# Linear BVPs and SIEs for Generalized Regular Functions in Clifford Analysis

Authors: Pingrun Li; Lixia Cao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study some properties of a regular function in Clifford analysis and generalize Liouville theorem and Plemelj formula with values in Clifford algebraAn(R). 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.


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## Body

## 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 Rn with values in Clifford algebra space An(R) 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 inAn(R). 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

LetAn(R) be a real Clifford algebra over an n-dimensional real vector space Rn with the orthogonal basis {e1,e2,…,en}, where e1=1 is an identity element. The product on Rn is defined by (1)e12=1,ei2=-1i=2,3,…,n,e1ei=eie1i=2,3,…,n,eiej+ejei=0,i≠j;-2,i=j2≤i,j≤n.Hence, any element x∈Rn has the type x=∑k=1n‍xkek(xk∈R). And An(R) is a 2n-1-dimensional real linear space, whose basis is {e1,e2,…,en;e2e3,…,en-1en;…;e2e3,…,en}. Each basis element of An(R) has a representation of the form eA=eα1α2⋯αr=eα1eα2⋯eαr, where A={α1,α2,…,αr}⊆{1,2,…,n} and 1≤α1<α2<⋯<αr≤n, and when A=1, α1=e1=1. Therefore, any element a∈An(R) can be denoted as a=∑A‍aAeA, where aA∈R is a real number. The norm for an element a∈An(R) is taken to be (2)a=a,a=a2=∑AaA2.It is easy to prove that, for anya,b∈An(R), we have (3)a+b≤a+b,ab≤2n-1ab.In fact, we denote a=∑A‍aAeA, b=∑A‍bAeA, then (4)a+b≤∑AaA+bA2≤∑AaA21/2+∑AbA21/2=a+b;ab≤∑A22naA2bA2≤2n-1∑AaA21/2∑AbA21/2=2n-1ab.Definition 1.
LetΩ⊂Rn be a nonempty, open, and connected set. A function f(x) defined in Ω and with values in Clifford algebra An(R) can be expressed as f(x)=∑A‍fA(x)eA; that is, f:Ω→An(R), where fA(x) is a real function.
LetC(r)(Ω) denote the set of all functions defined in Ω with derivatives of order r(≥1). We define the function class FΩ(r) in C(r)(Ω) as follows: (5)FΩr=f∣f:Ω⟶AnR,fx=∑AfAxeA,fAx∈CrΩ,x∈Ω,where Ω⊂Rn and r(≥1) is an integer.Definition 2.
Letf:Ω→An(R). If there exists a constant M>0 such that (6)fx1-fx2≤Mx1-x2α,0<α≤1,for any x1,x2∈Ω, then we call f to be Hölder continuous function in Ω, denoted by f∈HΩα, where M and α are called the Hölder constant and the Hölder index, respectively. And we define generalized Cauchy-Riemann operators (i.e., Dirac operator) D,D¯ in FΩ(r) as follows:(7)D=e1∂1-∑i=2nei∂i,D¯=∑i=1nei∂i.For any f∈FΩ(r)(r≥1), the operators D, D¯ act on function f being governed by the rules (8)Dxf=∑i=1n∑AeieA∂fA∂xi,fDx=∑A∑i=1neAei∂fA∂xi,then a solution of Dxf=0 is called left regular function in Ω. Similarly, a solution of fDx=0 is called right regular function. Generally the left regular function is called regular function in short.Let the boundary∂Ω of Ω be a smooth, directional, and compact Liapunov surface. We denote the set of all Hölder continuous function defined on ∂Ω as H(∂Ω,β) and define the norm as (9)fβ=Cf,∂Ω+Hf,∂Ω,β,where C(f,∂Ω)=maxt∈∂Ω|f(t)| is a continuous norm, and (10)Hf,∂Ω,β=supt1≠t2,t1,t2∈∂Ωft1-ft2t1-t2βis an Hölder norm. It is easy to prove that H(∂Ω,β) is a Banach space.Definition 3.
If there existsb (b∈An(R)) such that ab=ba=1, we call b as the inverse of a, denoted by a-1.In the following,x∈Rn can be expressed as x=∑i=1n‍xiei, x=xR+xnen, or x=(xR,xn), where xR=∑i=1n-1‍eixi. Thus, Rn can be divided into three parts: R+n={x∣xn>0}, R-n={x∣xn<0}, and R0n={x∣xn=0}. Obviously, R¯+n=R+n∪R0n and R¯-n=R-n∪R0n. However, R0n can also be represented by Rn-1, and R+n, R-n lie in the upper and lower half spaces of the hyperplane R0n, respectively.The conjugate of a numbera is the number a¯ given by a¯=∑A‍aAe¯A, where e¯A=e¯α1e¯α2⋯e¯αj, e¯j=-ej(j=1,2,…n). For any x∈Rn, we know that xx¯=-|x|2, and we can verify that x-1=-x¯/|x|2 is an inverse element of x≠0; that is, xx-1=x-1x=1.LetAn-1(R) be the subalgebra constructed by {e1,e2,…,en-1}, then An-1(R)⊆An(R) and An(R) has the decomposition An(R)=An-1(R)+enAn-1(R). It is clear that the decomposition is the generalization of the classical representation of a complex number. Thus, any x=∑A‍xAeA∈Rn(A) can be decomposed as x=b+cen, where b,c∈An-1(R). We may define operators P, Q as Px=b, Qx=c.Definition 4.
Letf(x) be defined in Rn, a∈Rn. If Dxf(a)=0, then we call a to be a regular point. If Dxf(a)≠0, but there always exists b in any neighborhood of a such that Dxf(b)=0, then we call a to be a singular point. If there is a hollow neighborhood U(a) of a such that Dxf(x)=0 for any x∈U(a¯)=U(a)-{a}, we call a to be an isolated point of f(x).Definition 5.
Iff(x) is a regular function in Rn, that is, Df(x)=0 and f(x)∈FΩ(r), then we say that f(x) is an entire function in Rn, and x=∞ is the only singular point of f(x) in Rn.


## 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(11)Fx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∈Rn∖∂Ω,where wn-1=2πn/2/Γ(n/2) denotes the surface area of a unit sphere in Rn and θ(t)=∑j=1n‍ejcos⁡(m,ej). m is a normal vector at t∈∂Ω, and dSt is a Lebesgue measure on ∂Ω. f(t) is a bounded Hölder continuous function on ∂Ω, that is, f(t)∈H(∂Ω,β)(0<β≤1), then (12)F±x=±12fx+Pfx,x∈∂Ω,where (13)Pfx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∈∂Ω.Lemma 7.
Lett,x∈Rn(n≥2) and m≥0 be an integer. Then one has(14)t-x1¯t-x1n-t-x2¯t-x2n≤Bmx1,x2,tt-x1n-1t-x2n-2·x1-x2,where (15)Bmx1,x2,t=∑t=0mt-x1m-kt-x2k,m>0;1,m=0.Proof.
It can easily be proved by a direct calculation.Lemma 8.
Iff(x)∈H(∂Ω,β), then (Pf)(x)∈H(∂Ω,β) and F±(x)∈H(∂Ω,β).Proof.
In order to prove that(Pf)(x)∈H(∂Ω,β), we consider (16)Pfx1-Pfx2=1wn-1∫∂Ωt-x1¯t-x1nθtftdSt-1wn-1∫∂Ωt-x2¯t-x2nθtftdSt=1wn-1∫∂Ωt-x1¯t-x1n-t-x2¯t-x2nθtftdSt≤1wn-1∫∂ΩBmx1,x2,tftθtt-x1n-2t-x2n-2dSt·x1-x2β=Mx1-x2β,where (17)M=1wn-1∫∂ΩBmx1,x2,tftθtt-x1n-2t-x2n-2dSt;that is, there exists a constant M>0 such that |(Pf)(x1)-(Pf)(x2)|≤M|x1-x2|β. Therefore, we get (Pf)(x)∈H(∂Ω,β).
Similarly, by Hile lemma (see [12]) we can prove ±f/2+Pfβ≤Cfβ, thus, F±(x)∈H(∂Ω,β).
The proof of Lemma8 is complete.Lemma 9 (see [13]).
LetΩ be a bounded domain in Rn, then the integral (18)P.V.1wn-1∫∂Ωt-x¯t-xnθtdSt,x∈∂Ω;that is, (19)1wn-1∫∂Ωt-x¯t-xnθtdStexists under the meaning of Cauchy principal value, and its value is equal to 1/2, where ∂Ω and θ(t) are the same as before.Lemma 10 (see [14]).
Letf(x)∈H(∂Ω,β), then the Cauchy type integral (20)Fx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∉∂Ωis a regular function in Ω, and DxF(x)=0, F(∞)=0.The following Lemmas11–13 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).
Letf(x) be a regular function in Rn and f(x)∈FΩ(r)(r≥1); that is, f(x) is an entire function. Then one has the following:(1)
Iff(∞)=0, then f(x)≡0.(2)
Iff(∞)=C, then f(x)≡C, where C is a Clifford constant.(3)
If|f(x)|≤M (constant) for any x∈Rn, then f(x) is a constant in Rn.(4)
Iflimx→∞f(x) exists, then f(x) is a constant in Rn.Lemma 12 (generalized Cauchy integral theorem).
Letf(x) be a regular function in Ω⊂Rn and Ω~ be a single connected domain of Ω. For any smooth, closed, and rectifiable curve, γ∈Ω~. Then (21)1wn-1∫γt-x¯t-xnθtftdSt=0,x∈Ω~.Lemma 13 (generalized Cauchy integral formula).
Letf(x) be a regular function in Ω⊂Rn and U⊂Ω be a bounded domain. ∂U is a smooth, differentiable, and compact surfaces. Then (22)fz=1wn-1∫∂Ut-z¯t-znθtftdStfor any z∈U, where U∪∂U=U¯⊂Ω.


## 4. BVP in Clifford Analysis

LetΩ⊂Rn be an open and connected set and the boundary ∂Ω of Ω be a smooth, compact, oriented, and closed Liapunov surface. Rn is divided into two domains Ω+=Ω and Ω-=Rn∖Ω by ∂Ω, with ∞∈Ω-. Our goal is to obtain an An(R)-valued function f(t) such that it is regular in Ω± and continuous on Ω±+∂Ω, and we have the following boundary value condition: (23)f+t=f-tB+ht,t∈∂Ω,where B∈An(R) is a given Clifford constant with B≠0 and B right inverse B-1. A known function h(t)∈H(∂Ω,β). f±(t) are the boundary values of f(t) on ∂Ω, respectively. If the order of f(t) is m at infinity, such a problem can be denoted as Rm. Actually, the problem R0 and problem R-1 are frequently discussed. On the problem R0, f(∞) is supposed to be finite and nonzero. On R-1, f(∞) is assumed to be zero. In order that a solution of (23) exists, we require f(t)∈H(∂Ω,β) (i.e., f±(t)∈H(∂Ω,β)) and the following condition is fulfilled: (24)1wn-1∫∂Ωft1+t21/2n-1dSt<+∞,where dSt is a unit area element on ∂Ω. Consider (25)Fx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∈Rn∖∂Ω,where wn-1 is the surface area of a unit sphere in Rn and θ(t)=∑j=1n‍ejcos⁡(m,ej); m is an outer normal vector at t∈∂Ω. By Lemmas 6–8 we have (26)F±x=±12fx+Pfx,x∈∂Ω,where (27)Pfx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∈∂Ω.Under condition (24), F(x) is regular for all x∈Rn∖Ω and DxF(x)=0. And (Pf)(x) is a singular integral and it is convergent under the meaning of Cauchy principal value. Moreover, F(x) is continuously extended to the boundary ∂Ω form Ω and Rn∖Ω, respectively, and then obtain the boundary values F+(x) and F-(x). Note that (28)Pfx=1wn-1∫∂Ωt-x¯t-xnθtftdSt=limδ→0+1wn-1∫t-x≥δt-x¯t-xnθtftdSt,x∈∂Ω.Particularly, when f(t)=1, we have (29)P1x=limδ→0+1wn-1∫t-x≥δt-x¯t-xnθtdSt=0,x∈∂Ω.Moreover, by [15–17] and Lemma 9, we can obtain (30)
1
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n
-
1
∫
∂
Ω
t
-
x
¯
t
-
x
n
θ
t
d
S
t
=
1
2
,
x
∈
Ω
;
-
1
2
,
x
∈
R
n
∖
Ω
.Since B∈An(R) and B(B≠0) present right inverse element B-1, we have BB-1=B-1B=I.Define(31)Yx=B,x∈Ω,1x∈Rn∖Ω.Obviously, Y(x) is a regular function and there exists an inverse element Y-1(x), and (32)Y+x=B,Y-x=1,Y+xY-1x=B,DxYx=0,x∈Rn∖Ω.Therefore, (23) can be transformed into (33)F+xY+x-1=F-xY-x-1+htY+x-1,x∈∂Ω.Denote(34)Hx=1wn-1∫∂Ωt-x¯t-xnθthtB-1dSt;it is easily seen that H(∞)=0, DxH(x)=0(x∈Rn∖Ω). By applying Lemmas 6 and 10 for H(x) in (34), we obtain (35)H+x-H-x=hxB-1.Combining (33) with (35), we have (36)F+xY+x-1-H+x=F-xY-x-1-H-x,x∈∂Ω.It follows from Painleve’s theorem (see [18]) that F(x)Y(x)-H(x) is a regular function in Rn. Since (23) find a solution in R0, thus, by Lemma 11, F(x)Y(x)-H(x) is a constant at x=∞; that is, there exists a Clifford constant C such that (37)FxYx-Hx=C;therefore, we obtain (38)Fx=YxHx+C=YxHx+YxC.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 F(x) of (23) is a nonzero constant at ∞, then, its solution is of the form (39)Fx=Yxwn-1∫∂Ωt-x¯t-xnθthtB-1dSt+YxC,x∈Rn∖Ω,where Y(x) is determined by (32).By Theorem14 and Lemmas 11–13, we obtain the following.Theorem 15.
Suppose that a solutionF(x) of (23) is zero at ∞, then (23) find a solution in problem R-1, and its solution is given by (40)Fx=Yxwn-1∫∂Ωt-x¯t-xnθthtB-1dSt,x∈Rn∖Ω.


## 5. SIE in Clifford Analysis

In Clifford analysis we solve following SIE of the form(41)K0f≡fxP+1wn-1∫∂Ωt-x¯t-xnθtftdSt·Q=gx,x∈∂Ω,where g(t)∈H(∂Ω,β)(0<β<1) and P, Q are Clifford constants. P+Q and P-Q have inverse elements and denoted them as (P+Q)-1 and (P-Q)-1, respectively. wn-1 is the same as the before. Ω⊂Rn is a bounded domain and its boundary ∂Ω is a smooth, oriented, and compact Lyapunov surface. And f(t)∈H(∂Ω,β)(0<β<1) is an unknown function. When g(t)≡0, we call that (41) is a homogeneous equation; otherwise (41) is nonhomogeneous. Since P+Q and P-Q are reversible on ∂Ω, (41) is called a regular type SIE with Cauchy kernel, and K0 is called the corresponding singular operator.Define a sectionally holomorphic function as follows:(42)Fx=1wn-1∫∂Ωt-x¯t-xnθtftdSt,x∉∂Ω;by applying Lemma 6 for (42), we obtain (43)fx=F+x-F-x,1wn-1∫∂Ωt-x¯t-xnθtftdSt=F+x+F-x,x∈∂Ω.Putting (43) into (41), one has (44)F+x=F-xP-QP+Q-1+gxP+Q-1.It follows from (42) that F(∞)=0. If (41) has a solution f(t)∈H(∂Ω,β), then a function F(x) defined by (42) is a solution of (44) in class R-1. Conversely, if (44) has a solution F(x) in class R-1, then f(x) obtained by the first equality of (43) is a solution of (41). Indeed, by the first equality of (43) and F(∞)=0, we can know that (42) and the second equality of (43) are fulfilled. Substituting (43) into (44), we easily obtain that f(t) satisfies (41). Thus, the problem (41) is equivalent to solving the problem (44) in class R-1.In order to solve (41), we define two functions Y(x) and H(x) as follows: (45)Yx=P-QP+Q-1,x∈Ω1,x∈Rn∖Ω,Hx=1wn-1∫∂Ωt-x¯t-xnθtgtP+Q-1dSt,x∈Rn∖Ω.The remaining discussions are the same as in Section4. We will not elaborate on that here.Thus, we have the following conclusions.Theorem 16.
Suppose thatP+Q, P-Q are reversible, then (41) has a solution on H(∂Ω,β), and its solution is obtained by (46)fx=F+x-F-x,where (47)Fx=gxa~+2wn-1∫∂Ωt-x¯t-xnθtgtb~dSt·c~,a~=PP+Q-1P-Q-1,b~=P+Q-1,c~=QP-Q-1.Note that, in Theorem16, if P=0, then Q-1 must exist. Therefore, we have the following.Corollary 17.
WhenP=0, (41) is solvable, its solution is still (46), where F(x) in (46) is determined by (48)Fx=2wn-1∫∂Ωt-x¯t-xnθtgtQ-1dSt,t∈∂Ω.Proof.
It can be proved directly from Theorem16.


## 6. BVP in Quaternion Analysis

Quaternion analysis researches analytical theory in quaternion algebraH. Quaternion algebra H is a four-dimensional real vector space with the basis elements 1, i, j, k, which satisfy the following multiplication rules: (49)i2=j2=k2=-1;ij=-ji=k;jk=-kj=i;ki=-ik=j.Therefore, H can be written as (50)H=x∣x=x1+ix2+jx3+kx4∣xt∈R,t=1,2,3,4.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 domainΩ⊂H, its boundary ∂Ω is a smooth, compact, and oriented Liapunov surface. Denote Ω+=Ω, Ω-=H∖Ω. We want to get a function f(t) such that it is regular in Ω± and continuous on Ω++∂Ω (or Ω-+∂Ω), and we have the following boundary value condition: (51)f+t=f-tB+ht,t∈∂Ω,B≠0,where B∈H is a known Clifford constant and B  shows the inverse element B-1. The known function h(t) is Hölder-continuous on ∂Ω. If f(∞)=C (constant) with C≠0, BVP (51) is denoted as R0; if f(∞)=0, this case is denoted as R-1; if the order of f(t) is m at ∞, denote it as Rm. By Theorems 14 and 16, we obtain a solution of (51) in Rm: (52)Fq=Yqwn-1∫∂Ωt-q¯t-qnθthtB-1dSt+∑l=0mqlal,where al∈H, q∈H∖∂Ω. Y(q), wn-1, and θ(t) 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 inAn(R), 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.



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