Monogenic functions taking values in generalized Clifford algebras
UDC 512.579Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.In this paper we introduce a new type of generalized Clifford algebra such that all components of a monogenic functionare solutions of an elliptic partial differential eq...
Gespeichert in:
| Datum: | 2021 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1033 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507125517647872 |
|---|---|
| author | Dinh, Doan Cong Doan Cong Dinh, Doan Cong |
| author_facet | Dinh, Doan Cong Doan Cong Dinh, Doan Cong |
| author_sort | Dinh, Doan Cong |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:46:33Z |
| description | UDC 512.579Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.In this paper we introduce a new type of generalized Clifford algebra such that all components of a monogenic functionare solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations inframework of Clifford analysis. We shall prove some Cauchy integral representation formulae for monogenic functions inthose cases. |
| doi_str_mv | 10.37863/umzh.v73i11.1033 |
| first_indexed | 2026-03-24T02:04:21Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i11.1033
UDC 512.579
Doan Cong Dinh (School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology, Vietnam)
MONOGENIC FUNCTIONS TAKING VALUES
IN GENERALIZED CLIFFORD ALGEBRAS
МОНОГЕННI ФУНКЦIЇ ЗI ЗНАЧЕННЯМИ
В УЗАГАЛЬНЕНИХ АЛГЕБРАХ КЛIФФОРДА
Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.
In this paper, we introduce a new type of generalized Clifford algebra such that all components of a monogenic function
are solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations in
framework of Clifford analysis. We shall prove some Cauchy integral representation formulae for monogenic functions in
those cases.
Узагальненi алгебри Клiффорда будуються рiзними методами i мають певнi застосування в математицi та фiзицi.
У цiй роботi введено новий тип узагальненої алгебри Клiффорда, такий, що всi компоненти моногенної функцiї є
розв’язками елiптичного диференцiального рiвняння з частинними похiдними. Однiєю з цiлей є охоплення бiльш
широкого класу диференцiальних рiвнянь з частинними похiдними в рамках аналiзу Клiффорда. У вiдповiдних
випадках доведено деякi формули iнтегрального зображення Кошi для моногенних функцiй.
1. Introduction. In the classical Clifford analysis [1], Clifford algebra \scrA n is generated by the
imaginary elements e1, e2, . . . , en with the following multiplication rules:
e2i = - 1, i = 1, 2, . . . , n,
eiej + ejei = 0, i \not = j.
A basis of \scrA n is \scrB =
\bigl\{
e0 = 1; ei1i2...ik = ei1ei2 . . . eik | 1 \leq i1 < i2 < . . . < ik \leq n
\bigr\}
. An
\scrA n-valued function f has 2n real-valued components f(x) =
\sum
A
fA(x)eA, fA(x) \in \BbbR . The
Cauchy – Riemann operator D and its adjoin D are
D =
\partial
\partial x0
+
n\sum
i=1
ei
\partial
\partial xi
, D =
\partial
\partial x0
-
n\sum
i=1
ei
\partial
\partial xi
.
The operator D applies from the left or from the right to a function f as follows:
Df =
\partial f
\partial x0
+
n\sum
i=1
ei
\partial f
\partial xi
, fD =
\partial f
\partial x0
+
n\sum
i=1
\partial f
\partial xi
ei.
A function f is called a left monogenic function if Df = 0 and a right monogenic function if fD =
= 0. We also call a left monogenic function shortly a monogenic function. Since DD = DD = \Delta ,
all components of a left (right) monogenic function satisfy the Laplace equation.
Generalized Clifford algebras are constructed by various methods and have some important appli-
cations [2 – 8]. We introduce a new type of generalized Clifford algebra \scrA n(2ki, \alpha ij). It is generated
by the imaginary elements e1, e2, . . . , en with the following multiplication rules:
c\bigcirc DOAN CONG DINH, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11 1483
1484 DOAN CONG DINH
e2kii =
2ki - 1\sum
j=0
\alpha ije
j
i , i = 1, 2, . . . , n,
eiej + ejei = 0, i \not = j,
where k1, k2, . . . , kn \in \BbbN \ast , \alpha ij \in \BbbR .
A basis of \scrA n(2ki, \alpha ij) is \scrB =
\bigl\{
e\lambda 1
1 e\lambda 2
2 . . . e\lambda n
n | 0 \leq \lambda i \leq 2ki - 1, 1 \leq i \leq n
\bigr\}
. With
\lambda = (\lambda 1, \lambda 2, . . . , \lambda n), denote e\lambda = e\lambda 1
1 e\lambda 2
2 . . . e\lambda n
n . An element w \in \scrA n(2ki, \alpha ij) is in the form
w =
\sum
e\lambda \in \scrB
w\lambda e\lambda , w\lambda \in \BbbR .
The Cauchy – Riemann operator D is defined by D =
\partial
\partial x0
+
\sum n
i=1
ei
\partial
\partial xi
.
In this paper, we introduce some generalized Clifford algebras such that all components of a
monogenic function are solutions of an elliptic partial differential equation. One of our aims is to
cover more partial differential equations in framework of Clifford analysis. This ideal is in [9], there
various partial differential equations are treated. There is another approach to this problem. Solu-
tions of some partial differential equations by means of monogenic functions given in commutative
algebras are investigated in [10 – 13]. In [12, 13], a theory of monogenic functions is developed
in finite-dimensional commutative algebras. In particular, integral theorems for these functions are
proved and their relations with partial differential equations are studied.
In Section 2, we introduce \scrA 1(2m) and get the equation
\partial 2mf
\partial x2m0
+
\partial 2mf
\partial x2m1
= 0 in \BbbR 2. In Section 3,
we introduce \scrA \prime
n and get the equation
\biggl( \sum n - 1
i=0
\partial 2
\partial x2i
\biggr) 2
f +
\partial 4f
\partial x4n
= 0. In Section 4, we introduce
\scrA \prime \prime
n and get the biharmonic equation in \BbbR n+1. We shall prove some Cauchy integral representation
formulae for monogenic functions in those cases.
2. Hypercomplex algebra \bfscrA 1(\bftwo \bfitm ). Instead of the equality i2 = - 1 in complex numbers,
we give the rule e2m = - 1 for a new imaginary element e. It generates a hypercomplex algebra
\scrA 1(2m), m \in \BbbN \ast . A basis of \scrA 1(2m) is
\scrB =
\bigl\{
1, e, e2, . . . , e2m - 1
\bigr\}
.
An element w \in \scrA 1(2m) has the form w =
\sum 2m - 1
k=0
wke
k, wk \in \BbbR . Define a norm \| w\| =
=
\sqrt{}
w2
0 + w2
1 + . . .+ w2
2m - 1.
In [14] the algebra with the basis \scrB 1 = \{ 1, e, e2\} for which e3 = - 1 is considered, and an
algebra isomorphic to this algebra is considered in [15].
The Cauchy – Riemann operator is D =
\partial
\partial x0
+ e
\partial
\partial x1
. The adjoin Cauchy – Riemann is defined
by
D =
\biggl(
\partial
\partial x0
- e
\partial
\partial x1
\biggr) \biggl(
\partial 2m - 2
\partial x2m - 2
0
+ e2
\partial 2m - 2
\partial x2m - 4
0 \partial x21
+ . . .+ e2m - 2 \partial 2m - 2
\partial x2m - 2
1
\biggr)
.
We have DD = DD =
\partial 2m
\partial x2m0
+
\partial 2m
\partial x2m1
.
We consider a special function z = ex0 - x1. It satisfies the equation Dz = 0. Then
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
MONOGENIC FUNCTIONS TAKING VALUES IN GENERALIZED CLIFFORD ALGEBRAS 1485
1
ex0 - x1
= - x1 + ex0
x21 - e2x20
= -
(x1 + ex0)
\bigl(
x2m - 2
1 + e2x2m - 4
1 x20 + . . .+ e2m - 2x2m - 2
0
\bigr)
x2m0 + x2m1
with x20 + x21 > 0.
The function
1
ex0 - x1
is considered as a Cauchy kernel of the operator D.
Lemma 2.1. Let \Omega be a domain in \BbbR 2 and \Omega 1 be a bounded domain with \scrC 1-boundary, \Omega 1 \subset \Omega .
Suppose that f \in \scrC 1(\Omega ,\scrA 1(2m)), then\oint
\partial \Omega 1
f(x0, x1)dz = -
\int \int
\Omega 1
\biggl(
\partial f
\partial x0
+ e
\partial f
\partial x1
\biggr)
dx0dx1 = -
\int \int
\Omega 1
Dfdx0dx1.
Lemma 2.2.
I =
\oint
x2
0+x2
1=\epsilon 2
edx0 - dx1
ex0 - x1
=
2\pi
m
m - 1\sum
k=0
1
\mathrm{s}\mathrm{i}\mathrm{n} (2k+1)\pi
2m
e2k+1 \forall \epsilon > 0,
I - 1 =
- 1
2m\pi
m - 1\sum
k=0
1
\mathrm{s}\mathrm{i}\mathrm{n}
(2k + 1)\pi
2m
e2k+1.
Proof. We have
I =
\oint
x2
0+x2
1=\epsilon 2
edx0 - dx1
ex0 - x1
=
=
\oint
x2
0+x2
1=\epsilon 2
(dx1 - edx0)(x1 + ex0)
\bigl(
x2m - 2
1 + e2x2m - 4
1 x20 + . . .+ e2m - 2x2m - 2
0
\bigr)
x2m0 + x2m1
,
x = \epsilon \mathrm{c}\mathrm{o}\mathrm{s} t, y = \epsilon \mathrm{s}\mathrm{i}\mathrm{n} t,
I =
\pi \int
- \pi
(\mathrm{c}\mathrm{o}\mathrm{s} t+ e \mathrm{s}\mathrm{i}\mathrm{n} t)(\mathrm{s}\mathrm{i}\mathrm{n} t+ e \mathrm{c}\mathrm{o}\mathrm{s} t)
\sum m - 1
k=0
e2k \mathrm{s}\mathrm{i}\mathrm{n}2m - 2k - 2 t \mathrm{c}\mathrm{o}\mathrm{s}2k t
\mathrm{c}\mathrm{o}\mathrm{s}2m t+ \mathrm{s}\mathrm{i}\mathrm{n}2m t
dt =
= e
\pi \int
- \pi
\sum m - 1
k=0
e2k \mathrm{s}\mathrm{i}\mathrm{n}2m - 2k - 2 t \mathrm{c}\mathrm{o}\mathrm{s}2k t
\mathrm{c}\mathrm{o}\mathrm{s}2m t+ \mathrm{s}\mathrm{i}\mathrm{n}2m t
dt =
= 4e
m - 1\sum
k=0
e2k
\pi
2\int
0
\mathrm{s}\mathrm{i}\mathrm{n}2m - 2k - 2 t \mathrm{c}\mathrm{o}\mathrm{s}2k t
\mathrm{c}\mathrm{o}\mathrm{s}2m t+ \mathrm{s}\mathrm{i}\mathrm{n}2m t
dt =
= 4e
m - 1\sum
k=0
e2k
+\infty \int
0
u2m - 2k - 2du
u2m + 1
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1486 DOAN CONG DINH
=
2e
m
m - 1\sum
k=0
e2k
+\infty \int
0
v
2m - 2k - 1
2m
- 1dv
v + 1
=
=
2e
m
m - 1\sum
k=0
B
\biggl(
2m - 2k - 1
2m
,
2k + 1
2m
\biggr)
e2k =
=
2\pi
m
m - 1\sum
k=0
1
\mathrm{s}\mathrm{i}\mathrm{n}
(2k + 1)\pi
2m
e2k+1,
where B( , ) is the Beta function.
Denote a polynomial P (x) =
\sum m - 1
k=0
1
\mathrm{s}\mathrm{i}\mathrm{n}
(2k + 1)\pi
2m
x2k+1. We obtain
P
\biggl[
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
\pi
2m
+ k
2\pi
2m
\biggr)
+ i \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
\pi
2m
+ k
2\pi
2m
\biggr) \biggr]
= mi (\forall k = 0, 1, . . . , 2m - 1) \Rightarrow
\Rightarrow [P (x)]2 +m2 = (x2m + 1)Q(x) (for some polynomial Q(x) of order 2m - 2) \Rightarrow
\Rightarrow [P (e)]2 = - m2,
I =
2\pi
m
P (e) \Rightarrow I - 1 = - P (e)
2m\pi
=
- 1
2m\pi
m - 1\sum
k=0
1
\mathrm{s}\mathrm{i}\mathrm{n}
(2k + 1)\pi
2m
e2k+1.
Example 2.1. We get
m = 1 \Rightarrow I - 1 =
1
2\pi i
,
m = 2 \Rightarrow I - 1 =
-
\surd
2
4\pi
\bigl(
e+ e3
\bigr)
,
m = 3 \Rightarrow I - 1 =
- 1
6\pi
\bigl(
2e+ e3 + 2e5
\bigr)
,
m = 4 \Rightarrow I - 1 =
- 1
8\pi
\biggl[ \sqrt{}
4 + 2
\surd
2(e+ e7) +
\sqrt{}
4 - 2
\surd
2(e3 + e5)
\biggr]
,
m = 5 \Rightarrow I - 1 =
- 1
10\pi
\Bigl[
(
\surd
5 + 1)(e+ e9) + e5 + (
\surd
5 - 1)(e3 + e7)
\Bigr]
.
Apply Lemmas 2.1 and 2.2 we get the following Cauchy integral representation formula for
monogenic functions taking value in \scrA 1(2m).
Theorem 2.1. Let \Omega be an open domain in \BbbR 2 and \Omega 1 be a bounded domain with \scrC 1-boundary,
\Omega 1 \subset \Omega . Suppose that f \in \scrC 1(\Omega ,\scrA 1(2m)) and Df = 0 in \Omega , then
f(y) =
- 1
2m\pi
m - 1\sum
k=0
e2k+1
\mathrm{s}\mathrm{i}\mathrm{n}
(2k + 1)\pi
2m
\oint
\partial \Omega 1
f(x)(edx0 - dx1)
e(x0 - y0) - (x1 - y1)
\forall y \in \Omega 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
MONOGENIC FUNCTIONS TAKING VALUES IN GENERALIZED CLIFFORD ALGEBRAS 1487
3. Generalized Clifford algebra \bfscrA \prime
\bfitn . We consider the generalized Clifford algebra \scrA \prime
n which
is generated by n imaginary elements e1, e2, . . . , en with the following multiplication rules:
e2j = - 1, j = 1, 2, . . . , n - 1,
e4n = - 1,
eiej + ejei = 0, i \not = j.
A basis of \scrA \prime
n is \scrB =
\bigl\{
ei11 e
i1
2 . . . einn | ik \in \{ 0, 1\} , 1 \leq k \leq n - 1, in \in \{ 0, 1, 2, 3\}
\bigr\}
. The dimension
of \scrA n is 2n+1. With \lambda = (\lambda 1, \lambda 2, . . . , \lambda n) denote e\lambda = e\lambda 1
1 e\lambda 2
2 . . . e\lambda n
n , an element w \in \scrA \prime
n has the
form w =
\sum
e\lambda \in \scrB
w\lambda e\lambda , w\lambda \in \BbbR .
The generalized Cauchy – Riemann operator and its adjoin are given by
D =
\partial
\partial x0
+
n\sum
i=1
ei
\partial
\partial xi
,
D =
\Biggl(
\partial
\partial x0
-
n\sum
i=1
ei
\partial
\partial xi
\Biggr) \Biggl(
n - 1\sum
i=0
\partial 2
\partial x2i
+ e2n
\partial 2
\partial x2n
\Biggr)
,
DD = DD =
\Biggl(
n - 1\sum
i=0
\partial 2
\partial x2i
\Biggr) 2
+
\partial 4
\partial x4n
.
Example 3.1. Matrix representation of \scrA \prime
2 :
E1 =
\left[
0 - 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 - 1 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 - 1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 - 1
0 0 0 0 0 0 1 0
\right]
,
E2 =
\left[
0 0 0 0 0 0 - 1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 - 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 - 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 - 1 0 0
\right]
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1488 DOAN CONG DINH
We have E2
1 = - I, E4
2 = - I, E1E2 + E2E1 = 0, \scrA \prime
2 can be considered as a subalgebra of the
space of matrices M8\times 8, \scrA \prime
2 is generated by E1, E2.
4. Generalized Clifford algebra \bfscrA \prime \prime
\bfitn . We change the multiplication rules of the imaginary
elements in \scrA \prime \prime
n as the following:
e2j = - 1, j = 1, 2, . . . , n - 1,
e4n = - 2e2n - 1,
eiej + ejei = 0, i \not = j.
The generalized Cauchy – Riemann operator and its adjoin are given by
D =
\partial
\partial x0
+
n\sum
i=1
ei
\partial
\partial xi
,
D =
\Biggl(
\partial
\partial x0
-
n\sum
i=1
ei
\partial
\partial xi
\Biggr) \Biggl(
n\sum
i=0
\partial 2
\partial x2i
+ (e2n + 1)
\partial 2
\partial x2n
\Biggr)
,
DD = DD =
\Biggl(
n\sum
i=0
\partial 2
\partial x2i
- (e2n + 1)
\partial 2
\partial x2n
\Biggr) \Biggl(
n\sum
i=0
\partial 2
\partial x2i
+ (e2n + 1)
\partial 2
\partial x2n
\Biggr)
=
=
\Biggl(
n\sum
i=0
\partial 2
\partial x2i
\Biggr) 2
- (e2n + 1)2
\partial 4
\partial x4n
=
\Biggl(
n\sum
i=0
\partial 2
\partial x2i
\Biggr) 2
= \Delta 2.
Example 4.1. Matrix representation of \scrA \prime \prime
2 :
E1 =
\left[
0 - 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 - 1 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 - 1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 - 1
0 0 0 0 0 0 1 0
\right]
,
E2 =
\left[
0 0 0 0 0 0 - 1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 - 1 0 0 0 0 0 0
0 0 1 0 0 0 - 2 0
0 0 0 - 1 0 0 0 2
0 0 0 0 1 0 0 0
0 0 0 0 0 - 1 0 0
\right]
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
MONOGENIC FUNCTIONS TAKING VALUES IN GENERALIZED CLIFFORD ALGEBRAS 1489
We have E2
1 = - I, E4
2 = - 2E2
2 - I, E1E2 + E2E1 = 0. \scrA \prime \prime
2 can be considered as a subalgebra of
the space of matrices M8\times 8, \scrA \prime \prime
2 is generated by E1, E2.
Definition 4.1. Define a Cauchy kernel of the operator D by
E(x, y) = DH(x, y),
where H(x, y) is the fundamental solution of the biharmonic equation in \BbbR n+1.
In the following lemma we consider the case n \geq 4:
H(x, y) =
1
2(n - 1)(n - 3)\omega n| x - y| n - 3
,
where \omega n is the surface area of the unit sphere in \BbbR n+1.
Lemma 4.1.
I =
\int
| x - y| =\epsilon
E(x, y)\bfn (x)dS(x) = 1 \forall \epsilon > 0,
where \bfn (x) = \nu 0 + e1\nu 1 + . . . + en\nu n,
- \rightarrow n = (\nu 0, \nu 1, . . . , \nu n) is the outer unit normal vector of the
boundary.
Proof. We have
\partial H
\partial xi
=
- (xi - yi)
2(n - 1)\omega n| x - y| n - 1
,
\partial 2H
\partial x2i
=
(xi - yi)
2
2\omega n| x - y| n+1
- 1
2(n - 1)\omega n| x - y| n - 1
,
\Phi (x, y) = \Delta H + (e2n + 1)
\partial 2H
\partial x2n
=
- (e2n + 3)
2(n - 1)\omega n| x - y| n - 1
+ (e2n + 1)
(xn - yn)
2
2\omega n| x - y| n+1
,
\int
| x - y| =\epsilon
E(x, y)\bfn (x)dS(x) =
\int
| x - y| =\epsilon
\Biggl(
\partial
\partial x0
-
n\sum
i=1
ei
\partial
\partial xi
\Biggr)
\Phi (x, y)\bfn (x)dS(x) =
=
\int
| x - y| =\epsilon
\Biggl(
\nu 0
\partial \Phi
\partial x0
-
n\sum
i=1
e2i \nu i
\partial \Phi
\partial xi
\Biggr)
dS(x) =
=
\int
| x - y| =\epsilon
\Biggl[ \Biggl(
\nu 0
\partial \Phi
\partial x0
+
n\sum
i=1
\nu i
\partial \Phi
\partial xi
\Biggr)
- (e2n + 1)\nu n
\partial \Phi
\partial xn
\Biggr]
dS(x),
\int
| x - y| =\epsilon
\Biggl(
\nu 0
\partial
\partial x0
+
n\sum
i=1
\nu i
\partial
\partial xi
\Biggr)
1
| x - y| n - 1
dS(x) = - (n - 1)\omega n,
\int
| x - y| =\epsilon
\Biggl(
\nu 0
\partial
\partial x0
+
n\sum
i=1
\nu i
\partial
\partial xi
\Biggr)
(xn - yn)
2
| x - y| n+1
dS(x) =
- (n - 1)
n+ 1
\omega n,
(e2n + 1)
\int
| x - y| =\epsilon
\nu n
\partial \Phi
\partial xn
dS(x) =
e2n + 1
n+ 1
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1490 DOAN CONG DINH
I =
e2n + 3
2
- (e2n + 1)
(n - 1)
2(n+ 1)
- e2n + 1
n+ 1
= 1.
Remark 4.1. The fundamental solutions of the biharmonic equation in the cases n = 2, n = 3
are given by
H(x, y) =
- | x - y|
8\pi
, n = 2,
H(x, y) =
- \mathrm{l}\mathrm{n} | x - y|
8\pi 2
, n = 3.
In these cases, Lemma 4.1 can be proved similarly.
Lemma 4.2. Let \Omega be a domain in \BbbR n+1, n \geq 4, and \Omega 1 be a bounded domain with \scrC 1-
boundary, \Omega 1 \subset \Omega . Suppose that f, g \in \scrC 1(\Omega ,\scrA \prime \prime
n), then\int
\partial \Omega 1
g(x)\bfn (x)f(x)dS(x) =
\int
\Omega 1
\bigl[
g(x)D.f(x) + g(x).Df(x)
\bigr]
dx.
Apply Lemmas 4.1 and 4.2 we get the following Cauchy integral representation formula for
monogenic functions taking value in \scrA \prime \prime
n.
Theorem 4.1. Let \Omega be a domain in \BbbR n+1 and \Omega 1 be a bounded domain with \scrC 1-boundary,
\Omega 1 \subset \Omega . Suppose that f \in \scrC 1(\Omega ,\scrA \prime \prime
n) and Df = 0 in \Omega , then
f(y) =
\int
\partial \Omega 1
E(x, y)\bfn (x)f(x)dS(x) \forall y \in \Omega 1.
With the similar purpose, the biharmonic equation is investigated in Clifford analysis [16] and in
the theory of monogenic functions talking values in commutative algebras [17 – 19].
An open problem. There is an open question: Consider a given partial differential equation.
Could we find a suitable generalized Clifford algebra such that all components of a monogenic
function are solutions of the given partial differential equation? For example, we can answer this
question in a simple case:
Let
\partial 2mf
\partial x2m0
-
2m - 1\sum
k=0
ak
\partial 2mf
\partial xk0\partial x
2m - k
1
= 0
be an elliptic equation in \BbbR 2, ak \in \BbbR . The imaginary element e must obey the multiplication rule
e2m =
\sum 2m - 1
k=0
( - 1)kake
k.
References
1. F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Res. Notes Math., vol. 76, Pitman, Boston, MA (1982).
2. A. O. Morris, On a generalized Clifford algebra, Quart. J. Math., 18, № 1, 7 – 12 (1967); DOI: https://doi.org/10.1093/
qmath/18.1.7.
3. W. Tutschke, An elementary approach to Clifford analysis, Functional Analytic Methods in Complex Analysis and
Applications to Rartial Differential Equations, 402 – 408 (1995).
4. W. Tutschke, C. J. Vanegas, Clifford algebras depending on parameters and their applications to partial differential
equations, Some Topics on Value Distribution and Differentiability in Complex and p-Adic Analysis, Sci. Press
(2008).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
MONOGENIC FUNCTIONS TAKING VALUES IN GENERALIZED CLIFFORD ALGEBRAS 1491
5. W. Tutschke, C. J. Vanegas, A boundary value problem for monogenic functions in parameter-depending Cli-
fford algebras, Complex Var. and Elliptic Equat., 56, № 1 – 4, 113 – 118 (2011); DOI: https://doi.org/10.1080/
17476930903394762.
6. A. R. M. Granik, On a new basis for a generalized Clifford algebra and its application to quantum mechanics,
Clifford Algebras with Numeric and Symbolic Computations, Birkhäuser, Boston, MA (1996).
7. S. Barry, Generalized Clifford algebras and their representations, Clifford Algebras and their Applications in
Mathematical Physics, 133 – 141 (2011).
8. R. Jagannathan, On generalized Clifford algebras and their physical applications, The Legacy of Alladi Ramakrishnan
in the Mathematical Sciences, Springer, New York (2010), p. 465 – 489.
9. E. Obolashvili, Higher order partial differential equations in Clifford analysis, Progr. Math. Phys., vol. 28, Birkhäuser,
Basel (2002).
10. M. N. Rosculet, Functii monogene pe algebre comutative, Acad. Rep. Soc. Romania, Bucuresti (1975).
11. S. A. Plaksa, Commutative algebras associated with classic equations of mathematical physics, Adv. Appl. Anal.,
Springer, Basel (2012), p. 177 – 223.
12. V. S. Shpakivskyi, Constructive description of monogenic functions in a finite-dimensional commutative associative
algebra, Adv. Pure and Appl. Math., 7, № 1, 63 – 75 (2016).
13. V. S. Shpakivskyi, Curvilinear integral theorems for monogenic functions in commutative associative algebras, Adv.
Appl. Clifford Algebras, 26, № 1, 417 – 434 (2016); DOI: https://doi.org/10.1007/s00006-015-0561-x.
14. D. Alpay, A. Vajiac, M. B. Vajiac, Gleason’s problem associated to a real ternary algebra and applications, Adv.
Appl. Clifford Algebras, 28, № 2 (2018); DOI: https://doi.org/10.1007/s00006-018-0857-8.
15. M. N. Rosculet, O teorie a functiilor de o variabila hipercomplexa in spaţiul cu trei dimensiuni, Stud. Cerc. Mat., 5,
№ 3-4, 361 – 401 (1954).
16. K. Gur̈lebeck, U. Kähler, On a boundary value problem of the biharmonic equation, Math. Methods Appl. Sci., 20,
№ 10, 867 – 883 (1997).
17. L. Sobrero, Nuovo metodo per lo studio dei problemi di elasticita, con applicazione al problema della piastra forata,
Ric. Ingegn., 13, № 2, 255 – 264 (1934).
18. V. F. Kovalev, I. P. Melnichenko, Biharmonic functions on biharmonic plane, Dopov. Akad. Nauk Ukr. Ser A, 8,
25 – 27 (1981).
19. S. V. Gryshchuk, S. A. Plaksa, Monogenic functions in the biharmonic boundary value problem, Math. Methods
Appl. Sci., 38, № 11, 2939 – 2952 (2016).
Received 07.08.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
|
| id | umjimathkievua-article-1033 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:21Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f3/3590366589e1a5dc343c14e49b36c9f3.pdf |
| spelling | umjimathkievua-article-10332025-03-31T08:46:33Z Monogenic functions taking values in generalized Clifford algebras Monogenic functions taking values in generalized Clifford algebras Dinh, Doan Cong Doan Cong Dinh, Doan Cong Generalized Clifford algebras ; Generalized Clifford analysis Monogenic functions Cauchy integral representation formula Generalized Clifford algebras ; Generalized Clifford analysis Monogenic functions Cauchy integral representation formula UDC 512.579Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics.In this paper we introduce a new type of generalized Clifford algebra such that all components of a monogenic functionare solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations inframework of Clifford analysis. We shall prove some Cauchy integral representation formulae for monogenic functions inthose cases. UDC 512.579 Моногеннi функцiї зi значеннямив узагальнених алгебрах Клiффорда Узагальненi алгебри Клiффорда будуються рiзними методами i мають певнi застосування в математицi та фiзицi. У цiй роботi вводиться новий тип узагальненої алгебри Клiффорда, такий, що всi компоненти моногенної функцiї є розв’язками елiптичного диференцiального рiвняння з частинними похiдними. Однiєю з цiлей є охоплення бiльш широкого класу диференцiальних рiвнянь з частинними похiдними в рамках аналiзу Клiффорда. У вiдповiдних випадках доведено деякi формули iнтегрального представлення Кошi для моногенних функцiй. Institute of Mathematics, NAS of Ukraine 2021-11-23 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1033 10.37863/umzh.v73i11.1033 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 11 (2021); 1483 - 1491 Український математичний журнал; Том 73 № 11 (2021); 1483 - 1491 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1033/9146 Copyright (c) 2021 C. D. Dinh |
| spellingShingle | Dinh, Doan Cong Doan Cong Dinh, Doan Cong Monogenic functions taking values in generalized Clifford algebras |
| title | Monogenic functions taking values in generalized Clifford algebras |
| title_alt | Monogenic functions taking values in generalized Clifford algebras |
| title_full | Monogenic functions taking values in generalized Clifford algebras |
| title_fullStr | Monogenic functions taking values in generalized Clifford algebras |
| title_full_unstemmed | Monogenic functions taking values in generalized Clifford algebras |
| title_short | Monogenic functions taking values in generalized Clifford algebras |
| title_sort | monogenic functions taking values in generalized clifford algebras |
| topic_facet | Generalized Clifford algebras Generalized Clifford analysis Monogenic functions Cauchy integral representation formula Generalized Clifford algebras Generalized Clifford analysis Monogenic functions Cauchy integral representation formula |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1033 |
| work_keys_str_mv | AT dinhdoancong monogenicfunctionstakingvaluesingeneralizedcliffordalgebras AT doancong monogenicfunctionstakingvaluesingeneralizedcliffordalgebras AT dinhdoancong monogenicfunctionstakingvaluesingeneralizedcliffordalgebras |