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Integral Representations and Its Applications in 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 2 Λ(f; Ω), 1 j Λ(f; Ω) = fjf 2 C (Ω;C(Vn;n)); max D f(x) = x2Ω = O(M j)(j ! +1); for some M; 0 < M < +1 : 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 com- plex valued, quaternionic and Clifford algebra valued functions have been well developed in [1-9, 12-19, 21, 24, 25 etc.]. These integral representa- tion 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! = f, 4k! = g and @4k! = 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 prop- erties of Cauchy-Pompeiu operator, higher order Cauchy-Pompeiu operator and Π operator for f 2 Λ(f; Ω), where 1 j Λ(f; Ω) = fjf 2 C (Ω;C(Vn;n)); max D f(x) = x2Ω = O(M j)(j ! +1); for some M; 0 < M < +1 ; 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 fe1; e2; ··· ; eng, C(Vn;s) be the 2 {dimensional real linear space with basis feA;A = fh1; ··· ; hrg2PN; 1 ≤ h1 < ··· < hr ≤ ng ; where N stands for the set f1; ··· ; ng 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 = fh1; ··· ; hrg 2 PN. The product on C(Vn;s) is defined by 8(1) > #((A\B)nS) P (A;B) <> eAeB = (−1) (−1) eA4B; if A; B 2 PN; > P P P P :> λµ = λAµBeAeB; if λ= λAeA; µ= µBeB: A2PN B2PN A2PN B2PN where S stands for the set f1; ··· ; sg, #(A) is the cardinal number of the P set A, the number P (A; B) = P (A; j), P (A; j) = #fi; i 2 A; i > jg, the j2B symmetric difference set A4B is also order-preserving in the above way, and λA 2 R is the coefficient of the eA{component of the Clifford number λ. We also denote λA as [λ]A, for abbreviaty, we denote λfig 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, 8 > 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 2 R , which n n n means to remove z from R . In particular R0 = R n f0g. 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 8 T > σ(A)+#(A S) <> eA = (−1) eA; if A 2 PN; (3) > P P :> λ = λAeA; if λ = λAeA; A2PN A2PN where σ(A) = #(A)(#(A) + 1)=2. From (1) and (3), we have 8 > > ei = ei; if i = 0; 1; ··· ; s; <> (4) e = −e ; if j = s + 1; ··· ; n; > j j > :> λµ = µλ, for any λ, µ 2 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 2 Λ(f; Ω), where 1 j Λ(f; Ω) = fjf 2 C (Ω;C(Vn;n)); max D f(x) = x2Ω = O(M j)(j ! +1); for some M; 0 < M < +1 : In [5], [24] the kernel functions (5) 8 A xj > j > n ; n is odd; > !n ρ (x) > > A xj > j > n ; 1 ≤ j < n; n is even; < !n ρ (x) ∗ Hj (x) = > Aj−1 2 > log(x ); j = n; n is even; > 2!n > ! > Xl−1 > An−1 l 2 Ci+1;0 :> Cl;0x log(x ) − 2 ; j = n + l; l > 0; n is even; 2! 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 ;!n 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 2 N (n is odd); [ 2 ] j−1 j−1 Q [ 2 ] 2 [ 2 ]! (2r − n) r=1 86 Zhang Zhongxiang 8 > 1; j = 0; > <> 1 (7) Cj;0 = ; j 2 N∗ = Nnf0g: > 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 2 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 2M (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 2 N , f 2 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 2 L (Ω;C(Vn;n)); p ≥ 1; then T f 2 p − n Cα(Ω;C(V )); α = :T f provides a particular weak solution to the n;n p k inhomogeneous equation Dk! = f(weak) in Ω. In this section, we shall 1 ∗ show that, if f 2 Λ(f; Ω), then Tif 2 C (Ω;C(Vn;n)), i 2 N and Tkf provides a particular solution to the inhomogeneous equation Dk! = 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 2 Λ(f; Ω). Then, for z 2 Ω X1 Z j+i ∗ j (10) Tif(z) = (−1) Hj+i+1(x − z)dσxD f(x); i 2 N: j=0 @Ω Proof. Step 1. For f 2 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 2 N, z 2 Ω. It is obvious that (11) is the direct result of Lemma 1 for i = 0.
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