# A Boundary Value Problem for Bihypermonogenic Functions in Clifford Analysis

Authors: Xiaoli Bian; Yuying Qiao
Journal: Abstract and Applied Analysis
 (2014)

---

## 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.


---

## Body

## 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 inRn+1 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

LetCl0,n be a real Clifford algebra over an n+1-dimensional real vector space Rn+1 with orthogonal basis e:={e0,e1,…,en}, satisfying the relation eiej+ejei=-2δij  (i,j=1,…,n), where δij is the usual Kronecker delta. Then Cl0,n has its basis e0=1,e1,…,en;e1e2,…,en-1en;…;e1⋯en. Hence the real Clifford algebra is formed by the elements presented as a=∑A‍xAeA,  xA∈R, where A={i1,i2,…,ik∣1≤i1<i2<⋯<ik≤n} or A=∅ and e∅=e0.Fora∈Cl0,n, we give some calculations as follows:
(1)a′=∑A‍xAeA′,
where eA′=(-1)|A|eA and |A|=nA is the cardinality of A; that is, when A=∅,  |A|=0 and when A={α1,α2,…,αh}≠∅, then |A|=h; e0′=1, ei′=-ei, i=1,…,n.Recall that any elementx∈Cl0,n may be uniquely decomposed as x=b+cen, for b,c∈Cl0,n-1  (the Clifford algebra generated by e0,…,en-1). Using this decomposition, we define the mappings P:Cl0,n→Cl0,n-1 and Q:Cl0,n→Cl0,n-1 by Px=b and Qx=c. Note that if x=∑A‍xAeA∈Cl0,n, then
(2)Px=∑n∉A‍xAeA,Qx=∑n∈A‍xAeA∖{n},P′x=∑n∉A‍xAeA′,Q′x=∑n∈A‍xAeA∖{n}′.We also introduce the Dirac operator(3)Dlf=∑i=0n‍ei∂f∂xi,Drf=∑i=0n‍∂f∂xiei,
and the modified Dirac operator
(4)Mlf(x)=Dlf(x)+(n-1)Q′fxn,Mrf(x)=Drf(x)+(n-1)Qfxn.Denote byΩ=Ω1×Ω2 an open connected set in the Euclidean space Rm+1×Rk+1, 1≤m≤n, 1≤k≤n. Define a set FΩ(r) to consist of all functions
(5)f(x,y)=∑A⊂{1,…,m}∑B⊂{m+1,…,m+k}‍fA,B(x,y)eAeB,
with values in Cl0,k+m for which fA,B(x,y)∈Cr(Ω).Definition 1.
Letf∈FΩ(1) and x∈Rm+1∖{xm=0}, y∈Rk+1∖{yk=0}. A function f(x,y) is calledbihypermonogenic on Ω, if
(6)Mxlf(x,y)=0,Myrf(x,y)=0,
for any (x,y)∈Ω, where
(7)Mxlf(x,y)=∂f∂x0(x,y)+∑i=1m‍ei∂f∂xi(x,y)+(m-1)Qx′(f(x,y))xm
is the left modified Dirac operator in Cl0,m calculated with respect to x∈Rm+1∖{xm=0} and
(8)Myrf(x,y)=∂f∂y0(x,y)+∑i=1k‍∂f∂yi(x,y)ei+m+(k-1)Qy(f(x,y))yk
is the right modified Dirac operator in the Clifford algebra generated by e0,em+1,…,em+k calculated with respect to y∈Rk+1∖{yk=0}, where
(9)Qx′f(x,y)=∑m∈A‍∑B‍fA,B(x,y)eA∖{m}′eB=∑m∈A‍∑B‍(-1)|A∖{m}|fA,B(x,y)eA∖{m}eB,Qyf(x,y)=∑A‍∑m+k∈B‍fA,B(x,y)eAeB∖{m+k}.


## 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 functionf(x,y):∂Ω1×∂Ω2→Cl0,n is said to be Hölder continuous on ∂Ω1×∂Ω2, if f(x,y) satisfies
(10)|f(x1,y1)-f(x2,y2)|≤M1|(x1,y1)-(x2,y2)|β,000(x1,y1),(x2,y2)∈∂Ω1×∂Ω2,(0<β<1).
Denote by H(∂Ω1×∂Ω2,β) the set of all Hölder continuous functions on ∂Ω1×∂Ω2 with the index β. For any f∈H(∂Ω1×∂Ω2,β), define the norm in H(∂Ω1×∂Ω2,β) as ∥f∥β=C(f,∂Ω1×∂Ω2)+H(f,∂Ω1×∂Ω2,β), where
(11)C(f,∂Ω1×∂Ω2)=sup⁡(ui,υi)∈∂Ω1×∂Ω2(u1,v1)≠(u2,υ2)|f(u1,v1)-f(u2,v2)||(u1,v1)-(u2,v2)|β,H(f,∂Ω1×∂Ω2,β)=max⁡(u,υ)∈∂Ω1×∂Ω2|f(u,υ)|,00000000000000000f∈H(∂Ω1×∂Ω2,β).
Furthermore, for any f,g∈H(∂Ω1×∂Ω2,β), we have
(12)∥fg∥β≤J0∥f∥β∥g∥β.Theorem 2 (see [15]).
LetΩ′ and Ω′′ be open subsets of R+m+1 and R+k+1, respectively. Suppose that Ω1 and Ω2 satisfy Ω1-⊂Ω′ and Ω2¯⊂Ω′′, respectively. The boundaries ∂Ω1, ∂Ω2 of Ω1, Ω2 are differentiable, oriented, compact Liapunov surfaces. If φ(x,y) is a bihypermonogenic function in Ω′×Ω′′, x∈Ω1,y∈Ω2, then
(13)φ(x,y)=λ∫∂Ω1×∂Ω2‍Em(u,x)dσm(u)φ(u,v)dσk(v)Ek(v,y)-λ∫∂Ω1×∂Ω2‍Em(u,x)dσm(u)φ(u,v)~dσk(v)~Fk(v,y)-λ∫∂Ω1×∂Ω2‍Fm(u,x)dσm(u)^φ(u,v)^dσk(v)Ek(v,y)+λ∫∂Ω1×∂Ω2‍Fm(u,x)dσm(u)^φ(u,v)^~dσk(v)~Fk(v,y),
where
(14)dσm=dx1∧dx2∧⋯dxm+∑i=1m(-1)ieidx0∧⋯∧dxi-1∧dxi+1∧⋯dxm,dσk=dy1∧dy2∧⋯dyk+∑i=1k(-1)iei+mdy0∧dy1∧⋯∧dyi-1∧dyi+1000000∧⋯dyk,v=v0+v1em+1+⋯+vkem+k,y=y0+y1em+1+⋯+ykem+k,λ=2m-1xmm-12k-1ykk-1ωm+1ωk+1,El(u,x)=(u-x)-1|u-x|l-1|u-x^|l-1,u,x∈Rm+1,Fl(u,x)=(u^-x)-1|u-x|l-1|u-x^|l-1,u,x∈Rm+1,El(v,y)=(v-y)-1|v-y|l-1|v-y~|l-1,Fl(v,y)=(v~-y)-1|v-y|l-1|v-y~|l-1,
and the involutions 0^ and 0~ are defined by
(15)ei^=ei,i∈{0,1,…,m+k}∖{m},em^=-em,ab^=a^b^,ei~=ei,i∈{0,1,…,m+k-1},em+k~=-em+k,ab~=a~b~.Definition 3 (see [15]).
The integral(16)ϕ(t1,t2)=λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)φ(u,v)dσk(v)Ek(v,t2)-λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)φ(u,v)~dσk(v)~Fk(v,t2)-λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^φ(u,v)^dσk(v)Ek(v,t2)+λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^φ(u,v)^~dσk(v)~Fk(v,t2)
is called a singular integral on ∂Ω1×∂Ω2, where λ, Em(u,t1), Ek(v,t2), Fm(u,t1), and Fk(v,t1) are given in Theorem 2.Definition 4 (see [15]).
Letδ>0 be a constant and λδ=B1(t1,δ)×B2(t2,δ), where Bi(ti,δ)  (i=1,2) are balls with the center at ti and the radius δ>0. Denote
(17)ϕδ(t1,t2)=λ∫∂Ω1×∂Ω2∖λδ‍Em(u,t1)dσm(u)φ(u,v)dσk(v)Ek(v,t2)-λ∫∂Ω1×∂Ω2∖λδ‍Em(u,t1)dσm(u)φ(u,v)~dσk(v)~Fk(v,t2)-λ∫∂Ω1×∂Ω2∖λδ‍Fm(u,t1)dσm(μ)^φ(u,v)^dσk(v)Ek(v,t2)+λ∫∂Ω1×∂Ω2∖λδ‍Fm(u,t1)dσm(u)^φ(u,v)^~dσk(v)~Fk(v,t2).
If lim⁡δ→0ϕδ(t1,t2)=I, then I is called the Cauchy principal value of a singular integral, denoted by I=ϕ(t1,t2).Lemma 5 (see [11]).
LetΩ be an open subset of R+n+1={x=(x0,x1,…,xn)∣xn>0} and let K be an n+1-chain satisfying K¯⊂Ω; then
(18)2n-1ynn-1ωn+1(∫∂K‍En(x,y)dσ(x)-∫∂K‍Fn(x,y)dσ(x)^)={1,y∈K,0,y∈R+n+1-K¯.Lemma 6 (see [11]).
LetΩ, K and ∂K be as in Lemma 5 and y∈∂K; then
(19)2n-1ynn-1ωn+1(∫∂K‍En(x,y)dσ(x)-∫∂K‍Fn(x,y)dσ(x)^)=12.Theorem 7 (see [15]).
Ifφ(u,v)∈H(∂Ω1×∂Ω2,β), then there exists the Cauchy principal value of singular integrals and
(20)ϕ(t1,t2)=-14φ(t1,t2)+X1(t1,t2)+X2(t1,t2)+X3(t1,t2)+X4(t1,t2)+14(P1φ+P2φ)+14(Q1φ+Q2φ),
where
(21)X1(t1,t2)=λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ1(u,v)dσk(v)0000000000×Ek(v,t2),X2(t1,t2)=-λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ2(u,v)dσk(v)~0000000000000×Fk(v,t2),X3(t1,t2)=-λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ3(u,v)dσk(v)00000000000×Ek(v,t2),X4(t1,t2)=λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ4(u,v)dσk(v)~0000000000×Fk(v,t2),ψ1(u,v)=φ(u,v)-φ(u,t2)-φ(t1,v)+φ(t1,t2),ψ2(u,v)=φ(u,v)~-φ(u,t2)-φ(t1,v)~+φ(t1,t2),ψ3(u,v)=φ(u,v)^-φ(u,t2)^-φ(t1,v)+φ(t1,t2),ψ4(u,v)=φ(u,v)^~-φ(u,t2)^-φ(t1,v)~+φ(t1,t2),P1φ=2λ1∫∂Ω1‍Em(u,t1)dσm(u)φ(u,t2),P2φ=2λ2∫∂Ω2‍φ(t1,v)dσk(v)Ek(v,t2),Q1φ=-2λ1∫∂Ω1‍Fm(u,t1)dσm(u)^φ(u,t2)^,Q2φ=-2λ2∫∂Ω2‍φ(t1,υ)~dσk(v)~Fk(v,t2),λ=λ1λ2,λ1=2m-1xmm-1ωm+1,λ2=2k-1ykk-1ωk+1.SetΩi+=Ωi,(i=1,2), Ω1-=R+m+1∖Ω1¯, Ω2-=R+k+1∖Ω2¯, and denote x(∈Ω1±)→t1∈∂Ω1 by x→t1±. Moreover denote y(∈Ω2±)→t2∈∂Ω2 by y→t2± and denote by ϕ±±(t1,t2) the limits of ϕ(x,y) when (x,y)→(t1±,t2±). Then we have the following important theorem.Theorem 8 (see [15]).
Ifφ(u,v)∈H(∂Ω1×∂Ω2,β), then
(22)ϕ++(t1,t2)=14[φ(t1,t2)+P1(φ)+P2(φ)+Q1(φ)+Q2(φ)000000+P3(φ)],ϕ+-(t1,t2)=14[-φ(t1,t2)-P1(φ)+P2(φ)-Q1(φ)+Q2(φ)000000+P3(φ)],ϕ-+(t1,t2)=14[-φ(t1,t2)+P1(φ)-P2(φ)+Q1(φ)-Q2(φ)000000+P3(φ)],ϕ--(t1,t2)=14[φ(t1,t2)-P1(φ)-P2(φ)-Q1(φ)-Q2(φ)000000+P3(φ)],
where (t1,t2)∈∂Ω1×∂Ω2,P3(φ)=4ϕ(t1,t2).


## 4. The Boundary Value Problem for Bihypermonogenic Functions

In this section, we consider the boundary value problem.Definition 9.
LetΩ1×Ω2 and H(∂Ω1×∂Ω2,β) be as before. We want to find a bihypermonogenic function ϕ(x,y) defined in R+m+1×R+k+1/∂Ω1×∂Ω2, which is continuous to ∂Ω1×∂Ω2 and ϕ+-(x,∞)=ϕ-+(∞,y)=ϕ--(∞,∞)=0 and satisfies the nonlinear boundary condition
(23)A(t1,t2)ϕ++(t1,t2)+B(t1,t2)ϕ+-(t1,t2)+C(t1,t2)ϕ-+(t1,t2)+D(t1,t2)ϕ--(t1,t2)=g(t1,t2)f(t1,t2,ϕ++(t1,t2),ϕ+-(t1,t2),ϕ-+(t1,t2),00000000000000ϕ--(t1,t2)),
in which A(t1,t2),B(t1,t2),C(t1,t2),D(t1,t2),g(t1,t2)∈H(∂Ω1×∂Ω2,β) and f are known functions. The above boundary value problem is called Problem R.
From Theorem8, we can transform the boundary condition of Problem R into an integral equation
(24)Fφ=φ,
where
(25)Fφ=(A+B)(φ+P1φ+P2φ+Q1φ+Q2φ+P3φ)+(C+D)(-φ+P1φ-P2φ+Q1φ-Q2φ+P3φ)+(B+D)(2φ-2P1φ-2Q1φ)+(1-4B)φ-4gf.Theorem 10 (see [9]).
LetΩ,∂Ω⊂R+l+1, and let H(∂Ω,β) be the set of Hölder continuous functions on ∂Ω with the index β. For φ∈H(∂Ω,β) and
(26)θφ=ϕφ-φ2,ϕφ=2l-1xll-1ωl+1[∫∂Ω‍El(t,x)dσ0(t)φ(t)00000000000000-Fl(t,x)dσ0(t)^φ(t)^∫∂ΩEl(t,x)dσ0(t)φ(t)],
where El(t,x),Fl(t,x) are as before, then θφ is a hypermonogenic function with
(27)∥θφ∥β≤J1∥φ∥β,
where J1 is a constant independent of φ.Lemma 11 (see [6]).
LetEm(u,x),Em(u,t1) be the same as in Theorem 2. If u∈Ω1¯,x∈Ω2¯,t1∈Ω2¯, then there exists a constant M such that
(28)|Em(u,x)-Em(u,t1)|≤M[∑i=0m-1‍|u-t1u-x|i|x-t1u-x|+|x-t1|]|u-t1|-m.Lemma 12.
Ifφ(t1,t2)∈H(∂Ω1×∂Ω2,β), then
(29)∥φ±(Piφ+Qiφ)∥β≤J2∥φ∥β,∥Piφ+Qiφ∥β≤J2∥φ∥β,i=1,2,
where J2 is a positive constant.Proof.
Using the following equations:(30)P1φ+Q1φ=2λ1∫∂Ω1‍Em(u,t1)dσm(u)φ(μ,t2)-2λ1∫∂Ω1‍Fm(u,t1)dσm(u)^φ(u,t2)^,P2φ+Q2φ=2λ2∫∂Ω2‍φ(t1,υ)dσk(υ)Ek(υ,t2)-2λ2∫∂Ω2‍φ(t1,υ)~dσk(υ)~Fk(υ,t2)
and Theorem 10, we can obtain the result.Theorem 13.
Suppose the boundaries∂Ω1, ∂Ω2 of Ω1, Ω2 be differentiable, oriented, compact Liapunov surfaces. If φ(t1,t2)∈H(∂Ω1×∂Ω2,β), then
(31)∥(P2φ+Q2φ)±P3φ∥β≤J3∥φ∥β,
where J3 is a positive constant which is independent of φ.Proof.
From (20), it follows that
(32)P2φ+Q2φ-P3φ=φ-4∑i=14Xi(t1,t2)-(P1φ+Q1φ).
Moreover, based on Lemma 12 we only need to prove ∥∑i=14‍Xi(t1,t2)∥β≤J4∥φ∥β. It is easy to prove |∑i=14‍Xi(t1,t2)|≤B1∥φ∥β. We rewrite ψi(u,v) as ψi0(t1,t2),(i=1,2,3,4). Now we consider H(∑i=14‍Xi(t1,t2),∂Ω1×∂Ω2,β) and write δ=|(t1,t2)-(t1′,t2′)|=δ12+δ22 for any (t1,t2), (t1′,t2′)∈∂Ω1×∂Ω2 and denote by ρ01,ρ02,ρ01′,ρ02′ the projections of |μ-t1|,|υ-t2|,|μ-t1′|,|υ-t2′| on the tangent plane of t1,t2,t1′,t2′, respectively. Moreover we construct spheres Oi(ti,3δi) with the center at ti and radius 3δi, where 6δi<di, δi<1, i=1,2, where di is a constant as in [5]. Denote by ∂Ωi1,∂Ωi2 the part of ∂Ωi lying inside the sphere Oi and its surplus part, respectively, and set
(33)R(∂Ω1×∂Ω2)=∑i=14‍Xi(t1,t2)-∑i=14‍Xi(t1′,t2′)=∑i=14‍Xi¯(∂Ω1×∂Ω2)-∑i=14‍Xi¯¯(∂Ω1×∂Ω2).
From
(34)|ψi0(t1,t2)|≤M∥φ∥β|u-t1|β,|ψi0(t1,t2)|≤M∥φ∥β|υ-t2|β,|ψi0(t1′,t2′)|≤M∥φ∥β|u-t1′|β,|ψi0(t1′,t2′)|≤M∥φ∥β|υ-t2′|β,00000000000000000i=1,2,3,|ψ40(t1,t2)|≤M∥φ∥β,|ψ40(t1′,t2′)|≤M∥φ∥β,
we obtain that
(35)|Em(u,t1)dσm(u)ψ10(t1,t2)dσk(υ)Ek(υ,t2)|≤M∥φ∥βρ01(β/2)-1dρ01ρ02(β/2)-1dρ02,(36)|Em(u,t1)dσm(u)ψ20(t1,t2)dσk(υ)~Fk(υ,t2)|≤M∥φ∥βρ01β-1dρ01dρ02,(37)|Fm(u,t1)dσm(u)^ψ30(t1,t2)dσk(υ)Ek(υ,t2)|≤M∥φ∥βdρ01ρ02β-1dρ02,(38)|Fm(u,t1)dσm(u)^ψ40(t1,t2)dσk(υ)~Fk(υ,t2)|≤M∥φ∥βdρ01dρ02.
Thus we have
(39)|R(∂Ω11×∂Ω21)|≤∑i=14|Xi¯(∂Ω11×∂Ω21)|+∑i=14|Xi¯¯(∂Ω11×∂Ω21)|≤B2∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Noting that|υ-t2′|≥2δ2,|υ-t2|≥3δ2>0 on ∂Ω22, we have
(40)|R(∂Ω11×∂Ω22)|≤B3∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Similarly, we can obtain|R(∂Ω12×∂Ω21)|≤B4∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Next we want to prove(41)|R(∂Ω12×∂Ω22)|≤C4∥φ∥β|(t1,t2)-(t1′,t2′)|β.
According to (33), we have
(42)X1¯(∂Ω12×∂Ω22)-X1¯¯(∂Ω12×∂Ω22)=λ∫∂Ω12×∂Ω22[Em(u,t1)-Em(m,t1′)]dσm(u)ψ10(t1,t2)000000000000×dσk(υ)Ek(υ,t2)+λ∫∂Ω12×∂Ω22Em(u,t1′)dσm(u)ψ10(t1,t2)dσk(υ)00000000000000×[Ek(υ,t2)-Ek(υ,t2′)]+λ∫∂Ω12×∂Ω22Em(u,t1′)dσm(u)00000000000000×[ψ10(t1,t2)-ψ10(t1′,t2′)]00000000000000×dσk(υ)Ek(υ,t2′)=E1+E2+E3.
Similarly, we can deal with
(43)X2¯(∂Ω12×∂Ω22)-X2¯¯(∂Ω12×∂Ω22)=F1+F2+F3,X3¯(∂Ω12×∂Ω22)-X3¯¯(∂Ω12×∂Ω22)=G1+G2+G3,X4¯(∂Ω12×∂Ω22)-X4¯¯(∂Ω12×∂Ω22)=H1+H2+H3.
From Lemma11 and (35) and by |υ-t2|≥3δ2>0,|υ-t2′|≥2δ2, we obtain |E2|≤C1∥φ∥β|(t1,t2)-(t1′,t2′)|β. Similarly, we can get the inequality estimation for E1 and G2. By (36) and (37), |υ-t2|≥3δ2>0,|υ-t2′|≥2δ2, we have |F2|≤C2∥φ∥β|(t1,t2)-(t1′,t2′)|β. Similarly, we can obtain the inequality estimation for F1,G1,H1, and H2.
Since(44)E3+F3+G3+H3=12λ1∫∂Ω12Em(u,t1′)dσm(u)[φ(u,t2′)-φ(u,t2)]-12λ1∫∂Ω12Fm(u,t1′)dσm(u)^[φ(u,t2′)^-φ(u,t2)^]+12λ2∫∂Ω22[φ(t1′,υ)-φ(t1,υ)]dσk(υ)Ek(υ,t2′)-12λ2∫∂Ω22[φ(t1′,υ)~-φ(t1,υ)~]dσk(υ)~Fk(υ,t2′)+14φ(t1,t2)-14φ(t1′,t2′),
by (35)–(38), we have |E3+F3+G3+H3|≤C3∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Summarizing the above discussion shows that(45)|R(∂Ω12×∂Ω22)|≤C4∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Thus we infer
(46)|R(∂Ω1×∂Ω2)|≤C5∥φ∥β|(t1,t2)-(t1′,t2′)|β.
Hence
(47)∥∑i=14‍Xi(t1,t2)∥β≤J4∥φ∥β.
This completes the proof.Corollary 14.
Ifφ(t1,t2)∈H(∂Ω1×∂Ω2,β), then
(48)∥ϕ++(t1,t2)∥β≤J5∥φ∥β,∥ϕ+-(t1,t2)∥β≤J5∥φ∥β,∥ϕ-+(t1,t2)∥β≤J5∥φ∥β,∥ϕ--(t1,t2)∥β≤J5∥φ∥β.Theorem 15.
LetA(x,y),B(x,y),C(x,y),D(x,y),g(x,y)∈H(∂Ω1×∂Ω2,β); then the function f(t1,t2,ϕ1,ϕ2,ϕ3,ϕ4) is a Hölder continuous function for (t1,t2)∈∂Ω1×∂Ω2 and satisfies the Lipschitz-condition for ϕ1,ϕ2,ϕ3,ϕ4 and any (t1,t2), namely,
(49)|f(t11,t21,ϕ11,ϕ12,ϕ13,ϕ14)-f(t12,t22,ϕ21,ϕ22,ϕ23,ϕ24)|≤J6|(t11,t21)-(t12,t22)|β+J7|ϕ11-ϕ21|+J8|ϕ12-ϕ22|+J9|ϕ13-ϕ23|+J10|ϕ14-ϕ24|,
where Ji   (i=6,…,10) is a positive constant and has nothing to do with t1j,t2j,ϕj1,…,ϕj4,j=1,2. If f(0,0,0,0,0,0)=0,∥A+B∥β<ε,∥C+D∥β<ε,∥D+B∥β<ε,∥1-4B∥β<ε,0<ε<1,0<μ=εJ0(2J5+J2+1)<1,∥g∥β<δ,0<δ<M(1-μ)/4·J0(J13+J14M), then the problem R has at least one solution, where M(∥φ∥β<M),J13,J14 are both positive constants satisfying ∥f∥β≤J13+J14∥φ∥β.Proof.
Suppose thatT={φ∣φ∈H(∂Ω1×∂Ω2,β),∥φ∥β<M} be denoted a subset of C(∂Ω1×∂Ω2). From Theorem 13, Corollary 14, and (38), we obtain C(f,∂Ω1×∂Ω2)≤J11+J12∥φ∥β. Similarly, we can get ∥f∥β≤J13+J14∥φ∥β. Hence, by (12) and Fφ=φ, ∥Fφ∥β≤M is derived. This shows that the operator F is the mapping of T→T.
Next we prove theF is a continuous mapping.
Suppose that the sequence of functions{φn}∈T uniformly converges to a function φ(t1,t2),(t1,t2)∈∂Ω1×∂Ω2; thus for arbitrary ε>0 and if n is large enough, then |(Pi+Qi)φn-(Pi+Qi)φ|<ε,(i=1,2).
Now we considerP3φn-P3φ by
(50)P3φn-P3φ=∑i=12‍Ai(∂Ω1×∂Ω2)+∑i=12‍Bi(∂Ω1×∂Ω2)+∑i=12‍Ci(∂Ω1×∂Ω2)+∑i=12‍Di(∂Ω1×∂Ω2)+∑i=13‍Ei,
where
(51)A1(∂Ω1×∂Ω2)=4λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ1n(u,υ)dσk(υ)000000000000×Ek(υ,t2),A2(∂Ω1×∂Ω2)=-4λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ1(u,υ)dσk(υ)000000000000×Ek(υ,t2),B1(∂Ω1×∂Ω2)=-4λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ2n(u,υ)dσk(υ)~000000000000×Fk(υ,t2),B2(∂Ω1×∂Ω2)=4λ∫∂Ω1×∂Ω2‍Em(u,t1)dσm(u)ψ2(u,υ)dσk(υ)~000000000000×Fk(υ,t2)C1(∂Ω1×∂Ω2)=-4λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ3n(u,υ)dσk(υ)000000000000×Ek(υ,t2),C2(∂Ω1×∂Ω2)=4λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ3(u,υ)dσk(υ)000000000000×Ek(υ,t2),D1(∂Ω1×∂Ω2)=4λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ4ndσk(υ)~Fk(υ,t2),D2(∂Ω1×∂Ω2)=-4λ∫∂Ω1×∂Ω2‍Fm(u,t1)dσm(u)^ψ4dσk(υ)~Fk(υ,t2),E1=φ(t1,t2)-φn(t1,t2),E2=(P1+Q1)φn-(P1+Q1)φ,E3=(P2+Q2)φn-(P2+Q2)φ.
Suppose6δ<di,i=1,2,δ>0,O((t1,t2),3δ) is the 3δ-neighborhood of (t1,t2) with the center at point (t1,t2)∈∂Ω1×∂Ω2 and the radius 3δ, ∂Ωi1×∂Ωi2 is as above; then
(52)⬜j(∂Ω1×∂Ω2)=⬜j(∂Ω11×∂Ω21)+⬜j(∂Ω11×∂Ω22)+⬜j(∂Ω12×∂Ω21)+⬜j(∂Ω12×∂Ω22),00000000000000000(⬜=A,B,C,D,j=1,2).
By (35), we can obtain that
(53)|A1(∂Ω11×∂Ω21)|≤J15δβ≤J16δβ/2,|A1(∂Ω12×∂Ω21)|≤J17δβ/2,|A1(∂Ω11×∂Ω22)|≤J18δβ/2.
Similarly, we can get the inequality estimations for A2(∂Ω11×∂Ω21),A2(∂Ω12×∂Ω21), and A2(∂Ω11×∂Ω22). By (35), (36), and (37), we can obtain the similar inequality estimations for Bi(∂Ω11×∂Ω21), Bi(∂Ω12×∂Ω21), Bi(∂Ω11×∂Ω22), Ci(∂Ω11×∂Ω21), Ci(∂Ω12×∂Ω21), Ci(∂Ω11×∂Ω22), Di(∂Ω11×∂Ω21), Di(∂Ω12×∂Ω21), and Di(∂Ω11×∂Ω22), i=1,2, respectively.
From(54)A1(∂Ω12×∂Ω22)+A2(∂Ω12×∂Ω22)=4λ∫∂Ω12×∂Ω22‍Em(u,t1)dσm(u)W1(u,υ)dσk(υ)00000000000000×Ek(υ,t2),
where
(55)W1(u,υ)={[φn(u,υ)-φ(u,υ)]-[φn(t1,υ)-φ(t1,υ)]}+{[φn(t1,t2)-φ(t1,t2)]-[φn(u,t2)-φ(u,t2)},
since |W1(u,υ)|≤2∥φn-φ∥β|u-t1|β/2|υ-t2|β/2 and from (35), we have
(56)|A1(∂Ω12×∂Ω22)+A2(∂Ω12×∂Ω22)|≤J19∥φn-φ∥β.
Similarly, we can get the inequality estimations for B1(∂Ω12×∂Ω22)+B2(∂Ω12×∂Ω22),C1(∂Ω12×∂Ω22)+C2(∂Ω12×∂Ω22).
From(57)D1(∂Ω12×∂Ω22)+D2(∂Ω12×∂Ω22)=4λ∫∂Ω12×∂Ω22‍Fm(u,t2)dσm(u)^W4(u,υ)dσk(υ)~Fk(υ,t2),
where
(58)W4(u,υ)={[φn(u,υ)^~-φ(u,υ)^~]-[φn(t1,υ)~-φ(t1,υ)~]}+{[φn(t1,t2)-φ(t1,t2)]-[φn(u,t2)^-φ(u,t2)^]},
since |W4(u,υ)|≤4∥φn-φ∥β and from (38), we obtain that
(59)|D1(∂Ω12×∂Ω22)+D2(∂Ω12×∂Ω22)|≤J21∥φn-φ∥β.
Summarizing the above discussion, we conclude|P3φn-P3φ|≤J22(ε+δβ/2+∥φn-φ∥β). Then for arbitrary ε>0, we first choose a sufficiently small number δ and next select a sufficiently large positive integer n; we have
(60)|P3φn-P3φ|<Gε,
where G is a positive constant.
Finally, we can choosen large enough such that |Fφn-Fφ|<Wε (W is a positive constant). Hence we can obtain F:T→T is a continuous mapping. According to Ascoli-Arzela Theorem, T is a compact set in the space C(∂Ω1×∂Ω2). Based on the Schauder fixed point principle, there exists a function φ∈H(∂Ω1×∂Ω2,β) satisfying the equation Fφ=φ.Corollary 16.
Iff≡1 in Theorem 15, then the problem R has the unique solution.Proof.
This corollary is not difficult to verify by the contraction mapping principle whenf≡1 in Theorem 15.



---

*Source: 974714-2014-07-20.xml*