# On the Solutionn-Dimensional of the Product ⊗k Operator and Diamond Bessel Operator

Authors: Wanchak Satsanit; Amnuay Kananthai
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

Firstly, we studied the solution of the equation⊗k◊Bku(x)=f(x) where u(x) is an unknown unknown function for  x=(x1,x2,…,xn)∈ℝn, f(x) is the generalized function, k is a positive integer. Finally, we have studied the solution of the nonlinear
equation ⊗k◊Bku(x)=f(x,□k−1LkΔBk□Bku(x)). It was found that the existence of the solution u(x) of such an
equation depends on the condition of f and □k−1LkΔBk□Bku(x). Moreover such solution u(x) is related to the inhomogeneous wave equation depending on the conditions of p, q, and k.


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## Body

## 1. Introduction

The operator◊k has been first introduced by Kananthai  (see [1]), is named as the Diamond operator iterated k-times, and is defined by(1.1)◊k=((∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2)k,p+q=n.n is the dimension of the space ℝn, for x=(x1,x2,…,xn)∈ℝn and k is a nonnegative integer. The operator ◊k can be expressed in the form ◊k=Δk□k=□kΔk, where Δk is the Laplacian operator itrerated k-times defined by(1.2)Δk=(∂2∂x12+∂2∂x22+⋯+∂2∂xn2)k,
and □k is the ultrahyperbolic operator iterated k-times defined by(1.3)□k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k.Kananthai (see [1, Theorem  3.1, page 33]) has shown that the convolution (-1)kR2ke(x)*R2kH(x) is an elementary solution of the operator ◊k, that is, (1.4)◊k((-1)kR2ke(x)*R2kH(x))=δ(x).
Next, Kananthai (see [2]) has studied the linear equation(1.5)◊ku(x)=f(x).
This equation is the generalization of the ultrahyperbolic equation and it can be applied to the wave equation. We obtain u(x)=(-1)kM2k,2k(x)*f(x) as a solution of such an equation (1.5) where (1.6)M2k,2k=R2kH(x)*R2ke(x).
The function R2kH(x) is called the ultrahyperbolic kernel defined by (2.2) and R2ke(x) is called the elliptic kernel defined by (2.8), with α=2k.Furthermore, Yıldırım et al. (see [3]) first introduced the ◊Bk operator that is named as Diamond Bessel operator, where ◊Bk is defined by(1.7)◊Bk=[(∑i=1pBxi)2-(∑j=p+1p+qBxj)2]k,
and Bxi=∂2/∂xi2+(2υi/xi)(∂/∂xi),2υi=2αi+1,αi>-1/2,xi>0. The operator ◊Bk can be expressed by ◊Bk=ΔBk□Bk=□BkΔBk, where(1.8)ΔBk=(∑i=1nBxi)k.(1.9)□Bk=(∑i=1pBxi-∑j=p+1p+qBxj)k.
Next, W. Satsanit has first introduced ⊗k operator and ⊗k is defined by(1.10)⊗k=[(∑i=1p∂2∂xi2)3-(∑j=p+1p+q∂2∂xj2)3]k=(∑i=1p∂2∂xi2-∑j=p+1p+q∂2∂xj2)k[(∑i=1p∂2∂xi2)2+(∑i=1p∂2∂xi2)·(∑j=p+1p+q∂2∂xj2)+(∑j=p+1p+q∂2∂xj2)2]k=□k(Δ2-14(Δ+□)(Δ-□))k=(34◊Δ+14□3)k,
where ◊,Δ, and □ are defined by (1.1), (1.2), and (1.3) with k=1, respectively.Now, firstly, the purpose of this work is to study the equation(1.11)⊗k◊Bku(x)=f(x),
where the operator ⊗k is defined by (1.10) and ◊Bk defined by (1.7), f(x) is a generalized function and u(x) is an unknown function. Finally we study the equation(1.12)⊗k◊Bku(x)=f(x,□k-1LkΔBk□Bku(x))
with f having a continuous first derivative for all x∈Ω∪∂Ω, where Ω is an open subset of Rn, and ∂Ω denotes the boundary of Ω,  f is bounded on Ω, that is, |f|≤N,N is constant, as well as □k-1,Lk,ΔBk, and □Bk are defined by (1.3), (2.46), (1.8) and (1.9), respectively.We can find the solutionu(x) of (1.12) that is unique under the boundary condition □k-1LkΔBk□Bku(x)=0 for x∈∂Ω. By [4, page 369] there exists a unique solution W(x) of the equation □W(x)=f(x,W(x)) for all x∈Ω with the boundary condition W(x)=0 for all x∈∂Ω where W(x)=□k-1LkΔBk□Bku(x).Moreover, if we putp=k=1 in □k□BkM(x)=W(x), then we found that M(x)=I2H(x)*I2(x)*W(x) is a solution of the inhomogeneous equation where I2H(x) and I2(x) are defined by (2.6) and (2.20) with α=2,γ=2, respectively.Before going into details, the following definitions and some important concepts are needed.


## 2. Preliminaries

Definition 2.1.
Letx=(x1,x2,…,xn) be a point of the n-dimensional Euclidean space ℝn, denoted by
(2.1)υ=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2.
The nondegenerated quadratic form p+q=n is the dimension of the space ℝn. Let Γ+={x∈ℝn:x1>0andu>0} be the interior of forward cone and let Γ¯+ denote its closure. For any complex number α, define the function
(2.2)RαH(υ)={υ(α-n)/2Kn(α),forx∈Γ+,0,forx∉Γ+,
where the constant Kn(α) is given by the formula
(2.3)Kn(α)=π(n-1)/2Γ((2+α-n)/2)Γ((1-α)/2)Γ(α)Γ((2+α-p)/2)Γ((p-α)/2).
The function RαH(υ) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki (see [5]).It is well known thatRαH(υ) is an ordinary function if Re(α)≥n and is a distribution of α if Re(α)<n. Let supp RαH(υ) denote the support of RαH(υ) and suppose that supp RαH(υ)⊂Γ̅+, that is, supp RαH(υ) is compact.From Trione (see [6, page 11]), R2kH(υ) is an elementary solution of the operator □k, that is,(2.4)□kR2kH(υ)=δ(x).By puttingp=1 in R2kH(υ) and taking into account Legendre's duplication formula for Γ(z), that is (2.5)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
we obtain(2.6)IαH(υ)=υ(α-n)/2Hn(α),
and υ=x12-x22-x32⋯-xn2 where (2.7)Hn(α)=π(n-2)/22α-1Γ(α+2-n2)Γ(α2).IαH(υ) is the hyperbolic kernel of Marcel Riesz.Definition 2.2.
Letx=(x1,x2,…,xn) be a point of ℝn and ω=x12+x22+⋯+xn2, then the function Rαe(ω) denoted the elliptic kernel of Marcel Riesz and is defined by
(2.8)Rαe(ω)=ω(α-n)/2Wn(α),
where
(2.9)ω=x12+x22+⋯+xn2,(2.10)Wn(α)=πn/22αΓ(α/2)Γ((n-α)/2)
where α is a complex parameter and n is the dimension of ℝn.It can be shown thatR-2ke(x)=(-1)kΔkδ(x) where Δk is defined by (1.2). It follows that R0e(x)=δ(x), (see [7, page 118]).Moreover, we obtain(-1)kR2ke(x) is an elementary solution of the operator Δk (see [8, Lemma  2.4, page 31]). That is(2.11)Δk((-1)kR2ke(x))=δ(x).
By (2.2) and (2.3) with q=0, then υ(α-n)/2 reduces to ωp(α-p)/2 where ωp=x12+x22+⋯+xp2 and Kn(α) reduces to Kp(α)=(π(p-1)/2Γ((1-α)/2)Γ(α))/Γ((p-α)/2). By using the formula
(2.12)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),Γ(12+z)Γ(12-z)=πsec(πz),
we obtain (2.13)Kp(α)=12sec(πα2)Wp(α),
where Wp(α) is defined by (2.10) with n=p. Thus, for q=0,(2.14)RαH(υ)=υ(α-p)/2Kp(α)=2cos(πα2)υ(α-p)/2Wp(α)=2cos(πα2)Rαe(ωp),
where ωp=x12+x22+⋯+xp2. Thus, if α=2k, then (2.15)R2kH(ωp)=2(-1)kR2ke(ωp)
for q=0 and ωp=x12+x22+⋯+xp2.Definition 2.3.
Letx=(x1,x2,…,xn),ν=(ν1,ν2,…,νn)∈ℝn+. For any complex number α, we define the function Sα(x) by
(2.16)Sα(x)=2n+2|ν|-2αΓ((n+2|ν|-α)/2)|x|α-n-2|ν|∏i=1n2νi-1/2Γ(νi+1/2).Definition 2.4.
Letx=(x1,x2,…,xn),ν=(ν1,ν2,…,νn)∈ℝn+, and V=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2 the nondegenerated quadratic form. Denote the interior of the forward cone by Γ+={x∈ℝn+:x1>0,x2>0,…,xn>0,V>0}. The function Rγ(x) is defined by
(2.17)Rγ(x)=V(γ-n-2|ν|)/2Kn(γ),
where
(2.18)Kn(γ)=π(n+2|ν|-1)/2Γ((2+γ-n-2|ν|)/2)Γ((1-γ)/2)Γ(γ)Γ((2+γ-p-2|ν|)/2)Γ((p-2|ν|-γ)/2),
and γ is a complex number. By putting p=1 in Rγ(x) and taking into account Legendre's duplication formula for Γ(z), that is,
(2.19)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12)
we obtain
(2.20)Iγ(x)=V(γ-n-2|ν|)/2Nn(γ),
and V=x12-x22-x32⋯-xn2 where
(2.21)Nn(γ)=π(n+2|ν|-1)/222k-1Γ(2+γ-n-2|ν|2)Γ(γ2).Lemma 2.5.
Given the equationΔBku(x)=δ(x) for x∈ℝn+, where ΔBk is defined by (1.8), then
(2.22)u(x)=(-1)kS2k(x),
where S2k(x) is defined by (2.16), with α=2k.Proof.
(See [3, page 379] and [9]).Lemma 2.6.
Given the equation□Bku(x)=δ(x) for x∈ℝn+, where □Bk is defined by (1.9), then
(2.23)u(x)=R2k(x),
where R2k(x) is defined by (2.17), with γ=2k.Proof.
(See [3, page 379] and [9]).Lemma 2.7.
Given thatP is a hyperfunction, then
(2.24)Pδk(p)+kδ(k-1)(p)=0,
where δ(k) is the Dirac-delta distribution with k-derivatives.Proof.
(See [8, page 233]).Lemma 2.8.
Given the equation(2.25)□ku(x)=0,
where □k is defined by (1.3) and x=(x1,x2,…,xn)∈ℝn, then u(x)=(R2(k-1)H(υ))(m) is a solution of (2.25) with m=(n-4)/2,n≥4 and n is even dimension. The function (R2(k-1)H(υ))(m) is defined by (2.2) with m-derivatives, α=2(k-1), and υ being defined by (2.1).Proof.
We first show the generalized functionδ(m)(r2-s2) where r2=x12+x22+⋯+xp2 and s2=xp+12+xp+22+⋯+xp+q2,p+q=n, is a solution of the equation
(2.26)□u(x)=0,
where □ is defined by (1.3) with k=1 and x=(x1,x2,…,xn)∈ℝn,
(2.27)∂∂xiδ(m)(r2-s2)=2xiδ(m+1)(r2-s2),∂2∂xi2δ(m)(r2-s2)=2δ(m+1)(r2-s2)+4xi2δ(m+2)(r2-s2),□δ(m)(r2-s2)=∑i=1p∂2∂xi2δ(m)(r2-s2)=2pδ(m+1)(r2-s2)+4r2δ(m+2)(r2-s2)=2pδ(m+1)(r2-s2)+4(r2-s2)δ(m+2)(r2-s2)+4s2δ(m+2)(r2-s2)=2pδ(m+1)(r2-s2)-4(m+2)δ(m+1)(r2-s2)+4s2δ(m+2)(r2-s2)=(2p-4(m+2))δ(m+1)(r2-s2)+4s2δ(m+2)(r2-s2).
By Lemma 2.5 with P=r2-s2, similarly, (2.28)∑j=p+1p+q∂2∂xj2δ(m)(r2-s2)=(-2q+4(m+2))δ(m+1)(r2-s2)+4r2δ(m+2)(r2-s2).
Thus
(2.29)□δ(m)(r2-s2)=∑i=1p∂2∂xi2δ(m)(r2-s2)-∑j=p+1p+q∂2∂xi2δ(m)(r2-s2)=(2(p+q)-8(m+2))δ(m+1)(r2-s2)-4(r2-s2)δ(m+2)(r2-s2)=(2n-8(m+2))δ(m+1)(r2-s2)+4(m+2)δ(m+1)(r2-s2)=(2n-4(m+2))δ(m+1)(r2-s2).
If 2n-4(m+2)=0, then we have □δ(m)(r2-s2)=0. That is, u(x)=δ(m)(r2-s2) is a solution of (2.26) with m=(n-4)/2,n≥4 and n is even dimension. We write
(2.30)ku(x)=□(k-1u(x))=0,
and from the above proof we have □k-1u(x)=δ(m)(r2-s2) with m=(n-4)/2,n≥4 and n is even dimension. Convolving the above equation by R2(k-1)H(υ), we obtain
(2.31)R2(k-1)H(υ)*□k-1u(x)=R2(k-1)H(υ)*δ(m)(r2-s2)□k-1(R2(k-1)H(υ))*u(x)=(R2(k-1)H(υ))(m),whereυ=(r2-s2)δ*u(x)=u(x)=(R2(k-1)H(υ))(m)
by (2.2), and υ=r2-s2 is defined by Definition (2.1).
Thusu(x)=(R2(k-1)H(υ))(m) is a solution of (2.25) with m=(n-4)/2,n≥4 and n is even dimension.Lemma 2.9.
Given the equation(2.32)⊗kG(x)=δ(x),
then
(2.33)G(x)=(R6kH(x)*(-1)2kR4ke(x))*(O*k(x))*-1
is an elementary solution for the ⊗k operator iterated k-times where ⊗k is defined by (1.10), and
(2.34)O(x)=34R4H(x)+14(-1)2R4e(x)
where O*k(x) denotes the convolution of O(x) itself k-times  and (O*k(x))*-1 denotes the inverse of O*k(x) in the convolution algebra. Moreover G(x) is a tempered distribution.Proof.
From (3.1), we have
(2.35)⊗kG(x)=(34◊Δ+143)kG(x)=δ(x),
or we can write
(2.36)(34◊Δ+14□3)(34◊Δ+14□3)k-1G(x)=δ(x).
Convolving both sides of the above equation by R6H(x)*(-1)2R4e(x),
(2.37)(34◊Δ+14□3)*(R6H(x)*(-1)2R4e(x))(34◊Δ+143)k-1G(x)=δ(x)*R6H(x)*(-1)2R4e(x),
or
(2.38)(34□(R2H(x))*Δ2(-1)2R4e(x)*R4H(x)+14□3R6H(x)*(-1)2R4e(x))*(34◊Δ+14□3)k-1G(x)=δ(x)*R6H(x)*(-1)2R4e(x).
By (2.4) and (2.8), we obtain
(2.39)(34δ*δ*R4H(x)+14δ*(-1)2R4e(x))*(34◊Δ+14□3)k-1G(x)=δ(x)*R6H(x)*(-1)2R4e(x).
Thus
(2.40)(34R4H(x)+14(-1)2R4e(x))*(34◊Δ+143)k-1G(x)=R6H(x)*(-1)2R4e(x).
Keeping on convolving both sides of the above equation by R6H(x)*(-1)2R4e(x) up to k-1 times, we obtain
(2.41)O*k(x)*G(x)=(R6H(x)*(-1)2R4e(x))*k
where the symbol *k denotes the convolution of itself k-times. By properties of Rα(x), we have
(2.42)(R6H(x)*(-1)2R4e(x))*k=R6kH(x)*(-1)2kR4ke(x).
Thus,
(2.43)O*k(x)*G(x)=R6kH(x)*(-1)2kR4ke(x).
Now, consider the functionO*k(x), since R6H(x)*(-1)2R4e(x) is a tempered distribution. Thus O(x) defined by (2.34) is a tempered distribution, and we obtain that O*k(x) is a tempered distribution and R6kH(x)*(-1)2kR4ke(x)∈𝒮′ is the space of tempered distribution. Choose 𝒮′⊂𝒟ℛ′ where 𝒟ℛ′ is the right-side distribution which is a subspace of 𝒟′ of distribution.
ThusR6kH(x)*(-1)2kR4ke(x)∈𝒟ℛ′. It follows that R6kH(x)*(-1)2kR4ke(x) is an element of convolution algebra, since 𝒟ℛ′ is a convolution algebra. Hence by the method of Zemanian (see [10]), (2.33) has a unique solution
(2.44)G(x)=(R6kH(x)*(-1)2kR4ke(x))*(O*k(x))*-1,
where (O*k(x))*-1 is an inverse of O*k(x) in the convolution algebra and G(x) is called the Green function of the ⊗k operator.Lemma 2.10.
Given the equation(2.45)LkK(x)=δ(x),
where Lk is the operator defined by
(2.46)Lk=(34Δ2+14□2)k
and Δ and □ are defined by (1.2) and (1.3) with k=1, respectively, one obtains that K(x) is an elementary solution of the Lk operator where
(2.47)K(x)=(R4kH(x)*(-1)2kR4ke(x))*(O*k(x))*-1,O(x)=34R4H(x)+14(-1)2R4e(x),
where O*k(x) denotes the convolution of O(x) itself k-times and (O*k(x))*-1 denotes the inverse of O*k(x) in the convolution algebra. Moreover K(x) is a tempered distribution.Proof.
The proof of Lemma2.10 is similar to the proof of Lemma 2.9.Lemma 2.11.
Given the equation(2.48)□u(x)=f(x,u(x)),
where f is defined and has continuous first derivatives for all x∈Ω∪∂Ω, where Ω is an open subset of Rn and ∂Ω is the boundary of Ω, assume that f is bounded, that is, |f(x,u(x))|≤N for all x∈Ω. Then one obtains a continuous function u(x) as unique solution of (2.48) with the boundary condition u(x)=0 for x∈∂Ω.Proof.
We can prove the existence of the solutionu(x) of (2.48) by the method of iterations and Schuder's estimates. The details of the proof are given by Courant and Hilbert; (see [4, pages 369–372]).Lemma 2.12.
The functionR-2kH(x) and S-2k(x) are the inverse of the convolution algebra of R2kH and S2k, respectively, that is,
(2.49)R-2kH(x)*R2kH(x)=R-2k+2kH(x)=R0H(x)=δ,S-2k(x)*S2k(x)=S-2k+2k(x)=S0(x)=δ.Proof.
(See [7, page 158] and [11]).


## 3. Main Results

Theorem 3.1.
Given the equation(3.1)⊗k◊Bku(x)=0,
where ⊗k is the Otimes operator iterated k-times and ◊Bk is Diamond Bessel operator iterated k- times defined by (1.10) and (1.7), respectively, and u(x) is an unknown function, one obtains that u(x) is a solution of (3.1) where
(3.2)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(-1)k-1(R2(k-1)H(υ))(m)
where K(x) is defined by (2.47), as well as S2k(x), R2k(x), and (R2(k-1)H(υ))m are defined by (2.16),(2.17), and (2.2) with α=2k, γ=2k and α=2(k-1), respectively.Proof.
Since(3.3)⊗k=(34◊Δ+14□3)k,◊Bk=ΔBk□Bk.
Consider the homogeneous equation(3.4)⊗k◊Bku(x)=0.
The above equation can be written as(3.5)(34◊Δ+14□3)kΔBk□Bku(x)=0,
or
(3.6)□k(34Δ2+14□2)kΔBk□Bku(x)=0.
That is,
(3.7)□kLkΔBk□Bku(x)=0,
where □k,Lk,ΔBk, and □Bk are defined by (1.3), (2.46), (1.8), and (1.9), respectively.  By Lemma 2.8, we obtain
(3.8)LkΔBk□Bku(x)=(R2(k-1)H(υ))(m).
Since(-1)kS2k(x),R2k(x) are the elementary solution of the operators ΔBk and □Bk, respectively, and by Lemma 2.10, we have that K(x) is an elementary of the operator Lk defined by (2.46), that is,
(3.9)ΔBk(-1)kS2k(x)=δ(x),□BkR2k(x)=δ(x),LkK(x)=δ(x).
Convolving both sides of (3.8) by K(x)*(-1)kS2k(x)*R2k(x), we obtain
(3.10)K(x)*(-1)kS2k(x)*R2k(x)*ŁkΔBk□Bku(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m).
By properties of convolution
(3.11)ŁkK(x)*ΔBk(-1)kS2k(x)*□BkR2k(x)*u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m).
By Lemmas2.10, 2.5, and 2.6, we obtain(3.12)δ(x)*δ(x)*δ(x)*u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m).
Thus(3.13)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m)
is the solution of (3.1).Theorem 3.2.
Given the equation(3.14)⊗k◊Bku(x)=f(x),
where ⊗k is the Otimes operator iterated k-times defined by (1.10), and ◊Bk is the Diamond Bessel operator iterated k-times defined by (1.7), f(x) is the generalized function, u(x) is an unknown function, x=(x1,x2,…,xn)∈ℝn and n is even,
One obtains that(3.15)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m)+G(x)*(-1)kS2k(x)*R2k(x)*f(x)
is a general solution of (3.14) and G(x) is defined by (2.33), K(x) is defined by (2.47), as well as S2k(x) and R2k(x) are defined by (2.16) and (2.17) with α=2k and γ=2k, respectively.Proof.
Consider the equation(3.16)⊗k◊Bku(x)=f(x)
or
(3.17)⊗kΔBk□Bku(x)=f(x).
Convolving both sides of (3.14) by G(x)*(-1)kS2k(x)*R2k(x), we obtain(3.18)G(x)*(-1)kS2k(x)*R2k(x)*⊗kΔBk□Bku(x)=G(x)*(-1)kS2k(x)*R2k(x)*f(x).
By properties of convolution,
(3.19)⊗kG(x)*ΔBk(-1)kS2k(x)*□BkR2k(x)*u(x)=G(x)*(-1)kS2k(x)*R2k(x)*f(x).
By Lemmas 2.9, 2.5, and 2.6, we obtain
(3.20)δ(x)*δ(x)*δ(x)*u(x)=G(x)*(-1)kS2k(x)*R2k(x)*f(x).
Thus
(3.21)u(x)=G(x)*(-1)kS2k(x)*R2k(x)*f(x).
Consider the homogeneous equation
(3.22)⊗k◊Bku(x)=0.
By Theorem 3.1, we have a homogeneous solution
(3.23)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m).
Thus, the general solution of (3.14) is
(3.24)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-1)H(υ))(m)+G(x)*(-1)kS2k(x)*R2k(x)*f(x),
as required.Theorem 3.3.
Consider the nonlinear equation(3.25)⊗k◊Bku(x)=f(x,□k-1LkΔBk□Bku(x))
where ⊗k,◊Bk,□k-1,Lk,ΔBk, and □Bk are defined by (1.10), (1.7),(1.3),(2.44), and (1.9), respectively. Let f be defined, and having continuous first derivative for all x∈Ω∪∂Ω,Ω is an open subset of Rn and ∂Ω denotes the boundary function, that is,
(3.26)|f(x,□k-1LkΔBk□Bku(x))|≤N
for all x∈Ω and the boundary condition
(3.27)□k-1LkΔBk□Bku(x)=0
for all x∈∂Ω. Then one obtains
(3.28)u(x)=R2(k-1)H(x)*G(x)*(-1)kS2k(x)*R2k(x)*W(x)
as a solution of (3.25) with the boundary condition
(3.29)u(x)=(R2(k-2)H(υ))(m)*G(x)*(-1)kS2k(x)*R2k(x)
for all x∈∂Ω,m=(n-4)/2, and W(x) is a continuous function for x∈Ω∪∂Ω,whileR2(k-2)(υ)H,S2k(x), and R2k(x) are given by (2.2), (2.16), and (2.17) with α=2(k-2),α=2k, and γ=2k, respectively. Moreover, for k=1 one obtains
(3.30)M(x)=(R-4H(x)*(-1)2R-4e(x))*(O*1(x))*(-1)kS-2(x)*u(x)
as a solution of the inhomogeneous equation
(3.31)□□BM(x)=W(x),
where □ and □B are defined by (1.3) and (1.9) with k=1, respectively, and u(x) is obtained from (3.28). Furthermore, If one puts p=k=1, then the operators □k and □Bk reduce to
(3.32)∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2,Bx1-Bx2-Bx3-⋯-Bxn,
respectively, and the solution M(x)=I2H(x)*I2(x)*W(x) is the inhomogeneous wave equation
(3.33)(∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2)·(Bx1-Bx2-Bx3-⋯-Bxn)M(x)=W(x),
where I2H(x) is defined by (2.6) with α=2 and I2(x) is defined by (2.20) with γ=2.Proof.
Since(3.34)⊗k◊Bku(x)=□□k-1LkΔBk□Bku(x)=f(x,□k-1LkΔBk□Bku(x)),u(x) has continuous derivative up to order 6k for k=1,2,3,…, and □k-1LkΔBk□Bku(x) exists as the generalized function. Thus we can assume that
(3.35)□k-1LkΔBk□Bku(x)=W(x),∀x∈Ω.
Then (3.34) can be written in the form
(3.36)⊗k◊Bku(x)=□W(x)=f(x,W(x)).
By (3.26)
(3.37)|f(x,W(x))|≤N,x∈Ω,
and by (3.27) W(x)=0,x∈∂Ω, or
(3.38)□k-1LkΔBk□Bku(x)=0,∀x∈∂Ω.
We obtain a unique solution of (3.28) which satisfies (3.27) by Lemma 2.8.
SinceR2(k-1)H(x),(-1)kS2k(x), and R2k(x) are the elementary solution of the operators □k-1,ΔBk, and □Bk, respectively, and by Lemma 2.10, we have that K(x) is an elementary of the operator Lk where Lk=((3/4)Δ2+(1/4)□2)k, that is,
(3.39)□k-1R2(k-1)H(x)=δ,ΔBk(-1)kS2k(x)=δ,BkR2k(x)=δ,LkK(x)=δ.
From (3.35), we have
(3.40)□k-1LkΔBk□Bku(x)=W(x).
Convolving the above equation by
(3.41)R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x),
we obtain
(3.42)(R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x))*(□k-1LkΔBk□Bku(x))=(R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x))*W(x).
By properties of convolution, we obtain
(3.43)(□k-1R2(k-1)H(x))*(ŁkK(x))*(ΔBk*(-1)kS2k)*(□BkR2k)*u(x)=(R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x))*W(x).
By (3.39) we obtain
(3.44)δ*δ*δ*δ*u(x)=(R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x))*W(x).
Thus
(3.45)u(x)=(R2(k-1)H(x)*K(x)*(-1)kS2k(x)*R2k(x))*W(x),
as a solution of (3.25).
Next, consider the boundary condition (3.38). From
(3.46)□k-1LkΔBk□Bku(x)=0,
by Lemma 2.8, we have
(3.47)LkΔBk□Bku(x)=(R2(k-2)H(υ))(m),
where m=(n-4)/2,n≥4 and n is even. Convolving both sides of (3.47) by
(3.48)K(x)*(-1)kS2k(x)*R2k(x),
we obtain
(3.49)(K(x)*(-1)kS2k(x)*R2k(x))*(LkΔBk□Bk)*u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-2)H(υ))(m).
By the properties of convolution, we obtain
(3.50)(LkK(x))*(ΔBk(-1)kS2k)*(□BkR2k)*u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-2)H(υ))(m).
By (3.39), we obtain
(3.51)δ*δ*δ*u(x)=(K(x)*(-1)kS2k(x)*R2k(x))*(R2(k-2)H(υ))(m).
Thus, for x∈∂Ω and k=2,3,4,5,….,(3.52)u(x)=K(x)*(-1)kS2k(x)*R2k(x)*(R2(k-2)H(υ))(m),
as required.
Now, fork=1 in (3.28), we have
(3.53)u(x)=δ(x)*G(x)*(-1)S2(x)*R2(x)*W(x).
By (2.47), we have
(3.54)G(x)=(R6H(x)*(-1)2R4e(x))*(O*1(x))*-1.
Taking into account (3.53), we obtain
(3.55)u(x)=(R6H(x)*(-1)2R4e(x))*(O*1(x))*-1*(-1)1S2(x)*R2(x)*W(x)
as a solution of (3.25) for k=1.
Convolving both sides of (3.55) by
(3.56)(R-4H(x)*(-1)2R-4e(x))*(O*1(x))*(-1)S-2(x),
by Lemma 2.12, we obtain
(3.57)(R-4H(x)*(-1)2R-4e(x))*(O*1(x))*(-1)S-2(x)*u(x)=R2H(x)*R2(x)*W(x).
By Lemma 2.6, we obtain
(3.58)M(x)=(R-4H(x)*(-1)2R-4e(x))*(O*1(x))*(-1)S-2(x)*u(x)
as a solution of the inhomogeneous equation
(3.59)□□BM(x)=W(x).
Now, consider the boundary condition for k=1 in (3.27); we have
(3.60)LΔB□Bu(x)=0,or□BLΔBu(x)=0
for x∈∂Ω. Thus by Lemma 2.8, for k=1, we have
(3.61)LΔBu(x)=δ(m)(υ)forx∈∂Ω,
where δ(m)(x)=R0H(x). Convolving the above equation by K(x)*(-1)S2(x) where K(x) is defined by (2.47) with k=1 and S2(x) is defined by (2.16) with α=2, we obtain
(3.62)K(x)*(-1)S2(x)*(LΔBu(x))=δ(m)(υ)*K(x)*(-1)S2(x).
By properties of convolution,
(3.63)LK(x)*ΔB(-1)S2(x)*u(x)=δ(m)(υ)*K(x)*(-1)S2(x).
By Lemmas 2.10 and 2.5, we obtain
(3.64)δ(x)*δ(x)*u(x)=δ(m)(υ)*K(x)*(-1)S2(x).
It follows that
(3.65)u(x)=δ(m)(υ)*K(x)*(-1)S2(x).
By (2.47) with k=1, we have
(3.66)K(x)=(R4H(x)*(-1)2R4e(x))*(O*1(x))*-1.
Taking into account (3.65), we obtain
(3.67)u(x)=δ(m)(υ)*(R4H(x)*(-1)2R4e(x))*(O*1(x))*-1*(-1)S2(x)forx∈∂Ω.
Now consider the casek=1,p=1, and q=n-1, that is, from (3.59), R2H(x) reduced to I2H(x) where I2H(x) is defined by (2.2) with α=2 and R2(x) reduced to I2(x) where I2(x) is defined by (2.17) with γ=2, and then the operator □ defined by (1.3) reduces to the wave operator
(3.68)□*=∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2,□B defined by (1.9) reduces to the Bessel wave operator
(3.69)□B*=Bx1-Bx2-Bx3-⋯-Bxn,
and then the solution M(x) reduced to
(3.70)M(x)=I2H(x)*I2(x)*W(x),
which is the solution of inhomogeneous wave equation
(3.71)□*□B*M(x)=W(x),
or
(3.72)(∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2)·(Bx1-Bx2-Bx3-⋯-Bxn)M(x)=W(x).
With the boundary condition for x∈∂Ω,
(3.73)L*□B*ΔBu(x)=0,
where L*=(3/4)Δ2+(1/4)(□*)2 and □* is defined by (3.68), or for x∈∂Ω and by (3.65), we obtain
(3.74)u(x)=δ(m)(s)*(I4H(x)*(-1)2R4e(x))*(D*1(x))*-1*(-1)S2(x),
where I4(x) is defined by (2.20) with γ=4, s=x12-x22-x32-⋯-xn2, and D(x) reduced from O(x) where it is defined by (2.34), that is, D(x)=(3/4)I4H(x)+(1/2)(-1)2R4e(x).



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*Source: 482467-2010-04-29.xml*