Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line
We consider a generalized Cauchy–Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions rel...
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irk-123456789-1694402020-06-14T01:26:40Z Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line Plaksa, S.A. We consider a generalized Cauchy–Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given boundary functions. 2019 Article Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line / S.A. Plaksa // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 200-214. — Бібліогр.: 34 назв. — англ. 1810-3200 2010 MSC. 30G20, 35J70, 35J56, 31A10 http://dspace.nbuv.gov.ua/handle/123456789/169440 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We consider a generalized Cauchy–Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given boundary functions. |
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Plaksa, S.A. Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line Український математичний вісник |
| author_facet |
Plaksa, S.A. |
| author_sort |
Plaksa, S.A. |
| title |
Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line |
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Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line |
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Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line |
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Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line |
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Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line |
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schwarz boundary-value problems for solutions of a generalized cauchy-riemann system with a singular line |
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Інститут прикладної математики і механіки НАН України |
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Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line / S.A. Plaksa // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 200-214. — Бібліогр.: 34 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT plaksasa schwarzboundaryvalueproblemsforsolutionsofageneralizedcauchyriemannsystemwithasingularline |
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2025-07-15T04:15:40Z |
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2025-07-15T04:15:40Z |
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1837684952443912192 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 2, 200 – 214
Schwarz boundary value problems for solutions
of a generalized Cauchy–Riemann system
with a singular line
Sergiy A. Plaksa
(Presented by V. Gutlyanskĭı)
Dedicated to memory of Professor Bogdan Bojarski
Abstract. We consider a generalized Cauchy–Riemann system with
a rectilinear singular interval of the real axis. Schwarz boundary value
problems for generalized analytic functions which satisfy the mentioned
system are reduced to the Fredholm integral equations of the second
kind under natural assumptions relating to the boundary of a domain
and the given boundary functions.
2010 MSC. 30G20, 35J70, 35J56, 31A10.
Key words and phrases. Axisymmetric potential; Stokes flow func-
tion; Beltrami equation; generalized Cauchy–Riemann system; general-
ized analytic function; Schwarz boundary value problem.
1. Introduction
1.1. Degenerated elliptic equations associated with potential
fields
It is well-known that a spatial potential solenoid field symmetric with
respect to the axis Ox is described in a meridian plane xOy in terms of
the axisymmetric potential φ and the Stokes flow function ψ satisfying
the following system of equations:
y
∂φ(x, y)
∂x
=
∂ψ(x, y)
∂y
, y
∂φ(x, y)
∂y
= −∂ψ(x, y)
∂x
. (1.1)
Received 26.05.2019
This research is partially supported by the State Program of Ukraine (Project
No. 0117U004077)
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
S. A. Plaksa 201
Under the condition that there exist continuous second-order partial
derivatives of the functions φ(x, y) and ψ(x, y), system (1.1) implies the
following equations for the axisymmetric potential and the Stokes flow
function:
y
(
∂2
∂x2
+
∂2
∂y2
)
φ(x, y) +
∂φ(x, y)
∂y
= 0 , (1.2)
y
(
∂2
∂x2
+
∂2
∂y2
)
ψ(x, y)− ∂ψ(x, y)
∂y
= 0 , (1.3)
that are degenerated on the axis Ox , as well as equations (1.1).
In the plane xOy, we consider a bounded simply connected domain
D symmetric with respect to the axis Ox. By Dz we denote the domain
in the complex plane C congruent to the domain D under the correspon-
dence z = x+ iy, (x, y) ∈ D, where i is the imaginary complex unit. Let
∂D and ∂Dz denote the boundaries of domains D and Dz, respectively.
By b1 and b2 we denote the points at which the boundary ∂Dz crosses
the real axis R. We assume that b1 < b2.
In what follows, z = x + iy , and we shall consider the solutions φ
and ψ of system (1.1) given in the domain D. In this case, the com-
plex potential H(z) := φ(x, y) + iψ(x, y) satisfies the following Beltrami
equation:
∂H(z)
∂z̄
=
1− Im z
1 + Im z
∂H(z)
∂z
, z ∈ Dz : Im z > 0 , (1.4)
where ∂
∂z̄ := 1
2
(
∂
∂x + i ∂∂y
)
and ∂
∂z := 1
2
(
∂
∂x − i ∂∂y
)
.
At the same time, the function W (z) = φ(x, y) + iv(x, y), where
v(x, y) := ψ(x,y)
y , satisfies the equation
2
∂W (z)
∂z̄
− 1
z − z̄
(
W (z)−W (z)
)
= 0 , z ∈ Dz : Im z > 0 , (1.5)
that is the complex form of a generalized Cauchy–Riemann system with
a rectilinear singular interval (b1, b2) of the real axis.
After substituting the function ψ(x, y) = y v(x, y) into equation (1.3)
we obtain the following equation for the function v:
y2
(
∂2
∂x2
+
∂2
∂y2
)
v(x, y) + y
∂v(x, y)
∂y
− v(x, y) = 0 . (1.6)
202 Schwarz boundary value problems for solutions...
The theory of mappings being solutions of elliptic equations is de-
veloped in the papers by M. A. Lavrentiev [1], I. N. Vekua [2], B. Bo-
jarski [3, 4], L. I. Volkovyski [5], L. G. Mikhailov [6] and many other au-
thors. Let us note that the theory which is developed in the mentioned
papers describes properties of solutions of equations (1.4) and (1.5), as
well as equations (1.1)–(1.3) and (1.6), only outside of neighbourhoods
of the real axis.
1.2. Some special methods for solving boundary value
problems for elliptic equations with a degeneration
on the real axis
Boundary value problems for elliptic equations are considered by
I. N. Vekua [2], B. Bojarski [7], A. V. Bitsadze [8], V. N. Monakhov [9],
B. Bojarski, V. Gutlyanski and V. Ryazanov [10] et al.
M. V. Keldysh [11] describes some correct statements of boundary
value problems for an elliptic equation with a degeneration on a straight
line. It shows that there are essential differences with boundary value
problems for elliptic equations without degeneration. Some special meth-
ods for researching boundary value problems for elliptic equations degen-
erating along the line are developed by I. I. Daniliuk [12], S. A. Tersenov
[13], R. P. Gilbert [14], L. G. Mikhailov and N. Radzhabov [15], A. Ya-
nushauskas [16], S. Rutkauskas [17,18].
One of the ways for solving boundary value problems for axisym-
metric potential solenoid fields is based on integral expressions of ax-
isymmetric potentials via analytic functions of a complex variable (cf.
E. T. Whittaker and G. N. Watson [19], H. Bateman [20], P. Henrici
[21], A. G. Mackie [22], Yu. P. Krivenkov [23], N. R. Radzhabov [24],
G. N. Polozhii [25], G. N. Polozhii and A.F. Ulitko [26], A.A. Kap-
shivyi [27], A. Ya. Aleksandrov and Yu. P. Soloviev [28], I. P. Mel’nichenko
and S. A. Plaksa [29]).
In particular, in the paper [29], we developed a method for the re-
duction of the Dirichlet problems for the axisymmetric potential and the
Stokes flow function to the Fredholm integral equations of the second
kind. It is made in the case where the boundary of a simply connected
domain belongs to a class being wider than the class of Lyapunov curves.
These results are used below for solving the Schwarz boundary value
problems for generalized analytic functions that are solutions of equation
(1.5).
S. A. Plaksa 203
2. Schwarz boundary value problems for generalized
analytic functions satisfying equation (1.5)
2.1. The statement of the problems
Let GAsym(Dz) be the class of generalized analytic functionsW :Dz−→
C satisfying the following conditions:
• W (x + iy) has the continuous partial derivatives of the first order
with respect to x and y in the domain {z ∈ Dz : Im z > 0} and
satisfies equation (1.5);
• W submits to Schwarz reflection principle, i.e,
W (z̄) =W (z) ∀ z ∈ Dz . (2.1)
Thus, for a functionW ∈ GAsym(Dz), the function φ(x, y) = ReW (z)
is even with respect to the variable y , and the function v(x, y) = ImW (z)
is odd with respect to the variable y .
Consider two Schwarz boundary value problems for solutions of equa-
tion (1.5):
Schwarz BVP–I: to find a function W ∈ GAsym(Dz) which is con-
tinuously extended onto the boundary ∂Dz, when values of its real part
are given on ∂Dz, i.e.,
ReW (z) = u1(x, y) ∀ z ∈ ∂Dz , (2.2)
where u1 : ∂D −→ R is a given continuous function even with respect to
the variable y ;
Schwarz BVP–II: to find a function W ∈ GAsym(Dz) which is
continuously extended onto the set ∂Dz \ {b1, b2}, when values of its
imaginary part are given on ∂Dz \ {b1, b2}, i.e.,
ImW (z) = u2(x, y) ∀ z ∈ ∂Dz \ {b1, b2} , (2.3)
where u2 :
(
∂D \ {(b1, 0), (b2, 0)}
)
−→ R is a given continuous function
odd with respect to the variable y . In addition, it is required that the
function W should satisfy the estimate
|W (z)| ≤ c
(
|z − b1|−βW + |z − b2|−βW
)
∀ z ∈ Dz , (2.4)
where the constant c does not depend on z and the constant βW ∈ (0; 1)
is dependent only of the function W .
In contradistinction to the classical Schwarz boundary value problem
for analytic functions of a complex variable, Schwarz BVP–II is not re-
duced to Schwarz BVP–I because, at least, the real and imaginary parts
204 Schwarz boundary value problems for solutions...
of a generalized analytic function W ∈ GAsym(Dz) satisfy the different
equations, viz., the function φ(x, y) = ReW (z) satisfies equation (1.2)
while the function v(x, y) = ImW (z) satisfies equation (1.6).
2.2. Preliminary notes
For every z ∈ Dz with Im z ̸= 0, we fix an arbitrary Jordan rectifiable
curve Γzz̄ in the domain Dz which is symmetric with respect to the real
axis R and connects the points z and z̄. For every z ∈ ∂Dz with Im z ̸= 0
by Γzz̄ we denote the Jordan subarc of the boundary ∂Dz with the end
points z and z̄ which contains the point b1.
Denote by Dz the closure of domain Dz .
For z ∈ Dz , Im z ̸= 0 , let
√
(t− z)(t− z̄) be that continuous branch
of the function G(t) =
√
(t− z)(t− z̄) analytic outside of the cut along
Γzz̄ for which G(b2) > 0 . We define
√
(t− z)(t− z̄) := t − z for each
z ∈ Dz with Im z = 0 .
It is well-known that any function W (x + iy) which belongs to the
class GAsym(Dz) is also analytic with respect to the variable x and y
outside of any neighbourhood of the real axis. Therefore, as a conse-
quence of integral expressions for the axisymmetric potential and the
Stokes flow function obtained in Theorem 1 in [30] and Theorem 1 in [31]
(cf. also Theorems 3.4 and 3.5 in [29]), one can conclude that there ex-
ists a unique analytic function F : Dz −→ C which submits to Schwarz
reflection principle of the type (2.1) and such that the equality
W (z) =
1
2πi
∫
γ
F (t)√
(t− z)(t− z̄)
(
1− i
(t− x)
y
)
dt (2.5)
is fulfilled for all z ∈ Dz with Im z ̸= 0 , where γ is an arbitrary closed
Jordan rectifiable curve in Dz which surrounds Γzz̄.
Let us note that if the function F is continuously extended onto the
set ∂Dz \ {b1, b2} and satisfies an estimate of the type (2.4), then the
formula (2.5) can be transformed to the form
W (z) =
1
2πi
∫
∂Dz
F (t)√
(t− z)(t− z̄)
(
1− i
(t− x)
y
)
dt (2.6)
for all z ∈ Dz with Im z ̸= 0 . For all z ∈ Dz with Im z = 0, equality
(2.6) is transformed by continuity into the equality W (z) = F (z) .
Our immediate purpose is to use integral expression (2.6) of any func-
tion W ∈ GAsym(Dz) for solving Schwarz BVP–I and Schwarz BVP–II.
S. A. Plaksa 205
We assume that the given function u1 belongs to the set H̃α(∂D) of
functions u : ∂D → R satisfying the following condition
|u(x1, y1)− u(x2, y2)|
≤ c (max{|z1 − b1||z1 − b2|, |z2 − b1||z2 − b2|})−ν |z1 − z2|α
∀ (x1, y1), (x2, y2) ∈ ∂D,
where z1 = x1 + iy1, z2 = x2 + iy2, α ∈ (1/2; 1], ν ∈ [0;α) and the
constant c does not depend on x1, y1, x2, y2.
We assume also that the given function u2 is of the form u2(x, y) ≡
ũ2(x, y)/y , where ũ2 ∈ H̃α(∂D) and ũ2(b1, 0) = ũ2(b2, 0) = 0 .
We shall formulate conditions on the boundary ∂Dz in terms of the
conformal mapping σ(Z) which maps the unit disk {Z ∈ C : |Z| < 1}
onto the domain Dz in such a way that σ(−1) = b1, σ(1) = b2 and
Im σ(i) > 0. Moreover,
σ(Z̄) = σ(Z) ∀Z ∈ {Z ∈ C : |Z| ≤ 1} .
We assume that the conformal mapping σ has the nonvanishing contin-
uous contour derivative on the unit circle and its modulus of continuity
ω(σ′, ε) := sup
|Z1−Z2|≤ε
|σ′(Z1)− σ′(Z2)|
satisfies the following condition:
1∫
0
ω(σ′, η)
η
ln3
2
η
dη <∞ . (2.7)
One can observe that if the boundary ∂D is the Lyapunov curve, then
the condition (2.7) is satisfied. It follows from the Kellog theorem. The
condition (2.7) is also satisfied in a more general case where the boundary
∂D is a smooth curve and, furthermore, the tangent angle θ(s) as function
of the arc length s has the modulus of continuity ω(θ, ε) satisfying the
following condition
1∫
0
ω(θ, η)
η
ln4
2
η
dη <∞.
The last statement follows from an estimate for the modulus of continuity
of conformal mapping of the unit disk given in the papers [32,33].
206 Schwarz boundary value problems for solutions...
2.3. Reduction of Schwarz BVP–I to the Fredholm integral
equation
Preparatory to formulate a result on the reduction of Schwarz BVP–I
to the Fredholm integral equation, let us introduce some denotations.
We introduce the function
M(Z, T ) :=
√
(T − Z)(T − Z̄)
(σ(T )− σ(Z))(σ(T )− σ(Z̄))
.
For each Z ̸= −1 it is understood as a continuous branch of the function
analytic with respect to the variable T in the unit disk and satisfying the
condition M(Z,−1) > 0.
Now, consider the conformal mapping Z = ξ−i
ξ+i of the complex plane.
This mapping assigns the points Z and Z̄ of the unit circle to the points
ξ and −ξ of the real axis, respectively. Let us introduce the function
m(ξ, τ) :=M(Z, T )
of real variables ξ and τ , where T = τ−i
τ+i . Denote
A(ξ, τ) := 2 Rem(ξ, τ), B(ξ, τ) := 2 Imm(ξ, τ).
Consider the functions
m̃(ξ, τ) :=
2ξ
π
ξ∫
τ
s(m(s,τ)−m(ξ,τ))
(ξ2−s2)3/2
√
s2−τ2 ds , when ξτ > 0 , |τ | < |ξ|,
0, when ξτ < 0 or ξτ > 0 , |τ | > |ξ| ,
Ã(ξ, τ) := 2 Re m̃(ξ, τ), B̃(ξ, τ) := 2 Im m̃(ξ, τ),
kp(ξ, τ) := − ξ
|ξ|
Ã(ξ, τ) +
1
π
ξ∫
0
B̃(ξ, η)
η − τ
dη ,
P (ξ) :=
√
σ′
(
ξ − i
ξ + i
)
−
√
σ′
(
ξ + i
ξ − i
)
and the integral operators
(kpf)(ξ) :=
∞∫
−∞
kp(ξ, τ)√
τ2 + 1
f(τ) dτ,
S. A. Plaksa 207
(Rf)(ξ) :=
√
ξ2 + 1
(
A(ξ, ξ)
(A(ξ, ξ))2 + (B(ξ, ξ))2
f(|ξ|)
+
P (ξ)
4πi
∞∫
−∞
f(|τ |)
√
(τ2 + 1)
∣∣Imσ( τ−iτ+i)
∣∣
2 |τ |
dτ
τ − ξ
)
.
For a given function u1 : ∂D −→ R, we define the function
f∗(ξ) := u∗1(ξ)− ξ
ξ∫
0
s(u∗1(s)− u∗1(ξ))
(ξ2 − s2)3/2
ds ,
where the function u∗1 is expressed via the given function u1 in the fol-
lowing form:
u∗1(ξ) :=
1√
ξ2 + 1
(
u1(x, y)− u1(b2, 0)
)
, x+ iy = σ
(
ξ − i
ξ + i
)
.
Let C(R) be the Banach space of continuous functions u∗ : (R ∪
{∞}) → C with the norm ∥u∗∥C(R) := sup
τ∈R
|u∗(τ)| and Ce(R) be the
subspace of C(R) containing all even continuous functions.
Denote by D(R) the set of functions u∗ ∈ C(R) for which the modulus
of continuity
ωR(u
∗, ε) := sup
τ1,τ2∈R,|τ1−τ2|≤ε
|u∗(τ1)− u∗(τ2)|
and the local centralized (with respect to the infinitely remote point)
modulus of continuity
ωR,∞(u∗, ε) := sup
τ∈R,|τ |≥1/ε
|u∗(τ)− u∗(∞)|
satisfy the Dini conditions
1∫
0
ωR(u
∗, η)
η
dη <∞,
1∫
0
ωR,∞(u∗, η)
η
dη <∞ .
We denote also by De(R) the set of even functions from D(R).
The following theorem establishes sufficient conditions for the reduc-
tion of Schwarz BVP–I to the Fredholm integral equation.
208 Schwarz boundary value problems for solutions...
Theorem 2.1. Suppose that u1 ∈ H̃α(∂D), and the conformal mapping
σ(Z) has the nonvanishing continuous contour derivative on the unit
circle, and its modulus of continuity satisfies condition (2.7). Then the
solution of the Schwarz BVP–I is given by formula (2.6), in which
F (z) = u1(b2, 0)−
2(ξ + i)
π σ′( ξ−iξ+i)
∞∫
−∞
U0(τ)√
τ2 + 1 (τ − ξ)
dτ,
z = σ
(
ξ + i
ξ − i
)
∀ ξ ∈ C : Im ξ > 0 , (2.8)
where U0 is a solution of the Fredholm integral equation
U0(ξ) + (R(kpU0))(ξ) = (Rf∗)(ξ) ∀ ξ ∈ R (2.9)
in the space Ce(R). Moreover, the operator Rkp is compact in the space
Ce(R), and equation (2.9) has a unique solution U0 ∈ Ce(R) which be-
longs necessarily to the set De(R).
Proof. Let us use Theorem 3.16 in [29] on reduction of Dirichlet boundary
value problem for the axisymmetric potential φ(x, y) = ReW (z) to the
Fredholm integral equation (2.9). It follows from Theorem 3.16 in [29]
that
φ(x, y) = ReW (z) =
1
2πi
∫
∂Dz
F (t)√
(t− z)(t− z̄)
dt ∀ z ∈ Dz , (2.10)
where F is defined by equality (2.8), and the function (2.10) is continu-
ously extended onto ∂Dz , and the boundary condition (2.2) is satisfied.
Further, it follows from Theorem 3.10 in [29] (cf. also Theorem 6
in [34]) that the function
v(x, y) = ImW (z) = − 1
2πiy
∫
∂Dz
F (t)(t− x)√
(t− z)(t− z̄)
dt (2.11)
is continuously extended from Dz onto the set ∂Dz \ {b1, b2} .
To complete the proof, it remains to prove that ImW (z) → 0 when
z ∈ Dz, Im z ̸= 0 and z → bj , j = 1, 2 .
Since ∂Dz is a smooth curve, there exists δ0 > 0 such that each of the
circles {t ∈ C : |t−bj | = δ} , j = 1, 2 , crosses ∂Dz only at two points for
all δ < δ0 . Then for all z ∈ Dz with Im z ̸= 0 and |z− bj | = δ , j = 1, 2 ,
S. A. Plaksa 209
we can replace the cut Γzz̄ by another cut Γ′
zz̄ of the plane C without
changing the values of the function
√
(t− z)(t− z̄) for all t ∈ ∂Dz .
Moreover, we can take Γ′
zz̄ as the arc of the circle {t ∈ C : |t − bj | = δ}
with the end points z and z̄ that is located in Dz.
Now, for z ∈ Dz, Im z ̸= 0 and |z − bj | = δ , j = 1, 2 , using the
Cauchy integral theorem, we obtain the equalities
ImW (z) = − 1
2πiy
∫
∂Dz
(
F (t)− F (bj)
)
(t− x)√
(t− z)(t− z̄)
dt =
= − 1
πiy
∫
Γ′
zz̄
(
F (t)− F (bj)
)
(t− x)(√
(t− z)(t− z̄)
)+ dt ,
where
(√
(t− z)(t− z̄)
)+
denotes the values of the function√
(t− z)(t− z̄) for t ∈ Γ′
zz̄ that are taken on the right side of the cut
Γ′
zz̄.
Since ∂Dz is a smooth curve, there exists a constant c0 ∈ (0; 1) such
that in the case z ∈ Dz : |z − bj | < δ0 and |Im z| ≤ c0 |z − bj | , j = 1, 2 ,
the two-sided inequality b1 < Re z < b2 is fulfilled. Therefore, in this
case |t − x|/|y| ≤ 1 for all t ∈ Γ′
zz̄ . In the case z ∈ Dz : |z − bj | < δ0
and |Im z| > c0 |z − bj | , j = 1, 2 , one has the relations |t − x|/|y| ≤
2 |z−bj |/(c0 |z−bj |) = 2/c0 for all t ∈ Γ′
zz̄ . Thus, the quotient |t−x|/|y|
is bounded for all t ∈ Γ′
zz̄ .
Therefore, we obtain the relations
|ImW (z)| ≤ c1 max
t∈Γ′
zz̄
∣∣F (t)− F (bj)
∣∣ ∫
Γ′
zz̄
|dt|√
|t− z||t− z̄|
≤
≤ c2 max
t∈Γ′
zz̄
∣∣F (t)− F (bj)
∣∣→ 0 , z → bj , j = 1, 2 ,
where the constants c1 , c2 are independent of z .
2.4. Reduction of Schwarz BVP–II to the Fredholm integral
equation
Preparatory to formulate a result on the reduction of Schwarz BVP–II
to the Fredholm integral equation, let us introduce some denotations.
Consider the function Π∗(ξ) := |ξ|β0 (|ξ|+ 1)β∞−β0 , where the num-
bers β0, β∞ satisfy the inequalities 1 − α + ν < β0 < 1 and 0 <
β∞ < min{α− ν, 1/2}, in which α and ν are the same numbers as in the
definition of the set H̃α(∂D) .
210 Schwarz boundary value problems for solutions...
Let us introduce the function
n(ξ, τ) :=M(Z, T ) (σ(T )− Reσ(Z))
of real variables ξ and τ , where Z = ξ−i
ξ+i and T = τ−i
τ+i .
Consider also the functions
ñ(ξ, τ) :=
2ξ
π
ξ∫
τ
s(n(s,τ)−n(ξ,τ))
(ξ2−s2)3/2
√
s2−τ2 ds , when ξτ > 0 , |τ | < |ξ| ,
0 , when ξτ < 0 or ξτ > 0 , |τ | > |ξ| ,
C̃(ξ, τ) := 2 Re ñ(ξ, τ), D̃(ξ, τ) := 2 Im ñ(ξ, τ),
kf (ξ, τ) := − ξ
|ξ|
D̃(ξ, τ)− 1
π
ξ∫
0
C̃(ξ, η)
η − τ
dη
and the integral operators
(kfg)(ξ) :=
∞∫
−∞
kf (ξ, τ)
Π∗(τ)
g(τ) dτ ,
(Qg)(ξ) := Π∗(ξ)
(
A(ξ, ξ)
(A(ξ, ξ))2 + (B(ξ, ξ))2
g(|ξ|)
Imσ( ξ−iξ+i)
+
P (ξ)
4πi
∞∫
−∞
g(|τ |)
Imσ( τ−iτ+i)
√
(τ2 + 1)
∣∣Imσ( τ−iτ+i)
∣∣
2 |τ |
dτ
τ − ξ
)
.
For a given function u2 :
(
∂D \ {(b1, 0), (b2, 0)}
)
−→ R, we define the
function
g∗(ξ) := u∗2(ξ)− ξ
ξ∫
0
s(u∗2(s)− u∗2(ξ))
(ξ2 − s2)3/2
ds ,
where the function u∗2 is expressed via the given function u2 in the fol-
lowing form:
u∗2(ξ) :=
y u2(x, y)√
ξ2 + 1
, x+ iy = σ
(
ξ − i
ξ + i
)
.
Denote by Cu(R) the subspace of C(R) including all odd continuous
functions. Denote also by Du(R) the set of odd functions from D(R).
The following theorem establishes sufficient conditions for the reduc-
tion of Schwarz BVP–II to the Fredholm integral equation.
S. A. Plaksa 211
Theorem 2.2. Suppose that the given function u2 is of the form
u2(x, y) ≡ ũ2(x, y)/y, where ũ2 ∈ H̃α(∂D) and ũ2(b1, 0) = ũ2(b2, 0) = 0.
Suppose also that the conformal mapping σ(Z) has the nonvanishing con-
tinuous contour derivative on the unit circle, and its modulus of conti-
nuity satisfies condition (2.7). Then the solution of the Schwarz BVP–II
is given by formula (2.6), in which
F (z) =
2(ξ + i)
πi σ′( ξ−iξ+i)
∞∫
−∞
V0(ξ)
Π∗(ξ)(τ − ξ)
dτ + C ,
z = σ
(
ξ + i
ξ − i
)
∀ ξ ∈ C : Im ξ > 0 , (2.12)
where C is a real constant, V0 is a solution of the Fredholm integral
equation
V0(ξ) + (Q(kfV0))(ξ) = (Qg∗)(ξ) ∀ ξ ∈ R (2.13)
in the space Cu(R). Moreover, the operator Qkf is compact in the space
Cu(R), and equation (2.13) has a unique solution V0 ∈ Cu(R) which
belongs necessarily to the set Du(R) and satisfies the equality
V0(0) = V0(∞) = 0 .
Proof. Let us use Theorem 3.27 in [29] on reduction of Dirichlet boundary
value problem for the Stokes flow function ψ(x, y) = y ImW (z) with
the given boundary function ũ2(x, y) to the Fredholm integral equation
(2.13). As a result, we obtain that the function (2.11), where F is defined
by equality (2.12), is continuously extended from Dz onto the set ∂Dz \
{b1, b2} , and the boundary condition (2.3) is satisfied. Moreover, for the
function ImW (z) as well as for the function F an estimate of the type
(2.4) is fulfilled. Finally, it is easy to establish that the function (2.10)
satisfies an estimate of the type (2.4) if the function F satisfies such an
estimate.
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Contact information
Sergiy A. Plaksa Department of Complex Analysis and
Potential Theory, Institute of Mathematics
of the National Academy of Science of
Ukraine, Kyiv, Ukraine
E-Mail: plaksa62@gmail.com
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