The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators

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Advances in Applied Clifford Algebras

The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators Doan Cong Dinh∗ Communicated by Rafal Ablamowicz Abstract. This paper deals with the static Maxwell system  → − div(Φ E ) = 0, → − curl E = 0, (x0 , x1 , x2 ) ∈ R3 . The system is reformulated in quaternion analysis by Kravchenko in the form LF = 0 with LF = DF + F α. We consider special cases of the coefficient function Φ = Φ0 (x0 )Φ1 (x1 )Φ2 (x2 ) and prove the existence of four generalized Cauchy kernels of the operator L. We construct four 2m 2n explicit generalized Cauchy kernels in the case Φ = x2p 0 x1 x2 . Mathematics Subject Classification. 30A05, 30G30, 35Q61, 30G20, 78A30. Keywords. Clifford analysis, Generalized monogenic functions, Static Maxwell system, Generalized Cauchy kernel, Integral representation of solutions.

1. Introduction The static Maxwell system in a three dimensional inhomogeneous isotropic media provided by a coefficient Φ = Φ(x0 , x1 , x2 ) > 0 is in the form  → − div(Φ E ) = 0, (1.1) → − curl E = 0, → − where E = (E1 , E2 , E3 ) is the electric field strength. The static Maxwell system is investigated with some different forms of the coefficient Φ in [2,3,6]. The system (1.1) is reformulated in quaternion analysis by Kravchenko [8,9] in the following way. We recall the notation of the quaternion algebra H = {q0 + q1 i + q2 j + q3 k| q0 , q1 , q2 , q3 ∈ R}, ∗ Corresponding

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2

Page 2 of 12

D. C. Dinh

Adv. Appl. Clifford Algebras

where the quaternion units i, j, and k obey the multiplication rules i2 = j2 = k2 = ijk = −1. The Dirac operator in quaternion analysis is given by D=i

∂ ∂ ∂ +j +k . ∂x0 ∂x1 ∂x2

→ − The vector E = (E1 , E2 , E3 ) is identified with iE1 + jE2 + kE3 ∈ H. Defining √ → − → − F = Φ E , the system (1.1) is equivalent to the equation → − − → D F + F α = 0, 1 ∂Φ , i = 0, 1, 2. 2Φ ∂xi Kravchenko introduces a generalized Dirac operator L by

where α = iα0 + jα1 + kα2 , αi =

LF = DF + F α,

(1.2)

where one considers functions F of the form F (x) = F0 + iF1 + jF2 + kF3 whose domain of definition is in R3 and values are quaternions. The operator of type L appeared in [12] for the first time using the matrix notation with constant coefficients. In this paper, we consider Φ = Φ0 (x0 )Φ1 (x1 )Φ2 (x2 ) which is introduced in [7] together with some physical models. That form of Φ corresponds to quite interesting, nontrivial situations. If Φ = Φ(x2 ) then we deal with electrostatic models in plane-layered inhomogeneous media [10,13]. In Sect. 2, with some additional conditions on the coefficients, we prove the existence of four Cauchy kernels of the operator L by using principal fundamental solutions of associated Schr¨ odinger operators. In Sect. 3, we consider a spe2m 2n cial function Φ = x2p 0 x1 x2 and construct four explicit generalized Cauchy kernels of the operator L.

2. Construction of four Cauchy Kernels Suppose that Φ = Φ0 (x0 )Φ1 (x1 )Φ2 (x2 ), Φi (xi ) ∈ C 2 , Φi (xi ) > 0, ∀xi ∈ R, i = 0, 1, 2. Define LF = DF − F α(x), F R = F D − α(