Cauchy Integral Decomposition of Multi-Vector Valued Functions on ...

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130 R. Abreu Blaya, J. Bory Reyes, R. Delanghe and F. Sommen CMFT where F (l) , l = k, k − 1, are l-multi-vector fields over (ε 1 , . . . , ε m−1 ). Applying Lemma 5.1 to (25) we get η k = ΘF k = ΘF (k) + Θ(nF (k−1) ) 〈 ( = 1 m−1 ) k 〉〈 ( ∑ F (k) , ε k! j φ j + kdν m−1 ) k−1 〉 ∑ nF (k−1) , n ε k! j φ j j=1 j=1 = η (k) + dν η (k−1) . Here η (l) , l = k, k − 1, are l-forms over (dω 1 , . . . , dω m−1 ). Now suppose η k = ΘF k is harmonic in Ω, i.e. it satisfies the Hodge-de Rham system (23). As η k is assumed to be extendable to C ∞ (Σ), by restricting to the boundary, we immediately obtain Dyn’kin’s Condition (A) (see [7]), namely { dηk = 0 (A1) d ∗ η k = 0 (A2) . We now claim that in Clifford analysis terms, Condition (A) is equivalent to the (CL)-condition for the boundary value F + k = F k| Σ <strong>of</strong> the harmonic k-multi-vector field F k (see (15)) n(∂ ω F + k )n + F + k ∂ ω = 0. Let us prove this statement in the case where k is odd, the pro<strong>of</strong> <strong>of</strong> the case k even being similar. In order to avoid tedious calculations, we only give a sketch <strong>of</strong> the pro<strong>of</strong>. Let us first show how the (CL)-condition may be derived from Condition (A). In fact, we shall establish separately conditions upon F + k which are equivalent to, respectively, the Conditions (A1) and (A2). Let us start by translating Condition (A1). As we have seen (see (24)), Condition (A1) also reads: As k is odd, d(ΘF k ) ∣ ∣ Σ = (Θ(∂ x ∧ F k )) ∣ ∣ Σ = 0. ∂ x ∧ F k = 1 2 (∂ xF k − F k ∂ x ) = 1 2 [∂ x, F k ]. Moreover, as [∂ x , F k ] is a (k + 1)-multi-vector field, we may write it in the form (see (25)) (26) [∂ x , F k ] = F (k+1) + nF (k) where F (k+1) and F (k) are constructed over (ε 1 , . . . , ε m−1 ). By virtue <strong>of</strong> (26) (Θ[∂ x , F k ]) ∣ ∣ Σ = ( ΘF (k+1) + dνΘF (k)) ∣ ∣ Σ = ( ΘF (k+1)) ∣ ∣ Σ = Θ(F (k+1)∣ ∣ Σ ). But, k being odd, we have that nF (k+1) = F (k+1) n, nF (k) = −F (k) n,
5 (2005), No. 1 <strong>Cauchy</strong> <strong>Integral</strong> <strong>Decomposition</strong> <strong>of</strong> <strong>Multi</strong>-<strong>Vector</strong> <strong>Valued</strong> <strong>Functions</strong> 131 whence from (26) we get (27) F (k+1)∣ ∣ Σ = Condition (A1), namely is thus equivalent to or to F (k+1) | Σ = 0. ( ) ∣ −n ∣∣∣Σ 2 {n, [∂ x, F k ]} . d(ΘF k ) ∣ ∣ Σ = 0 Θ(F (k+1) | Σ ) = 0 In view <strong>of</strong> (27) it is hence also equivalent to {n, [∂ x , F k ]} ∣ ∣ Σ = 0. Now writing F k in the form (see also (25)) F k = ˜F (k) + n ˜F (k−1) , by using the relation (7), straightforward calculations lead to where (CL1) is given by (A1) ⇐⇒ (CL1) (28) 0 = {n, [∂ x , F k ]} ∣ ∣ Σ = {n, [∂ ‖x , F k ]} ∣ ∣ Σ = {n, [∂ ω , F + k ]}. Analogously it may be shown that where (CL2) is given by (A2) ⇐⇒ (CL2) (29) 0 = [n, {∂ x , F k }] ∣ ∣ Σ = [n, {∂ ‖x , F k }] ∣ ∣ Σ = [n, {∂ ω , F + k }]. Obviously, (28) and (29) imply the (CL)-condition n(∂ ω F + k )n + F + k ∂ ω = 0. Conversely, suppose that F k is a harmonic k-multi-vector field in Ω which extends to C ∞ (Σ) and which is such that its trace on the boundary F + k = F k| Σ satisfies the (CL)-condition n(∂ ω F + k )n + F + k ∂ ω = 0 or equivalently, (30) n(∂ ω F + k ) = (F + k ∂ ω)n. is R(k) As F + k 0,m-valued, ∂ ω F + k , which splits into a (k+1)- and a (k−1)-multivector, may be written as (31) ∂ ω F + k = A + nB, where A = A (k+1) + A (k−1) , B = B (k) + B (k−2) .
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