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A Boundary Value Problem for Bihypermonogenic Functions in Clifford Analysis
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.
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 . 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 , 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  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.  obtained the integral formulas and Plemelj formula for bihypermonogenic functions. Yang et al.  studied a kind of boundary value problem for hypermonogenic function vector. Zhang and Gürlebeck  studied Riemann boundary value problems in Clifford analysis.
In this paper, based on the integral formulas and Plemelj formula for bihypermonogenic functions presented in , 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.
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 . 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 ). 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 4 (see ). 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 ). Let be an open subset of and let be an -chain satisfying ; then
Theorem 7 (see ). 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 ). 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 ). 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 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
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