The Riemann boundary value problem related to the time-harmonic Maxwell equations

*Correspondence: wlpxjj@163.com; xuzl@ruc.edu.cn 1School of Mathematics, Renmin University of China, Beijing, 100872, P.R. China 2School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei Province, 050024, P.R. China Abstract In this paper, we first give the definition of Teodorescu operator related to theN matrix operator and discuss a series of properties of this operator, such as uniform boundedness, Hölder continuity and so on. Then we propose the Riemann boundary value problem related to theN matrix operator. Finally, using the intimate relationship of the corresponding Cauchy-type integral between theN matrix operator and the time-harmonic Maxwell equations, we investigate the Riemann boundary value problem related to the time-harmonic Maxwell equations and obtain the integral representation of the solution.


Introduction
The boundary value problem for partial differential equations is a very meaningful research subject which has important applications in physics, chemistry, financial mathematics and many other fields. Teodorescu operator is a generalized solution of the inhomogeneous Dirac equation and it has been widely used in solving the boundary value problem of partial differential equations. Therefore many experts and scholars studied the properties of the Teodorescu operator and corresponding boundary value problem, for example, Vekua N [1] first discussed some properties of the Teodorescu operator on the plane and its application in the shell theory and gas dynamics. Hile GN [2] and Gilbert RP et al. [3] studied some properties of the Teodorescu operator in n-dimensional Euclidean space and higher-dimensional complex spaces. Du JY, Yang PW, Qiao YY, Taira K and Wang LP et al. studied some properties and boundary value problems associated with the Teodorescu operator in quaternion analysis and Clifford analysis (see [4][5][6][7][8][9][10][11][12][13][14][15][16]).
Quaternion analysis is an important branch of modern analysis, which studies the functions defined in the domain of n-dimensional Euclidean space with values in quaternion spaces. It is an important tool for the solution of boundary value problems of highdimensional partial differential equations, including Maxwell equations. The Maxwell equations are a set of partial differential equations describing the relationship between the electric field, the magnetic field and the charge density, which is the basic equation of the electromagnetism in physics. The properties of the singular integral operator and boundary value problems related to the Maxwell equations have been studied by many scholars, for example, Mcintosh A and Mitrea M [17] discussed the problems related to the Maxwell equations in Lipschitz domains. Schneider B and Shapiro M [18,19] studied the Cauchy-type integral of time-harmonic electromagnetic fields in the case of a piecewise Liapunov surface of integration. Kravchenko VV and Shapiro MV [20][21][22] discussed the Cauchy-type integral associated with Maxwell's equations, and obtained some important integral formulas. Moreover, Kravchenko VV considered quaternionic reformulations of Maxwell's equations and discussed the Dirichlet boundary value problem. Russell DL [23] studied the Dirichlet-Neumann boundary problem associated with the control theory of Maxwell's equations. Yang PW et al. [24] investigated an initial-boundary value problem for Maxwell equations and obtained the general solutions. Colton D and Kress R [25] discussed the boundary value problem for the time-harmonic Maxwell equations and the vector Helmholtz equation. Abreu-Blaya R et al. [26] presented a new definition of Cauchy integral associated with Maxwell equations on 3-dimensional domains with fractal boundaries.
Time-harmonic Maxwell equations in physics are the fundamental equations of electromagnetism and can be rewritten as Helmholtz equations by using the quaternion analysis. (1.1) The system is called the time-harmonic Maxwell equations. ( − → E , − → H ) is called a timeharmonic electromagnetic field. It is easy to prove that they satisfy the homogeneous Helmholtz equation where λ = iωμσ ∈ C. In this paper, we will study the Riemann boundary value problem related to the time-harmonic Maxwell equations in quaternion analysis. For the above purpose, we introduce the N matrix operator which establishes the relationship between the Helmholtz equation and the time-harmonic Maxwell equations. In [15], we discuss some properties of Teodorescu operator and the Riemann boundary value problem related to the Helmholtz equation. By using the N matrix operator and the conclusions in [15], we give the integral representation of the solution for the Riemann boundary value problem related to the time-harmonic Maxwell equations. The structure of this paper is as follows: In Sect. 2, we review some basic knowledge of quaternion analysis and introduce some necessary notions for the understanding of this article. In Sect. 3, we first discuss some properties of the singular integral operator T N ,α related to the N matrix operator, such as uniform boundedness, Hölder continuity and so on. Secondly, we give the integral representation of the solution for the Riemann boundary value problem related to the N matrix. In Sect. 4, we first introduce the time-harmonic Maxwell equations. Then, using the corresponding Cauchy-type integral relationship between the N matrix operator and the time-harmonic Maxwell equations, we investigate the Riemann boundary value problem related to the time-harmonic Maxwell equations and obtain the integral representation of the solution.

Preliminaries
Let {i 1 , i 2 , i 3 } be an orthogonal basis of the Euclidean space R 3 , and H(C) be the set of complex quaternions with basis {i 0 , i 1 , i 2 , i 3 }. Then an arbitrary quaternion a can be written as a = 3 k=0 a k i k , a k ∈ C, where i 0 is the unit, i 1 , i 2 , i 3 are the quaternionic imaginary units with the properties The norm for an element a ∈ H(C) is taken to be |a| = 3 k=0 |a k | 2 . The conjugate operation in H(C) is governed by the rules For any complex quaternions a, b, we have where a, b , [ a, b] stand for usual scalar product and vector product. In particular, Suppose ⊂ R 3 is a domain with a Liapunov boundary ∂ . Then the function which is defined in and valued in H(C) can be expressed as f = 3 We define the differential operators as follows: Let λ ∈ C\{0}, and α be its complex square root, α 2 = λ. For the above α, let us introduce the operators These are called the left (right) mutually conjugate (ψ, α)-hyperholomorphic Cauchy-Riemann operators. We have the equalities where λ is the 3-dimensional Helmholtz operator with a complex parameter λ.
In both cases it is a fundamental solution of the Helmholtz equation with λ = α 2 . Then the fundamental solution to the operator ψ D α , K ψ,α is given by the formula An analogous representation holds for If f is a Hölder function, then its α-hyperholomorphic Cauchy-type integral is defined by In [15], we introduce the Teodorescu operator related to the Helmholtz equation as follows: In [15], we studied the properties of the above integral operators and obtained the integral representation of the solution for the Riemann boundary value problem related to the Helmholtz equation. The specific results are as follows.

Lemma 2.3 ([15]) Let B be as stated above. Find a quaternion-valued function u(x) satisfying the system
and vanishing at infinity with the boundary condition where u ± (τ ) = lim x∈B ± ,x→τ u(x), G is a quaternion constant, G -1 exists, and f ∈ H ν ∂B (0 < ν < 1). Then the solution can be expressed as

The relevant definitions and symbols
We will consider the following matrix operator: where α 2 = iωμσ . We shall consider it on the set C (1) ( , Mat 2×2 (H(C))), Mat 2×2 (H(C)) being the set of 2 × 2 matrices with entries from H(C). Let where " * " stands for matrix multiplication. Analogously, let For α ∈ C\{0}, let be the quaternionic Cauchy-Maxwell kernel, which is the fundamental solution of N operator. The reasons are as follows. By the definition of N operator, we have be an analog of the Cauchy-type integral in the theory of the integral representations with the quaternionic Cauchy-Maxwell kernel, where f : ∂ → Mat 2×2 (H(C)) and is taken to be f ∞ = max 1≤i,j≤2 |f ij |. From equality (3.1), we know there exists a direct connection between K N ,α and the corresponding hyperholomorphic Cauchy kernels K ψ,α , K ψ,α .

Some properties of the Teodorescu operator T N ,α related to the N matrix operator
In this section, we will discuss some properties of the following singular integral operators: Proof (1) From (3.2), we can obtain where By Lemma 2.1 and Lemma 2.2, we have Therefore where Q 1 (p) = max 1≤i≤8 {Q (i) 1 (p)}. (2) From (3.3), we can obtain For each x 1 , x 2 ∈ , by Lemma 2.1 and Lemma 2.2, we have Therefore Proof This case is similar to Theorem 3.1.
Thus, from Theorem 3.1 and Theorem 3.2, we obtain the following results.

The Riemann boundary value problem related to the N matrix operator
Proof Let B 2 be as above and its inverse B -1 2 = -σ α σ α . Then, for g = B -1 2 u = Then Theorem 3.5 Let B be as stated above, and g = g 11 g 12 g 21 g 22 with entries belonging to L p,3 (R 3 , H(C)), 3 < p < +∞. Find w = w 11 w 12 w 21 w 22 ∈ Mat 2×2 (H(C)) satisfying the following system: Proof By Theorem 3.3, we know The boundary condition (3.5) becomes Again from Theorem 3.3, we know that (T N ,α [g])(x) has continuity in ⊂ R 3 . Thus . Thus we can obtain , then (3.7) has the following form:

Let
− → E , − → H : → C 3 be a pair of complex-valued vector fields.
i.e. E 0 = 0, H 0 = 0. The following system: is called the time-harmonic Maxwell equations. ( − → E , − → H ) is called a time-harmonic electromagnetic field, where σ is a complex electrical conductivity and μ is a magnetic permeability. It is known that they satisfy the homogeneous Helmholtz equation For k ∈ Z + , set The operator i.e. the restriction of M ontoĈ (1) , will be termed the time-harmonic Maxwell operator. Then (4.1) and (4.2) become , which are identified naturally with columns a b . We shall not distinguish them in this paper.
(iii) By the definition of vector product, we have