Generalized $$(k_i)$$ ( k i ) -Monogenic Func
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Advances in Applied Clifford Algebras
Generalized (ki)-Monogenic Functions Doan Cong Dinh∗ Communicated by Swanhild Bernstein Abstract. In this paper we introduce generalized (ki )-monogenic functions in Clifford analysis. They are the general types of the k-hypermonogenic functions founded by Leutwiler and Eriksson. Each component of a generalized (ki )-monogenic function is a solution of a generalized Weinstein’s equation. We will construct 2n generalized Cauchy kernels and give an integral representation of the generalized (ki )-monogenic functions. Mathematics Subject Classification. 30A05, 30G30, 30G35, 15A66, 35A08. Keywords. Clifford analysis, k-Hypermonogenic functions, Generalized monogenic functions, Generalized Cauchy kernel, Integral representation of solutions, Weinstein’s equations.
1. Introduction Let us revisit the well-known theory of k-hypermonogenic functions founded by Leutwiler and Eriksson. In a new view, we can consider it as a natural development of the generalized Stokes–Beltrami system in the plane to Clifford analysis. Let Φ(x, y) and Ψ(x, y) be a pair of functions which possess in the halfplane y > 0 continuous second derivatives and satisfy the system ⎧ ∂Ψ ∂Φ ⎪ ⎨ yp = ∂x ∂y (1.1) ∂Φ ∂Ψ ⎪ ⎩ yp =− . ∂y ∂x The system (1.1) is the generalized Stokes–Beltrami system which was introduced by Weinstein in [12,13]. If p = 0 the system (1.1) is the Cauchy– Riemann system in the plane. The system (1.1) was first considered by Beltrami [1] with p = 1, and later by Bers and Gelbart [3] for any positive values ∗ Corresponding
author.
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Adv. Appl. Clifford Algebras
of p. If Φ(x, y) and Ψ(x, y) satisfy the system (1.1) then p ∂Φ = 0, y ∂y p ∂Ψ ΔΨ − = 0. y ∂y ΔΦ +
(1.2)
Weinstein constructed fundamental solutions of the equation of type (1.2) in [12,13] and [6] for general case. An equation of type (1.2) is called the Weinstein equation. It plays an important role in hydrodynamic, elasticity, and transonic flow. In the concluding remarks of [12], Weinstein pointed out an important extension of the theory to higher dimensions. We will consider a similar situation in Clifford analysis. Clifford algebra An is generated by e0 = 1, e1 , e2 , . . . , en with the relations ei ej + ej ei = −2δij (the Kronecker symbol), i, j ∈ {1, . . . , n}. Denote
B = ∅, (i1 , i2 , . . . , ik )|ij ∈ N, 1 ≤ i1 < i2 < · · · < ik ≤ n .
Denote e∅ = 1, eα = ei1 ei2 . . . eik for each α = (i1 , i2 , . . . , ik ) ∈ B. A basis of An is {eα | α ∈ B}. An An -valued function w has 2n real-valued components w(x) = wα (x)eα , wα (x) ∈ R. α∈B
The Cauchy–Riemann operator D and its adjoint D are given by n
D :=
∂ ∂ + ei ∂x0 i=1 ∂xi
n
and
D :=
∂ ∂ − ei , ∂x0 i=1 ∂xi
with x = (x0 , x1 , . . . , xn ) ∈ Rn+1 . A function w is called a left monogenic function if Dw = 0 and a right monogenic function if wD = 0. We also call a left monogenic function shortly a monogenic function. Since DD = DD = Δ, all components of a left (right) monogenic function satisfy the Laplace equation. We refer the readers to th
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