Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations
Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhot...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509527896489984 |
|---|---|
| author | Schneider, B. Шнайдер, Б. |
| author_facet | Schneider, B. Шнайдер, Б. |
| author_sort | Schneider, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:30Z |
| description | Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis. |
| first_indexed | 2026-03-24T02:42:32Z |
| format | Article |
| fulltext |
UDC 517.5
B. Schneider (Izmir Univ. Econ., Turkey)
SOME PROPERTIES OF THE CAUCHY-TYPE INTEGRAL
FOR THE MOISIL – THEODORESCO SYSTEM
OF PARTIAL DIFFERENTIAL EQUATIONS
DEQKI VLASTYVOSTI INTEHRALIV TYPU KOÍI
DLQ SYSTEM MOISIL – TEODORESKO
DYFERENCIAL\NYX RIVNQN\
Z ÇASTYNNYMY POXIDNYMY
Our main interest is the analog of the Cauchy-type integral for the theory of Moisil – Theodoresco system of
differential equations in the case of a piecewise Liapunov surface of integration. The topics of the paper concern
theorems which cover basic properties of that Cauchy-type integral: the Sokhotski – Plemelj theorem for it as
well as the necessary and sufficient condition for the possibility to extend a given Hölder function from such a
surface up to a solution of Moisil – Theodoresco system of partial differential equations in a domain. A formula
for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate
relations between the theory of Moisil – Theodoresco system of partial diferential equations and some versions
of quaternionic analysis.
Robotu v osnovnomu prysvqçeno vyvçenng analoha intehrala typu Koßi dlq teori] system Moisil –
Teodoresko dyferencial\nyx rivnqn\ u vypadku kuskovo] poverxni intehruvannq Lqpunova. Rozhlqda-
gt\sq teoremy, wo oxoplggt\ bazovi vlastyvosti c\oho intehrala typu Koßi, a same teorema Soxoc\ko-
ho – Plemel\ dlq n\oho, a takoΩ neobxidna i dostatnq umova prodovΩuvanosti zadano] funkci] Hel\dera
z nazvano] vywe poverxni do rozv’qzku systemy Moisil – Teodoresko dyferencial\nyx rivnqn\ z çastyn-
nymy poxidnymy v oblasti. Navedeno formulu kvadrata synhulqrnoho intehrala typu Koßi. Dovedennq
vsix cyx faktiv bazu[t\sq na blyz\kyx zv’qzkax miΩ teori[g system Moisil – Teodoresko dyferen-
cial\nyx rivnqn\ z çastynnymy poxidnymy i deqkymy versiqmy kvaternionnoho analizu.
1. Introduction. As is well known, the role of the Cauchy-type integral in holomorphic
function theory of one complex variable is very important. In this article, we investigate
the properties of the Cauchy-type integral for the first order elliptic system in R
3. Let Ω
be a domain in R
3. Suppose that f = f0 + �f ∈ C1(Ω, R4). The homogeneous system
divf = 0,
gradf0 + rot�f = 0
is called Moisil – Theodoresco system and is the simplest analog of the Cauchy – Riemann
system in the three-dimensional case. Thus, the theory of solutions of the Moisil – Theodo-
resco system of differential equations reduces, in some degenerate cases, to that of com-
plex holomorphic functions. Hence, one may consider the former to be a generalization
of the latter.
Note that if f0 = 0, we have
divf = 0,
rotf = 0.
(1)
Solutions to system (1) are called solenoidal and irrotational vector fields (cf. [1], where
some applications to geophysics are given. It is known that solutions of (1) satisfy the
Laplace equation and are sometimes called Laplacian or harmonic vector fields. In [2], we
studied some properties of the Cauchy-type integral for the Laplace vector fields theory,
also.
c© B. SCHNEIDER, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1 105
106 B. SCHNEIDER
In the present paper, we follow the approach presented in paper [3] in which we stud-
ied the analog of the Cauchy-type integral for the theory of time-harmonic solutions of
the relativistic Dirac equation in the case of a piecewise Liapunov surface of integration.
The paper is organized as follows. In Section 2, we formulate a series of theorems which
cover basic properties of the Cauchy-type integral for the theory of Moisil – Theodoresco
system of differential equations in the case of a piecewise Liapunov surface of integra-
tion. The proofs of all of them one can find in Section 4 in the form of more or less direct
corollaries of the corresponding facts valid for hyperholomorphic function theory, which
is developed in Section 3 and [4].
2. Moisil – Theodoresco system of partial differential equations and the Cauchy –
Moisil – Theodoresco integral. 2.1. Let Ω denote a domain in R
3 and let Γ := ∂Ω be
its boundary. For Ω ⊂ R
3 consider an R
4-valued function f = (f0, f1, f2, f3), which
satisfies the following system of partial differential equations:
0 +
∂f1
∂x1
+
∂f2
∂x2
+
∂f3
∂x3
= 0,
∂f0
∂x1
+ 0 − ∂f2
∂x3
+
∂f3
∂x2
= 0,
∂f0
∂x2
+
∂f1
∂x3
+ 0 − ∂f3
∂x1
= 0,
∂f0
∂x3
− ∂f1
∂x2
+
∂f2
∂x1
+ 0 = 0.
It is usually called a Moisil – Theodoresco system. Let Vst :=
(
1 0
0 a
)
with a =
= (δk
j )3j,k=1 (δk
j is the Kronecker symbol), x = (0, x1, x2, x3)T , and dx̂ = (0, dx[1],
−dx[2], dx[3])T , where dx[k] denotes, as usual, the differential form dx1∧dx2∧dx3 with
the factor dxk omitted. The integral
VstKΓ[f ](x) :=
1
4π
∫
Γ
1
|τ − x|3 Bl(V T
st · (τ − x))Bl(V T
st · dτ̂)f(τ), x /∈ Γ,
plays the role of an analog of the Cauchy-type integral in the theoryof the Moisil –
Theodoresco system of partial differential equations with f : Γ → R
4 (see [5]). We
shall call it the Cauchy – Moisil – Theodoresco-type integral.
2.2. For reader’s convenience, we collect here some definitions which we use in
the sequel. Let Hµ(Γ, R4) denote the class of functions satisfying the Hölder condition
{f ∈ R
4| |f(t1)− f(t2)| ≤ Lf |t1 − t2|µ ∀{t1, t2} ⊂ Γ, Lf = const} with the exponent
0 < µ ≤ 1. Here, |f | means the Euclidean norm in R
4 while |t| is the Euclidean norm in
R
3. We say (see, e.g., [6]) that the surface Γ in R
3 is a Liapunov surface if the following
conditions are satisfied:
1. At each point t ∈ Γ, there is the tangential hyperplane.
2. There exists a constant number R > 0 such that for any point t ∈ Γ, the set
Γ ∩ B
3(t, R) is connected and lines, that are parallel to the normal �n(t) to the surface Γ
at the point t, intersect Γ∩B
3(t, R) at not more than one point. Here, B
3(t, R) is an open
ball in R
3 centered at the point t and with radius R.
3. The normal vector field �n : Γ → R
3 satisfies the Hölder condition.
A conical surface in R
3 is a surface generated by a straight line (the generator), which
passes through a fixed point (the vertex or conical point) and moves along a fixed curve
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
SOME PROPERTIES OF THE CAUCHY-TYPE INTEGRAL ... 107
(the directing curve). A solid angle in R
3 is a part of the space R
3 bounded by some
conical surface. A tangential conical surface to Γ at the point t0 is the conical surface
generated by straight tangent lines to surface Γ at point t0 (the conical point of tangential
conical surface). In particular, for a smooth point, the tangential conical surface is its
tangential plane. The measure of a solid angle in R
3 is the surface area cut out by the
solid angle from the unit sphere having its center in the vertex; the value of the measure
is defined in accordance with the orientation of the conical surface.
Let l be a smooth, closed, and simple curve on the surface Γ ⊂ R
3 such that Γ \ l is
a Liapunov surface. Then the curve l is called an edge of the surface Γ and Γ is called a
Liapunov surface with edge.
For l as above, let t0 ∈ l. Then the normal plane to the curve at the point t0 intersects
the surface Γ by the curve lt0 . The curve lt0 is a smooth curve except, possibly, t0.
Assume that the curve lt0 has both one-sided tangents P1 and P2 at t0. Let p be a tangent
line to the curve l itself at point t0. Then the plane T1, passing through P1 and p, and the
plane T2, passing through P2 and p, generate a dihedral angle which is called tangential
dihedral angle.
A linear measure of the tangential dihedral angle is the value of the angle formed by
the one-sided tangents P1 and P2. Denote it by η(t). In the sequel, we take η(t0) = const
on l, the constant being different from 0 and 2π. If η(t) = π on l, then Γ is a smooth
surface. In particular, for a smooth surface, any closed, smooth, and simple curve is an
edge.
A solid measure of the tangential dihedral angle is the surface area cut out by the
planes T1 and T2 from the unit sphere having its center at the point t0 ∈ l; the value of
the measure is defined in accordance with the orientation of the surface with edge.
Let Γ be a surface in R
3 which contains a finite number of conical points and a finite
number of nonintersecting edges such that none of the edges contain any of conical points.
If the complement (in Γ) of the union of conical points and edges is a Liapunov surface,
then we shall refer to Γ as a piecewise Liapunov surface in R
3.
2.3. Theorem (Sokhotski – Plemelj formulas for the Cauchy – Moisil – Theodoresco-
type integral with the piecewise Liapunov surface of integration). Let Ω be a bounded
domain in R
3 with the piecewise Liapunov boundary. Let f ∈ Hµ(Γ, R4). Then the
following limits exist:
lim
Ω±�x→t∈Γ
VstKΓ[f ](x) =: VstKΓ[f ]±(t);
moreover, the following identities hold:
VstKΓ[f ]+(t) =
(
1 − γ(t)
4π
)
f(t) + VstKΓ[f ](t) :=
(
1 − γ(t)
4π
)
f(t) +
1
2
VstSΓ[f ](t),
VstKΓ[f ]−(t) = −γ(t)
4π
f(t) + VstKΓ[f ](t) := −γ(t)
4π
f(t) +
1
2
VstSΓ[f ](t)
for all t ∈ Γ, where
VstSΓ[f ](t) := 2 VstKΓ[f ](t),
the integrals being understood in the sense of the Cauchy principal value, γ(t) is the
measure of a solid angle of the tangential conical surface at the point t or is the solid
measure of the tangential dihedral angle at the point t.
2.4. We shall call the operator VstSΓ the singular Cauchy – Moisil – Theodoresco in-
tegral operator. It’s appeared that many properties which are of interest for us, can be
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
108 B. SCHNEIDER
expressed better in terms of another operator
Vst S̆Γ[f ](t) :=
2π − γ(t)
2π
f(t) + VstSΓ[f ](t)
for any t ∈ Γ. We shall call Vst S̆Γ the modified singular Cauchy – Moisil – Theodoresco
integral operator.
2.5. Theorem (Plemelj – Privalov’s-type theorem for the Moisil – Theodoresco sys-
tem of partial differential equations theory). Let Ω be a bounded domain in R
3 with
piecewise Liapunov boundary. Then
f ∈ Hµ(Γ, R4) ⇒ Vst S̆Γ[f ](t) ∈ Hµ(Γ, R4) (2)
for 0 < µ < 1.
2.6. Theorem (extension of a Hölder function given on Γ up to solution of the Moisil –
Theodoresco system of partial differential equations). Let Ω be a bounded domain in R
3
with piecewise Liapunov boundary.
1. In order that a function f ∈ Hµ(Γ, R4) be a boundary value of a function f̃
which satisfies a Moisil – Theodoresco system of partial differential equations in Ω+ and
is continuous in Ω+, it is necessary and sufficient that
f(t) = Vst S̆Γ[f ](t) ∀t ∈ Γ.
2. In order that a function f ∈ Hµ(Γ, R4) be a boundary value of a function f̃
which satisfies a Moisil – Theodoresco system of partial differential equations in Ω− and
is continuous in Ω− and vanishes at infinity, it is necessary and sufficient that
f(t) = −Vst S̆Γ[f ](t) ∀t ∈ Γ.
2.7. Theorem (on the square of the operators VstSΓ and Vst S̆Γ). If Γ is a piecewise
Liapunov surface, then we have the following formulas for f ∈ Hµ(Γ, R4), 0 < µ < 1:
VstS2
Γ[f ](t) = a1(t)f(t) + a2(t)VstSΓ[f ](t) + VstSΓ[a3f ](t), (3)
Vst S̆2
Γ[f ](t) = f(t) (4)
for all t ∈ Γ, i.e., the modified singular Cauchy – Moisil – Theodoresco integral operator
Vst S̆Γ is an involution on Hµ(Γ, R4), 0 < µ < 1,
Vst S̆2
Γ = I,
where
a1(t) :=
γ(t)
π
− γ2(t)
4π2
, a2(t) :=
γ(t)
2π
− 2, a3(t) :=
γ(t)
2π
.
The proofs of these theorems can be found in Section 4.
3. Hyperholomorphic function theory: general information. In this section, we
provide some background on quaternionic analysis needed in this paper. For more infor-
mation, we refer the reader to [7 – 9].
3.1. We consider the skew-field of real quaternions H:
H := {x = x0i0 + x1i1 + x2i2 + x3i3; (x0, x1, x2, x3)T ∈ R
4},
where i0 is the unit, and i1, i2, i3 are the quaternionic imaginary units with the properties:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
SOME PROPERTIES OF THE CAUCHY-TYPE INTEGRAL ... 109
i20 = i0 = −i2k, i0ik = iki0 = ik, k ∈ N3;
i1i2 = −i2i1 = i3, i2i3 = −i3i2 = i1, i3i1 = −i1i3 = i2.
Let x =
∑3
k=0 xkik ∈ H. Then
x0 =: Sc(x) and �x :=
3∑
k=1
xk · ik =: Vect(x)
are called, respectively, the scalar and the vector part of a quaternion. We can write
x = x0 + �x.
In vector terms, the multiplication of two arbitrary real quaternions x, y can be rewritten
as follows:
x · y =
(
x0 + �x
)
·
(
y0 + �y
)
= x0 · y0 − 〈�x, �y〉 + x0�y + y0�x + [�x, �y],
where 〈·, ·〉 and [·, ·] denote the usual scalar and vector products of three-dimensional
vectors. In particular, if x0 = y0 = 0, then we have
x · y = −〈�x, �y〉 + [�x, �y ].
The quaternionic conjugation of x = x0i0 + x1i1 + x2i2 + x3i3 is given by
x̄ := x0i0 − x1i1 − x2i2 − x3i3.
We use the Euclidean norm |x| in H, defined by
|x| :=
√
xx̄ =
√
x2
0 + x2
1 + x2
2 + x2
3.
An important property is that
|xy| = |x| · |y|.
3.2. Let the matrix
Bl(b) :=
b0 −b1 −b2 −b3
b1 b0 −b3 b2
b2 b3 b0 −b1
b3 −b2 b1 b0
(5)
be the left regular representation of real quaternion b, and, respectively, let the matrix
Br(b) :=
b0 −b1 −b2 −b3
b1 b0 b3 −b2
b2 −b3 b0 b1
b3 b2 −b1 b0
be the right regular representation of real quaternion b. Then H can be identified as a
skew-field with Bl := {Bl(b) | b ∈ H}. The same holds for Br := {Br(b) | b ∈
∈ H} and H. Moreover, the left-multiplication by the real quaternion b corresponds to the
multiplication by the matrix Bl(b), i.e.,
b · x ↔ Bl(b) · (x0, x1, x2, x3)T ,
where (x0, x1, x2, x3)T :=
x0
x1
x2
x3
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
110 B. SCHNEIDER
3.3. We shall consider functions ranged in H and defined in a domain Ω ⊂ R
3.
Notations Cp(Ω, H), p ∈ N ∪ {0}, have the usual componentwise meaning. A function
f is called left-hyperholomorphic if
D[f ] :=
3∑
k=1
ik
∂f
∂xk
=:
3∑
k=1
ik∂k[f ] = 0
holds in Ω. Let θ = − 1
4π
1
|x| be the fundamental solution of the Laplace operator. Then
the fundamental solution to the operator D, K, is given by the formula (see [9])
K(x) := −D[θ](x) =
1
4π
3∑
k=1
īk
xk
|x|3
=
1
4π
1
|x|3
Bl(V T
st · x), (6)
where st := {i1, i2, i3}. Set
σx := i1dx[1] − i2dx[2] + i3dx[3],
where dx[k] denotes, as usual, the differential form dx1 ∧ dx2 ∧ dx3 with the factor dxk
omitted. Note that if Γ is a piecewise smooth surface in R
3 and if �n(τ) = (n1(τ), n2(τ),
n3(τ)) is the outward unit normal to surface Γ at τ, then
σ |Γ= �n(τ)dsτ =:
3∑
k=1
nk(τ)ikdsτ ,
where ds is the differential form of the two-dimensional surface Γ in R
3. Let Ω = Ω+ be
a domain in R
3 with the boundary Γ which is assumed to be a piecewise Liapunov surface;
denote Ω− := R
3 \ (Ω+ ∪ Γ). If f is a Hölder function, then its left-hyperholomorphic
Cauchy-type integral is defined as follows:
KΓ[f ](x) :=
∫
Γ
K(τ − x) · στ · f(τ), x ∈ Ω±.
For more information about hyperholomorphic functions, we refer to [7 – 10] (see
also [11]).
4. Proofs of the theorems from Section 2. In this section, we prove all theorems
from Section 2 using the relations between the Moisil – Theodoresco system of partial
differential equations theory and the theory of hyperholomorphic functions.
4.1. We start this section with a brief description of the relations between the Moisil –
Theodoresco system of partial differential equations theory and the theory of hyperholo-
morphic functions.
On the set C1(Ω, H), the well-known Moisil – Theodoresco operator is defined by the
formula
D :=
3∑
k=1
ik
∂
∂xk
.
Using matrix (5), the equality D[f ] = 0 (the Moisil – Theodoresco system) can be also
rewritten as
Bl
(
3∑
k=1
ik
∂
∂xk
)
fT = 0
with
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
SOME PROPERTIES OF THE CAUCHY-TYPE INTEGRAL ... 111
Bl
(
3∑
k=1
ik
∂
∂xk
)
=
0 − ∂
∂x1
− ∂
∂x2
− ∂
∂x3
∂
∂x1
0 − ∂
∂x3
∂
∂x2
∂
∂x2
∂
∂x3
0 − ∂
∂x1
∂
∂x3
− ∂
∂x2
∂
∂x1
0
.
Thus,
D[f ] = 0 ⇐⇒
0 − ∂
∂x1
− ∂
∂x2
− ∂
∂x3
∂
∂x1
0 − ∂
∂x3
∂
∂x2
∂
∂x2
∂
∂x3
0 − ∂
∂x1
∂
∂x3
− ∂
∂x2
∂
∂x1
0
f0
f1
f2
f3
= 0,
i.e., one can identify the class of the solutions of the elliptic system of the partial differ-
ential equations with the constant coefficients by the set of hyperholomorphic functions.
By the equality (6) for R
4-valued function f, we have
KΓ[f ](x) :=
1
4π
∫
Γ
1
|τ − x|3 Bl(V T
st · (τ − x))Bl(V T
st · dτ̂)f(τ), x /∈ Γ.
So, the integral KΓ[f ](x) coincides with VstKΓ[f ](x). In the same way,
SΓ[f ](t) := 2KΓ[f ](t) =
=
1
2π
∫
Γ
1
|τ − t|3 Bl(V T
st · (τ − t))Bl(V T
st · dτ̂)f(τ) ∀t ∈ Γ,
so, the integral SΓ for f ∈ Hµ(Γ, R4) coincides with VstSΓ[f ].
4.2. Proof of Theorem 2.3. Let f ∈ Hµ(Γ, R4). Consider VstKΓ[f ](x). It was
proved that
VstKΓ[f ](x) = KΓ[f ](x).
By [4] (Theorem 2.1 for α = 0), see also [7, 8], there exists KΓ[f ]±(t) and
KΓ[f ]+(t) =
(
1 − γ(t)
4π
)
f(t) + KΓ[f ](t) =:
(
1 − γ(t)
4π
)
f(t) +
1
2
SΓ[f ](t),
KΓ[f ]−(t) = −γ(t)
4π
f(t) + KΓ[f ](t) =: −γ(t)
4π
f(t) +
1
2
SΓ[f ](t).
Hence, there exists VstKΓ[f ]±(t) and, after not complicated computation, we obtain the
required result. Set
S̆Γ[f ](t) :=
2π − γ(t)
2π
f(t) + SΓ[f ](t)
for any t ∈ Γ.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
112 B. SCHNEIDER
4.3. Proof of Theorem 2.5. Let f ∈ Hµ(Γ, R4), consider VstKΓ[f ](x). By Theo-
rem 2.3, there exists VstKΓ[f ]±(t) and
VstKΓ[f ]+(t) =
1
2
[f(t) + Vst S̆Γ[f ](t)],
VstKΓ[f ]−(t) =
1
2
[−f(t) + Vst S̆Γ[f ](t)],
where Vst S̆Γ was defined in Subsection 2.4. By Subsection 4.1, f ∈ Hµ(Γ, R4), hence,
on Γ, Vst S̆Γ[f ] = S̆Γ[f ]. In [4] (Subsection 2.2 for α = 0), it was proved that S̆Γ satisfy
the Hölder condition. So, recalling the relationship between the operators S̆Γ and Vst S̆Γ,
we have that Vst S̆Γ[f ] ∈ Hµ(Γ, R4).
4.4. Proof of Theorem 2.6. This proof follows from [4] (Theorem 2.3 for α =
= 0) taking into account the above relation between the class of solutions of the Moisil –
Theodoresco system of partial differential equations and the set of hyperholomorphic
functions.
4.5. Proof of Theorem 2.7. Let f ∈ Hµ(Γ, R3). Consider VstKΓ[f ]. In Subsec-
tion 4.1, it was proved that
f ∈ Hµ(Γ, R4) =⇒ f ∈ Hµ(Γ, R4).
So, we obtain (3) after taking into account [4] (Theorem 2.4 for α = 0), see also [8],
combined with a straightforward calculation. Using the definition of the modified singular
operator VstSΓ, we obtain (4).
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Received 16.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
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| id | umjimathkievua-article-3436 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:32Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/32/65967c1f83594942674a97810c03a232.pdf |
| spelling | umjimathkievua-article-34362020-03-18T19:54:30Z Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations Деякі властивості інтегралів типу Коші для систем Моісіл - Теодореско диференціальних рівнянь з частинними похідними Schneider, B. Шнайдер, Б. Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis. Роботу в основному присвячено вивченню аналога інтеграла типу Коші для теорії систем Моісіл-Теодореско диференціальних рівнянь у випадку кускової поверхні інтегрування Ляпунова. Розглядаються теореми, що охоплюють базові властивості цього інтеграла типу Коші, а саме теорема Сохоцького - Племель для нього, а також необхідна і достатня умова продовжуваності заданої функції Гельдера з названої вище поверхні до розв'язку системи Моісіл - Теодореско диференціальних рівнянь з частинними похідними в області. Наведено формулу квадрата сингулярного інтеграла типу Коші. Доведення всіх цих фактів базується на близьких зв'язках між теорією систем Моісіл - Теодореско диференціальних рівнянь з частинними похідними і деякими версіями кватерніонного аналізу. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3436 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 105–112 Український математичний журнал; Том 58 № 1 (2006); 105–112 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3436/3612 https://umj.imath.kiev.ua/index.php/umj/article/view/3436/3613 Copyright (c) 2006 Schneider B. |
| spellingShingle | Schneider, B. Шнайдер, Б. Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title | Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title_alt | Деякі властивості інтегралів типу Коші для систем Моісіл - Теодореско диференціальних рівнянь з частинними похідними |
| title_full | Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title_fullStr | Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title_full_unstemmed | Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title_short | Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations |
| title_sort | some properties of a cauchy-type integral for the moisil-theodoresco system of partial differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3436 |
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