Integral Representations and Its Applications in Clifford Analysis

Home , Clifford analysis

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),

∞ j Λ(f, Ω) = f|f ∈ C (Ω,C(Vn,n)), max D f(x) = x∈Ω

= O(M j)(j → +∞), for some M, 0 < M < +∞ .

The integral representations of Tif are also given. Some properties

of Tif and Πf are shown. As applications of the higher order Pom- peiu formula, we get the solutions of the Dirichlet problem and the inhomogeneous equations Dku = f.

2000 Mathematics Subject Classiﬁcation: 30G35, 45J05. Keywords: Universal Cliﬀord algebra, Integral representation,

Ti−operator, Π−operator.

81 82 Zhang Zhongxiang

1 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 ∂kω = f, 4kω = g and ∂4kω = h. In [5, 24], the higher order Cauchy- Pompeiu 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,

k k the weak solutions of the inhomogeneous equations L u = f and L∗u = f, Pn k ≥ 1, are obtained, where Lu = Du + uh and L∗u = uD − hu, h = hiei, 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

∞ j Λ(f, Ω) = f|f ∈ C (Ω,C(Vn,n)), max D f(x) = x∈Ω

= O(M j)(j → +∞), for some M, 0 < M < +∞ , the integral representations of Tif are given, some properties of Tif 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 Dku = 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 n basis {e1, e2, ··· , en}, C(Vn,s) be the 2 –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) P (A,B)  eAeB = (−1) (−1) eA4B, if A, B ∈ PN,  P P P P  λµ = λAµBeAeB, if λ= λAeA, µ= µBeB. 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,   2  ei = 1, if i = 1, ··· , s,    2  ej = −1, if j = s + 1, ··· , n, (2)   eiej = −ejei, if 1 ≤ i < j ≤ n,    eh1 eh2 ··· ehr = eh1h2···hr , 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. n n Frequent use will be made of the notation Rz where z ∈ R , which n n n means to remove z from R . In particular R0 = R \{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 Xn ∂ (r) (r−1) D = ek : C (Ω,C(Vn,n)) → C (Ω,C(Vn,n)). ∂xk k=1

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 Xn X Xn X ∂fA ∂fA D[f] = ekeA , [f]D = eAek , ∂xk ∂xk k=1 A k=1 A An involution is deﬁned by  T  σ(A)+#(A S)  eA = (−1) eA, if A ∈ PN, (3)  P P  λ = λAeA, if λ = λAeA, A∈PN A∈PN where σ(A) = #(A)(#(A) + 1)/2. From (1) and (3), we have    ei = ei, if i = 0, 1, ··· , s,  (4) e = −e , if j = s + 1, ··· , n,  j j   λµ = µλ, for any λ, µ ∈ C(Vn,s).

The C (Vn.n)–valued (n − 1)–diﬀerential form Xn k−1 N dσ = (−1) ekdxbk k=1 Integral Representations and its Applications in Cliﬀord Analysis 85 is exact, where

N 1 k−1 k+1 n dxbk = dx ∧ · · · ∧ dx ∧ dx ∧ · · · ∧ dx .

2 Integral Representations

In this section, we shall give the integral representations for f and Tif, i ≥ 1, f ∈ Λ(f, Ω), where

∞ j Λ(f, Ω) = f|f ∈ C (Ω,C(Vn,n)), max D f(x) = x∈Ω

= O(M j)(j → +∞), for some M, 0 < M < +∞ .

In [5], [24] the kernel functions (5)  A xj  j  n , n is odd;  ωn ρ (x)   A xj  j  n , 1 ≤ j < n, n is even;  ωn ρ (x) ∗ Hj (x) =  Aj−1 2  log(x ), j = n, n is even;  2ωn  !  Xl−1  An−1 l 2 Ci+1,0  Cl,0x log(x ) − 2 , j = n + l, l > 0, n is even; 2ω C n i=0 i,0

1 n n 2 P P 2 are constructed for any j ≥ 1, where x = xkek, ρ(x) = xk , ωn k=1 k=1 denotes the area of the unit sphere in Rn, and

1 ∗ (6) Aj = j , 1 ≤ j < n(n is even), j ∈ N (n is odd), [ 2 ] j−1 j−1 Q [ 2 ] 2 [ 2 ]! (2r − n) r=1 86 Zhang Zhongxiang

  1, j = 0,   1 (7) Cj,0 = , j ∈ N∗ = N\{0}.  j−1  [ 2 ]  [ j ] j Q  2  2 [ 2 ]! (n + 2µ) µ=0

Lemma 1.(Higher order Cauchy-Pompeiu formula) (see [24])Suppose that M is an n–dimensional diﬀerentiable compact oriented manifold con-

n (r) tained in some open non empty subset Ω ⊂ R , f ∈ C (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) Xk−1 Z Z j ∗ j k ∗ k N f(z) = (−1) Hj+1(x − z)dσxD f(x) + (−1) Hk (x − z)D f(x)dx . 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 i ∗ N (9) Tif(z) = (−1) Hi (x − z)f(x)dx Ω

∗ ∗ p where Hi (x) is denoted as in (5), i ∈ N , f ∈ L (Ω,C(Vn,n)), p ≥ 1. The operator T1 is the Pompeiu operator T . Especially, we denote f as T0f.

p In [25], it is shown that, if f ∈ L (Ω,C(Vn,n)), p ≥ 1, then T f ∈ p − n Cα(Ω,C(V )), α = .T f provides a particular weak solution to the n,n p k inhomogeneous equation Dkω = f(weak) in Ω. In this section, we shall

∞ ∗ show that, if f ∈ Λ(f, Ω), then Tif ∈ C (Ω,C(Vn,n)), i ∈ N and Tkf provides a particular solution to the inhomogeneous equation Dkω = 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 Li- apunov boundary ∂Ω, f ∈ Λ(f, Ω). Then, for z ∈ Ω

X∞ Z j+i ∗ j (10) Tif(z) = (−1) Hj+i+1(x − z)dσxD f(x), i ∈ N. j=0 ∂Ω

Proof. Step 1. For f ∈ Λ(f, Ω), we shall ﬁrstly prove Z k P j+i ∗ j Tif(z) = (−1) Hj+i+1(x − z)dσxD f(x) j=0 (11) Z∂Ω i+k+1 ∗ k+1 N +(−1) Hi+k+1(x − z)D f(x)dx , 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) ∗ ∗ ∗ n D Hj+1(x − z) = Hj+1(x − z) D = Hj (x − z), x ∈ Rz , for any j ≥ 1.

Combining Stokes formulas with (12), we have

(13) Z Z k i ∗ N P j+i ∗ j (−1) Hi (x − z)f(x)dx = (−1) Hj+i+1(x − z)dσxD f(x) j=0 Ω\B(z,ε) ∂(Ω\ZB(z,ε)) i+k+1 ∗ k+1 N +(−1) Hi+k+1(x − z)D f(x)dx . Ω\B(z,ε)

For i ≥ 1 and j ≥ 0, it is easy to check that, Z ∗ j (14) lim Hj+i+1(x − z)dσxD 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

∗ 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

k k (16) max D f(x) ≤ C0M . x∈Ω Case 1. n is odd. For any k ≥ N, we have

∗ k+1 N n k+1 i+k+1−n (17) Hi+k+1(x − z)D f(x)dx ≤ 2 Ai+k+1C0V (Ω)M δ ,

Ω where δ = sup ρ(x1 − x2),V (Ω) denotes the volume of Ω. It is obvious x1,x2∈Ω that the series X∞ n k+1 i+k+1−n (18) 2 Ai+k+1C0V (Ω)M δ k=1 converges. Then

n k+1 i+k+1−n (19) lim 2 Ai+k+1C0V (Ω)M δ = 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 Xk−1 Z j+i ∗ j (20) Tif(z) = (−1) Hj+i+1(x − z)dσxD f(x), i ∈ N. j=0 ∂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+1f] = Tif, i ∈ N.

Remark 2.Corollary 2 implies that Tkf provides a particular solution to the inhomogeneous equation Dkω = 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 Tkf 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 Pn in U, where Lu = Du + uh, h = hiei, hi ∈ R or h be a real (complex) i=1 number. Then for z ∈ Ω X∞ Z j+i ∗ j (22) Tif(z) = (−1) Hj+i+1(x − z)dσxD f(x), i ∈ N. j=0 ∂Ω 90 Zhang Zhongxiang

Proof. Obviously, if f is a solution of equation Lu = 0 in U, where Lu = Pn Du + uh, h = hiei or h is a real (complex) number, then f ∈ Λ(f, Ω). i=1 By Theorem 1, the result follows.

∞ k P (αx e ) 4 i i αxiei Example 1.Suppose ui(x) = = e , i = 1, ··· , n, where α k=0 k! is a real number. Clearly, Dui(x) = αui(x). Thus for ui(x), z ∈ Ω, by Corollary 3, (22) holds. r Pn Pn 2 Example 2.Suppose h = hiei, hi ∈ R. Denote R = |h| = hi . i=1 i=1 Obviously, eRxiei satisﬁes Du − Ru = 0, thus eRxiei is also a solution of the Helmholtz equation 4u − R2u = 0. Then eRxiei (R − h) is a solution of equation Du +uh = 0. For eRxiei (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  Z   K(x − z)f(x)dxN , z ∈ Ω,  (23) Πf(z) = Ω Z  N  lim K(x − ξ)f(x)dx z ∈ ∂Ω,  ξ→z ξ∈Ω Ω where 1 (2 − n)e1 nxe1x n (24) K(x) = n − n+2 , x ∈ R0 . ωn ρ (x) ρ (x) α f ∈ H (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.

α For u ∈ H (∂Ω,C(Vn,n)), 0 < α ≤ 1, denote Z ∗ n (25) (F∂Ωu)(x) = H1 (y − x)dσyu(y), x ∈ R \ ∂Ω. ∂Ω Integral Representations and its Applications in Cliﬀord Analysis 91

Z ∗ (26) (S∂Ωu)(x) = H1 (y − x)dσyu(y), x ∈ ∂Ω. ∂Ω

  +  (F∂Ωu)(x), x ∈ Ω , (27) F + u (x) = ∂Ω  1  u(x) + (S u)(x) x ∈ ∂Ω. 2 ∂Ω Theorem 2.Let Ω be an open non empty bounded subset of Rn with a Li-

1 apunov boundary ∂Ω, f ∈ C (Ω,C(Vn,n)), Πf is deﬁned as in (23). Then 2 − n (28) Πf(z) = F + (αe αf) (z) + T (e D [f]) (z) − e f(z), z ∈ Ω, ∂Ω 1 1 n 1 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\BZ(z,ε) ∗ N = lim [H1 (x − z)e1] Df(x)dx ε→0 (29) Ω\B(Zz,ε) ∗ = lim H1 (x − z)e1dσxf(x) + T (e1D [f]) (z) ε→0 Z ∂(Ω\B(z,ε)) 2 − n = H∗(x − z)e dσ f(x) + T (e D [f]) (z) − e f(z). 1 1 x 1 n 1 ∂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 n − 2 (30) D [Πf] = e D [f] + D [e f] . 1 n 1 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 Cauchy- Pompeiu formula. The solutions of Dirichlet problems as well as the inho-

k mogeneous equations D u = f are obtained. In the sequel, Kn denotes the unit ball in Rn (n ≥ 3), more clearly,

n Kn = {x|x = (x1, x2, ··· , xn) ∈ R , |x| < 1} .

Denote (32) 1 1 G(y, x) = n−2 − y , x ∈ Kn, y ∈ Kn, x 6= y. ρ (y − x) |y|n−2ρn−2( − x) |y|2 Remark 3.G(y, x) has the following properties:

(1) 4xG(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.

2 Theorem 3.Suppose f ∈ C (Kn,C(Vn,n)), then for x ∈ Kn Z 2 Z 1 1 − |x| 1 N (33) f(x) = n f(y)dSy + G(y, x)4yf(y)dy . ωn ρ (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 Z 1 y − x 1 1 f(x) = n dσyf(y) − n−2 dσyD[f](y) ωn ρ (y − x) (2 − n)ωn ρ (y − x) ∂Kn Z ∂Kn 1 1 N + n−2 4yf(y)dy . (2 − n)ωn ρ (y − x) Kn

By Stokes formula, for x ∈ Kn and x 6= 0, we have (35) x Z y − Z 1 |x|2 1 1 0 = x dσyf(y) − x dσyD[f](y) ωn ρn(y − ) (2 − n)ωn ρn−2(y − ) ∂Kn Z |x|2 ∂Kn |x|2 1 1 N + x 4yf(y)dy . (2 − n)ωn ρn−2(y − ) Kn |x|2 (35) can be rewritten as x Z |x|2 y − 1 |x|2 (36) 0 = x dσyf(y)− ωn |x|nρn(y − ) ∂Kn |x|2 Z 1 1 − x dσyD[f](y)+ (2 − n)ωn |x|n−2ρn−2(y − ) ∂Kn |x|2 Z 1 1 N + x 4yf(y)dy . (2 − n)ωn |x|n−2ρn−2(y − ) Kn |x|2 In view of x y (37) |x|kρk(y − ) = |y|kρk( − x), k ∈ N∗, |x|2 |y|2 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

2 Remark 4.Suppose f ∈ C (Kn,C(Vn,n)), moreover, f is harmonic in Kn.

Then for x ∈ Kn Z 1 1 − |x|2 (38) f(x) = n f(y)dSy. ωn ρ (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 2 Z 1 1 − |x| 1 N (39) u(x) = n γ(y)dSy + G(y, x)f(y)dy . 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 X∞ Xk−1 X N j N (40) f(x ) = f(0) + Cj,p−jx Vl1,··· ,lp−j (x )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

k D 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

X∞ Xk−1 X j N (41) u = C0 + Cj,p−jx Vl1,··· ,lp−j (x )Cl1,··· ,lp−j + Tkf.

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 sugges- tions and discussions.

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School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China E–mail: [email protected]