We mainly deal with the boundary value problem for triharmonic function with value in a universal Clifford algebra: Δ3[u](x)=0, x∈Rn∖∂Ω, u+(x)=u-(x)G(x)+g(x), x∈∂Ω, (Dju)+(x)=(Dju)-(x)Aj+fj(x), x∈∂Ω, u(∞)=0, where (j=1,…,5)∂Ω is a Lyapunov surface in Rn, D=∑k=1nek(∂/∂xk) is the Dirac operator, and u(x)=∑AeAuA(x) are unknown functions with values in a universal Clifford algebra Cl(Vn,n). Under some hypotheses, it is proved that the boundary value problem has a unique solution.

1. Introduction and Preliminaries

The theory of Riemann boundary value problems in complex plane has been systematically developed in [1, 2]. It is an interesting topic to generalize the classical Riemann boundary value problems theory to Clifford analysis. In [3–6], and so forth, many interesting results about boundary value problem and Riemann Hilbert problems for monogenic functions in Clifford analysis are presented. In [7], Green’s function for the Dirichlet problem for polyharmonic equations was studied. The aim of this paper is to study the Riemann boundary value problem for triharmonic functions. At first, based on the higher order Cauchy integral representation formulas in [8, 9] and the Plemelj formula, we give some properties of triharmonic functions in Clifford analysis, for example, the mean value theorem, the Painlevé theorem, and so forth. Furthermore, on the basis of the above results, we consider the following Riemann boundary value problems:
(1)Δ3[u](x)=0,x∈Rn∖∂Ω,u+(x)=u-(x)A+g(x),x∈∂Ω,(Dju)+(x)=(Dju)-(x)Aj+fj(x),x∈∂Ω,|u(∞)|≤C*,(2)Δ3[u](x)=0,x∈Rn∖∂Ω,u+(x)=u-(x)G(x)+g(x),x∈∂Ω,(Dju)+(x)=(Dju)-(x)Aj+fj(x),x∈∂Ω,u(∞)=0,
where (j=1,…,5).

In (1) and (2), A and Aj are invertible constants; we denote the inverse elements as A-1 and Aj-1.u(x), (Dju)(x), g(x), fj(x)∈Hβ(∂Ω,Cl(Vn,n)), j=1,…,5, 0<β≤1. The explicit solutions for (1) are given and the boundary value problem (2) is shown to have a unique solution under some hypotheses.

Let Vn,s (0≤s≤n) be an n-dimensional (n≥1) real linear space with basis {e1,e2,…,en}, and Cl(Vn,s) the universal Clifford algebra over Vn,s. For more information on Cl(Vn,s) (0≤s≤n), we refer to [10–12].

Throughout this paper, suppose Ω is an open, bounded nonempty subset of Rn with a Lyapunov boundary ∂Ω, denoting Ω+=Ω, Ω-=Rn∖Ω¯. In this paper, for simplicity, we will only consider the case of s=n. The operator D is given as
(3)D=∑k=1nek∂∂xk:Cr(Ω,Cl(Vn,n))⟶Cr-1(Ω,Cl(Vn,n)).

Let u be a function with value in Cl(Vn,n), defined in Ω, and the operator D acts on the function u from the left and from the right, which is being governed by the following rule:
(4)D[u]=∑k=1n∑AekeA∂uA∂xk,[u]D=∑k=1n∑AeAek∂uA∂xk.

Definition 1.

A compact surface Γ is called Lyapunov surface with Hölder exponent α, if the following conditions are satisfied.

At each point x∈Γ there is a tangential space.

There exists a number r, such that for any point x∈Γ the set Γ∩Br(x) (Lyapunov ball) is connected and parallel lines to the outer normal α(x) intersect at not more than one point.

The normal α(x) is Hölder continuous on Γ; that is, there are constants C>0 and 0<α≤1 such that for x,y∈Γ(5)|α(x)-α(y)|≤C|x-y|α.

Let Ω be an open nonempty subset of Rn with a Lyapunov boundary, u(x)=∑AeAuA(x), where uA(x) are real functions; u(x) is called a Hölder continuous function on Ω¯ if the following condition is satisfied:
(6)|u(x1)-u(x2)|=[∑A|uA(x1)-uA(x2)|2]1/2≤C|x1-x2|α,
where for any x1,x2∈Ω¯, x1≠x2, 0<α≤1, C is a positive constant independent of x1,x2.

Let Hα(∂Ω,Cl(Vn,n)) denote the set of Hölder continuous functions with values in Cl(Vn,n) on ∂Ω (the Hölder exponent is α, 0<α<1). We denote the norm in Hα(∂Ω,Cl(Vn,n)) as
(7)∥u∥(α,∂Ω)=∥u∥∞+∥u∥α,
where
(8)∥u∥∞∶=supx∈∂Ω|u(x)|,∥u∥α∶=supx1,x2∈∂Ωx1≠x2|u(x1)-u(x2)||x1-x2|α.

Lemma 2.

The Hölder space Hα(∂Ω,Cl(Vn,n)) is a Banach space with norm (7).

Denote the fundamental solutions of Dj (j=1,2,…,6) by
(9)H1(x)=1ωnxρn(x),H2(x)=12-n1ωn1ρn-2(x),H3(x)=12(2-n)1ωnxρn-2(x),H4(x)=12(2-n)(4-n)1ωn1ρn-4(x),H5(x)=18(2-n)(4-n)1ωnxρn-4(x),H6(x)=18(2-n)(4-n)(6-n)1ωn1ρn-6(x),
where ρ(x)=(∑i=1nxi2)1/2 and ωn denotes the area of the unit sphere in Rn (n≥3, n≠4, n≠6).

We will introduce the following operators:
(10)(F∂Ωu)(x)=∫∂Ω2H1(y-x)dσyu(y),x∈Rn∖∂Ω,(11)(S∂Ωu)(x)=∫∂Ω2H1(y-x)dσyu(y),x∈∂Ω,
where u∈Hα(∂Ω,Cl(Vn,n)).

Lemma 3 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

The integral operator S∂Ω in (11) is a bounded linear operator mapping the function space Hα(∂Ω,Cl(Vn,n)) into itself; that is, there exists a positive constant M such that, for all u∈Hα(∂Ω;Cl(Vn,n)),
(12)∥S∂Ωu∥(α,∂Ω)≤M∥u∥(α,Cl(Vn,n).

2. Some Properties for Triharmonic FunctionsTheorem 4 (Gauss-mean value formula for triharmonic functions, see [<xref ref-type="bibr" rid="B12">6</xref>, <xref ref-type="bibr" rid="B13">13</xref>]).

Suppose Δ3[u]=0 in Rn; then, for any x∈Rn,
(13)u(x)=1ωnRn-1∫∂B(x,R)u(y)dS-R22nΔ[u](x)-R48n(n+2)Δ2[u](x),
or
(14)u(x)=nωnRn∫B(x,R)u(y)dV-R22(n+2)Δ[u](x)-R48(n+2)(n+4)Δ2[u](x),
where ωn denotes the area of the unit sphere in Rn.

Corollary 5.

Suppose Δ3[u]=0 in Rn and |u(x)|=O(|x|) (|x|→∞); then Δ[u]=0 and Δ2[u]=0 in Rn.

Corollary 6.

Suppose Δ3[u]=0 in Rn and u(x) is bounded in Rn; then u(x)≡C.

Theorem 7.

Suppose Δ3[u]=0 in Rn∖∂Ω, and for x∈∂Ω, u∈C6(Rn∖∂Ω,Cl(Vn,n)), [u]+(x)=[u]-(x)∈C5(∂Ω,Cl(Vn,n)), and, moreover, Dj[u]+(x)=Dj[u]-(x)∈Hαj(∂Ω,Cl(Vn,n)), 0<αj≤1, where j=0,1,…,5. Then Δ3[u]=0 in Rn.

Theorem 8.

Let u∈C6(Ω-,Cl(Vn,n))∩C5(Ω-¯,Cl(Vn,n)), Δ3[u]=0 in Ω-, Dj[u](x)∈Hαj(∂Ω,Cl(Vn,n)), 0<αj≤1 (j=0,1,…,5), and |u(x)|=O(1) (|x|→∞); then for x∈Ω-(15)u(x)=∑j=05(-1)j+1∫∂ΩHj+1(y-x)dσyDj[u](y)+C,
where Hj(y-x) is as in (9) and C is a constant.

Proof.

For y∈∂Ω, denote Dj[u](y)=-Ψj(y), (j=0,1,…,5). For x∈Rn∖∂Ω, denoting
(16)Θ(x)=∑j=05(-1)j∫∂ΩHj+1(y-x)dσyΨj(y),u*(x)={-Θ(x),x∈Ω+u(x)-Θ(x),x∈Ω-,
then Δ3[u*]=0 in Rn∖∂Ω. By using Plemelj formula, combining with weak singularity of Hj(x) (j≥2), we obtain the following:
(17)Dj[u*]+(x)=Dj[u*]-(x)∈Hαj′(∂Ω,Cl(Vn,n)),
where 0<αj′≤1 (j=0,1,…,5). By using Theorem 7, we have Δ3[u*]=0 in Rn. It is clear that we have |u*(x)|=O(1) (|x|→∞). In view of Corollary 6, then the results follow.

Corollary 9.

Let u∈C6(Ω-,Cl(Vn,n))∩C5(Ω-¯,Cl(Vn,n)), Δ3[u]=0 in Ω-, Dj[u](x)∈Hαj(∂Ω,Cl(Vn,n)), 0<αj≤1 (j=0,1,…,5), and |u(x)|=O(1) (|x|→∞), then
(18)Dj[u](∞)=0,j=1,…,5,

Remark 10.

When the condition |u(x)|=O(1) (|x|→∞) in Corollary 9 is replaced by u(∞)=0, then the results in Corollary 9 are still valid.

3. Riemann Boundary Value Problem for Triharmonic Functions

In this section, we will consider the Riemann boundary value problem (1); the explicit expression of the solution is given.

Theorem 11.

The Riemann boundary value problem (1) is solvable and the solution can be written as
(19)u(x)={∑i=16Ψi(x)+C,x∈Ω,∑i=16Ψi(x)+CA-1,x∈Ω-,
where
(20)Ψ1(x)={-∫∂ΩH6(y-x)dσyf5(y),x∈Ω+-∫∂ΩH6(y-x)dσyf5(y)A5-1,x∈Ω-(21)Ψ2(x)={∫∂ΩH5(y-x)dσyf~4(y),x∈Ω+∫∂ΩH5(y-x)dσyf~4(y)A4-1,x∈Ω-(22)Ψ3(x)={-∫∂ΩH4(y-x)dσyf~3(y),x∈Ω+-∫∂ΩH4(y-x)dσyf~3(y)A3-1,x∈Ω-(23)Ψ4(x)={∫∂ΩH3(y-x)dσyf~2(y),x∈Ω+∫∂ΩH3(y-x)dσyf~2(y)A2-1,x∈Ω-(24)Ψ5(x)={-∫∂ΩH2(y-x)dσyf~1(y),x∈Ω+-∫∂ΩH2(y-x)dσyf~1(y)A1-1,x∈Ω-(25)Ψ6(x)={∫∂ΩH1(y-x)dσyg~(y),x∈Ω+∫∂ΩH1(y-x)dσyg~(y)A-1,x∈Ω-(26)f~4(x)=f4(x)-∫∂ΩH2(y-x)dσyf5(y)(-1+A5-1A4),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhihx∈∂Ω,(27)f~3(x)=f3(x)+∫∂ΩH3(y-x)dσyf5(y)(-1+A5-1A3)-∫∂ΩH2(y-x)dσyf~4(y)(-1+A4-1A3),mmmmmmmmmmmmmmmmmmmmx∈∂Ω,(28)f~2(x)=f2(x)-∫∂ΩH4(y-x)dσyf5(y)(-1+A5-1A2)+∫∂ΩH3(y-x)dσyf~4(y)(-1+A4-1A2)-∫∂ΩH2(y-x)dσyf~3(y)(-1+A3-1A2),mmmmmmmmmmmmmmimmmmmmx∈∂Ω,(29)f~1(x)=f1(x)+∫∂ΩH5(y-x)dσyf5(y)(-1+A5-1A1)-∫∂ΩH4(y-x)dσyf~4(y)(-1+A4-1A1)+∫∂ΩH3(y-x)dσyf~3(y)(-1+A3-1A1)-∫∂ΩH2(y-x)dσyf~2(y)(-1+A2-1A1),mmmmmmmmmmmmmmmmmmx∈∂Ω,(30)g~(x)=g(x)-∫∂ΩH6(y-x)dσyf5(y)(-1+A5-1A)+∫∂ΩH5(y-x)dσyf~4(y)(-1+A4-1A)-∫∂ΩH4(y-x)dσyf~3(y)(-1+A3-1A)+∫∂ΩH3(y-x)dσyf~2(y)(-1+A2-1A)-∫∂ΩH2(y-x)dσyf~1(y)(-1+A1-1A),mmmmmmmmmmmimmmmmmx∈∂Ω.

4. Existence of Solutions for Riemann Boundary Value Problem for Triharmonic FunctionsTheorem 12.

Suppose G(x)∈Hα(∂Ω,Cl(Vn,n))0<α<1 and G(x) satisfies the following condition:
(31)2n-2∥1-G(x)∥(α,∂Ω)(M+1)<1,
where M is the positive constant mentioned in Lemma 3. Then (2) admits a unique solution.

Proof.

Denoting w(x)=D5[u](x) then w+(x)=w-(x)A5+f5(x), x∈∂Ω. Moreover, by D5[u](∞)=0. we get the following:
(32)ω(x)={∫∂ΩH1(y-x)dσyf5(y),x∈Ω+∫∂ΩH1(y-x)dσyf5(y)A5-1,x∈Ω-,
With Ψ1(x) being as in (20), it is easy to check that
(33)D5(u-Ψ1)(x)=0,x∈Rn∖∂Ω.
Denoting Δ2u(x)-Δ2Ψ1(x)∶=φ1(x), x∈Rn∖∂Ω, and using (Δ2u)+(x)=(Δ2u)-(x)A4+f4(x), x∈∂Ω, we conclude that
(34)φ1+(x)=φ1-(x)A4+f~4(x),x∈∂Ω,
where f~4(x)∈Hβ~(∂Ω,Cl(Vn,n)), 0<β~≤1 being as in (26). By Corollary 9, it is clear that φ1(∞)=0, and we then get the following representation formula:
(35)φ1(x)={∫∂ΩH1(y-x)dσyf~4(y),x∈Ω+∫∂ΩH1(y-x)dσyf~4(y)A4-1,x∈Ω-.
Analogously, we find with Ψ2(x) from (21) that Δ2u(x)-Δ2Ψ1(x)-Δ2Ψ2(x)=0, x∈Rn∖∂Ω. Denote
(36)D3u(x)-D3Ψ1(x)-D3Ψ2(x)∶=φ2(x),hhhhhhhhhhhhhhhhhhhhhix∈Rn∖∂Ω.
Using the condition (D3u)+(x)=(D3u)-(x)A3+f3(x), x∈∂Ω, we obtain that
(37)φ2+(x)=φ2-(x)A3+f~3(x),x∈∂Ω,
where f~3(x)∈Hβ~(∂Ω,Cl(Vn,n)), 0<β~≤1 being as in (27). By Corollary 9, it is clear that φ2(∞)=0, and we then get the following representation formula:
(38)φ2(x)={∫∂ΩH1(y-x)dσyf~3(y),x∈Ω+∫∂ΩH1(y-x)dσyf~3(y)A3-1,x∈Ω-.

Here Ψ3(x) being as in (22), then we have D3u(x)-D3Ψ1(x)-D3Ψ2(x)-D3Ψ3(x)=0, x∈Rn∖∂Ω. We denote
(39)Δu(x)-ΔΨ1(x)-ΔΨ2(x)-ΔΨ3(x)∶=φ3(x)
and use the condition (Δu)+(x)=(Δu)-(x)A2+f2(x), x∈∂Ω. We conclude that
(40)φ3+(x)=φ3-(x)A2+f~2(x),x∈∂Ω,
where f~2(x)∈Hβ~(∂Ω,Cl(Vn,n)), 0<β~≤1 being as in (28). Corollary 9 ensures that φ3(∞)=0; then
(41)φ3(x)={∫∂ΩH1(y-x)dσyf~2(y),x∈Ω+∫∂ΩH1(y-x)dσyf~2(y)A2-1,x∈Ω-.
Use the same way again, Ψ4(x) being as in (23), that Δu(x)-ΔΨ1(x)-ΔΨ2(x)-ΔΨ3(x)-ΔΨ4(x)=0, x∈Rn∖∂Ω. Denoting
(42)Du(x)-∑i=14DΨi(x)∶=φ4(x),x∈Rn∖∂Ω
and using (Du)+(x)=(Du)-(x)A1+f1(x), x∈∂Ω, we get
(43)φ4+(x)=φ4-(x)A1+f~1(x),x∈∂Ω,
where f~1(x)∈Hβ~(∂Ω,Cl(Vn,n)), 0<β~≤1 being taken from (29). By Corollary 9, it is clear that φ4(∞)=0, and we then get the following representation formula:
(44)φ4(x)={∫∂ΩH1(y-x)dσyf~1(y),x∈Ω+∫∂ΩH1(y-x)dσyf~1(y)A1-1,x∈Ω-.

Finally, we use Ψ5(x) as defined in (24) and get Du(x)-∑i=14DΨi(x)=0, x∈Rn∖∂Ω. Define
(45)u(x)-∑i=15Ψi(x)∶=φ5(x),x∈Rn∖∂Ω.
Working with the condition u+(x)=u-(x)G(x)+g(x) we arrive at
(46)φ5+(x)=φ5-(x)G(x)+g~(x),x∈∂Ω,
where g~(x)∈Hβ~(∂Ω,Cl(Vn,n)), 0<β~≤1 being taken from (30). It is clear that φ5(∞)=0. We obtain that
(47)Dφ5=0,Rn∖∂Ωφ5+(x)=φ5-(x)G(x)+g~(x),x∈∂Ωφ5(∞)=0.

We only need to consider the existence of solutions to (47). The solution to this problem may be written in the form
(48)φ5(x)=∫∂ΩH1(y-x)dσyφ6(y),
where φ6(y) is a Hölder continuous function to be determined on ∂Ω. Then, by using Plemelj formula, (47) can be reduced to an equivalent singular integral equation for φ6,
(49)φ6(x)=[φ6(x)2-∫∂ΩH1(y-x)dσyφ6(y)](1-G(x))+g~(x),mmmmmmmmmmmmmmmmmmmmmmmmmmx∈∂Ω.

Letting T denote the integral operator defined by the right hand side of (49), we get
(50)(Tφ6)(x)=[φ6(x)-(S∂Ωφ6)(x)](1-G(x))2+g(x)~.
For any ω1,ω2∈Hα(∂Ω,Cl(Vn,n)), we have
(51)∥Tω1-Tω2∥(α,∂Ω)≤2n-12∥ω1-ω2∥(α,∂Ω)∥1-G∥(α,∂Ω)(1+M).
Under the condition (31), the integral operator T is a contraction operator mapping the Banach space Hα(∂Ω,Cl(Vn,n)) into itself, which has a unique fixed point for the operator T. Thus, there exists a unique solution to (47). The proof is done.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (11271175), the AMEP, and DYSP of Linyi University.

LuJ. K.MuskhelishvilliN. I.GongY.DuJ.A kind of Riemann and Hilbert boundary value problem for left monogenic functions in Rm(m≥2)ZhangZ.GürlebeckK.Some Riemann boundary value problems in Clifford analysis (I)XuZ. Y.On linear and nonlinear Riemann-Hilbert problems for regular function with values in a Clifford algebraGuL.DuJ.ZhangZ.Riemann boundary value problems for triharmonic functions in Clifford analysisKalmenovT. S.KoshanovD. B.Representation for the Green's function of the Dirichlet problem for polyharmonic equations in a ballZhangZ.On k-regular functions with values in a universal Clifford algebraBegehrH.DuJ.ZhangZ.On higher order Cauchy-Pompeiu formula in CLIfford analysis and its applicationsDelangheR.On regular-analytic functions with values in a Clifford algebraDelangheR.On the singularities of functions with values in a Clifford algebraBrackxF.DelangheR.SommenF.GuL.ZhangZ.Gauss-mean value formula for triharmonic functions and its applications in Clifford analysis