Abstract and Applied Analysis

Volume 2014 (2014), Article ID 974714, 9 pages

http://dx.doi.org/10.1155/2014/974714

Research Article

A Boundary Value Problem for Bihypermonogenic Functions in Clifford Analysis

Xiaoli Bian¹ and Yuying Qiao²

¹School of Science, Tianjin University of Technology and Education, Tianjin 300222, China

²College of Mathematics and Information, Hebei Normal University, Shijiazhuang 050016, China

Received 8 May 2014; Revised 25 June 2014; Accepted 1 July 2014; Published 20 July 2014

Academic Editor: Guo-Cheng Wu

Copyright © 2014 Xiaoli Bian and Yuying Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a nonlinear boundary value problem for bihypermonogenic functions in Clifford analysis. The integrals of quasi-Cauchy’s type and Plemelj formula for bihypermonogenic functions are firstly reviewed briefly. The nonlinear Riemmann boundary value problem for bihypermonogenic functions is discussed and the existence of solutions is obtained, which also indicates that the linear boundary value problem has a unique solution.

1. Introduction

Clifford algebra is an associative and noncommutative algebraic structure that was set up at the beginning of the twentieth century. Clifford analysis is an important branch of modern analysis, which studies the functions defined in with the value in Clifford algebra space [1]. Clifford analysis possesses not only important theoretical value but also applicable value, which plays an important role in many fields, such as quantum mechanics, Maxwell equation, and Yang-Mills field. Since 1987, Xu [2, 3], Wen [4], Huang [5, 6], Qiao [7–9], and so forth have done a lot of work on boundary value problems for monogenic functions and biregular functions in Clifford analysis. Eriksson and Leutwiler [10–12] introduced hypermonogenic functions in Clifford analysis, studied some of its properties, and discussed the integral representation for hypermonogenic functions. Qiao [9] investigated the boundary value problems of hypermonogenic functions. In recent years, Zhang and Du [13, 14] discussed Riemann boundary value problems and singular integral equations in Clifford analysis. Bian et al. [15] obtained the integral formulas and Plemelj formula for bihypermonogenic functions. Yang et al. [16] studied a kind of boundary value problem for hypermonogenic function vector. Zhang and Gürlebeck [17] studied Riemann boundary value problems in Clifford analysis.

In this paper, based on the integral formulas and Plemelj formula for bihypermonogenic functions presented in [15], we study a nonlinear Riemann boundary value problem for bihypermongenic functions. We first review briefly the integrals of quasi-Cauchy's type and Plemelj formula for bihypermonogenic functions and then prove the existence of solutions of a nonlinear Riemann boundary value problem and derive the unique solution of the corresponding linear Riemann boundary value problem.

2. Preliminaries

Let be a real Clifford algebra over an -dimensional real vector space with orthogonal basis , satisfying the relation   , where is the usual Kronecker delta. Then has its basis . Hence the real Clifford algebra is formed by the elements presented as ,  , where or and .

For , we give some calculations as follows:

where and is the cardinality of ; that is, when ,   and when then ; , , .

Recall that any element may be uniquely decomposed as , for   (the Clifford algebra generated by ). Using this decomposition, we define the mappings and by and . Note that if , then

We also introduce the Dirac operator

and the modified Dirac operator

Denote by an open connected set in the Euclidean space , , . Define a set to consist of all functions

with values in for which .

Definition 1. Let and , . A function is called bihypermonogenic on , if

for any , where

is the left modified Dirac operator in calculated with respect to and

is the right modified Dirac operator in the Clifford algebra generated by calculated with respect to , where

3. The Cauchy Integral Formula and Plemelj Formula

In this section, we give some simple review on the Cauchy integral formula and Plemelj formula for bihypermonogenic functions obtained by us and presented in [15]. We first give some notations which will be used in the following analysis.

A function is said to be Hölder continuous on , if satisfies

Denote by the set of all Hölder continuous functions on with the index . For any , define the norm in as , where

Furthermore, for any , we have

Theorem 2 (see [15]). Let and be open subsets of and , respectively. Suppose that and satisfy and , respectively. The boundaries , of , are differentiable, oriented, compact Liapunov surfaces. If is a bihypermonogenic function in , , then

where

and the involutions and are defined by

Definition 3 (see [15]). The integral

is called a singular integral on , where , , , , and are given in Theorem 2.

Definition 4 (see [15]). Let be a constant and , where    are balls with the center at and the radius . Denote

If , then is called the Cauchy principal value of a singular integral, denoted by .

Lemma 5 (see [11]). Let be an open subset of and let be an -chain satisfying ; then

Lemma 6 (see [11]). Let , and be as in Lemma 5 and ; then

Theorem 7 (see [15]). If , then there exists the Cauchy principal value of singular integrals and

where

Set , , , and denote by . Moreover denote by and denote by the limits of when . Then we have the following important theorem.

Theorem 8 (see [15]). If , then

where .

4. The Boundary Value Problem for Bihypermonogenic Functions

In this section, we consider the boundary value problem.

Definition 9. Let and be as before. We want to find a bihypermonogenic function defined in , which is continuous to and and satisfies the nonlinear boundary condition

in which and are known functions. The above boundary value problem is called Problem R.

From Theorem 8, we can transform the boundary condition of Problem R into an integral equation

where

Theorem 10 (see [9]). Let , and let be the set of Hölder continuous functions on with the index . For and

where are as before, then is a hypermonogenic function with

where is a constant independent of .

Lemma 11 (see [6]). Let be the same as in Theorem 2. If , then there exists a constant such that

Lemma 12. If , then

where is a positive constant.

Proof. Using the following equations:

and Theorem 10, we can obtain the result.

Theorem 13. Suppose the boundaries , of , be differentiable, oriented, compact Liapunov surfaces. If , then

where is a positive constant which is independent of .

Proof. From (20), it follows that

Moreover, based on Lemma 12 we only need to prove . It is easy to prove . We rewrite as . Now we consider and write for any , and denote by the projections of on the tangent plane of , respectively. Moreover we construct spheres with the center at and radius , where , , , where is a constant as in . Denote by the part of lying inside the sphere and its surplus part, respectively, and set

From

we obtain that

Thus we have

Noting that on , we have

Similarly, we can obtain .

Next we want to prove

According to (33), we have

Similarly, we can deal with

From Lemma 11 and (35) and by , we obtain . Similarly, we can get the inequality estimation for and . By (36) and (37), , we have . Similarly, we can obtain the inequality estimation for , and .

Since

by (35)–(38), we have .

Summarizing the above discussion shows that

Thus we infer

Hence

This completes the proof.

Corollary 14. If , then

Theorem 15. Let ; then the function is a Hölder continuous function for and satisfies the Lipschitz-condition for and any , namely,

where    () is a positive constant and has nothing to do with . If , then the problem R has at least one solution, where are both positive constants satisfying .

Proof. Suppose that be denoted a subset of . From Theorem 13, Corollary 14, and (38), we obtain . Similarly, we can get . Hence, by (12) and , is derived. This shows that the operator is the mapping of .

Next we prove the is a continuous mapping.

Suppose that the sequence of functions uniformly converges to a function ; thus for arbitrary and if is large enough, then ,.

Now we consider by

where

Suppose is the -neighborhood of with the center at point and the radius , is as above; then

By (35), we can obtain that

Similarly, we can get the inequality estimations for , and . By (35), (36), and (37), we can obtain the similar inequality estimations for , , , , , , , , and , , respectively.

From

where

since and from (35), we have

Similarly, we can get the inequality estimations for ,.

From

where

since and from (38), we obtain that

Summarizing the above discussion, we conclude . Then for arbitrary , we first choose a sufficiently small number and next select a sufficiently large positive integer ; we have

where is a positive constant.

Finally, we can choose large enough such that ( is a positive constant). Hence we can obtain is a continuous mapping. According to Ascoli-Arzela Theorem, is a compact set in the space . Based on the Schauder fixed point principle, there exists a function satisfying the equation .

Corollary 16. If in Theorem 15, then the problem R has the unique solution.

Proof. This corollary is not difficult to verify by the contraction mapping principle when in Theorem 15.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was completed with the support of the National Natural Science Foundation of China (NNSFC) through Grants no. 10671207 and no. 11301136.

References

F. Brackx, R. Delanghe, and F. Sommen, “Clifford analysis,” in Research Note in Mathematics, vol. 76, Pitman Book, London, UK, 1982.

Z. Z. Xu, “Riemann boundary value problem for regular function in Clifford algebra,” Chinese Science Bulletin, vol. 32, pp. 476–477, 1987.

Z. Z. Xu, “On linear and nonlinear Riemann-Hilbert problems for regular function with values in a Clifford algebra,” Chinese Annals of Mathematics B, vol. 11, no. 3, pp. 349–358, 1990.

G. C. Wen, Clifford Annalysis and Elliptic System, Hyperbolic Systems of First Order Equations, World Scientific Publishing, Singapore, 1991.

S. Huang, “Nonlinear boundary value problem for biregular functions in Clifford analysis,” Science in China A, vol. 39, no. 11, pp. 1152–1163, 1996.

S. Huang, Y. Y. Qiao, and G. C. Wen, Real and Complex Clifford Analysis, Springer, New York, NY, USA, 2005.

Y. Y. Qiao, “A nonlinear boundary value problem with a Haseman shift for biregular function,” Journal of Systems Science and Mathematical Sciences, vol. 19, no. 4, pp. 484–489, 1999.

Y. Y. Qiao, “A nonlinear boundary value problem with a shift for generalized biregular functions,” Journal of Systems Science and Mathematical Sciences, vol. 22, no. 1, pp. 43–49, 2002.

Y. Y. Qiao, “A boundary value problem for hypermonogenic functions in Clifford analysis,” Science in China A: Mathematics, vol. 48, pp. 324–332, 2005.

S.-L. Eriksson and H. Leutwiler, “Hypermonogenic functions,” in Clifford Algebras and Their Applications in Mathematical Physics, vol. 2, pp. 287–302, Birkhäauser, Boston, Mas, USA, 2000.

S.-L. Eriksson, “Integral formulas for hypermonogenic functions,” Bulletin of the Belgian Mathematical Society, vol. 11, no. 5, pp. 705–718, 2004.

S. L. Eriksson and H. Leutwiler, “An improved Cauchy formula for hypermonogenic functions,” Advances in Applied Clifford Algebras, vol. 19, no. 2, pp. 269–282, 2009.

Z. Z. Zhang and J. Y. Du, “On certain Riemann boundary value problems and singular integral equations in Clifford analysis,” Chinese Journal of Contemporary Mathematics, vol. 22, pp. 237–244, 2001.

Z. Z. Zhang and J. Y. Du, “Laurent expansion and residue theorems in universal Clifford analysis,” Acta Mathematica Scientia A, vol. 23, no. 6, pp. 692–703, 2003.

X. L. Bian, S.-L. Eriksson, J. Li, and Y. Y. Qiao, “Cauchy integral formula and Plemelj formula of bihypermonogenic functions in real Clifford analysis,” Complex Variables and Elliptic Equations, vol. 54, no. 10, pp. 957–976, 2009.

H. J. Yang, Y. H. Xie, and Y. Y. Qiao, “A kind of boundary value problem for hypermonogenic function vectors,” Journal of Mathematical Research and Exposition, vol. 31, no. 3, pp. 490–496, 2011.

Z. Z. Zhang and K. Gürlebeck, “Some Riemann boundary value problems in Clifford analysis (I),” Complex Variables and Elliptic Equations, vol. 58, no. 7, pp. 991–1003, 2013.