hilbert problem (11) with an infinite index and two logarithmic vorticity points for the
analytic function. Since the Hilbert boundary value problem with such a vorticity character
has not yet been studied, we will carry out the solution of the problem in detail. The boundary
condition of the problem, taking into account the formulas (9), (12)-(15) and the equality ln|t
– t̅| = 2ln|t – t0| + 2ln|t – t1| – 2ln2, we will rewrite in the form:
Re
{
e
Φ(t)+2a0(t)ln((t – τ0)(t – τ1))
e
Γ+(t)
ϕ
(
t
)] =
f*
(
t
)
e
Γ0(t)
e
Re[Φ(t) + 2a0(t)ln((t – τ0)(t – τ1))]
,
(16)
where Γ+(t) – is the limit value on the contour L of the integral:
z
dt
z
L
)(1
=)(
5
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STCCE – 2021
with a density φ(t) = α(t) + Im[Ω(t) – Φ(t) – 2a0(t)(ln(t – τ0) + ln(t – τ1))] continuous
throughout on L:
.
)(1
=)(
0t
dt
t
L
We will first consider a homogeneous problem:
.=,= ,0,=)]([ 10
)(
))
1
(ln)
0
(ln)((
0
2)(
ttLtteeRe t
tttat
(17)
We will introduce an analytic function in D:
.)(:=)( )(
))
1
(ln)
0
(ln)((
0
2)( zeiezF z
zzzaz
(18)
This function on L by virtue of (17) satisfies the condition:
ImF
+
(
t
) = 0.
(19)
We will express from equality (18) the desired function:
).(=)( )(
))
1
(ln)
0
(ln)((
0
2)( zFeiez z
zzzaz
(20)
It is clear that if function (20) is a bounded solution to problem (17), then there must be
a function F(z), analytic in D. This function satisfies the inequality:
1,= 0= , ,,|)(| ||ln
2
||
ln2
1jилиjzDzCezF j
j
zC
j
zC
(21)
That is, this function is an exact growth of T in a semi-neighborhood of the point T, and
its boundary values must satisfy condition (19) and the inequality:
|
F
(
t
)|
≤
Ce
ReΩ(t)
,
t
∈
L
,
t
≠
τ
j
,
j
= 0,
j
= 1.
(22)
The validity of the inverse statement follows from the generalized maximum principle for
analytic functions ([32] p. 456, 457, see also [33]). Thus, it is proved.
3.1 Theorem 1
In order for the solution ϕ(z) of problem (17) to be bounded in the domain D, it is necessary
and sufficient that the function F(z), which is included in the formula of the general solution
(20), satisfies in D the growth constraints (21) and on the boundary conditions (19) and (22).
It is clear that the existence and set of bounded solutions to problem (17) depends on the
existence and set of analytic functions in D, and they satisfy conditions (19), (21), and (22).
In [35], it is proved that a homogeneous Hilbert problem with n points of vorticity is solvable
if and only if all n homogeneous problems with a single point of vorticity are solvable, that
is, the solvability of the problem is affected only by the parameters of each point of vorticity.
In this case, the solution of the problem with n points of vorticity can be represented as the
products of the solutions of these problems with a single point of vorticity. Thus, the solution
of problem (17) ϕ(z) = ϕ0(z)ϕ1(z) where ϕj(z) is the general solution of a homogeneous
problem with a single point of vorticity τj, j = 0, j = 1, is determined by the formula:
ϕ
j
(
z
) =
–
ie
–Φj(z) – 2a0(z)ln(z – τj)
e
–Γj
+
(z)
F
j
(
z
),
(23)
where Fj(z) is an analytic function in the domain D that satisfies the conditions:
. ,|)(|, ,|)(| 0,=)( ||ln
2
||
ln2
1
)(
j
j
zC
j
zC
j
tRe
jj zCezFLtCetFtFIm
(24)
Besides F(z) = F0(z)F1(z). If we use these remarks, the following theorem will be proved.
3.2 Theorem 2
Homogeneous boundary value problem (17)
a) has no nontrivial bounded solutions if:
β0– – β0+ < 0, or β1– – β1+ < 0,
β0– – β0+ = 0 and Δ0+ > 0, or β0– – β0+ = 0 and Δ0– > 0,
β0– – β1+ = 0 and Δ1+ > 0, or β1– – β1+ = 0 and Δ1– > 0;
b) has a single solution of the form:
0,= ,= ,=)( )(
))
1
(ln)
0
(ln)((
0
)( ImAconstAAeiezz
zzzaz
if one of the following two conditions is met β0– – β0+ = 0, Δ0+ = 0, Δ0– < 0 or β1– – β1+ = 0
and Δ1– > 0 or β0– – β0+ = 0, Δ0– = 0, Δ0+ < 0, and one of the conditions is β1– – β1+ = 0, Δ1+
= 0, Δ1– < 0 or β1– – β1+ = 0, Δ1– = 0, Δ1+ < 0;
c) has an infinite set of solutions of the form (20), where F(z) is an analytic function in D
that satisfies the conditions (19)-(22), if one of the following two conditions is met β0– – β0+
> 0, or β0– – β0+ = 0, Δ0+ < 0, Δ0– < 0 and one of the following conditions β1– – β1+ > 0, or β1–
– β1+ = 0, Δ1+ < 0, Δ1– < 0.
We will prove point a) of the theorem. We will assume that βj– – βj+ < 0 and problem (17)
are solvable. The solvability of the homogeneous problem (17) is equivalent to [35] the
solvability of the homogeneous Hilbert problem with a point of vorticity τ1 and the
homogeneous Hilbert problem with a point of vorticity τ0. The general solution of a
homogeneous problem with one singular point τj, j = 0.1 is represented by the formula (23),
in which the function Fj(z) is subject to the conditions (24). Now we will consider the
question of the existence of the function Fj(z), for which we will transfer it to the upper half-
plane H+. We will get the function:
.=,
1
1
=)(),(=)( 11111
i
i
i
zzFf
This function must meet the following three conditions:
Imf1(ξ) = 0, – ∞ < ξ < +∞, |f1(ζ)| ≤ CeC1ln2|ζ| + C2ln|ζ|, |ζ| → +∞,
1.>,ln2ln
2
ln
4
exp
1,1,
1,<,||ln2ln
2
||
ln
4
exp
|)(|
11
1
2
11
1
2
11
1
jj
C
C
C
f
(25)
It follows from these conditions that the function f1(ζ) is a narrowing to the upper half-
plane of the whole function:
,,)(
,),(
=)(
~
1
1
1Hf
Hf
f
a refined zero-order ρ(|ζ|) ≤ ln ln2|ζ |/ln|ζ | with a restriction on the growth of its boundary
values in the form of inequalities (25). For a function )(
~
1
f and a plane with a cut along the
real semiaxis, we apply the Phragmén – Lindelöf theorem and deduce that under condition
βj– – βj+ < 0 follows f1(ζ) ≡ 0, F1(z) ≡ 0, that is, problem (17) has only a trivial solution.
Similarly, we consider the case of the function и F0(z).
Now let βj– – βj+ = 0. We will consider in detail the case j = 1. Conditions (21), (22) after
passing to the half-plane using the mapping z1(ζ) = τ1(iζ + 1)/(iζ – 1) now take the form of
the inequalities:
,||,|)(|||ln
1
1
C
Cef
1.>},ln{exp
1,1,
1,<|},|ln{exp
|)(|
1
1
1
C
C
C
f
(26)
From this, as above, we deduce that if Δ1+ > 0 or Δ1– > 0, then the problem has only a
trivial solution.
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STCCE – 2021
with a density φ(t) = α(t) + Im[Ω(t) – Φ(t) – 2a0(t)(ln(t – τ0) + ln(t – τ1))] continuous
throughout on L:
.
)(1
=)(
0t
dt
t
L
We will first consider a homogeneous problem:
.=,= ,0,=)]([ 10
)(
))
1
(ln)
0
(ln)((
0
2)(
ttLtteeRe t
tttat
(17)
We will introduce an analytic function in D:
.)(:=)( )(
))
1
(ln)
0
(ln)((
0
2)( zeiezF z
zzzaz
(18)
This function on L by virtue of (17) satisfies the condition:
ImF
+
(
t
) = 0.
(19)
We will express from equality (18) the desired function:
).(=)( )(
))
1
(ln)
0
(ln)((
0
2)( zFeiez z
zzzaz
(20)
It is clear that if function (20) is a bounded solution to problem (17), then there must be
a function F(z), analytic in D. This function satisfies the inequality:
1,= 0= , ,,|)(| ||ln
2
||
ln2
1jилиjzDzCezF j
j
zC
j
zC
(21)
That is, this function is an exact growth of T in a semi-neighborhood of the point T, and
its boundary values must satisfy condition (19) and the inequality:
|
F
(
t
)|
≤
Ce
ReΩ(t)
,
t
∈
L
,
t
≠
τ
j
,
j
= 0,
j
= 1.
(22)
The validity of the inverse statement follows from the generalized maximum principle for
analytic functions ([32] p. 456, 457, see also [33]). Thus, it is proved.
3.1 Theorem 1
In order for the solution ϕ(z) of problem (17) to be bounded in the domain D, it is necessary
and sufficient that the function F(z), which is included in the formula of the general solution
(20), satisfies in D the growth constraints (21) and on the boundary conditions (19) and (22).
It is clear that the existence and set of bounded solutions to problem (17) depends on the
existence and set of analytic functions in D, and they satisfy conditions (19), (21), and (22).
In [35], it is proved that a homogeneous Hilbert problem with n points of vorticity is solvable
if and only if all n homogeneous problems with a single point of vorticity are solvable, that
is, the solvability of the problem is affected only by the parameters of each point of vorticity.
In this case, the solution of the problem with n points of vorticity can be represented as the
products of the solutions of these problems with a single point of vorticity. Thus, the solution
of problem (17) ϕ(z) = ϕ0(z)ϕ1(z) where ϕj(z) is the general solution of a homogeneous
problem with a single point of vorticity τj, j = 0, j = 1, is determined by the formula:
ϕ
j
(
z
) =
–
ie
–Φj(z) – 2a0(z)ln(z – τj)
e
–Γj
+
(z)
F
j
(
z
),
(23)
where Fj(z) is an analytic function in the domain D that satisfies the conditions:
. ,|)(|, ,|)(| 0,=)( ||ln
2
||
ln2
1
)(
j
j
zC
j
zC
j
tRe
jj zCezFLtCetFtFIm
(24)
Besides F(z) = F0(z)F1(z). If we use these remarks, the following theorem will be proved.
3.2 Theorem 2
Homogeneous boundary value problem (17)
a) has no nontrivial bounded solutions if:
β0– – β0+ < 0, or β1– – β1+ < 0,
β0– – β0+ = 0 and Δ0+ > 0, or β0– – β0+ = 0 and Δ0– > 0,
β0– – β1+ = 0 and Δ1+ > 0, or β1– – β1+ = 0 and Δ1– > 0;
b) has a single solution of the form:
0,= ,= ,=)( )(
))
1
(ln)
0
(ln)((
0
)( ImAconstAAeiezz
zzzaz
if one of the following two conditions is met β0– – β0+ = 0, Δ0+ = 0, Δ0– < 0 or β1– – β1+ = 0
and Δ1– > 0 or β0– – β0+ = 0, Δ0– = 0, Δ0+ < 0, and one of the conditions is β1– – β1+ = 0, Δ1+
= 0, Δ1– < 0 or β1– – β1+ = 0, Δ1– = 0, Δ1+ < 0;
c) has an infinite set of solutions of the form (20), where F(z) is an analytic function in D
that satisfies the conditions (19)-(22), if one of the following two conditions is met β0– – β0+
> 0, or β0– – β0+ = 0, Δ0+ < 0, Δ0– < 0 and one of the following conditions β1– – β1+ > 0, or β1–
– β1+ = 0, Δ1+ < 0, Δ1– < 0.
We will prove point a) of the theorem. We will assume that βj– – βj+ < 0 and problem (17)
are solvable. The solvability of the homogeneous problem (17) is equivalent to [35] the
solvability of the homogeneous Hilbert problem with a point of vorticity τ1 and the
homogeneous Hilbert problem with a point of vorticity τ0. The general solution of a
homogeneous problem with one singular point τj, j = 0.1 is represented by the formula (23),
in which the function Fj(z) is subject to the conditions (24). Now we will consider the
question of the existence of the function Fj(z), for which we will transfer it to the upper half-
plane H+. We will get the function:
.=,
1
1
=)(),(=)( 11111
i
i
i
zzFf
This function must meet the following three conditions:
Imf1(ξ) = 0, – ∞ < ξ < +∞, |f1(ζ)| ≤ CeC1ln2|ζ| + C2ln|ζ|, |ζ| → +∞,
1.>,ln2ln
2
ln
4
exp
1,1,
1,<,||ln2ln
2
||
ln
4
exp
|)(|
11
1
2
11
1
2
11
1
jj
C
C
C
f
(25)
It follows from these conditions that the function f1(ζ) is a narrowing to the upper half-
plane of the whole function:
,,)(
,),(
=)(
~
1
1
1Hf
Hf
f
a refined zero-order ρ(|ζ|) ≤ ln ln2|ζ |/ln|ζ | with a restriction on the growth of its boundary
values in the form of inequalities (25). For a function )(
~
1
f and a plane with a cut along the
real semiaxis, we apply the Phragmén – Lindelöf theorem and deduce that under condition
βj– – βj+ < 0 follows f1(ζ) ≡ 0, F1(z) ≡ 0, that is, problem (17) has only a trivial solution.
Similarly, we consider the case of the function и F0(z).
Now let βj– – βj+ = 0. We will consider in detail the case j = 1. Conditions (21), (22) after
passing to the half-plane using the mapping z1(ζ) = τ1(iζ + 1)/(iζ – 1) now take the form of
the inequalities:
,||,|)(|||ln
1
1
C
Cef
1.>},ln{exp
1,1,
1,<|},|ln{exp
|)(|
1
1
1
C
C
C
f
(26)
From this, as above, we deduce that if Δ1+ > 0 or Δ1– > 0, then the problem has only a
trivial solution.
7
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STCCE – 2021
We prove point b) of Theorem 2. Let the condition β1– – β1+ = 0, Δ1+ = 0, Δ1– < 0, or
condition β1– – β1+ = 0, Δ1– = 0, Δ1+ < 0 be satisfied. We introduce, as above, the function
f1(ζ), ζ H+, and the function )(
~
1
f. By virtue of the mirror symmetry of the function )(
~
1
f
and inequality (26) according to the Phragmén-Lindelöf principle for a plane cut along the
real semiaxis with the subsequent application of Liouville’s theorem, we obtain f1(ζ) ≡ A, A
= const. But then F1(z) ≡ A, that is, we get a unique, up to a constant factor, solution to the
homogeneous problem with a vorticity at the point τ1. The uniqueness of the solution of the
problem with a vorticity at the point is also proved τ0. The product of these solutions gives a
unique solution to problem (17).
We will prove point с) of the theorem 2. Let the condition βj– – βj+ > 0 be satisfied. To
prove the theorem, we will make sure that formula (20) is meaningful. To do this, we will
make sure that there is a function F(z) analytic in the disk D that satisfies conditions (19),
(21), and (22). For this we need a whole function:
∞
(27)
This function is constructed in [34] on the basis of the example of P.G. Yurov [36]. Here
rj,k = exp{((k – ½)/λj)1/γ}, λj > 0, γ > 0, θj, 0 ≤ θj ≤ π – is a fixed quantity. With this choice of
parameters for the function )(
ˆ
j
f in [20], an asymptotic formula is derived
, |),|
ln
(||
ln
|=)(
ˆ
|ln 11
Of jj
It follows from this formula that for γ = 2 the exact growth of the function at infinity
||
ln
=|)(| 2
~
j
f
h and the refined zero order ||ln/||
ln
ln=|)(| 2
~
j
f. The
asymptotic formula for its values on the real axis will take the form:
.||(1),||
ln
|=)(
ˆ
|ln 2
Of jj
Using the fractional-linear mapping of the unit disc to the upper half-plane:
,=,=)( j
i
j
j
j
je
z
z
iz
(28)
we construct a regular function in the disc 0,1.= ),(
ˆ
=)(
ˆjzfzF jjj
For the function )(
ˆzFj, we can easily obtain an asymptotic representation in a semi-
neighborhood of the point τj:
1.=0,= ,(1)},
||
1
ln4ln||
ln
{exp|=)(
ˆ
|2jjzO
z
zzF j
j
jjjj
(29)
From the formula (29) immediately follows the fulfillment of the inequality (21). It
follows from formulas (12), (14) that if we choose:
,
4ln
},{min
,
4
min
jjjj
j
(30)
in formula (27), then the inequality (22) holds for the function )(
ˆ
)(
ˆ
=)(
ˆ10
FFF . It is
obvious that the function (27) takes real values on the real axis and, therefore, the functions
)(
ˆ
),(
ˆ10 zFzF satisfy the condition (19). Then, using the formula (23) , we find ϕ0(z), ϕ1(z) and
the desired solution of the homogeneous problem ϕ(z) = ϕ0(z)ϕ1(z).
Finally we consider the case β1– – β1+ = 0, Δ1– < 0, Δ1+ < 0. To prove the theorem in this
subsection, it suffices to verify the existence of a solution ϕ1(z) to the homogeneous problem
with vorticity at the point τ1. As above, this is equivalent to the existence in H+ of an analytic
function f1(ζ). The refined zero order of this function satisfies the inequality
||ln/||lnln|)(|
1
f, This function takes real values on the boundary and satisfying
condition (26). As such a function, we can take 1
1=)(
~
k
f
if we choose a positive integer
]}[],{[min=111
k. Similarly, but using the conditions β0– – β0+ = 0, Δ0+ < 0, Δ0– < 0,
the existence of the solution ϕ0(z) is justified.
We now turn to the solution of the inhomogeneous Hilbert problem (11). We will assume
that the conditions © of Theorem 2 are satisfied, under which the homogeneous boundary
value problem (17) is solvable (the situation with condition (b) is elementary).
We will look for the general solution of the inhomogeneous problem in the form of the sum
of the general solution of the corresponding homogeneous problem and the particular solution
of the inhomogeneous problem. To find the latter, we need some solution of the homogeneous
problem (17). For this solution, the function F, which is included in the formula (20) and is
defined by the conditions (19), (21) and (22), has the following additional properties:
a) everywhere (except, perhaps, for the points τ1, τ2) on L, the condition F is satisfied
0;=)(
~
tF
b) the limit )(
~
lim
)(
tF
etRe
j
t
exists or goes to infinity of power order kj < 1;
c) the function )(
~
)(
tF
etRe
is Helder continuous on open arcs х L+, L–.
We will search for the function )(
~
zF in the form ),(
~
)(
~
=)(
~
21 zFzFzF where the
functions )(
~
),(
~
21 zFzF satisfy the conditions (24). We will construct these functions in the
following two cases. We will first consider the case of βj– – βj+ > 0, j = 0.1. We introduce a
function )(
ˆ
j
f, which we define in the complex plane ζ by formula (27). We'll take λj = (βj–
– βj+)/4π.
Next, using the mapping (28) , we construct a function:
),(
ˆ
=)(
ˆ
1
0=
zfzF jj
j
j
for the module of which the asymptotic representation (29) is valid. Now we will denote the
integer part of the number x with the symbol x. We will consider the following restrictions
fulfilled:
.=4ln=4ln jjjjj r
(31)
Let .4ln= jjjj r
We define the function:
2.=1,= ,)(
ˆ
=)(
~jj
z
z
izFzF
j
r
j
j
jj
It is easy to check that this function takes real values at the points of the unit circle, and
if we compare formulas (29) and (12), (14), it is not difficult to verify the validity of the other
two conditions (24). Obviously, also for the constructed function, the fulfillment of additional
constraints. Moreover, the function
j
j
j
j
tRe
ttO
tF
e
,|(|=
)(
~
)(
is bounded only when
the difference jjj r 4ln
is an integer.
8
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
We prove point b) of Theorem 2. Let the condition β1– – β1+ = 0, Δ1+ = 0, Δ1– < 0, or
condition β1– – β1+ = 0, Δ1– = 0, Δ1+ < 0 be satisfied. We introduce, as above, the function
f1(ζ), ζ H+, and the function )(
~
1
f. By virtue of the mirror symmetry of the function )(
~
1
f
and inequality (26) according to the Phragmén-Lindelöf principle for a plane cut along the
real semiaxis with the subsequent application of Liouville’s theorem, we obtain f1(ζ) ≡ A, A
= const. But then F1(z) ≡ A, that is, we get a unique, up to a constant factor, solution to the
homogeneous problem with a vorticity at the point τ1. The uniqueness of the solution of the
problem with a vorticity at the point is also proved τ0. The product of these solutions gives a
unique solution to problem (17).
We will prove point с) of the theorem 2. Let the condition βj– – βj+ > 0 be satisfied. To
prove the theorem, we will make sure that formula (20) is meaningful. To do this, we will
make sure that there is a function F(z) analytic in the disk D that satisfies conditions (19),
(21), and (22). For this we need a whole function:
∞
(27)
This function is constructed in [34] on the basis of the example of P.G. Yurov [36]. Here
rj,k = exp{((k – ½)/λj)1/γ}, λj > 0, γ > 0, θj, 0 ≤ θj ≤ π – is a fixed quantity. With this choice of
parameters for the function )(
ˆ
j
f in [20], an asymptotic formula is derived
, |),|
ln
(||
ln
|=)(
ˆ
|ln 11
Of jj
It follows from this formula that for γ = 2 the exact growth of the function at infinity
||
ln
=|)(| 2
~
j
f
h and the refined zero order ||ln/||
ln
ln=|)(| 2
~
j
f. The
asymptotic formula for its values on the real axis will take the form:
.||(1),||
ln
|=)(
ˆ
|ln 2
Of jj
Using the fractional-linear mapping of the unit disc to the upper half-plane:
,=,=)( j
i
j
j
j
je
z
z
iz
(28)
we construct a regular function in the disc 0,1.= ),(
ˆ
=)(
ˆjzfzF jjj
For the function )(
ˆzFj, we can easily obtain an asymptotic representation in a semi-
neighborhood of the point τj:
1.=0,= ,(1)},
||
1
ln4ln||
ln
{exp|=)(
ˆ
|2jjzO
z
zzF j
j
jjjj
(29)
From the formula (29) immediately follows the fulfillment of the inequality (21). It
follows from formulas (12), (14) that if we choose:
,
4ln
},{min
,
4
min
jjjj
j
(30)
in formula (27), then the inequality (22) holds for the function )(
ˆ
)(
ˆ
=)(
ˆ10
FFF . It is
obvious that the function (27) takes real values on the real axis and, therefore, the functions
)(
ˆ
),(
ˆ10 zFzF satisfy the condition (19). Then, using the formula (23) , we find ϕ0(z), ϕ1(z) and
the desired solution of the homogeneous problem ϕ(z) = ϕ0(z)ϕ1(z).
Finally we consider the case β1– – β1+ = 0, Δ1– < 0, Δ1+ < 0. To prove the theorem in this
subsection, it suffices to verify the existence of a solution ϕ1(z) to the homogeneous problem
with vorticity at the point τ1. As above, this is equivalent to the existence in H+ of an analytic
function f1(ζ). The refined zero order of this function satisfies the inequality
||ln/||lnln|)(|
1
f, This function takes real values on the boundary and satisfying
condition (26). As such a function, we can take 1
1=)(
~
k
f
if we choose a positive integer
]}[],{[min=111
k. Similarly, but using the conditions β0– – β0+ = 0, Δ0+ < 0, Δ0– < 0,
the existence of the solution ϕ0(z) is justified.
We now turn to the solution of the inhomogeneous Hilbert problem (11). We will assume
that the conditions © of Theorem 2 are satisfied, under which the homogeneous boundary
value problem (17) is solvable (the situation with condition (b) is elementary).
We will look for the general solution of the inhomogeneous problem in the form of the sum
of the general solution of the corresponding homogeneous problem and the particular solution
of the inhomogeneous problem. To find the latter, we need some solution of the homogeneous
problem (17). For this solution, the function F, which is included in the formula (20) and is
defined by the conditions (19), (21) and (22), has the following additional properties:
a) everywhere (except, perhaps, for the points τ1, τ2) on L, the condition F is satisfied
0;=)(
~
tF
b) the limit )(
~
lim
)(
tF
etRe
j
t
exists or goes to infinity of power order kj < 1;
c) the function )(
~
)(
tF
etRe
is Helder continuous on open arcs х L+, L–.
We will search for the function )(
~
zF in the form ),(
~
)(
~
=)(
~
21 zFzFzF where the
functions )(
~
),(
~
21 zFzF satisfy the conditions (24). We will construct these functions in the
following two cases. We will first consider the case of βj– – βj+ > 0, j = 0.1. We introduce a
function )(
ˆ
j
f, which we define in the complex plane ζ by formula (27). We'll take λj = (βj–
– βj+)/4π.
Next, using the mapping (28) , we construct a function:
),(
ˆ
=)(
ˆ
1
0=
zfzF jj
j
j
for the module of which the asymptotic representation (29) is valid. Now we will denote the
integer part of the number x with the symbol x. We will consider the following restrictions
fulfilled:
.=4ln=4ln jjjjj r
(31)
Let .4ln= jjjj r
We define the function:
2.=1,= ,)(
ˆ
=)(
~jj
z
z
izFzF
j
r
j
j
jj
It is easy to check that this function takes real values at the points of the unit circle, and
if we compare formulas (29) and (12), (14), it is not difficult to verify the validity of the other
two conditions (24). Obviously, also for the constructed function, the fulfillment of additional
constraints. Moreover, the function
j
j
j
j
tRe
ttO
tF
e
,|(|=
)(
~
)(
is bounded only when
the difference jjj r 4ln
is an integer.
9
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
Now we will consider the case when βj– – βj+ = 0, and the numbers Δj+ ≤ 0, Δj– ≤ 0, and at
least one of the inequalities is strict. Here we put 2,=1,= ,=)(
~jj
z
z
izF
j
r
j
j
j
where rj
is defined as above.
Finally, we find a function )(
~
)(
~
=)(
~
21 zFzFzF that satisfies the conditions (19), (21), and
(22) and the additional constraints a), b), c). По формуле (20) мы построим каноническое
решение однородной задачи (17).
).(
~
=)(
~
)(
))
1
(ln)
0
(ln)((
0
2)( zFeiez z
zzzaz
Since the function )(
~
*ziei
takes real values on the boundary circle, we divide the
boundary condition (11) of the inhomogeneous problem by )(
~
*tiei
. Finally, we will get:
.
)(
~
)(
=)(
~
),(
~
=
)(
~)( )(
0
))]
1
(ln)
0
(ln)((
0
2)([
*
tF
eetf
tftf
t
t
iRe
ttttatRe
(32)
We will look for a particular solution to the inhomogeneous problem. This problem has
the same sequences of zeros as the function )(
~
zF and hence ).(
~
z
Therefore, the relation )(
~
)/(zz
for the desired function will be an analytic and
bounded function in D, with the exception, perhaps, of the points τ0, τ1, in which a power
singularity of order less than one is allowed. Since ,)()(
~
LHtf and at the points of τ1, τ2
can have power-law singularities of order less than one, the function )(
~
)/(zzi
can be
represented by the Schwarz formula, therefore:
.
)(1
)(
~
)}({exp=)( 1
1=
)(
zt
dttc
zFzigiez
L
j
n
j
zi
(33)
The last formula gives a particular solution to the inhomogeneous boundary value problem
(11), the general solution of which is represented as the sum of the general solution of the
corresponding homogeneous problem and the given particular solution. The following is true.
3.3 Theorem 3
An inhomogeneous problem (11) is solvable in the class A of analytic functions in D if the
corresponding homogeneous problem (17) is solvable. The general solution of the
inhomogeneous problem is represented as the sum of the general solution (20) of the
homogeneous problem and the particular solution (33) of the inhomogeneous problem.
After we substitute the found solution of the inhomogeneous problem (11) in the formula
(6), we get the general solution of the boundary value problem (2).
4 Discussion
The article contains a solution and a study of the solvability of the Hilbert boundary value
problem for the generalized Cauchy-Riemann equation with a singular coefficient of the form
(3). The solution of the boundary value problem is based on the construction of the formula
(6) of the general solution of equation (3). This formula allows us to reduce the solution of
the boundary value problem (2) with a finite index for a generalized analytic function to the
boundary value problem (11) with an infinite index for an analytic function. The main content
of our paper is the solution and investigation of the solvability of the boundary value problem
(11) with an infinite index and two points of a new type of vorticity. The picture of the
solvability of problem (11) is contained in Theorems 2, 3. It is described in terms of the
characteristics of the singularities of the coefficient and is a picture of the solvability of the
boundary value problem (2). If we use the works [20] and [37-39], then we can get a similar
result for the Riemann problem.
5 Conclusions
We have formulated and solved the inhomogeneous Hilbert boundary value problem for an
important special case (3) of the generalized Cauchy-Riemann equation with a singular
coefficient. We have obtained a formula for the general solution of this problem. We have
found the conditions for the existence and uniqueness of the solution of the boundary value
problem. In the case of non-uniqueness of the solution, a complete description of the set of
solutions is given. The solution of the Hilbert boundary value problem with an infinite index
of the theory of analytic functions is also of independent importance.
References
1. I.N. Vekua, Bull. Math. Soc. Math. Phys R.P.R 1 2, 233-247 (1957).
2. S.I. Bezrodnykh, Math. Surveys 73 6, 941-1031 (2018).
3. I.N. Vekua, Generalized analytical functions. Nauka, Russian, (1959)
4. I.N. Vekua, Mat Sbornik N S 31 73, 217-314 (1952).
5. W. Haack, II, Math. Nachr, 1-30 (1952).
6. S.N. Timergaliev, Mathematics, 45-61 (2019).
7. S.I. Bezrodnykh, V. Vlasov, Computational Mathematics and Mathematical Physics 60
11, 1898-1914 (2020).
8. S.N. Timergaliev, R.S. Yakushev, Mathematics (2019).
9. A.V. Bitsadze, Some classes of partial differential equations. Nauka, Russian, (1981).
10. M.I. Višik, O.A. Ladyženskaya, Uspehi Mat. Nauki 11 6 (72), 41-97 (1956).
11. V.A. Solonnikov, Zap. Nauchn. Sem. 444, 133-156 (2016).
12. N.A. Zhura, A.P. Soldatov, Izv. Math 81 3, 542-567 (2017).
13. N.A. Zhura, A.P. Soldatov, Differential Equations 55 6, 815-823 (2019).
14. J. Nitsche, Math. Nachr 7 1, 31-33 (1952).
15. B.V. Boyarskii, Russian Mat. Sb. N.S. 43 85, 451-503 (1957).
16. L.G. Mihailov, Russian Dokl. Akad. Nauk SSSR (N.S.) 112, 13-15 (1957).
17. A.P. Soldatov, Journal of Mathematical Sciences 239 3, 381-411 (2019).
18. A.P. Soldatov, Math. Notes 108 2, 272-276 (2020).
19. Z.D. Usmanov, Generalized Cauchy-Riemann systems with a singular point. Dushanbe,
ed. AN, Taj.SSR, (1992).
20. A.B. Rasulov, Math. Notes 108 5, 756-760 (2020).
21. N.R. Radzhabov, Introduction to the theory of partial differential equations with
supersingular coefficients. Dushanbe, (1992).
22. A.P. Soldatov, A.B. Rasulov, Differential Equations 54 2, 239-249 (2018).
23. A.B. Rasulov, Differential Equations 53 6, 809-817 (2017).
24. N.R. Radjabov, Siberian Math. J. 13 4, 666 (1972).
25. A.B. Rasulov, M.A. Bobodzhanova, Y.S. Fedorov, Journal of Mathematical Sciences 241
3, 327-339 (2019).
26. A.B. Rasulov, A.P. Soldatov, Complex Variables and Elliptic Equations 64 8, 1275-
1284 (2019).
27. U.S. Fedorov, A.B. Rasulov, Differential Equations 57 1, 140-144 (2021).
10
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
Now we will consider the case when βj– – βj+ = 0, and the numbers Δj+ ≤ 0, Δj– ≤ 0, and at
least one of the inequalities is strict. Here we put 2,=1,= ,=)(
~jj
z
z
izF
j
r
j
j
j
where rj
is defined as above.
Finally, we find a function )(
~
)(
~
=)(
~
21 zFzFzF that satisfies the conditions (19), (21), and
(22) and the additional constraints a), b), c). По формуле (20) мы построим каноническое
решение однородной задачи (17).
).(
~
=)(
~
)(
))
1
(ln)
0
(ln)((
0
2)( zFeiez z
zzzaz
Since the function )(
~
*ziei
takes real values on the boundary circle, we divide the
boundary condition (11) of the inhomogeneous problem by )(
~
*tiei
. Finally, we will get:
.
)(
~
)(
=)(
~
),(
~
=
)(
~)( )(
0
))]
1
(ln)
0
(ln)((
0
2)([
*
tF
eetf
tftf
t
t
iRe
ttttatRe
(32)
We will look for a particular solution to the inhomogeneous problem. This problem has
the same sequences of zeros as the function )(
~
zF and hence ).(
~
z
Therefore, the relation )(
~
)/(zz
for the desired function will be an analytic and
bounded function in D, with the exception, perhaps, of the points τ0, τ1, in which a power
singularity of order less than one is allowed. Since ,)()(
~
LHtf and at the points of τ1, τ2
can have power-law singularities of order less than one, the function )(
~
)/(zzi
can be
represented by the Schwarz formula, therefore:
.
)(1
)(
~
)}({exp=)( 1
1=
)(
zt
dttc
zFzigiez
L
j
n
j
zi
(33)
The last formula gives a particular solution to the inhomogeneous boundary value problem
(11), the general solution of which is represented as the sum of the general solution of the
corresponding homogeneous problem and the given particular solution. The following is true.
3.3 Theorem 3
An inhomogeneous problem (11) is solvable in the class A of analytic functions in D if the
corresponding homogeneous problem (17) is solvable. The general solution of the
inhomogeneous problem is represented as the sum of the general solution (20) of the
homogeneous problem and the particular solution (33) of the inhomogeneous problem.
After we substitute the found solution of the inhomogeneous problem (11) in the formula
(6), we get the general solution of the boundary value problem (2).
4 Discussion
The article contains a solution and a study of the solvability of the Hilbert boundary value
problem for the generalized Cauchy-Riemann equation with a singular coefficient of the form
(3). The solution of the boundary value problem is based on the construction of the formula
(6) of the general solution of equation (3). This formula allows us to reduce the solution of
the boundary value problem (2) with a finite index for a generalized analytic function to the
boundary value problem (11) with an infinite index for an analytic function. The main content
of our paper is the solution and investigation of the solvability of the boundary value problem
(11) with an infinite index and two points of a new type of vorticity. The picture of the
solvability of problem (11) is contained in Theorems 2, 3. It is described in terms of the
characteristics of the singularities of the coefficient and is a picture of the solvability of the
boundary value problem (2). If we use the works [20] and [37-39], then we can get a similar
result for the Riemann problem.
5 Conclusions
We have formulated and solved the inhomogeneous Hilbert boundary value problem for an
important special case (3) of the generalized Cauchy-Riemann equation with a singular
coefficient. We have obtained a formula for the general solution of this problem. We have
found the conditions for the existence and uniqueness of the solution of the boundary value
problem. In the case of non-uniqueness of the solution, a complete description of the set of
solutions is given. The solution of the Hilbert boundary value problem with an infinite index
of the theory of analytic functions is also of independent importance.
References
1. I.N. Vekua, Bull. Math. Soc. Math. Phys R.P.R 1 2, 233-247 (1957).
2. S.I. Bezrodnykh, Math. Surveys 73 6, 941-1031 (2018).
3. I.N. Vekua, Generalized analytical functions. Nauka, Russian, (1959)
4. I.N. Vekua, Mat Sbornik N S 31 73, 217-314 (1952).
5. W. Haack, II, Math. Nachr, 1-30 (1952).
6. S.N. Timergaliev, Mathematics, 45-61 (2019).
7. S.I. Bezrodnykh, V. Vlasov, Computational Mathematics and Mathematical Physics 60
11, 1898-1914 (2020).
8. S.N. Timergaliev, R.S. Yakushev, Mathematics (2019).
9. A.V. Bitsadze, Some classes of partial differential equations. Nauka, Russian, (1981).
10. M.I. Višik, O.A. Ladyženskaya, Uspehi Mat. Nauki 11 6 (72), 41-97 (1956).
11. V.A. Solonnikov, Zap. Nauchn. Sem. 444, 133-156 (2016).
12. N.A. Zhura, A.P. Soldatov, Izv. Math 81 3, 542-567 (2017).
13. N.A. Zhura, A.P. Soldatov, Differential Equations 55 6, 815-823 (2019).
14. J. Nitsche, Math. Nachr 7 1, 31-33 (1952).
15. B.V. Boyarskii, Russian Mat. Sb. N.S. 43 85, 451-503 (1957).
16. L.G. Mihailov, Russian Dokl. Akad. Nauk SSSR (N.S.) 112, 13-15 (1957).
17. A.P. Soldatov, Journal of Mathematical Sciences 239 3, 381-411 (2019).
18. A.P. Soldatov, Math. Notes 108 2, 272-276 (2020).
19. Z.D. Usmanov, Generalized Cauchy-Riemann systems with a singular point. Dushanbe,
ed. AN, Taj.SSR, (1992).
20. A.B. Rasulov, Math. Notes 108 5, 756-760 (2020).
21. N.R. Radzhabov, Introduction to the theory of partial differential equations with
supersingular coefficients. Dushanbe, (1992).
22. A.P. Soldatov, A.B. Rasulov, Differential Equations 54 2, 239-249 (2018).
23. A.B. Rasulov, Differential Equations 53 6, 809-817 (2017).
24. N.R. Radjabov, Siberian Math. J. 13 4, 666 (1972).
25. A.B. Rasulov, M.A. Bobodzhanova, Y.S. Fedorov, Journal of Mathematical Sciences 241
3, 327-339 (2019).
26. A.B. Rasulov, A.P. Soldatov, Complex Variables and Elliptic Equations 64 8, 1275-
1284 (2019).
27. U.S. Fedorov, A.B. Rasulov, Differential Equations 57 1, 140-144 (2021).
11
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
28. I.N. Dorofeeva, A.B. Rasulov, Computational Mathematics and Mathematical Physics 60
10, 1679-1685 (2020).
29. A.B. Rasulov, Differential Equations 40 9, 1364-1366 (2004).
30. A.B. Rasulov, A.P. Soldatov, Differential Equations 52 5, 616-629 (2016).
31. I.M. Melnik, Publishing House of the Mathematical Institute 24, 149-162 (1957).
32. A. Hurwitz, R. Courant, Function theory. Nauka, Russian, (1968).
33. R.B. Salimov, A.Kh. Fatykhov, P.L. Shabalin, Lobachevskii J. Math 38 3, 414-419
(2017).
34. P.L. Shabalin1, A.Kh. Fatykhov, Russian Mathematics 65 1, 57-71 (2021).
35. A.Kh. Fatykhov, P.L. Shabalin, Probl. Anal. Issues Anal 7 25, 30-38 (2018).
36. P.G. Yurov, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 158-163 (1966).
37. E.N. Khasanova, P.L. Shabalin, Russian Math. 63 3, 31-44 (2019).
38. R.B. Salimov, A.Z. Suleimanov, Russian Math. 61 5, 61-65 (2017).
39. R.B. Salimov, Ufa Mathematical Journal 10 1, 80-93 (2018).
12
E3S Web of Conferences 274, 11003 (2021) https://doi.org/10.1051/e3sconf/202127411003
STCCE – 2021
