General Mathematics Vol. 13, No. 3 (2005), 81–98
Integral Representations and its
Applications in Clifford Analysis
Zhang Zhongxiang
Dedicated to Professor Dumitru Acu on his 60th anniversary
Abstract
In this paper, we mainly study the integral representations for
functions f with values in a universal Clifford algebra C(Vn,n ), where
f ∈ Λ(f, Ω),
Λ(f, Ω) = f |f ∈ C ∞ (Ω, C(Vn,n )), max Dj f (x) =
x∈Ω
= O(M j )(j → +∞), for some M, 0 < M < +∞ .
The integral representations of Ti f are also given. Some properties
of Ti f and Πf are shown. As applications of the higher order Pompeiu formula, we get the solutions of the Dirichlet problem and the
inhomogeneous equations Dk u = f .
2000 Mathematics Subject Classification: 30G35, 45J05.
Keywords: Universal Clifford algebra, Integral representation,
Ti −operator, Π−operator.
81
82
1
Zhang Zhongxiang
Introduction and Preliminaries
Integral representation formulas of Cauchy-Pompeiu type expressing complex valued, quaternionic and Clifford algebra valued functions have been
well developed in [1-9, 12-19, 21, 24, 25 etc.]. These integral representation formulas serve to solve boundary value problems for partial differential
equations. In [2, 3], H. Begehr gave the different integral representation
formulas for functions with values in a Clifford algebra C(Vn,0 ), the integral
operators provide particular weak solutions to the inhomogeneous equations
∂ k ω = f , 4k ω = g and ∂4k ω = h. In [5, 24], the higher order CauchyPompeiu formulas for functions with values in a universal Clifford algebra
C(Vn,n ) are obtained. In [16], G.N. Hile gave the detailed properties of the
..
T -operator by following the techniques of Vekua. In [14, 15], K. Gurlebeck
..
gave many properties of the Π-operator. In [18], H. Malonek and B. Muller
→
gave some properties of the vectorial integral operator Π. In [7, 19, 21],
the integral representations related with the Helmholtz operator are given,
the weak solutions of the inhomogeneous equations Lk u = f and Lk∗ u = f ,
n
P
hi ei ,
k ≥ 1, are obtained, where Lu = Du + uh and L∗ u = uD − hu, h =
i=1
D is the Dirac operator. In this paper, we shall continue to study the properties of Cauchy-Pompeiu operator, higher order Cauchy-Pompeiu operator
and Π operator for f ∈ Λ(f, Ω), where
Λ(f, Ω) = f |f ∈ C ∞ (Ω, C(Vn,n )), max Dj f (x) =
x∈Ω
= O(M )(j → +∞), for some M, 0 < M < +∞ ,
j
the integral representations of Ti f are given, some properties of Ti f and Πf
are shown. As applications, we get the solutions of the Dirichlet problem
Integral Representations and its Applications in Clifford Analysis
83
and the inhomogeneous equations Dk u = f which are not in weak sense as
in [2, 25].
Let Vn,s (0 ≤ s ≤ n) be an n–dimensional (n ≥ 1) real linear space with
basis {e1 , e2 , · · · , en }, C(Vn,s ) be the 2n –dimensional real linear space with
basis
{eA , A = {h1 , · · · , hr } ∈ PN, 1 ≤ h1 < · · · < hr ≤ n} ,
where N stands for the set {1, · · · , n} and PN denotes the family of all
order-preserving subsets of N in the above way. We denote e∅ as e0 and eA
as eh1 ···hr for A = {h1 , · · · , hr } ∈ PN . The product on C(Vn,s ) is defined by
(1)

#((A∩B)\S)

(−1)P (A,B) eA4B , if A, B ∈ PN,

 eA eB = (−1)
P P
P
P


λA µB eA eB ,
if λ =
λA eA , µ =
µB eB .
 λµ =
A∈PN B∈PN
A∈PN
B∈PN
where S stands for the set {1, · · · , s}, #(A) is the cardinal number of the
P
set A, the number P (A, B) =
P (A, j), P (A, j) = #{i, i ∈ A, i > j}, the
j∈B
symmetric difference set A4B is also order-preserving in the above way, and
λA ∈ R is the coefficient of the eA –component of the Clifford number λ. We
also denote λA as [λ]A , for abbreviaty, we denote λ{i} as [λ]i . It follows at
once from the multiplication rule (1) that e0 is the identity element written
now as 1 and in particular,


e2i = 1,








 e2j = −1,
(2)



ei ej = −ej ei ,







eh1 eh2 · · · ehr = eh1 h2 ···hr ,
if i = 1, · · · , s,
if j = s + 1, · · · , n,
if 1 ≤ i < j ≤ n,
if 1 ≤ h1 < h2 · · · , < hr ≤ n.
84
Zhang Zhongxiang
Thus C(Vn,s ) is a real linear, associative, but non-commutative algebra
and it is called the universal Clifford algebra over Vn,s .
Frequent use will be made of the notation Rnz where z ∈ Rn , which
means to remove z from Rn . In particular Rn0 = Rn \ {0}.
Let Ω be an open non empty subset of Rn , since we shall only consider
the case of s = n in this paper, we shall only consider the operator D which
is written as
D=
n
X
k=1
ek
∂
: C (r) (Ω, C(Vn,n )) → C (r−1) (Ω, C(Vn,n )).
∂xk
Let f be a function with value in C(Vn,n ) defined in Ω, the operator D acts
on the function f from the left and from the right being governed by the
rule
D[f ] =
n X
X
k=1
A
n
XX
∂fA
∂fA
, [f ]D =
eA ek
,
ek eA
∂xk
∂xk
k=1 A
An involution is defined by

T

σ(A)+#(A S)

eA ,
 eA = (−1)
(3)
P

λ
=
λA eA ,


A∈PN
if A ∈ PN,
if λ =
P
A∈PN
λA eA ,
where σ(A) = #(A)(#(A) + 1)/2. From (1) and (3), we have


ei = ei ,
if i = 0, 1, · · · , s,




(4)
ej = −ej , if j = s + 1, · · · , n,





λµ = µλ, for any λ, µ ∈ C(Vn,s ).
The C (Vn.n )–valued (n − 1)–differential form
n
X
dσ =
(−1)k−1 ek db
xNk
k=1
Integral Representations and its Applications in Clifford Analysis
85
is exact, where
db
xNk = dx1 ∧ · · · ∧ dxk−1 ∧ dxk+1 ∧ · · · ∧ dxn .
2
Integral Representations
In this section, we shall give the integral representations for f and Ti f ,
i ≥ 1, f ∈ Λ(f, Ω), where
Λ(f, Ω) = f |f ∈ C ∞ (Ω, C(Vn,n )), max Dj f (x) =
x∈Ω
= O(M j )(j → +∞), for some M, 0 < M < +∞ .
In [5], [24] the kernel functions
(5)

A j xj


,


ωn ρn (x)





A j xj



 ωn ρn (x) ,
∗
Hj (x) =
Aj−1


log(x2 ),


2ωn


!


l−1

X

A
C
n−1
i+1,0


Cl,0 xl log(x2 ) − 2
,

2ωn
Ci,0
i=0
are constructed for any j ≥ 1, where x =
n
P
k=1
denotes the area of the unit sphere in Rn , and
1
(6) Aj =
2[
j−1
]
2
[ j−1
]!
2
[ 2j ]
Q
n is odd;
1 ≤ j < n, n is even;
j = n, n is even;
j = n + l, l > 0, n is even;
xk ek , ρ(x) =
n
P
k=1
x2k
12
, ωn
, 1 ≤ j < n(n is even), j ∈ N ∗ (n is odd),
(2r − n)
r=1
86
Zhang Zhongxiang



1,




(7)
Cj,0 =
j = 0,
1
,
j−1

[ 2 ]


Q
j

[ ] j

(n + 2µ)
 2 2 [ 2 ]!
j ∈ N∗ = N\{0}.
µ=0
Lemma 1.(Higher order Cauchy-Pompeiu formula) (see [24])Suppose
that M is an n–dimensional differentiable compact oriented manifold contained in some open non empty subset Ω ⊂ Rn , f ∈ C (r) (Ω, C (Vn,n )),
r ≥ k, moreover ∂M is given the induced orientation, for each j = 1, · · · , k,
◦
Hj∗ (x) is as above. Then, for z ∈M
(8)
Z
Z
k−1
X
j
∗
j
k
f (z) = (−1) Hj+1 (x − z)dσx D f (x) + (−1) Hk∗ (x − z)Dk f (x)dxN .
j=0
M
∂M
In the following, Ω is supposed to be an open non empty subset of Rn
with a Liapunov boundary ∂Ω. Denote
Z
(9)
Ti f (z) = (−1)
i
Hi∗ (x − z)f (x)dxN
Ω
where Hi∗ (x) is denoted as in (5), i ∈ N ∗ , f ∈ Lp (Ω, C(Vn,n )), p ≥ 1. The
operator T1 is the Pompeiu operator T . Especially, we denote f as T0 f .
In [25], it is shown that, if f ∈ Lp (Ω, C(Vn,n )), p ≥ 1, then T f ∈
p−n
C α (Ω, C(Vn,n )), α =
. Tk f provides a particular weak solution to the
p
inhomogeneous equation Dk ω = f (weak) in Ω. In this section, we shall
show that, if f ∈ Λ(f, Ω), then Ti f ∈ C ∞ (Ω, C(Vn,n )), i ∈ N∗ and Tk f
provides a particular solution to the inhomogeneous equation Dk ω = f in
Ω.
Integral Representations and its Applications in Clifford Analysis
87
Theorem 1.Let Ω be an open non empty bounded subset of Rn with a Liapunov boundary ∂Ω, f ∈ Λ(f, Ω). Then, for z ∈ Ω
(10)
Ti f (z) =
∞
X
Z
(−1)
j+i
j=0
∗
Hj+i+1
(x − z)dσx Dj f (x), i ∈ N.
∂Ω
Proof. Step 1. For f ∈ Λ(f, Ω), we shall firstly prove
Ti f (z) =
k
P
Z
j+i
(−1)
j=0
(11)
i+k+1
+(−1)
∗
Hj+i+1
(x − z)dσx Dj f (x)
Z∂Ω
∗
Hi+k+1
(x − z)Dk+1 f (x)dxN ,
Ω
where i, k ∈ N, z ∈ Ω. It is obvious that (11) is the direct result of Lemma
1 for i = 0.
For i ≥ 1, in view of the properties of the kernel functions of Hj∗ (x − z)
(12)
∗
∗
D Hj+1
(x − z) = Hj+1
(x − z) D = Hj∗ (x − z), x ∈ Rnz , for any j ≥ 1.
Combining Stokes formulas with (12), we have
(13)
Z
i
(−1)
Hi∗ (x
− z)f (x)dx
N
=
k
P
Z
(−1)
j=0
Ω\B(z,ε)
∗
Hj+i+1
(x − z)dσx Dj f (x)
j+i
∂(Ω\B(z,ε))
Z
+(−1)i+k+1
∗
Hi+k+1
(x − z)Dk+1 f (x)dxN .
Ω\B(z,ε)
For i ≥ 1 and j ≥ 0, it is easy to check that,
Z
∗
(14)
lim
Hj+i+1
(x − z)dσx Dj f (x) = 0.
ε→0
∂B(z,ε)
88
Zhang Zhongxiang
In view of the weak singularity of the kernel functions and (14), taking
limits as ε → 0 in (13), (11) holds.
Step 2. For f ∈ Λ(f, Ω), we shall show that
Z
k+1
N
∗
(15)
lim max Hi+k+1 (x − z)D f (x)dx = 0.
k→∞ z∈Ω Ω
Since f ∈ Λ(f, Ω), then there exist constants C0 , M , 0 < C0 , M < +∞, and
N ∈ N∗ , such that for any k ≥ N
max Dk f (x) ≤ C0 M k .
(16)
x∈Ω
Case 1. n is odd. For any k ≥ N , we have
Z
∗
k+1
N
(17) Hi+k+1 (x − z)D f (x)dx ≤ 2n Ai+k+1 C0 V (Ω)M k+1 δ i+k+1−n ,
Ω
where δ = sup ρ(x1 − x2 ), V (Ω) denotes the volume of Ω. It is obvious
x1 ,x2 ∈Ω
that the series
(18)
∞
X
2n Ai+k+1 C0 V (Ω)M k+1 δ i+k+1−n
k=1
converges. Then
(19)
lim 2n Ai+k+1 C0 V (Ω)M k+1 δ i+k+1−n = 0,
k→∞
thus (15) holds.
Case 2. n is even. In view of (5) and (7), it can be similarly proved that
(15) holds.
Combining (11) with (15), taking limits k → ∞ in (11), (10) follows.
By Theorem 1, we have
Integral Representations and its Applications in Clifford Analysis
89
Corollary 1.Suppose that f is k-regular in a domain U in Rn , Ω is an
open non empty bounded subset of U with a Liapunov boundary ∂Ω. Then,
for z ∈ Ω
(20)
Ti f (z) =
k−1
X
Z
(−1)
j+i
j=0
∗
Hj+i+1
(x − z)dσx Dj f (x), i ∈ N.
∂Ω
Remark 1.For i = 0, (20) is exactly the higher order Cauchy integral formula which has been obtained in [5, 24]. Analogous higher order Cauchy
integral formula can be also found in [2, 3, 12].
Corollary 2.Let Ω be an open non empty bounded subset of Rn with a
Liapunov boundary ∂Ω, f ∈ Λ(f, Ω). Then, for z ∈ Ω
(21)
D[Ti+1 f ] = Ti f, i ∈ N.
Remark 2.Corollary 2 implies that Tk f provides a particular solution to
the inhomogeneous equation Dk ω = f in Ω for f ∈ Λ(f, Ω). Especially,
suppose U is a domain in Rn , Ω is an open non empty bounded subset of U
with a Liapunov boundary ∂Ω, f is regular in U , then Tk f is (k + 1)-regular
in Ω. This result gives an improved result in [2, 25] under the assumption
of f ∈ Λ(f, Ω).
Corollary 3.Let U be a domain in Rn , Ω be an open non empty bounded
subset of U with a Liapunov boundary ∂Ω, f be a solution of equation Lu = 0
n
P
in U , where Lu = Du + uh, h =
hi ei , hi ∈ R or h be a real (complex)
i=1
number. Then for z ∈ Ω
(22)
Ti f (z) =
∞
X
j=0
Z
(−1)
∗
(x − z)dσx Dj f (x), i ∈ N.
Hj+i+1
j+i
∂Ω
90
Zhang Zhongxiang
Proof. Obviously, if f is a solution of equation Lu = 0 in U , where Lu =
n
P
hi ei or h is a real (complex) number, then f ∈ Λ(f, Ω).
Du + uh, h =
i=1
By Theorem 1, the result follows.
∞ (αx e )k
P
4
i i
= eαxi ei , i = 1, · · · , n, where α
k!
k=0
is a real number. Clearly, Dui (x) = αui (x). Thus for ui (x), z ∈ Ω, by
Example 1.Suppose ui (x) =
Corollary 3, (22) holds.
Example 2.Suppose h =
n
P
r
hi ei , hi ∈ R. Denote R = |h| =
i=1
n
P
i=1
h2i .
Obviously, eRxi ei satisfies Du − Ru = 0, thus eRxi ei is also a solution of
the Helmholtz equation 4u − R2 u = 0. Then eRxi ei (R − h) is a solution of
equation Du + uh = 0. For eRxi ei (R − h), z ∈ Ω, by Corollary 3, (22) holds.
Ω is supposed to be an open non empty subset of Rn with a Liapunov
boundary ∂Ω. Denote
(23)
 Z


K(x − z)f (x)dxN , z ∈ Ω,



Ω Z
Πf (z) =


lim K(x − ξ)f (x)dxN z ∈ ∂Ω,


ξ→z
 ξ∈Ω
Ω
where
(24)
1
K(x) =
ωn
(2 − n)e1
nxe1 x
− n+2
n
ρ (x)
ρ (x)
, x ∈ Rn0 .
f ∈ H α (Ω, C(Vn,n )), 0 < α ≤ 1, Πf is a singular integral to be taken in the
..
Cauchy principal sense. In [25], we have proved the existence and Holder
continuity of Πf in Ω.
For u ∈ H α (∂Ω, C(Vn,n )), 0 < α ≤ 1, denote
Z
(25)
(F∂Ω u)(x) = H1∗ (y − x)dσy u(y), x ∈ Rn \ ∂Ω.
∂Ω
Integral Representations and its Applications in Clifford Analysis
91
Z
(26)
H1∗ (y − x)dσy u(y), x ∈ ∂Ω.
(S∂Ω u)(x) =
∂Ω
(27)
+
F∂Ω
u (x) =


 (F∂Ω u) (x),
x ∈ Ω+ ,

 1 u(x) + (S u) (x)
∂Ω
2
x ∈ ∂Ω.
Theorem 2.Let Ω be an open non empty bounded subset of Rn with a Liapunov boundary ∂Ω, f ∈ C 1 (Ω, C(Vn,n )), Πf is defined as in (23). Then
2−n
+
e1 f (z), z ∈ Ω,
(28) Πf (z) = F∂Ω
(αe1 αf ) (z) + T (e1 D [f ]) (z) −
n
where α(x) denotes the unit outer normal of ∂Ω.
Proof. For z ∈ Ω, by Stokes formula, we have,
Z
Πf (z) = lim
K(x − z)f (x)dxN
ε→0
Ω\B(z,ε)
Z
= lim
ε→0
Ω\B(z,ε)
Z
(29)
= lim
[H1∗ (x − z)e1 ] Df (x)dxN
ε→0
Z ∂(Ω\B(z,ε))
H1∗ (x − z)e1 dσx f (x) + T (e1 D [f ]) (z)
H1∗ (x − z)e1 dσx f (x) + T (e1 D [f ]) (z) −
=
2−n
e1 f (z).
n
∂Ω
For z ∈ ∂Ω, taking limits in (29), (28) follows.
Corollary 4.Let Ω be an open non empty bounded subset of Rn with a
Liapunov boundary ∂Ω, f ∈ Λ(f, Ω), Πf is defined as in (23). Then in Ω
(30)
D [Πf ] = e1 D [f ] +
n−2
D [e1 f ] .
n
92
Zhang Zhongxiang
Corollary 5.Suppose that f is regular in a domain U in Rn , Ω is an open
non empty bounded subset of U with a Liapunov boundary ∂Ω. Πf is defined
as in (23). Then in Ω
(31)
4 [Πf ] = 0,
where 4 is the Laplace operator.
3
Some applications
In this section, we shall give some applications of the higher order CauchyPompeiu formula. The solutions of Dirichlet problems as well as the inhomogeneous equations Dk u = f are obtained. In the sequel, Kn denotes the
unit ball in Rn (n ≥ 3), more clearly,
Kn = {x|x = (x1 , x2 , · · · , xn ) ∈ Rn , |x| < 1} .
Denote
(32)
G(y, x) =
1
1
−
, x ∈ Kn , y ∈ Kn , x 6= y.
ρn−2 (y − x) |y|n−2 ρn−2 ( y − x)
|y|2
Remark 3.G(y, x) has the following properties:
(1) 4x G(y, x) = 0, x ∈ Kn \ {y}.
(2) G(y, x) = G(x, y), x, y ∈ Kn , x 6= y.
(3) G(y, x) = 0, y ∈ ∂Kn , x ∈ Kn .
Theorem 3.Suppose f ∈ C 2 (Kn , C(Vn,n )), then for x ∈ Kn
Z
Z
1
1
1 − |x|2
(33) f (x) =
f (y)dSy +
G(y, x)4y f (y)dy N .
ωn
ρn (y − x)
(2 − n)ωn
∂Kn
Kn
Integral Representations and its Applications in Clifford Analysis
93
Proof. By Lemma 1, for x ∈ Kn , we have
(34)
Z
y−x
1
1
dσy f (y) −
dσy D[f ](y)
n
n−2
ρ (y − x)
(2 − n)ωn
ρ (y − x)
∂Kn
∂Kn
Z
1
1
N
+
4y f (y)dy .
(2 − n)ωn
ρn−2 (y − x)
1
f (x) =
ωn
Z
Kn
By Stokes formula, for x ∈ Kn and x 6= 0, we have
(35)
x
Z
1
1
1
|x|2
0 =
dσy f (y) −
x
x dσy D[f ](y)
ωn
(2 − n)ωn
n (y −
n−2 (y −
ρ
)
ρ
)
2
∂Kn
∂Kn
|x|2
Z |x|
1
1
N
+
x 4y f (y)dy .
(2 − n)ωn
n−2
ρ (y − 2 )
Kn
|x|
Z
y−
(35) can be rewritten as
x
Z |x| y − 2
1
|x|
0=
x dσy f (y)−
ωn
n ρn (y −
|x|
)
∂Kn
|x|2
Z
1
1
−
x dσy D[f ](y)+
(2 − n)ωn
n−2 ρn−2 (y −
|x|
)
∂Kn
|x|2
Z
1
1
+
4y f (y)dy N .
x
(2 − n)ωn
|x|n−2 ρn−2 (y − 2 )
Kn
|x|
2
(36)
In view of
(37)
|x|k ρk (y −
x
y
) = |y|k ρk ( 2 − x), k ∈ N∗ ,
2
|x|
|y|
combining (34), (36) with (37), (33) follows.
For x = 0, by Stokes formula and (34), (33) still holds. Thus the result
is proved.
94
Zhang Zhongxiang
Remark 4.Suppose f ∈ C 2 (Kn , C(Vn,n )), moreover, f is harmonic in Kn .
Then for x ∈ Kn
1
f (x) =
ωn
(38)
Z
1 − |x|2
f (y)dSy .
ρn (y − x)
∂Kn
(38) is exactly the Poisson expression of harmonic functions.
Theorem 4.The solution of the Dirichlet problem for the Poisson equation
in the unit ball Kn
4u = f in Kn , u = γ on ∂Kn ,
for f ∈ Λ(f, Kn ) and γ ∈ C(∂Kn , C(Vn,n )) is uniquely given by
Z
Z
1
1 − |x|2
1
(39) u(x) =
γ(y)dSy +
G(y, x)f (y)dy N .
n
ωn
ρ (y − x)
(2 − n)ωn
Kn
∂Kn
Proof. It can be directly proved by Corollary 2, Theorem 3, Remark 3 and
Remark 4.
Lemma 2.(see [26]) If f is k-regular in an open neighborhood Ω of the
◦
origin, then in a suitable open ball B (0, R) ⊂ Ω
(40)
f (xN ) = f (0) +
∞ X
k−1
X
X
Cj,p−j xj Vl1 ,··· ,lp−j (xN )Cl1 ,··· ,lp−j ,
p=1 j=0 (l1 ,··· ,lp−j )
Cj,p−j and Cl1 ,··· ,lp−j are constants which are suitably chosen.
By Lemma 2 and Corollary 2, we have
Theorem 5.The solutions of inhomogeneous equations in the unit ball Kn
Dk u = f in Kn ,
Integral Representations and its Applications in Clifford Analysis
95
◦
for f ∈ Λ(f, Kn ) are given in a suitable open ball B (0, R) ⊂ Kn by
(41)
u = C0 +
∞ X
k−1
X
X
Cj,p−j xj Vl1 ,··· ,lp−j (xN )Cl1 ,··· ,lp−j + Tk f.
p=1 j=0 (l1 ,··· ,lp−j )
C0 , Cj,p−j and Cl1 ,··· ,lp−j are constants which are suitably chosen.
Acknowledgements
This paper is done while the author is visiting at Free University, Berlin
during 2004 and 2005 supported by a DAAD K. C. Wong Fellowship. The
author would like to thank sincerely Prof. H. Begehr for his helpful suggestions and discussions.
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School of Mathematics and Statistics,
Wuhan University,
Wuhan 430072, P. R. China
E–mail: zhangzx9@sohu.com