Some properties and applications of the Teodorescu operator associated to the Helmholtz equation

In this paper, we first define the Teodorescu operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\psi,\alpha }$\end{document}Tψ,α related to the Helmholtz equation and discuss its properties in quaternion analysis. Then we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.


Introduction
It is well known that the Helmholtz equation is an elliptic partial differential equation describing the electromagnetic wave, which has important applications in geophysics, medicine, engineering application, and many other fields. Many problems associated with the Helmholtz equation have been studied by many scholars, for example [-]. The boundary value problem for partial differential equations is an important and meaningful research topic. The singular integral operator is the core component of the solution of the boundary value problem for a partial differential system. The Teodorescu operator is a generalized solution of the inhomogeneous Dirac equation, which plays an important role in the integral representation of the general solution for the boundary value problem. Many experts and scholars have studied the properties of the Teodorescu operator. For example, Vekua [] first discussed some properties of the Teodorescu operator on the plane and its application in thin shell theory and gas dynamics. Hile [] and Gilbert [] studied some properties of the Teodorescu operator in n-dimensional Euclid space and high complex space, respectively. Yang [] and Gu [] studied the boundary value problem associated with the Teodorescu operator in quaternion analysis and Clifford analysis, respectively. Wang [-] studied the properties of many Teodorescu operators and related boundary value problems.
In this paper, we will study some properties of the singular integral operator and the Riemann boundary value problem associated to the Helmholtz equation using the quaternion analysis method. The structure of this paper is as follows: in Section , we review some basic knowledge of quaternion analysis. In Section , we first construct a singular integral operator T ψ,α related to the Helmholtz equation and study some of its properties.
In Section , we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.

Preliminaries
Let {i  , i  , i  } be an orthogonal basis of the Euclidean space R  and H(C) be the set of complex quaternions with basis where i  is the unit and i  , i  , i  are the quaternionic imaginary units with the following properties: Then an arbitrary quaternion a can be written as a =  k= a k i k , a k ∈ C. The quaternionic conjugation is defined byā = a  - k= a k · i k . The norm for an element a ∈ H(C) is taken to be |a| =  k= |a k |  . Moreover, if aā =āa = |a|  and |a| = , then we say that a is reversible. Obviously, its inverse element can be written as a - =ā |a|  .
Let λ ∈ C\{} and let α be its complex square root: α ∈ C, α  = λ. Suppose ⊂ R  is a domain and ∂ is its boundary. We shall consider functions f defined in ⊂ R  with values in H(C). Then f can be expressed as f =  k= f k (x)i k . Here f k (x) (k = , , , ) are complex functions defined on . Let We define the operators as follows: For the above stated α, let us introduce the following operators: Let α ∈ C\{} and Im α = . For x ∈ R  \{}, we introduce the following notation: In both cases it is a fundamental solution of the Helmholtz equation with λ = α  . Then the fundamental solution to the operator ψ D α , K ψ,α is given by Definition . Suppose that the functions u, v, ϕ are defined in with values in H(C) and u, v ∈ L  ( , H(C)). If, for arbitrary ϕ ∈ C ∞  ( , H(C)), u, v satisfy

Some properties of the singular integral operator T ψ,α for the Helmholtz equation
In this section, we will discuss some properties of the singular integral operators as follows: Theorem . Assume B to be as stated above, α = a + ib, b > . If f ∈ L p (B, H(C)),  < p < +∞, then (i) By the Taylor series, we have |e iα|y-x| | = |e i(a+ib)|y-x| | = e -b|y-x| ≤  b|y-x| . By the Hölder inequality, we have When x ∈ R  \B, by Lemma . and the generalized spherical coordinate, we have where ρ = |y -x|, d  = d(x, B). Therefore, for arbitrary x ∈ R  , we obtain Obviously, e -b|y-x| ≤ . By the Hölder inequality, we have Then, by inequality (.) and (.), we have (iii) This case is similar to (ii). We obtain Let us consider e iα|y-x| . For arbitrary x ∈ R  , it is easy to prove |e iα|y-x| | ≤  and satisfy (i) For arbitrary x  , x  ∈ R  , by the Hölder inequality, we have By the Hölder inequality, we have By the Hölder inequality and the Hile lemma, we have We suppose α = q, β = q. As  < q <   , we have α = q < , β = q < , α + β = q > . Hence, by Lemma ., we have So we have where  < β = p- p < . By inequality (.) and (.), we have Similar to I ()  , we have By the Hölder inequality and the Hile lemma, we have By inequalities (.) and (.), we have where M ()  (p) = J  + J  . By inequalities (.), (.) and (.), we have Hence, in the sense of generalized derivatives, Theorem . Assume B to be as stated above and α = a As the first step, by the Hölder inequality, we have We suppose α = q, β = q. As  < q <   , we have  < α < ,  < β < , α + β = q > . Thus, by Lemma ., we have Therefore, by (.)-(.), we have As the second step, by the Hölder inequality, we have As the third step, similar to I  , we have By inequalities (.), (.), and (.), Firstly, we discuss I  . We have By the Hölder inequality, we have By (.) and (.), we have O  (x) ≤ max{C  , C  }. Therefore By the Taylor series, we have |e By (.), we have In the following, we discuss O  (x) in four cases.

Integral representation of solution of Riemann boundary problem to inhomogeneous partial differential system
In this section, we will discuss the inhomogeneous partial differential system of first order equations as follows: where w j (x), c j (x) (j = , , , ) are real-value functions.
Problem P Let B ⊂ R  be as stated above. The Riemann boundary value problem for system (.) is to find a solution w(x) of (.) that satisfies the boundary condition where w ± (τ ) = lim x∈B ± ,x→τ w(x), B + = B, B -= R  \B, G is a quaternion constant, G - exists, and f ∈ H ν ∂B ( < ν < ).
In fact, By (.) and (.), the inhomogeneous partial differential system (.) can be transformed to the following equation: (.) Therefore Problem P as stated above can be transformed to Problem Q.
Problem Q Let B ⊂ R  be as stated above. The Riemann boundary value problem for system (.) is to find a solution w(x) of (.) that satisfies the boundary condition where w ± (τ ) = lim x∈B ± ,x→τ w(x), B + = B, B -= R  \B, G is a quaternion constant, G - exists, and f ∈ H ν ∂B ( < ν < ).