Linear BVPs and SIEs for Generalized Regular Functions in Clifford Analysis

Abstract

We study some properties of a regular function in Clifford analysis and generalize Liouville theorem and Plemelj formula with values in Clifford algebra . By means of the classical Riemann boundary value problem and of the theory of a regular function, we discuss some boundary value problems and singular integral equations in Clifford analysis and obtain the explicit solutions and the conditions of solvability. Thus, the results in this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.

1. Introduction

As we know, boundary value problems (BVPs) for holomorphic function and singular integral equations (SIEs) are one of important branches in classical holomorphic function theory of one complex variable, and they have important applications in many fields, such as mechanics, physics, and engineering. Many practical problems can often be transformed into BVPs and SIEs. In recent decades, some kinds of BVPs and SIEs have been well studied and a lot of results were obtained (see, e.g., [1–6]). Clifford analysis is an important field of modern mathematics which studies the functions defined in with values in Clifford algebra space and possesses both theoretical and applicable values, such as physics, quantum mechanics, Maxwell equation, theory of Yang-Mills field, and other branches of mathematics (see, e.g., [7–10]). The results of theory of Clifford regular function are generalizations of the classical theory of holomorphic functions in complex analysis. A lot of results of the classical theory of holomorphic functions can be extended to Clifford analysis. In recent decades, many mathematical workers devote to study the BVPs and SIEs in Clifford analysis, and there were many important and significant results (see, e.g., [11–13]).

Based on the above-mentioned work, we further solve some kinds of BVPs and SIEs in Clifford analysis as well as BVP in quaternion analysis. We first study some properties of a regular function and then generalize Cauchy integral theorem and Cauchy integral formula in . By means of the Riemann boundary value problem and of the properties of the Cauchy principal value integral we obtain the explicit expressions of general solution and their solvability conditions for these equations. Thus, this paper generalizes the theory of integral equations and the classical boundary value problems for analytic function.

We begin by developing the necessary preliminaries in Clifford algebra and analysis that we require here.

2. Preliminaries

Let be a real Clifford algebra over an -dimensional real vector space with the orthogonal basis , where is an identity element. The product on is defined by Hence, any element has the type . And is a -dimensional real linear space, whose basis is . Each basis element of has a representation of the form , where and , and when , . Therefore, any element can be denoted as , where is a real number. The norm for an element is taken to be

It is easy to prove that, for any , we have In fact, we denote , , then

Definition 1. Let be a nonempty, open, and connected set. A function defined in and with values in Clifford algebra can be expressed as ; that is, , where is a real function.
Let denote the set of all functions defined in with derivatives of order (≥1). We define the function class in as follows: where and (≥1) is an integer.

Definition 2. Let . If there exists a constant such that for any , then we call to be Hölder continuous function in , denoted by , where and are called the Hölder constant and the Hölder index, respectively. And we define generalized Cauchy-Riemann operators (i.e., Dirac operator) in as follows:For any , the operators , act on function being governed by the rules then a solution of is called left regular function in . Similarly, a solution of is called right regular function. Generally the left regular function is called regular function in short.

Let the boundary of be a smooth, directional, and compact Liapunov surface. We denote the set of all Hölder continuous function defined on as and define the norm as where is a continuous norm, and is an Hölder norm. It is easy to prove that is a Banach space.

Definition 3. If there exists () such that , we call as the inverse of , denoted by .

In the following, can be expressed as , , or , where . Thus, can be divided into three parts: , , and . Obviously, and . However, can also be represented by , and , lie in the upper and lower half spaces of the hyperplane , respectively.

The conjugate of a number is the number given by , where , . For any , we know that , and we can verify that is an inverse element of ; that is,

Let be the subalgebra constructed by , then and has the decomposition It is clear that the decomposition is the generalization of the classical representation of a complex number. Thus, any can be decomposed as , where . We may define operators , as , .

Definition 4. Let be defined in , . If , then we call to be a regular point. If , but there always exists in any neighborhood of such that , then we call to be a singular point. If there is a hollow neighborhood of such that for any , we call to be an isolated point of .

Definition 5. If is a regular function in , that is, and , then we say that is an entire function in , and is the only singular point of in .

3. Some Lemmas and Their Properties

In this section we present some lemmas, which are important to our results.

Lemma 6 ((Plemelj formula) (see [7])). Let where denotes the surface area of a unit sphere in and . is a normal vector at , and is a Lebesgue measure on . is a bounded Hölder continuous function on , that is, , then where

Lemma 7. Let and be an integer. Then one haswhere

Proof. It can easily be proved by a direct calculation.

Lemma 8. If , then and

Proof. In order to prove that , we consider where that is, there exists a constant such that Therefore, we get .
Similarly, by Hile lemma (see [12]) we can prove , thus,
The proof of Lemma 8 is complete.

Lemma 9 (see [13]). Let be a bounded domain in , then the integral that is, exists under the meaning of Cauchy principal value, and its value is equal to , where and are the same as before.

Lemma 10 (see [14]). Let , then the Cauchy type integral is a regular function in , and , .

The following Lemmas 11–13 are obvious facts, and their proofs are similar to that of the classical theorems. More details will be omitted here.

Lemma 11 (generalized Liouville theorem). Let be a regular function in and ; that is, is an entire function. Then one has the following:(1)If , then .(2)If , then , where is a Clifford constant.(3)If (constant) for any , then is a constant in .(4)If   exists, then is a constant in .

Lemma 12 (generalized Cauchy integral theorem). Let be a regular function in and be a single connected domain of . For any smooth, closed, and rectifiable curve, . Then

Lemma 13 (generalized Cauchy integral formula). Let be a regular function in and be a bounded domain. is a smooth, differentiable, and compact surfaces. Then for any , where .

4. BVP in Clifford Analysis

Let be an open and connected set and the boundary of be a smooth, compact, oriented, and closed Liapunov surface. is divided into two domains and by , with . Our goal is to obtain an -valued function such that it is regular in and continuous on , and we have the following boundary value condition: where is a given Clifford constant with and right inverse . A known function . are the boundary values of on , respectively. If the order of is at infinity, such a problem can be denoted as . Actually, the problem and problem are frequently discussed. On the problem , is supposed to be finite and nonzero. On , is assumed to be zero. In order that a solution of (23) exists, we require (i.e., ) and the following condition is fulfilled: where is a unit area element on . Consider where is the surface area of a unit sphere in and ; is an outer normal vector at . By Lemmas 6–8 we have where

Under condition (24), is regular for all and . And is a singular integral and it is convergent under the meaning of Cauchy principal value. Moreover, is continuously extended to the boundary form and , respectively, and then obtain the boundary values and . Note that Particularly, when , we have

Moreover, by [15–17] and Lemma 9, we can obtain Since and present right inverse element , we have

Define Obviously, is a regular function and there exists an inverse element , and Therefore, (23) can be transformed into

Denote it is easily seen that , . By applying Lemmas 6 and 10 for in (34), we obtain Combining (33) with (35), we have It follows from Painleve’s theorem (see [18]) that is a regular function in . Since (23) find a solution in , thus, by Lemma 11, is a constant at ; that is, there exists a Clifford constant such that therefore, we obtain

From the above discussion, we obtain the main result of this section.

Theorem 14. Under condition (24), BVP (23) has a solution. If a solution of (23) is a nonzero constant at , then, its solution is of the form where is determined by (32).

By Theorem 14 and Lemmas 11–13, we obtain the following.

Theorem 15. Suppose that a solution of (23) is zero at , then (23) find a solution in problem , and its solution is given by

5. SIE in Clifford Analysis

In Clifford analysis we solve following SIE of the form where and , are Clifford constants. and have inverse elements and denoted them as and , respectively. is the same as the before. is a bounded domain and its boundary is a smooth, oriented, and compact Lyapunov surface. And is an unknown function. When , we call that (41) is a homogeneous equation; otherwise (41) is nonhomogeneous. Since and are reversible on , (41) is called a regular type SIE with Cauchy kernel, and is called the corresponding singular operator.

Define a sectionally holomorphic function as follows: by applying Lemma 6 for (42), we obtain Putting (43) into (41), one has It follows from (42) that . If (41) has a solution , then a function defined by (42) is a solution of (44) in class . Conversely, if (44) has a solution in class , then obtained by the first equality of (43) is a solution of (41). Indeed, by the first equality of (43) and , we can know that (42) and the second equality of (43) are fulfilled. Substituting (43) into (44), we easily obtain that satisfies (41). Thus, the problem (41) is equivalent to solving the problem (44) in class .

In order to solve (41), we define two functions and as follows:

The remaining discussions are the same as in Section 4. We will not elaborate on that here.

Thus, we have the following conclusions.

Theorem 16. Suppose that , are reversible, then (41) has a solution on , and its solution is obtained by where

Note that, in Theorem 16, if , then must exist. Therefore, we have the following.

Corollary 17. When , (41) is solvable, its solution is still (46), where in (46) is determined by

Proof. It can be proved directly from Theorem 16.

6. BVP in Quaternion Analysis

Quaternion analysis researches analytical theory in quaternion algebra . Quaternion algebra is a four-dimensional real vector space with the basis elements , , , , which satisfy the following multiplication rules: Therefore, can be written as

Quaternion analysis offers another possibility of generalizing complex function theory to higher dimension, and it has wide applications in mathematical physics, Dirichlet BVPs of higher-order partial differential equations, and other fields. More details about quaternion analysis can be found in the literatures [19–22].

Boundary value problem for quaternion analysis is described as follows: given a bounded domain , its boundary is a smooth, compact, and oriented Liapunov surface. Denote , . We want to get a function such that it is regular in and continuous on (or ), and we have the following boundary value condition: where is a known Clifford constant and   shows the inverse element . The known function is Hölder-continuous on . If (constant) with , BVP (51) is denoted as ; if , this case is denoted as ; if the order of is at , denote it as . By Theorems 14 and 16, we obtain a solution of (51) in : where , . , , and are the same as in Section 4.

7. Conclusions

In this paper, we studied some properties of a regular function in Clifford analysis and generalized several classical theorems in , such as Liouville theorem, Plemelj formula, Cauchy integral theorem, and Cauchy integral formula. By means of these theorems and of the classical boundary value theory, we dealt with the solvability and the explicit solutions for some classes of BVPs and SIEs in Clifford analysis. Here, our method is different from the ones of the classical BVP, and it is novel and effective. Thus, this paper generalizes the theory of the classical BVPs and SIEs. By using our method, other classes of BVPs and SIEs are discussed further in Clifford analysis.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology Plan Project of Qufu Normal University (xkj 201606) and the author gratefully acknowledges this support.