New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions
The solutions of a boundary problem with non-local boundary conditions for a three-dimensional Laplace equation are studied. Here, the boundary conditions are the most common and linear. Further, we note that the singular integrals appearing in the necessary conditions are multi-dimensional. Therefo...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1405532018-07-11T01:22:58Z New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions Mustafayeva, Y.Y. Aliyev, N.A. The solutions of a boundary problem with non-local boundary conditions for a three-dimensional Laplace equation are studied. Here, the boundary conditions are the most common and linear. Further, we note that the singular integrals appearing in the necessary conditions are multi-dimensional. Therefore, the regularization of these singularities is much more di±cult than the regularization of one-dimensional singular integrals. After the regularization of singularities the Fredholm property of the problem is proved. 2016 Article New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions / Y.Y. Mustafayeva, N.A. Aliyev // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 3. — С. 185-204. — Бібліогр.: 25 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.03.185 Mathematics Subject Classification 2010: 35J05, 35J40 http://dspace.nbuv.gov.ua/handle/123456789/140553 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The solutions of a boundary problem with non-local boundary conditions for a three-dimensional Laplace equation are studied. Here, the boundary conditions are the most common and linear. Further, we note that the singular integrals appearing in the necessary conditions are multi-dimensional. Therefore, the regularization of these singularities is much more di±cult than the regularization of one-dimensional singular integrals. After the regularization of singularities the Fredholm property of the problem is proved. |
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Article |
| author |
Mustafayeva, Y.Y. Aliyev, N.A. |
| spellingShingle |
Mustafayeva, Y.Y. Aliyev, N.A. New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions Журнал математической физики, анализа, геометрии |
| author_facet |
Mustafayeva, Y.Y. Aliyev, N.A. |
| author_sort |
Mustafayeva, Y.Y. |
| title |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions |
| title_short |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions |
| title_full |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions |
| title_fullStr |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions |
| title_full_unstemmed |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions |
| title_sort |
new method of solvability of a three-dimensional laplace equation with nonlocal boundary conditions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/140553 |
| citation_txt |
New Method of Solvability of a Three-dimensional Laplace Equation with Nonlocal Boundary Conditions / Y.Y. Mustafayeva, N.A. Aliyev // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 3. — С. 185-204. — Бібліогр.: 25 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT mustafayevayy newmethodofsolvabilityofathreedimensionallaplaceequationwithnonlocalboundaryconditions AT aliyevna newmethodofsolvabilityofathreedimensionallaplaceequationwithnonlocalboundaryconditions |
| first_indexed |
2025-07-10T10:43:08Z |
| last_indexed |
2025-07-10T10:43:08Z |
| _version_ |
1837256349205921792 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 3, pp. 185–204
New Method of Solvability of a Three-dimensional
Laplace Equation with Nonlocal Boundary Conditions
Y.Y. Mustafayeva and N.A. Aliyev
Baku State University
23 Z. Khalilov Str., AZ 1148, Baku, Azerbaijan
E-mail: helenmust@rambler.ru
aliyev.nihan@mail.ru
Received November 9, 2014, revised October 23, 2015
The solutions of a boundary problem with non-local boundary conditions
for a three-dimensional Laplace equation are studied. Here, the boundary
conditions are the most common and linear. Further, we note that the sin-
gular integrals appearing in the necessary conditions are multi-dimensional.
Therefore, the regularization of these singularities is much more difficult
than the regularization of one-dimensional singular integrals. After the reg-
ularization of singularities the Fredholm property of the problem is proved.
Key words: non-local boundary conditions, three-dimensional Laplace
equation, multi-dimensional singular integral, necessary conditions, regular-
ization, Fredholm property.
Mathematics Subject Classification 2010: 35J05, 35J40.
1. Introduction
We have been studying boundary value problems and the related conditions
since the 1970s. After publishing the results on ODE’s, we began to study the
problems for partial differential equations. The first published work in this di-
rection was the problem for an elliptic equation of first order, i.e., the Cauchy–
Riemann equation [1], and the second was the problem for the Cauchy–Riemann
equation with nonlocal boundary conditions [2]. Note that if we consider the
Cauchy–Riemann equation with local boundary conditions (Dirichlet condition),
this problem is incorrect. Further we considered the Cauchy–Riemann equation
with non-local and global terms in the boundary conditions [3]. The extension of
that work on an integral-differential equation with the Cauchy–Riemann equation
as its principal part and non-local conditions of the general form was given in
c© Y.Y. Mustafayeva and N.A. Aliyev, 2016
Y.Y. Mustafayeva and N.A. Aliyev
[4]. A mixed-type equation of the first order with nonlocal boundary conditions
was considered in [5], and a Steklov problem for the Cauchy–Riemann equation
was considered in [6]. The inverse problem for the Cauchy–Riemann equation
in the Tikhonov–Lavrentiev sense was considered in [7]. Also there was studied
the effect of Carleman conditions on the Fredholm property of a problem for the
Cauchy–Riemann equation [8].
The results obtained for the Cauchy–Riemann equation were extended to a
boundary-value problem for the two-dimensional Laplace equation in [9], [10] and
the Steklov problem for this equation, in [11]. The methods proposed and used
in the above works were applied to the study of solutions to the mixed problem
for the equations of parabolic and hyperbolic types [12–15].
The boundary value problem for a two-dimensional integro-differential loaded
equation with boundary conditions containing both nonlocal and global terms
(integrals) was considered in [16], and boundary-value problems for an equation
of composite type with general linear nonlocal boundary conditions were studied
by this method in [17, 18]. A mixed problem for the Navier–Stokes equations was
considered in [19, 20]. The present work deals with the solutions to the boundary
value problem for the three-dimensional Laplace equation with nonlocal boundary
conditions. The analyses of boundary value problems for the Laplace equation
with nonlocal boundary conditions were conducted by a different method in [21].
A version of the new method of [21] was applied in [11] to the two-dimensional
Laplace equation. However, for the three-dimensional case the reasoning is more
complicated, and this is for the first time that we give a complete proof, in
particular, that of the Fredholm property.
The difference of the new method from the previous ones (e.g., the method
of successive approximations [22, p. 74], [23]) applied to a one- and a two-
dimensional equations [24] is as follows. In our case the singular integral equations
in the obtained necessary conditions are in the spectrum, i.e., if one applies only
one iteration, we obtain a singular integral equation with the same singularity. If
one applies the Poincare–Bertrand formula, the resulting jump will be eliminated
by the external term. As a result, the Fredholm integral equation of the first kind
is obtained.
The approach is based on the necessary conditions for the existence of the
solution to the problem which are obtained in the work. Note that the necessary
conditions for a linear ODE are obtained in the form of conventional boundary
conditions.
If n necessary conditions are obtained for an n-th order differential equation
with n boundary conditions, then we obtain a system of 2n linear algebraic equa-
tions (the unknowns are the boundary values of the unknown function and its
derivatives up to the (n−1)-th order inclusive). If we can determine the unknowns
from this system, then the solution of the problem is obtained from the Lagrange
186 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
formula. If the resulting system of linear algebraic equations is unsolvable, then
the boundary value problem has no solution either. If the system has an infinite
number of solutions, then the boundary value problem has an infinite number of
solutions too (if at least one boundary condition is a linear combination of the
necessary conditions or vice versa).
These necessary conditions are derived from the basic equation of the problem
and the fundamental solution of the adjoint equation using the second Green’s
formula (it suffices to use the Lagrange formula for an ODE).
Some necessary conditions (for partial differential equations) contain singu-
larities. The regularization of singular integrals cannot be obtained by usual
methods. That is why, in this paper, we present the new schemes of regular-
ization. We also derive below an analogue of the second Green’s formula for
derivatives and use it to obtain a complete system of necessary conditions.
Combining the regularized necessary conditions with the given boundary con-
ditions, we obtain sufficient conditions for the Fredholm property of the problem.
2. Problem Statement
Let us consider the three-dimensional Laplace equation in the domain D ⊂ R3,
convex in the direction x3, whose projection onto the plane Ox1x2 = Ox′ is the
domain S ⊂ Ox1x2, Γ is the boundary (surface) of the domain D:
Lu = ∆u(x) =
∂2u(x)
∂x2
1
+
∂2u(x)
∂x2
2
+
∂2u(x)
∂x2
3
= 0, (2.1)
x = (x1, x2, x3) ∈ D,
with non-local boundary conditions:
liu =
3∑
j=1
[
α
(1)
ij (x′)
∂u(x)
∂xj
∣∣
x3=γ1(x′) + α
(2)
ij (x′)
∂u(x)
∂xj
∣∣
x3=γ2(x′)
]
+α
(1)
i (x′)u(x′, γ1(x′)) + α
(2)
i (x′)u(x′, γ2(x′)) = fi(x′), (2.2)
i = 1, 2; x′ ∈ S,
u(x) = f0(x), x ∈ Γ̄1
⋂
Γ̄2. (2.3)
Here Γ1 and Γ2 are the upper and lower half-surfaces of the boundary Γ, respec-
tively, defined as Γk = {ξ = (ξ1, ξ2, ξ3) : ξ3 = γk(ξ′), ξ′ = (ξ1, ξ2) ∈ S = prξ3=0Γk} ,
k = 1, 2, where ξ3 = γk(ξ1, ξ2), k = 1, 2, are equations of the half-surfaces Γ1
and Γ2; the functions γk(ξ′), k = 1, 2, are twice differentiable with respect to
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 187
Y.Y. Mustafayeva and N.A. Aliyev
the variables (ξ1, ξ2) ; the coefficients α
(k)
ij (x′), α
(k)
i (x′), k = 1, 2, are continuous
functions in the domain S.
The fundamental solution for the three-dimensional Laplace equation has the
form of [25]:
U(x− ξ) = − 1
4π
1
|x− ξ| . (2.4)
3. Basic Relationships and Necessary Conditions
Multiplying Eq. (2.1) by the fundamental solution (2.4), integrating it over
the domain D and taking into account that ∆xU(x−ξ) = δ(x−ξ), where δ(x−ξ)
is the Dirac δ-function, we can get the first basic relationship:
−
3∑
j=1
∫
Γ
[(
∂u(x)
∂xj
U(x− ξ)− u(x)
∂U(x− ξ
∂xj
)
cos(ν, xj)dx
]
=
∫
D
u(x)δ(x− ξ)dx =
{
u(ξ), ξ ∈ D,
1
2u(ξ), ξ ∈ Γ.
(3.1)
Here the first relationship gives the representation of the general solution of equa-
tion (2.1), and the second expression in (3.1) is the first necessary condition.
Consider the first necessary condition (ξ ∈ Γ):
1
2
u(ξ) = −
3∑
j=1
∫
Γ
(
∂u(x)
∂xj
U(x− ξ)− u(x)
∂U(x− ξ)
∂xj
)
cos(νx, xj)dx
= −
∫
Γ
∂u(x)
∂ν
U(x− ξ)dx +
∫
Γ
u(x)
3∑
j=1
∂U(x− ξ)
∂xj
cos(νx, xj)dx. (3.2)
As ∂U(x−ξ)
∂xi
= − xi−ξi
4π|x−ξ|3 = − cos(x−ξ,xi)
4π|x−ξ|2 , all the integrands in (3.2) have a
weak singularity, e.i., the order of singularity does not exceed the multiplicity of
integrals.
Thus we have proved
Theorem 3.1. Let D ⊂ R3 be a bounded domain, convex in the x3−direction
and such that its boundary Γ is a Lyapunov surface. Then the obtained first
necessary condition (3.2) is regular.
188 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
Multiplying (2.1) by ∂U(x−ξ)
∂xi
, i = 1, 3, and integrating it over the domain D,
we obtain the rest of three basic relationships:
∫
Γ
∂u(x)
∂xi
∂U(x− ξ)
∂νx
dx
+
∫
Γ
∂u(x)
∂xm
[
∂U(x− ξ)
∂xi
cos(νx, xm)− ∂U(x− ξ)
∂xm
cos(νx, xi)
]
dx
+
∫
Γ
∂u(x)
∂xl
[
∂U(x− ξ)
∂xi
cos(νx, xl)− ∂U(x− ξ)
∂xl
cos(νx, xi)
]
dx
=
{
−∂u(ξ)
∂ξi
, ξ ∈ D,
−1
2
∂u(ξ)
∂ξi
, ξ ∈ Γ,
i = 1, 3 , (3.3)
where the triple i, m, l is a permutation of numbers 1,2,3.
The second expressions in (3.3) are the other three necessary conditions
( ξ ∈ Γ,i = 1, 3 ):
−1
2
∂u(ξ)
∂ξi
=
∫
Γ
∂u(x)
∂xi
∂U(x− ξ)
∂νx
dx
+
∫
Γ
∂u(x)
∂xm
[
∂U(x− ξ)
∂xi
cos(νx, xm)− ∂U(x− ξ)
∂xm
cos(νx, xi)
]
dx
+
∫
Γ
∂u(x)
∂xl
[
∂U(x− ξ)
∂xi
cos(νx, xl)− ∂U(x− ξ)
∂xl
cos(νx, xi)
]
dx . (3.4)
where the triple i,m,l is a permutation of numbers 1,2,3.
Taking into account that ∂U(x−ξ)
∂xi
= − xi−ξi
4π|x−ξ|3 = − cos(x−ξ,xi)
4π|x−ξ|2 and introducing
the designations
Kij(x, ξ) = (cos(x− ξ, xi) cos(νx, xj)− cos(x− ξ, xj) cos(νx, xi)) , (3.5)
we can write the 2nd, the 3rd and the 4th necessary conditions (3.4) in
the form:
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 189
Y.Y. Mustafayeva and N.A. Aliyev
−1
2
∂u(ξ)
∂ξi
=
∫
Γ
∂u(x)
∂xi
∂U(x− ξ)
∂νx
dx
+
∫
Γ
∂u(x)
∂xm
Kim(x, ξ)
4π |x− ξ|2 dx +
∫
Γ
∂u(x)
∂xl
Kil(x, ξ)
4π |x− ξ|2 dx , (3.6)
where the triple i,m,l is a permutation of numbers 1,2,3.
To introduce the second group of necessary conditions we write the two first
surface integrals in the (i + 1)-th relationship (3.6) (i = 1, 2, 3) over the upper
and lower half-surfaces Γk, k = 1, 2 in the form:
−1
2
∂u
∂ξi
∣∣
ξ3=γk(ξ′)
=
2∑
j=1
(−1)j−1
∫
S
∂u(x)
∂xm
∣∣∣x3=γj(x′)
Kim(x, ξ)
4π |x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γj(x′)
ξ3 = γk(ξ′)
dx′
cos(νx, x3)
+
2∑
j=1
(−1)j−1
∫
S
∂u(x)
∂xl
∣∣∣x3=γj(x′)
Kil(x, ξ)
4π |x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γj(x′)
ξ3 = γk(ξ′)
dx′
cos(νx, x3)
+
∫
Γ
∂u(x)
∂xi
∂U(x− ξ)
∂νx
∣∣
ξ3=γk(ξ′) dx. (3.7)
The singular terms for =1,2 in the 2nd, 3rd, 4th necessary conditions (i =
1, 3) are:
1
2
∂u
∂ξi
∣∣
ξ3=γk(ξ′) = (−1)k
∫
S
∂u(x)
∂xm
∣∣
x3=γk(x′)
Kim(x, ξ)
4π |x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
dx′
cos(νx, x3)
+(−1)k+1
∫
S
∂u(x)
∂xl
∣∣
x3=γk(x′)
Kil(x, ξ)
4π |x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
dx′
cos(νx, x3)
+. . . , k = 1, 2,
(3.8)
190 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
where three dots denote the sum of nonsingular terms.
R e m a r k 3.1. Three dots in (3.8) contain the derivatives ∂u(x)
∂xi
∣∣
x3=γk(x′) ,
l = 1, 2, 3; k = 1, 2, under the sing of integral and we will take it into consideratiob
later.
Let us introduce the designations:
K
(k)
ij (x′, ξ′) = Kij(x, ξ)
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
, k = 1, 2. (3.9)
We consider |x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
, k=1,2:
|x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
=
∣∣x′ − ξ′
∣∣2 + (γk(x′)− γk(ξ′))2
=
∣∣x′ − ξ′
∣∣2
[
1 +
2∑
m=1
(
∂γk(x′)
∂xm
)2
cos2(x′ − ξ′, xm) + O(
∣∣x′ − ξ′
∣∣)
]
.
Let us introduce the designations:
Pk(x′, ξ′) = 1 +
2∑
m=1
(
∂γk(x′)
∂xm
)2
cos2(x′ − ξ′, xm) + O(
∣∣x′ − ξ′
∣∣), (3.10)
whence we have that
|x− ξ|2
∣∣∣∣∣∣∣∣
x3 = γk(x′)
ξ3 = γk(ξ′)
=
∣∣x′ − ξ′
∣∣2 Pk(x′, ξ′).
R e m a r k 3.2 . Notice that for ξ′ = x′ we have
Pk(x′, x′) = 1 +
(
∂γk
∂x1
)2
+
(
∂γk
∂x2
)2
+ 2
∂γk
∂x1
∂γk
∂x2
6= 0, k = 1, 2.
By means of designations (3.9), (3.10) we can rewrite the necessary conditions
(3.8) for k=1,2 as follows:
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 191
Y.Y. Mustafayeva and N.A. Aliyev
1
2
∂u
∂ξi
∣∣
ξ3=γk(ξ′)
= (−1)(k)
∫
S
∂u(x)
∂xm
∣∣∣∣x3=γk(x′)
1
4π |x′ − ξ′|2
K
(k)
im (x′, ξ′)
Pk(x′, ξ′)
dx′
cos(νx, x3)
+(−1)(k+1)
∫
S
∂u(x)
∂xl
∣∣
x3=γk(x′)
1
4π |x′ − ξ|′2
K
(k)
il (x′, ξ′)
Pk(x′, ξ′)
dx′
cos(νx, x3)
+ . . . (3.11)
i = 1, 2, 3; k = 1, 2.
Theorem 3.2. Under assumptions of Theorem 3.1 necessary conditions
(3.11) are singular.
4. Regularization of the Necessary Conditions
Let us build a linear combination of necessary conditions (3.11) for k=1,2
(j=1,2,3):
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
=
∫
S
2∑
k=1
β
(k)
ij (ξ′)(−1)k
∂u(x)
∂xm
∣∣
x3=γk(x′)
K
(k)
jm(x, ξ)
Pk(x′, ξ′)
+
∂u(x)
∂xl
∣∣
x3=γk(x′)
K
(k)
jl (x, ξ)
Pk(x′, ξ′)
× 1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
+ . . . , (4.1)
where the triple j, m, l is a permutation of numbers 1,2,3.
Form a sum of (4.1) for j=1,2,3 and factor out the common factor 1
2π|x′−ξ′|2
under the sign of integral (i = 1, 2):
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
=
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
(−1)k
3∑
j=1
β
(k)
ij (ξ′)
192 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
×
∂u(x)
∂x2
∣∣
x3=γk(x′)
K
(k)
jm(x′, ξ′)
Pk(x′, ξ′)
+
∂u(x)
∂x3
∣∣
x3=γk(x′)
K
(k)
jl (x′, ξ′)
Pk(x′, ξ′)
+ . . . . (4.2)
Adding and subtracting β
(k)
ij (x′) to and from β
(k)
ij (ξ′) , k = 1, 2 in (4.2), we obtain
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
=
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
(−1)k
3∑
j=1
β
(k)
ij (x′)
×
∂u(x)
∂xm
∣∣
x3=γk(x′)
K
(k)
jm(x′, ξ′)
Pk(x′, ξ′)
+
∂u(x)
∂xl
∣∣
x3=γk(x′)
K
(k)
jl (x′, ξ′)
Pk(x′, ξ′)
+
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
(−1)k
3∑
j=1
[
β
(k)
ij (ξ′)− β
(k)
ij (x′)
]
×
∂u(x)
∂xm
∣∣
x3=γk(x′)
K
(k)
jm(x′, ξ′)
Pk(x′, ξ′)
+
∂u(x)
∂xl
∣∣
x3=γk(x′)
K
(k)
jl (x′, ξ′)
Pk(x′, ξ′)
+ . . . . (4.3)
The second integral in the right-hand side of (4.3) has a week singularity under the
condition that the functions β
(k)
ij (ξ′) satisfy Hölder’s condition. Before grouping
the derivatives in the first integral, we expand all the coefficients at the derivatives
by Taylor’s formula at the point ξ′ = x′:
K
(k)
ij (x′, ξ′)
Pk(x′, ξ′)
=
K
(k)
ij (x′, x′)
Pk(x′, x′)
+
∂
∂x1
(
K
(k)
ij (x′, x′)
Pk(x′, x′)
)
(x1 − ξ1)
+
∂
∂x2
(
K
(k)
ij (x′, x′)
Pk(x′, x′)
)
(x2 − ξ2) + ... .
All the terms except the first one reduce the order of singularity and make it
weaker for a double integral over the surface S. For this reason we will consider
only the first term of each expansion:
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 193
Y.Y. Mustafayeva and N.A. Aliyev
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
=
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
(−1)k
3∑
j=1
β
(k)
ij (x′)
×
∂u(x)
∂xm
∣∣
x3=γk(x′)
K
(k)
jm(x′, ξ′)
Pk(x′, ξ′)
+
∂u(x)
∂xl
∣∣
x3=γk(x′)
K
(k)
jl (x′, ξ′)
Pk(x′, ξ′)
+ . . .
=
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
(−1)k
3∑
j=1
∂u(x)
∂xj
× ∣∣
x3=γk(x′)
β
(k)
il (x′)
K
(k)
lj (x′, x′)
Pk(x′, x′)
+ β
(k)
im (x′)
K
(k)
mj (x
′, x′)
Pk(x′, x′)
+ . . . , (4.4)
where the triple j, l, m is a permutation of numbers 1,2,3; i=1,2.
To regularize the integral in the right-hand side of (4.4), we should impose
conditions on the coefficients β
(k)
ij (ξ′), i.e., make the coefficients at the derivatives
under the sign of the integral (4.4) equal to the coefficients α
(k)
ij (ξ′) from boundary
conditions (2.2). Then we get a system of 6 equations for each i=1,2:
(−1)kβ
(k)
il (x′)
K
(k)
lj (x′, x′)
Pk(x′, x′)
+ (−1)kβ
(k)
im (x′)
K
(k)
mj (x
′, x′)
Pk(x′, x′)
= α
(k)
ij (x′), (4.5)
k=1,2; j=1,2,3, where, as we have mentioned above, the triple j, l, m is a per-
mutation of numbers 1,2,3.
Systems (4.5) can be written in the matrix form for i=1,2:
0 0 K
(1)
12 (x,x)
P1(x′,x′) 0 K
(1)
31 (x,x)
P1(x′,x′) 0
0 0 0 −K
(2)
21 (x,x)
P2(x′,x′) 0 −K
(2)
31 (x,x)
P2(x′,x′)
K
(1)
12 (x,x)
P1(x′,x′) 0 0 0 K
(1)
32 (x,x)
P1(x′,x′) 0
0 −K
(2)
12 (x,x)
P2(x′,x′) 0 0 0 −K
(2)
32 (x,x)
P2(x′,x′)
K
(1)
13 (x,x)
P1(x′,x′) 0 K
(1)
23 (x,x)
P1(x′,x′) 0 0 0
0 −K
(2)
13 (x,x)
P2(x′,x′) 0 −K
(2)
23 (x,x)
P2(x′,x′) 0 0
194 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
×
β
(1)
i1 (x′)
β
(2)
i1 (x′)
β
(1)
i2 (x′)
β
(2)
i2 (x′)
β
(1)
i3 (x′)
β
(2)
i3 (x′)
=
α
(1)
i1 (x′)
α
(2)
i1 (x′)
α
(1)
i2 (x′)
α
(2)
i2 (x′)
α
(1)
i3 (x′)
α
(2)
i3 (x′)
. (4.6)
We reduce systems (4.6) to the triangular form:
1 0 0 0 K
(1)
32
K
(1)
12
0
0 1 0 0 0 K
(2)
31
K
(2)
12
0 0 1 0 K
(1)
31
K
(1)
21
0
0 0 0 1 0 K
(2)
31
K
(2)
21
0 0 0 0 −K
(1)
32
K
(1)
12
K
(1)
13
P1
− K
(1)
32
P1
0
0 0 0 0 0 K
(2)
32
K
(2)
12
K
(2)
13
P2
+ K
(2)
23
P2
K
(2)
31
K
(2)
21
β
(1)
i1
β
(2)
i1
β
(1)
i2
β
(2)
i2
β
(1)
i3
β
(2)
i3
=
P1
K
(1)
12
α
(1)
i2
− P2
K
(2)
12
α
(2)
i2
P1
K
(1)
21
α
(1)
i1
− P2
K
(1)
21
α
(2)
i1
α
(1)
i3 − α
(1)
i1 −K
(1)
13
α
(2)
i3 − K
(1)
13
K
(2)
12
α
(1)
i2 − K
(2)
23
K
(2)
21
α
(1)
i1
. (4.7)
We suppose that systems (4.6), or (4.7), have the solutions (β(1)
i1 , β
(2)
i1 , β
(1)
i2 , β
(2)
i2 , β
(1)
i3 , β
(2)
i3 )
for i=1, 2, respectively.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 195
Y.Y. Mustafayeva and N.A. Aliyev
R e m a r k 4.1. The obtain function β
(k)
ij , i, k = 1, 2; j = 1, 2, 3, are linear
functions of the given functions α
(k)
ij , i, k = 1, 2; j = 1, 2, 3, and, therefore satisfy
Hölder condition.
Then for further regularization, we replace the expression under the integral
sign in the right-hand side of (4.4) by using boundary conditions (2.2):
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
=
∫
S
fi(x′)
2π |x′ − ξ′|2
dx′
cos(νx, x3)
−
∫
S
1
2π |x′ − ξ′|2
[
2∑
k=1
α
(k)
i (x′)u(x′, γk(x′))
]
dx′
cos(νx, x3)
. . . . (4.8)
From necessary condition (3.2) for u(ξ) on Γk , k = 1 , 2, by discarding the term
with normal derivative ∂u
∂νx
in the integrand and leaving only weakly singular
terms, we get:
u(ξ)
∣∣
ξ3=γk(ξ′) =−
∫
S
u(x)
∣∣
x3=γk(x′)
2π |x′ − ξ′|2
cos(x− ξ, νx)
∣∣∣∣∣∣∣∣
ξ3 = γk(ξ′)
x3 = γk(x′)
Pk(x′, ξ′)
dx′
cos(νx, x3)
+ . . . .
(4.9)
Substituting necessary conditions (4.9) into (4.8), we obtain:
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
= −
∫
S
1
2π |x′ − ξ′|2
dx′
cos(νx, x3)
2∑
k=1
α
(k)
i (x′)
×
∫
S
u(ζ)
∣∣
ζ3=γk(ζ′)
2π |x′ − ζ ′|2
cos(ζ − ξ, νζ)
∣∣∣∣∣∣∣∣
ζ3 = γk(ζ ′)
x3 = γk(x′)
Pk(x′, ζ ′)
dζ ′
cos(νζ , ζ3)
196 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
+
∫
S
fi(x′)
2π |x′ − ξ′|2
dx′
cos(νx, x3)
+ . . . . (4.10)
But it is evident that if we change the order of integration in the first integral
on the right in (4.10), then the singularity vanishes (in Cauchy’s sense), i.e., we
obtain the regularized relationships:
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
= −
2∑
k=1
∫
S
u(ζ)
∣∣
ζ3=γk(ζ′) dζ ′
cos(νζ , ζ3)
∫
S
α
(k)
i (x′)
cos(ζ − ξ, νζ)
∣∣∣∣∣∣∣∣
ζ3 = γk(ζ ′)
x3 = γk(x′)
2π |x′ − ζ ′|2 Pk(x′, ζ ′)
dx′
cos(νx, x3)
+
∫
S
fi(x′)
2π |x′ − ξ′|2
dx′
cos(νx, x3)
+ . . . . (4.11)
Thus we have established the following
Theorem 4.1. Let the conditions of Theorem 3.1 hold true. If system
(4.6) is uniquely solvable, conditions (2.2) are linearly independent, the coef-
ficients α
(k)
ij (x′)for i = 1, 2; j = 1, 3; k = 1, 2, belong to some Hölder class
and the remaining coefficients and kernels are continuous functions, the func-
tions fi(x′), i = 1, 2, are continuously differentiable and vanish on the boundary
∂S = S̄\S, then the relationships (4.11) are regular.
5. Fredholm Property of the Problem
It is well known that
∂
∂xp
u(x1, x2, γk(x1, x2)) =
∂u(x)
∂xp
∣∣
x3=γk(x′) +
∂u(x)
∂x3
∣∣
x3=γk(x′)
∂γk(x′)
∂xp
,
k=1,2; p=1,2, whence we have
∂u(x)
∂xp
∣∣
x3=γk(x′) =
∂u(x′, γk(x′))
∂xp
− ∂u(x)
∂x3
∣∣
x3=γk(x′)
∂γk(x′)
∂xp
, (5.1)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 197
Y.Y. Mustafayeva and N.A. Aliyev
p=1,2; k=1,2. So, the derivatives ∂u(x)
∂x1
∣∣
x3=γk(x′) and ∂u(x)
∂x2
∣∣
x3=γk(x′) are defined
through the derivative ∂u(x)
∂x3
∣∣
x3=γk(x′) . Then we have two unknowns: the bound-
ary values u(x′, γ1(x′)) and u(x′, γ2(x′)) of unknown function. We substitute (5.1)
for ∂u(x)
∂x1
∣∣
x3=γk(x′) and ∂u(x)
∂x2
∣∣
x3=γk(x′) into boundary conditions (2.2):
liu =
3∑
j=1
2∑
m=1
α
(m)
ij (x′)
(
∂u(x′,γm(x′))
∂xj
− ∂u(x)
∂x3
∣∣
x3=γm(x′)
∂γm(x′)
∂xj
)
+α
(1)
i (x′)u(x′, γ1(x′)) + α
(2)
i (x′)u(x′, γ2(x′)) = fi(x′),
x′ ∈ S, i = 1, 2.
We group the unknowns to get the system
liu =
2∑
j=1
[
2∑
m=1
[
α
(m)
ij (x′)
(
∂u(x′, γm(x′))
∂xj
− ∂u(x)
∂x3
∣∣
x3=γm(x′)
∂γm(x′)
∂xj
)
+α
(m)
i3 (x′)
∂u(x)
∂x3
∣∣
x3=γm(x′) + α
(m)
i (x′)u(x′, γm(x′))
]
−
2∑
k=1
∂u(x)
∂x3
∣∣
x3=γk(x′)
[
2∑
m=1
α
(k)
im (x′)
∂γk(x′)
∂xm
− α
(k)
i3 (x′)
]
+
2∑
j=1
[
α
(1)
ij (x′)
∂u(x′, γ1(x′))
∂xj
+ α
(2)
ij (x′)
∂u(x′, γ2(x′))
∂xj
]
+α
(1)
i (x′)u(x′, γ1(x′)) + α
(2)
i (x′)u(x′, γ2(x′)) = fi(x′), (5.2)
x′ ∈ S, i = 1, 2.
Let us introduce the designations:
Aij(x′) = −
[
2∑
m=1
α
(1)
im(x′)
∂γ1(x′)
∂xm
− α
(1)
i3 (x′)
]
, i, j = 1, 2 .
Then system (5.2) can be written in the form
Ai1(x′)
∂u(x)
∂x3
∣∣
x3=γ1(x′) + Ai1(x′)
∂u(x)
∂x3
∣∣
x3=γ2(x′) = Fi(x′) , i = 1, 2, (5.3)
where the right-hand sides of system (4.11) have the form:
Fi(x′) = fi(x′)−
2∑
j=1
2∑
m=1
α
(m)
ij (x′)
∂u(x′, γm(x′))
∂xj
198 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
+
2∑
k=1
α
(k)
i (x′)u(x′, γk(x′)), x′ ∈ S, i = 1, 2. (5.4)
R e m a r k 5.1. Note that the right sides Fi(x′) of system (5.3) are
functionals of the boundary values of the desired function and partial derivatives
of these boundary values what follow from (5.4):
Fi(x′) = Fi(x′, u |Γ1 , u |Γ2 ,
∂u |Γ1
∂x1
,
∂u |Γ1
∂x2
,
∂u |Γ2
∂x1
,
∂u |Γ2
∂x2
), i = 1, 2. (5.5)
We will reduce system (5.3) to a normal form. For this purpose the determi-
nant of the system is required to be nonzero:
∆(x′) =
∣∣∣∣
A11(x′) A12(x′)
A21(x′) A22(x′)
∣∣∣∣ 6= 0. (5.6)
If the coefficients α
(k)
ij (x′), i, j, k = 1, 2, and the equations of the boundaries
γ1(x′) and γ2(x′) satisfy condition (5.6), then by Cramer’s formulas we have
∂u(x)
∂x3
∣∣
x3=γ1(x′) = 1
∆(x′)
∣∣∣∣
F1(x′) A12(x′)
F2(x′) A22(x′)
∣∣∣∣ ,
∂u(x)
∂x3
∣∣
x3=γ2(x′) = 1
∆(x′)
∣∣∣∣
A11(x′) F1(x′)
A21(x′) F2(x′)
∣∣∣∣ .
(5.7)
System (5.7) is a system of integro-differential Fredholm equations of the second
kind with a regular kernel. Solving it, we will obtain the solution in the form:
∂u(x)
∂x3
∣∣
x3=γk(x′) = Φk(u |Γ1 , u |Γ2 ,
∂u |Γ1
∂x1
,
∂u |Γ1
∂x2
,
∂u |Γ2
∂x1
,
∂u |Γ2
∂x2
), k = 1, 2.
(5.8)
Going back to regular boundaries conditions (4.7), let us substitute expressions
(5.1) for the derivatives ∂u(x)
∂xj
∣∣
x3=γk(x′) , j, k = 1, 2, into (4.11):
3∑
j=1
(
β
(1)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ1(ξ′) + β
(2)
ij (ξ′)
∂u(ξ)
∂ξj
∣∣
ξ3=γ2(ξ′)
)
=
2∑
j=1
2∑
m=1
β
(m)
ij (ξ′)
(
∂u(ξ′, γm(ξ′))
∂ξj
− ∂u(ξ)
∂ξ3
∣∣
ξ3=γm(ξ′)
∂γm(ξ′)
∂ξj
)
+β
(1)
i3 (ξ′)
∂u(ξ)
∂ξ3
∣∣
ξ3=γ1(ξ′) + β
(2)
i3 (ξ′)
∂u(ξ)
∂ξ3
∣∣
ξ3=γ2(ξ′)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 199
Y.Y. Mustafayeva and N.A. Aliyev
= −
∫
S
1
2π |x′ − ξ′|2
[
2∑
m=1
α
(m)
i (x′)u(x′, γm(x′))
]
dx′
cos(νx, x3)
+
∫
S
fi(x′)dx′
2π |x′ − ξ′|2 cos(νx, x3)
+ . . . , i = 1, 2. (5.9)
Now we group the terms both on the left and on the right in (5.9):
−
2∑
k=1
∂u(ξ)
∂ξ3
∣∣
ξ3=γk(ξ′)
[
2∑
m=1
β
(k)
im (ξ′)
∂γk(ξ′)
∂ξm
− β
(k)
i3 (ξ′)
]
+
2∑
j=1
[
β
(1)
ij (ξ′)
∂u(ξ′, γ1(ξ′))
∂ξj
+ β
(2)
ij (ξ′)
∂u(ξ′, γ2(ξ′))
∂ξj
]
= −
∫
S
dx′
2π |x′ − ξ′|2 cos(νx, x3)
2∑
m=1
α
(m)
i (x′)u(x′, γm(x′))
+
∫
S
fi(x′)dx′
2π |x′ − ξ′|2 cos(νx, x3)
+ . . . , x′ ∈ S, i = 1, 2. (5.10)
The terms in (5.10) are either with weakly singular kernels or with regular ones.
If we introduce the designations
Cij(ξ′) = −
[
2∑
m=1
β
(j)
im(ξ′)
∂γj(ξ′)
∂ξm
− β
(j)
i3 (ξ′)
]
, i, j = 1, 2,
Bi(ξ′) = −
2∑
j=1
2∑
m=1
β
(m)
ij (ξ′)
∂u(ξ′, γm(ξ′))
∂ξj
−
∫
S
dx′
2π |x′ − ξ′|2 cos(νx, x3)
2∑
m=1
α
(m)
i (x′)u(x′, γm(x′))
+
∫
S
fi(x′)dx′
2π |x′ − ξ′|2 cos(νx, x3)
+ ... , x′ ∈ S, i = 1, 2,
then system (5.10) can be written in the form
2∑
m=1
Cim(ξ′)
∂u(ξ)
∂ξ3
∣∣
ξ3=γm(ξ′) = Bi(ξ′) , i = 1, 2. (5.11)
200 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3
New Method of Solvability of a Three-dimensional Laplace Equation
Evidently, system (5.11) is a system of integro-differential Fredholm equations
of the second kind with respect to ∂u(ξ)
∂ξ3
∣∣
ξ3=γk(ξ′) , k = 1, 2, and thus, has the
unique solution
∂u(ξ)
∂ξ3
∣∣
ξ3=γk(ξ′) = Ψk(u |Γ1 , u |Γ2 ,
∂u |Γ1
∂ξ1
,
∂u |Γ2
∂ξ1
,
∂u |Γ1
∂ξ2
,
∂u |Γ2
∂ξ2
), (5.12)
where u |Γk
= u(ξ′, γk(ξ′)),
∂u|Γk
∂ξj
= ∂u(ξ′,γk(ξ′))
∂ξj
, j = 1, 2,k = 1, 2, are the bound-
ary values of the desired solution u(x) on the surfaces Γk , k = 1, 2, and the
derivatives of its boundary values, respectively.
The functionals Φk, Ψk , k = 1, 2, from (5.8) and (5.12) are linear with re-
spect to the unknown values u |Γ1 , u |Γ2 ,
∂u|Γ1
∂ξj
,
∂u|Γ2
∂ξj
, j = 1, 2:
Φk(u |Γ1 , u |Γ2 ,
∂u |Γ1
∂ξ1
,
∂u |Γ2
∂ξ1
,
∂u |Γ1
∂ξ2
,
∂u |Γ2
∂ξ2
)
=
2∑
i=1
a
(k)
i (ξ′)u |Γi +
2∑
i,j=1
b
(k)
ij (ξ′)
∂u |Γi
∂ξj
+
2∑
i=1
∫
S
c
(k)
i (ζ ′)u |Γi dζ
+
2∑
i,j=1
∫
S
d
(k)
ij (ζ ′)
∂u |Γi
∂ζj
dζ + ϕk(ξ′), k = 1, 2, (5.13)
Ψk(u |Γ1 , u |Γ2 ,
∂u |Γ1
∂ξ1
,
∂u |Γ2
∂ξ1
,
∂u |Γ1
∂ξ2
,
∂u |Γ2
∂ξ2
)
=
2∑
i=1
a
(l)
i (ξ′)u |Γi +
2∑
i,j=1
b
(l)
ij (ξ′)
∂u |Γi
∂ξj
+
2∑
i=1
∫
S
c
(l)
i (ζ ′)u |Γi dζ
+
2∑
i,j=1
∫
S
d
(l)
ij (ζ ′)
∂u |Γi
∂ζj
dζ + ϕl(ξ′), l = 3, 4; k = 1, 2. (5.14)
Excluding ∂u(ξ)
∂ξ3
∣∣
ξ3=γk(ξ′) , k = 1, 2, from system (5.13), (5.14), we will obtain a
system of linear integro-differential Fredholm equations of the second kind with
respect to u(ξ′, γk(ξ′)), k = 1, 2:
2∑
i=1
A
(k)
i (ξ′)u |Γi +
2∑
i,j=1
B
(k)
ij (ξ′)
∂u |Γi
∂ξj
+
2∑
i=1
∫
S
C
(k)
i (ζ ′)u |Γi dζ+
+
2∑
i,j=1
∫
S
D
(k)
ij (ζ ′)
∂u |Γi
∂ζj
dζ + gk(ξ′) = 0, k = 1, 2, (5.15)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 201
Y.Y. Mustafayeva and N.A. Aliyev
where
A
(k)
i (ξ′) = a
(k)
i (ξ′)− a
(k+2)
i (ξ′) , B
(k)
ij (ξ′) = b
(k)
ij (ξ′)− b
(k+2)
ij (ξ′),
C
(k)
i (ζ ′) = c
(k)
i (ζ ′)− c
(k+2)
i (ζ ′), D(k)
ij (ζ ′) = d
(k)
ij (ζ ′)− d
(k+2)
ij (ζ ′),
gk(ξ′) = ϕk(ξ′)− ϕk+2(ξ′), k = 1, 2.
Thus, we have come to a two-dimensional system of linear integro-differential
equations of the first order for which Dirichlet’s conditions (2.3) are given on the
boundary of a two-dimensional domain S. As the boundary is one-dimensional,
this Dirichlet’s condition does not restrict the generality because its co-dimension
is two units less than the dimension of the domain D.
Thus, we have established the following
Theorem 5.1. Let the assumptions of Theorem 4.1 and (5.6) hold true and
system (5.11) be uniquely solvable. Then boundary value problem (2.1)–(2.2) is
reduced to a two-dimensional system of linear integro-differential equations (5.15)
with Dirichlet’s condition (2.3) on the boundary ∂S = S\S.
Finally, there has been established
Theorem 5.2. Under the assumptions of Theorem 5.1 boundary value prob-
lem (2.1), (2.2), (2.3) possesses the Fredholm property.
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