# On the Dirichlet Problem for the Stokes System in Multiply Connected Domains

Authors: Alberto Cialdea; Vita Leonessa; Angelica Malaspina
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The Dirichlet problem for the Stokes system in a multiply connected domain ofℝn(n≥2) is considered in the present paper. We give the necessary and sufficient conditions for the representability of the solution by means of a simple layer hydrodynamic potential, instead of the classical double layer hydrodynamic potential.


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## Body

## 1. Introduction

Potential theory methods have been employed for a long time in the study of boundary value problems. In particular they were widely used in BVPs for the Stokes system, starting from [1, 2].Recently some papers have used the integral representations of solutions for studying some BVPs for the Stokes system also in multiply connected domains [3–8]. All these papers concern the double layer hydrodynamic potential approach for the Dirichlet problem and the simple layer hydrodynamic potential approach for the traction problem.The aim of the present paper is to investigate a different integral representation for the Dirichlet problem for the Stokes system in a multiply connected bounded domain ofℝn (n≥2). Namely, we consider the simple layer potential approach for the Dirichlet problem in a domain
(1)Ω=Ω0∖⋃j=1mΩ-j,
where Ωj (j=0,…,m) are suitable domains with connected boundaries in C1,λ, λ∈(0,1].We use a new method which hinges on a singular integral system in which the unknown is a usual vector valued function, while the data is a vector whose components are differential forms.The paper is organized as follows. In Section2 we give an outlook of the method with a brief description of some previous results.After the preliminary Section3, in Section 4 we study in detail the case n=2, where some particular phenomena appear.Section5 is devoted to determine the eigenspace of a certain singular integral system in which the unknowns are differential forms of degree n-2 on ∂Ω. In the same section, we recall some known results concerning the eigenspaces of some classical integral systems.In Section6 we construct a left reduction for the singular integral system under study. Such a singular integral system is equivalent in a precise sense to the Fredholm system obtained through the reduction.Finally, in the last section, we find the solution of the Dirichlet problem for the Stokes system in a multiply connected domain by means of a simple layer hydrodynamic potential.The main result is that, givenf∈[W1,p(∂Ω)]n, we can represent the solution of the Dirichlet problem
(2)μΔv=∇rinΩ,divv=0inΩ,v=fon∂Ω,
by means of a simple layer hydrodynamic potential if, and only if, the conditions
(3)∫∂Ωjf·νdσ=0,j=0,1,…,m
are satisfied (ν being the outwards unit normal on ∂Ω). Moreover, if the data f satisfies only the condition
(4)∫∂Ωf·νdσ=0
(which is necessary for the existence of a solution of the Dirichlet problem (2)) we show how to modify the integral representation of the solution (see Theorem 23).


## 2. Sketch of the Method

The aim of this section is to give a better understanding of the method we are going to use in the present paper.We will do that by considering the Dirichlet problem for Laplace equation in a bounded simply connected domainΩ⊂ℝn, whose boundary we denote by Σ as follows:
(5)Δu=0inΩ,u=gonΣ.Suppose thatg∈W1,p(Σ), 1<p<∞. If we want to find the solution in the form of a simple layer potential whose density belongs to Lp(Σ), we have to solve an integral equation of the first kind on Σ as follows:
(6)∫Σ‍φ(y)s(x,y)dσy=g(x),x∈Σ,
where s(x,y) is the fundamental solution of Laplace equation
(7)s(x,y)={-12πlog1|x-y|,ifn=2,-1ωn(n-2)1|x-y|n-2,ifn≥3.In [9] a new method for discussing such an equation was proposed. Namely, the first step is to consider the differential (in the sense of the theory of differential forms) of both sides in (6). In this way we obtain the equation
(8)∫Σφ(y)dx[s(x,y)]dσy=dg(x),x∈Σ,
in which we look for a solution φ∈Lp(Σ).The integral on the left hand side is a singular integral and it can be considered as a linear and continuous operator fromLp(Σ) to L1p(Σ) (we denote by Lhp(Σ) the space of the differential forms of degree h whose coefficients belong to Lp(Σ) in every local coordinate system).It must be remarked that, ifn≥3, the space in which we look for the solution of (8) and the space in which the data is given are different.We recall that, ifB and B′ are two Banach spaces and S:B→B′ is a continuous linear operator, S can be reduced on the left if there exists a continuous linear operator S′:B′→B such that S′S=I+T, where I stands for the identity operator on B, and T:B→B is compact. Analogously, one can define an operator S reducible on the right. One of the main properties of such operators is that the equation Sα=β has a solution if, and only if, 〈γ,β〉=0 for any γ such that S*γ=0, S* being the adjoint of S (for more details see, e.g., [10, 11]).Let us denote bySφ the left hand side of (8). In [9] a reducing operator S′was explicitly constructed. This implies that there exists a solution of (8) if, and only if, the compatibility conditions
(9)∫Σdg∧h=0
are satisfied for any h∈Ln-2q(Σ) (q=p/(p-1)) such that S*h=0. Moreover one can show that S*h=0 if, and only if, h is a weakly closed form. Therefore the compatibility conditions  (9) are satisfied, and there exists a solution φ∈Lp(Σ) of  (8).A left reduction is said to beequivalent if N(S′)={0}, where N(S′) denotes the kernel of S′ (see, e.g., [11, page 19-20]). Obviously this means that Sx=y if, and only if, S′Sx=S′y. In [12] it was remarked that if N(S′S)=N(S), we still have a kind of equivalence. Indeed the coincidence of these two kernels implies the following fact: if y is such that the equation Sx=y is solvable, then this equation is satisfied if, and only if, S′Sx=S′y.SinceN(S′S)=N(S), then we have (8) equivalent to the Fredholm equation S′Sφ=S′(dg). These results lead to a simple layer potential theory for the Dirichlet problem (5).As a consequence one can obtain also a double layer representation for the Neumann problem for Laplace equation [12].A characteristic of this method is that it uses neither the theory of pseudodifferential operators nor the concept of hypersingular integrals.This method has been used also for studying other BVPs. In particular in [13] it was used to study the Dirichlet and the Neumann problems in multiply connected domains. Among other things, an interesting by-product of these results was obtained as follows (see [13, Theorem 6.1]).Let
u
be a harmonic function of class
C
1
(
Ω
¯
)
, where
Ω
is the multiple connected domain(1). There exists a2-formvconjugate touinΩif, and only if,(10)∫∂Ωj∂u∂νdσ=0,j=0,1,…,m.An explicit integral expression forv was also given. We recall that the 2-form v is conjugate to u if du=δv,dv=0.The method has been applied to different BVPs for several PDEs (see [12–19]).


## 3. Preliminaries

In this paperΩ denotes an (m+1)-connected domain of ℝn (n≥2), that is an open-connected set of the form (1), where each Ωj (j=0,…,m) is a bounded domain of ℝn with connected boundaries Σj∈C1,λ (λ∈(0,1]), and such that Ω¯j⊂Ω0  and  Ω¯j∩Ω¯k=∅,  j,k=1,…,m,j≠k. Let ν be the outwards unit normal on the boundary Σ=∂Ω.We consider the classical Stokes system for the incompressible viscous fluid(11)μΔu=∇p,divu=0,inΩ,
where the unknowns u=(u1,…,un) and p=p(x) are the velocity and pressure of the fluid flow, respectively, and the constant μ>0 is the kinematic viscosity of the fluid. A fundamental solution for this system is given by the pair of fundamental velocity tensor and its associated pressure vector
(12)γij(x,y)={-14πμ[(xi-yi)(xj-yj)|x-y|2δijlog1|x-y|-14πμ+(xi-yi)(xj-yj)|x-y|2],ifn=2,-12ωnμ[(xi-yi)(xj-yj)|x-y|nδijn-21|x-y|n-2-12ωnμ+(xi-yi)(xj-yj)|x-y|n],ifn≥3,(13)εj(x,y)=-1ωnxj-yj|x-y|n,
(i,j=1,…,n), ωn being the hypersurface measure of the unit sphere in ℝn. For a solution (u,p) of (11) we consider the following classical boundary operators:
(14)Tju=[-δijp+μ(∂jui+∂iuj)]νi,Tj′u=[δijp+μ(∂jui+∂iuj)]νi,j=1,…,n.Through this paper,p indicates a real number such that 1<p<+∞. We denote by [Lp(Σ)]n the space of all measurable vector-valued functions u=(u1,…,un) such that |uj|p is integrable over Σ (j=1,…,n). If h is any nonnegative integer, Lhp(Σ) is the vector space of all differential forms of degree h (briefly h-forms) defined on Σ such that their components are integrable functions belonging to Lp(Σ) in a coordinate system of class C1 and consequently in every coordinate system of class C1. The space [Lhp(Σ)]n is constituted by the vectors (v1,…,vn) such that vj is a differential form of Lhp(Σ) (j=1,…,n). [W1,p(Σ)]n is the vector space of all measurable vector-valued functions u=(u1,…,un) such that uj belongs to the Sobolev space W1,p(Σ) (j=1,…,n).The pair(v,r) with components
(15)vi(x)=-∫Σγij(x,y)φj(y)dσy,i=1,…,n,x∈ℝn,(16)r(x)=-∫Σεj(x,y)φj(y)dσy,x∈ℝn
is the simple layer hydrodynamic potential with density φ.The pair(w,q) with components
(17)wi(x)=∫ΣTj,y′[γi(x,y)]ψj(y)dσy,i=1,…,n,x∈ℝn,(18)q(x)=2μ∫Σ∂∂νy[εj(x,y)]ψj(y)dσy,x∈ℝn
is the double layer hydrodynamic potential with density ψ.


## 4. On the Bidimensional Case

It is wellknown that there are some exceptional plane domains in which no every harmonic function can be represented by a simple layer potential. The simplest example of this kind is given by the unit disk, for which one has(19)∫|y|=1log|x-y|dsy=0,|x|<1.It is also known that such domains do not occur in higher dimensions. For similar questions for the Laplace equation and the elasticity system, see [13, Section 3] and [16, Section 4], respectively.In this section we show that also for the Stokes system there are similar domains. We say that the boundary of the domainΩ is exceptional if there exists some constant vector which cannot be represented in Ω by a simple layer potential.Denoting byΣR the circle of radius R centered at the origin, we have the following lemma.Lemma 1.
The circleΣR with R=exp(1/2) is exceptional for the Stokes system.Proof.
Keeping in mind that (see, e.g., [16, Section 4])
(20)∫ΣRlog|x-y|dsy=2πRlogR,∫ΣR(xi-yi)(xj-yj)|x-y|2dsy=δijπR,|x|<R,
we find
(21)∫ΣRγij(x,y)dsy=R4μδij(2logR-1),|x|<R.
TakingR=exp(1/2) we obtain the result.Let us consider now the exceptional boundaries of not simply connected domains.Proposition 2.
LetΩ⊂ℝ2 be an (m+1)-connected domain. Denote by 𝒫 the eigenspace in [Lp(Σ)]2 of the singular integral system
(22)∫Σφj(y)∂∂sxγij(x,y)dsy=0,a.e.x∈Σ,i=1,2.
Then dim𝒫=2(m+1).Proof.
As in the proof of [16, Lemma 12], one can show that
(23)∂∂sxγij(x,y)=14πμδij∂∂sxlog|x-y|+𝒪(|y-x|h-1),
deduce that system (22) can be regularized to a Fredholm one, and see that its index is zero. Since the vectors eiχΣj  (by χX we denote the characteristic function of the set X) (i=1,2, j=0,1,…,m) are the only eigensolutions of the adjoint system
(24)∫Σφj(y)∂∂syγij(x,y)dsy=0,a.e.x∈Σ,i=1,2,
we have dim𝒫=2(m+1).Theorem 3.
LetΩ⊂ℝ2 be an (m+1)-connected domain. The following conditions are equivalent(1)
There exists a Hölder continuous vector functionφ≢0 such that
(25)∫Σγ(x,y)φ(y)dsy=0,x∈Σ.(2)
There exists a constant vector which cannot be represented inΩ by a simple layer potential;(3)
Σ0 is exceptional.(4)
Letφ1,…,φ2m+2 be linearly independent vectors of 𝒫 (see Proposition 2), and let cjk=(αjk,βjk)∈ℝ2 be given by
(26)∫Σγ(x,y)φj(y)dsy=cjk,x∈Σk,j=1,…,2m+2,k=0,1,…,m.
Then det𝒞=0, where
(27)𝒞=(α1,0⋯α2m+2,0⋯⋯⋯α1,m⋯α2m+2,mβ1,0⋯β2m+2,0⋯⋯⋯β1,m⋯β2m+2,m).Proof.
The proof runs as in [16, Theorem 1] with obvious modifications. We omit the details.


## 5. Some Eigenspaces

We determine the structure of the kernel of a particular singular integral system. Namely, let us denote by𝒩p the space of ψ∈[Ln-2p(Σ)]n such that
(28)∫Σψj(y)∧dy[γij(x,y)]=0,a.e.onΣ,i=1,…,n.We begin by proving the following result.Lemma 4.
Letu∈[C0∞(ℝn)]n. Then, for any x∈ℝn,
(29)ui(x)=μ∫ℝnΔuj(y)γij(x,y)dy+∫ℝn∂2uj(y)∂yi∂yjs(x,y)dy,
where γ(x,y) and s(x,y) are given by (12) and (7), respectively.Proof.
By the well-known Stokes identity we have(30)ui(x)=∫ℝnΔui(y)s(x,y)dy=∫ℝnΔuj(y)δijs(x,y)dy.
Since, for every n≠2,4,
(31)(xi-yi)(xj-yj)|x-y|n=1(4-n)(2-n)∂2∂yi∂yj|x-y|4-n-δijωns(x,y),γij(x,y)=δij2μs(x,y)-12μωn(xi-yi)(xj-yj)|x-y|n,∀n≥2,
we can rewrite
(32)γij(x,y)=δijμs(x,y)-12μωn1(4-n)(2-n)∂2∂yi∂yj×|x-y|4-n.
Then(33)δijs(x,y)=μγij(x,y)+12ωn(4-n)(2-n)∂2∂yi∂yj|x-y|4-n,ui(x)=μ∫ℝnΔuj(y)γij(x,y)dyui(x)=+12ωn(4-n)(2-n)∫ℝnΔui(y)∂2∂yi∂yj|x-y|4-ndy.
Integrating by parts, it follows that the last integral is equal to
(34)12ωn(4-n)(2-n)∫ℝn∂2uj(y)∂yi∂yjΔy|x-y|4-ndy=∫ℝn∂2uj(y)∂yi∂yjs(x,y)dy,
since Δy|x-y|4-n=2(4-n)|x-y|2-n. Then the claim holds for n≠2,4.
In the same manner it is possible to show formula (29) for n=2 and n=4 after observing that, if n=2, we have
(35)(xi-yi)(xj-yj)|x-y|2=12∂2∂yi∂yj|x-y|2log|x-y|-δijlog|x-y|-12δij,Δy|x-y|2log|x-y|=4(log|x-y|+1), while, for n=4,
(36)(xi-yi)(xj-yj)|x-y|4=δij2|x-y|2+12∂2∂yi∂yjlog|x-y|,
Δylog|x-y|=2/|x-y|2.Lemma 5.
Letζ1,…,ζn be differential forms in Ln-2p(Σ) such that dζj=(-1)n-1νjdσ on Σ. One has ψ∈𝒩p if, and only if,
(37)ψj=∑h=0mchχΣhζj+ηj,j=1,…,n,
where c0,…,cm∈ℝ and η1…,ηn are weakly closed forms belonging to Ln-2p(Σ).Proof.
It is easy to construct the differential formsζ1,…,ζn. For example, one can take the restriction on Σ of the following forms: ζ1=(-1)n-1x2dx3⋯dxn, ζj=(-1)n-jx1dx2⋯j^⋯dxn (j=2,…,n). We remark that (37) holds if, and only if, the weak differentials dψj exist and
(38)dψj=(-1)n-1∑h=0mchχΣhνjdσ,j=1,…,n,
that is,
(39)∫Σψj∧duj=∑h=0mch∫Σhu·νdσ,∀u∈[C0∞(ℝn)]n.
Let us prove that (39) holds if, and only if,
(40)∫Σkψj∧duj=ck∫Σku·νdσ,∀u∈[C0∞(ℝn)]n,k=0,…,m.
It is obvious that (40) implies (39).
Conversely, suppose that  (39) is true. Define Ukε={x∈ℝn∣dist(x,Σk)<ε}, where 0<ε<min0≤h<k≤mdist(Σh,Σk). Let vk∈C0∞(Ukε) be such that vk=1 in Ukε/2. Since vku∈[C0∞(ℝn)]n, we may write
(41)∫Σψj∧d(vkuj)=∑h=0mch∫Σhvku·νdσ,
and (40) follows immediately.
Suppose now that (39) is true. From (40) it follows that
(42)∫Σkψj(y)∧dy[γij(x,y)]=ck∫Σkγij(x,y)νj(y)dσy,γij(x,y)νj(y)dσy∀x∉Σk.
An integration by parts shows that(43)∫Σkψj(y)∧dy[γij(x,y)]=0,∀x∉Ω¯k.
Taking the exterior angular boundary value (for the definition of internal (external) angular boundary values see, e.g., [20, page 53] or [21, page 293]), we have
(44)∫Σkψj(y)∧dy[γij(x,y)]=0
a.e. on Σk. Arguing as in [9, pages 189-190], this implies that
(45)∫Σkψj(y)∧dy[γij(x,y)]=0
also in Ωk. Summing over k we find
(46)∫Σψj(y)∧dy[γij(x,y)]=0,
for every x∈ℝn∖Σ and a.e. on Σ. In particular ψ is the solution of the singular integral system (28).
Conversely, suppose (28) holds. Arguing again as in [9, pages 189-190], from (28) it follows that
(47)∫Σψj(y)∧dy[γij(x,y)]=0,x∉Σ.
Since εj(x,y)=-∂xjs(x,y), system (11) implies that Δx[γij(x,y)]=-(1/μ)(∂2/∂xi∂xj)s(x,y). Hence,
(48)∂2∂xi∂xj∫Σψj(y)∧dy[s(x,y)]=0,x∉Σ.
Therefore, there exist some constants a0,a1,…,am such that
(49)∂jΨj(x)={-ahx∈Ωh,h=1,…,m,-a0x∈Ω,0x∈ℝn∖Ω¯,
where
(50)Ψj(x)=∫Σψj(y)∧dy[s(x,y)].
Then, on account of Lemma 4, for every u∈[C0∞(ℝn)]n,
(51)∫Σψj∧duj=μ∫ℝnΔuj(x)dx∫Σψj(y)∧dy[γij(x,y)]+∫ℝn∂2∂xi∂xjui(x)dx∫Σψj(y)∧dy[s(x,y)].
The first term of the right hand side vanishes because of (47). As far as the second one is concerned, integrating by parts we get
(52)∫ℝn∂2∂xi∂xjui(x)Ψj(x)dx=∑h=1m∫Ωh∂2∂xi∂xjui(x)Ψj(x)dx+∫Ω∂2∂xi∂xjui(x)Ψj(x)dx+∫ℝn∖Ω¯0∂2∂xi∂xjui(x)Ψj(x)dx=-∑h=1m∫Σh∂iuiΨjνjdσ+∑h=1m∫Ωh∂iui∂jΨjdx+∫Σ∂iuiΨjνjdσ-∫Ω∂iui∂jΨjdx-∫Σ0∂iuiΨjνjdσ+∫ℝn∖Ω¯0∂iui∂jΨjdx=∑h=1m‍∫Ωh∂iui∂jΨjdx-∫Ω∂iui∂jΨjdx.
Hence, by (49),
(53)∫ℝn∂2∂xi∂xjui(x)Ψj(x)dx=-∑h=1mah∫Ωh‍∂iuidx+a0∫Ω∂iuidx=∑h=1mah∫Σhu·νdσ+a0∫Σu·νdσ=a0∫Σ0u·νdσ+∑h=1m(a0+ah)∫Σhu·νdσ.
By setting c0=a0 and ch=a0+ah (h=1,…,m) we get the claim.Remark 6.
Lemma5 shows that the dimension of the kernel 𝒩p is infinite. However, if we consider the quotient space 𝒩p/Ξp, Ξp being the space of weakly closed differential forms in Ln-2p(Σ), we have dim(𝒩p/Ξp)=m+1.
We conclude this section by recalling some properties concerning the following eigenspaces:(54)𝒱±={φk∈Lp(Σ):±12φk(x)+∫ΣFki(x,y)φi(y)dσy=0,k=1,…,n},𝒲±={φk∈Lp(Σ):∓12φk(x)+∫ΣFik(y,x)φi(y)dσy=0,k=1,…,n},
where (see, e.g., [22])
(55)Fki(x,y):=Ti,y′[γk(x,y)]=-nωn(xk-yk)(xi-yi)(xj-yj)|x-y|n+2νj(y).
For the proofs of the following two results see [7, Lemma 3.3] and [8, Theorem 3.2], respectively.Proposition 7.
The sets𝒱+ and 𝒲- are linear subspaces of L1(Σ) and
(56)dim(𝒱+)=dim(𝒲-)=1+n(n+1)m2.
A basis of𝒲- is expressed by the fields {ψih,ν:i=1,…,n(n+1)/2,h=1,…,m}. The simple layer potentials vih whose densities are ψik such that: vih|Ω¯k=δhkρi, i=1,…,n(n+1)/2,h,k=1,…,m, where ρi are rigid displacement in ℝn, specifically ρi(x)=ei,i=1,…,n, and, for i=n+1,…,n(n+1)/2, ρi(x)=(eh∧ek)x, h=1,…,n-1,k=h+1,…,n,h[n-(h+1)/2]+ k=i.
In addition, everyψ∈𝒲- has the property that v|Σ0=0, where v is the simple layer potential with density ψ.Proposition 8.
The sets𝒱- and 𝒲+ are linear subspaces of L1(Σ) and
(57)dim(𝒱-)=dim(𝒲+)=n(n+1)2+m.
A basis for 𝒲+ is expressed by the fields {ψi,νχΣh:i=1,…,n(n+1)/2,h=1,…,m}, where ψi, i=1,…,n(n+1)/2 are zero on Σ∖Σ0, and such that the simple layer potentials with density ψi are n(n+1)/2 rigid displacement in Ω0 (linearly independent for n≥3).
Finally, every functionφ which is the restriction to Σ of a rigid displacement belongs to 𝒱-.One recalls that ifφ∈[L1(Σ)]n belongs to one of the eigenspaces 𝒱±,𝒲±, then φ∈[Cλ(Σ)]n. This follows from general results about integral equations (see [8, Lemma 31] and [7, page 81]).Remark 9.
We can make the statement of Proposition8 slightly more precise, saying that the simple layer potentials with densityψiaren(n+1)/2rigid displacement inΩ0linear independent for anyn≥2, unlessn=2andΣ0is exceptional. Indeed, let us show that if n=2 and Σ0 is not exceptional, such rigid displacements are linearly independent. Let ci be such that
(58)∑i=13ci∫Σ0ψi(y)γ(x,y)dσy=0,inΩ0.
We have also
(59)∫Σ0∑i=13ciψi(y)γ(x,y)dσy=0,a.e.x∈Σ0.
Let φ=∑i=13‍ciψi. In view of the equivalence between (1) and (3) of Theorem 3, φ has to vanish. Therefore ci=0 (i=1,2,3) because of the linearly independence of ψi. On the other hand, if n=2 and Σ0 is exceptional, Theorem 3 shows that the potentials with densities {ψi}i=1,2,3 are linearly dependent.


## 6. Reduction of a Certain Singular Integral Operator

For everyψ∈L1p(Σ), let Θh be the operator defined by
(60)Θh(ψ)(x)=*(∫Σdx[sn-2(x,y)]∧ψ(y)∧dxh),(y)∧dxhdx[sn-2(x,y)]∧ψx∈Ω,
where * and d denote the Hodge star operator and the exterior derivative, respectively, and sh(x,y) is the double h-form introduced by Hodge in [23] as follows:
(61)sh(x,y)=∑j1<⋯<jhs(x,y)dxj1⋯dxjhdyj1⋯dyjh.
Note that the operator Θh satisfies the equation
(62)∂h∫Σu(y)∂∂νys(x,y)dσy=-Θh(du),x∈Ω,
for each u∈W1,p(Σ), since (see [9, page 187])
(63)*d∫Σu(y)∂∂νys(x,y)dσy=dx∫Σdu(y)∧sn-2(x,y),x∈Ω.Moreover we introduce the operatorsℋjh defined as
(64)ℋjh(ψ)(x)=Θh(ψj)(x)-δlij3⋯jn123⋯n(n-2)!×∫Σ∂xhHlj(x,y)∧ψi(y)∧dyj3⋯∧dyjn,
for every ψ∈[L1p(Σ)]n, where
(65)Hlj(x,y)=1ωn(yl-xl)(yj-xj)|y-x|n.In the sequeldu denotes the vector (du1,…,dun) whose elements are 1-forms, and ψ=(ψ1,…,ψn)∈[L1p(Σ)]n.Lemma 10.
Let(w,q) be the double layer hydrodynamic potential of (17)-(18) with density u∈[W1,p(Σ)]n. Then, for x∉Σ,
(66)∂hwj(x)=ℋjh(du)(x),(67)q(x)=2μΘh(duh)(x),
where ℋjh and Θh are given by (60) and (64), respectively.Proof.
Note that, even if one could prove (66)-(67) directly, it seems easier to deduce them from the similar results we have already obtained for the elasticity system (see [16, Section 3]). For k>(n-2)/n, let w(k) be the double layer elastic potential with density u, that is,
(68)w(k)j(x)=∫Σui(y)Li,y(k)[Γj(k)(x,y)]dσy,
where L(k) and Γ(k) are the stress operator and the Kelvin's matrix associated to the Lamé system -Δu-k∇divu=0, respectively.
Thanks to [16, Lemma 1], we know that
(69)∂hw(k)j(x)=ℋ(k)jh(du)(x),
where
(70)ℋ(k)jh(ψ)(x)=Θh(ψj)(x)-δlij3⋯jn123⋯n(n-2)!×∫Σ∂xhH(k)lj(x,y)∧ψi(y)∧dyj3⋯dyjn,H(k)lj(x,y)=kωn(k+1)(yl-xl)(yj-xj)|y-x|n-1k+1δljs(x,y),
and Θh is given by  (60).
From [16, formula (5)] (where we set ξ=1), letting k→+∞, we get
(71)∂xh{Li,y(k)[Γj(k)(x,y)]}⟶-nωn∂xh{(yi-xi)(yj-xj)(yk-xk)|x-y|n+2νk(y)}=∂xhTi,y′[γj(x,y)],x∉Σ, from which ∂hw(k)→∂hw as k→+∞. Therefore we obtain formula (66) by letting k→+∞ in (69). Formula (67) is an immediate consequence of (62) because εj(x,y)=-∂xjs(x,y).For the next lemma it is convenient to recall here two jump formulas proved in [16, Lemmas 2 and 3].Letf∈L1(Σ). If η∈Σ is a Lebesgue point for f, we get
(72)limx→η∫Σf(y)∂xs(yl-xl)(yj-xj)|x-y|ndσy=ωn2(δlj-2νj(η)νl(η))νs(η)f(η)+∫Σ‍f(y)∂xs(yl-ηl)(yj-ηj)|x-y|ndσy,
where the limit has to be understood as an internal angular boundary value, and the integral in the right hand side is a singular integral.Further, letψ∈L1p(Σ) and write ψ as ψ=ψhdxh with
(73)νhψh=0.
Assumption (73) is not restrictive, because, given the 1-form ψ on Σ, there exist scalar functions ψh defined on Σ such that ψ=ψhdxh and (73) holds (see [24, page 41]). Then, for almost every η∈Σ,
(74)limx→ηΘh(ψ)(x)=-12ψh(η)+Θh(ψ)(η),
where Θh is given by (60), and the limit has to be understood again as an internal angular boundary value.Lemma 11.
Letψ∈L1p(Σ). Let one write ψ as ψ=ψhdxh and suppose that (73) holds. Then, for almost every η∈Σ,
(75)limx→η1(n-2)!δlij3⋯jn123⋯n∫Σ∂xsHlj(x,y)∧ψ(y)∧dyj3⋯∧dyjn=-12[νj(η)ψi(η)+νi(η)ψj(η)]νs(η)+1(n-2)!δlij3⋯jn123⋯n∫Σ∂xsHlj(η,y)∧ψ(y)∧dyj3⋯∧dyjn,
where Hlj is defined by (65), and the limit has to be understood as an internal angular boundary value.Proof.
We have(76)1(n-2)!δlij3⋯jn123⋯n∫Σ‍∂xsHlj(x,y)∧ψ(y)∧dyj3⋯∧dyjn=1(n-2)!δlij3⋯jn123⋯nδrhj3⋯jn123⋯n∫Σ‍∂xsHlj(x,y)ψh(y)νr(y)dσy=δrhli∫Σ‍∂xsHlj(x,y)ψh(y)νr(y)dσy.
Hence, by (65) and (72),
(77)limx→η1(n-2)!δlij3⋯jn123⋯n∫Σ‍∂xsHlj(x,y)∧ψ(y)∧dyj3⋯∧dyjn=δrhli2(δlj-2νj(η)νl(η))νs(η)νr(η)ψh(η)+1(n-2)!δlij3⋯jn123⋯n∫Σ‍∂xsHlj(η,y)∧ψ(y)∧dyj3⋯∧dyjn.
Keeping in mind (73), we find
(78)δrhli2(δlj-2νjνl)νsνrψh=(12δljνs-νjνlνs)(νlψi-νiψl)=-12νsνjψi-12νsνiψj,
and the result follows.Lemma 12.
Letψ=(ψ1,…,ψn)∈[L1p(Σ)]n. Then, for almost every η∈Σ,
(79)limx→ημ[2δijΘh(ψh)(x)+ℋij(ψ)(x)+ℋji(ψ)(x)]νi(x)=μ[2δijΘh(ψh)(η)+ℋij(ψ)(η)+ℋji(ψ)(η)]νi(η),Θh and ℋ being as in (60) and (64), respectively, and the limit has to be understood as an internal angular boundary value.Proof.
Let us writeψi as ψi=ψihdxh with
(80)νhψih=0,i=1,…,n.
On account of (72) and (74), we infer
(81)limx→ημ[2δijΘh(ψh)(x)+ℋij(ψ)(x)+ℋji(ψ)(x)]νi(x)=μΨij(ψ)(η)νi(η)+μ[2δijΘh(ψh)(η)+μ+ℋij(ψ)(η)+ℋji(ψ)(η)]νi(η),
where
(82)Ψij(ψ)=-δijψhh-12ψij+12(νiψss+νsψsi)νj-12ψji+12(νjψss+νsψsj)νi.
By (80) we get Ψij(ψ)νi=-ψhhνj-ψijνi/2+ψssνj+νsψsj/2=0.Remark 13.
Whenever we consider external boundary values, we have just to change the sign in the first term on the right hand sides in (72), (74), and (75), while (79) remains unchanged.Lemma 14.
Letw be the double layer potential (17) with density u∈[W1,p(Σ)]n. Then T+,jw=T-,jw=μ[2δijΘh(duh)+ℋij(du)+ℋji(du)]νi a.e. on Σ, where T+w and T-w denote the internal and the external angular boundary limits of Tw, respectively, and Θh is given by (60) and ℋ by (64).Proof.
It is an immediate consequence of (66), (67), (79), and Remark 13.Proposition 15.
LetR:[Lp(Σ)]n→[L1p(Σ)]n be the following singular integral operator
(83)Rφ(x)=-∫Σdx[γ(x,y)]φ(y)dσy.
Let one define R′:[L1p(Σ)]n→[Lp(Σ)]n to be the singular integral operator
(84)Rj′(ψ)(x)=μ[2δijΘh(ψh)(x)+ℋij(ψ)(x)μ+ℋji(ψ)(x)]νi(x).
Then
(85)R′Rφ=14φ-K2φ,
where
(86)Kφ(x)=-∫ΣTx[γ(x,y)]φ(y)dσy.Proof.
Letv be the simple layer potential (15) with density φ∈[Lp(Σ)]n. In view of Lemma 14, we have a.e. on Σ(87)Rj′(Rφ)=μ[2δijΘh(dvh)+ℋij(dv)+ℋji(dv)]νi=Tjw,
where w is the double layer potential (17) with density v. Moreover, if x∈Ω,
(88)wk(x)=∫Σvi(y)Ti,y′[γk(x,y)]dσy=vk(x)+∫Σγik(x,y)Ti[v(y)]dσy,
and then, on account of (86),
(89)Tw=12Tv-K(Tv)=12(12φ+Kφ)-K(12φ+Kφ)=14φ-K2φ.


## 7. The Dirichlet Problem

Let us consider the Dirichlet problem for the Stokes system(90)μΔv=∇rinΩ,divv=0inΩ,v=fonΣ,
where the given data f∈[W1,p(Σ)]n satisfies the compatibility condition (4).The aim of the present section is to study the representability of the solution of this problem by means of a simple layer hydrodynamic potential (15)-(16).By the symbol𝒮p we mean the class of the simple layer hydrodynamic potentials (15)-(16) with density in [Lp(Σ)]n. Whenever n=2 and Σ0 is exceptional (see Section 4), we say that (v,r) belongs to 𝒮p if, and only if,
(91)v(x)=-∫Σγ(x,y)φ(y)dσy+c,x∈Ω,r(x)=-∫Σεj(x,y)φj(y)dσy,x∈Ω,
where φ∈[Lp(Σ)]2 and c∈ℝ2.We will see that condition (4) is not sufficient to prove the existence of the solution in the class 𝒮p, but it must be satisfied on each Σj, j=0,1,…,m.We begin by proving the following result.Theorem 16.
Givenω∈[L1p(Σ)]n, there exists a solution φ∈[Lp(Σ)]n of the singular integral system
(92)-∫Σdx[γ(x,y)]φ(y)dσy=ω(x),a.e.x∈Σ,
if, and only if,
(93)∫Σψi∧ωi=0,
for every ψ=(ψ1,…,ψn)∈[Ln-2q(Σ)]n  (q=p/(p-1)) such that the weak differentials dψj exist and  (38) holds for some real constants c0,…,cm.Proof.
Consider the adjoint ofR (see (83)), R*:[Ln-2q(Σ)]n→[Lq(Σ)]n, that is, the operator whose components are given by
(94)Ri*ψ(x)=-∫Σψi(y)∧dy[γij(x,y)].
Proposition 15 implies that the integral system (92) has a solution φ∈[Lp(Σ)]n if, and only if,
(95)∫Σψi∧ωi=0,
for each ψ=(ψ1,…,ψn)∈[Ln-2q(Σ)]n such that R*ψ=0. The result follows from Lemma 5.Proposition 17.
Givenf∈[W1,p(Σ)]n, there exists a solution of the BVP
(96)(v~,r~)∈𝒮p,μΔv~=∇r~inΩ,divv~=0inΩ,dv~=dfonΣ,
if, and only if, conditions (3) are satisfied. The density φ of the pair (v~,r~) (see (15)-(16)) solves the singular integral system Rφ=df, where R is given by (83).Proof.
Clearly, there exists a solution of this BVP if, and only if, there exists a solutionφ∈[Lp(Σ)]n of the singular integral system
(97)-∫Σdx[γ(x,y)]φ(y)dσy=df(x),a.e.x∈Σ.
In view of Theorem16, there exists a solution φ of this system if, and only if,
(98)∫Σψi∧dfi=0,
for every ψ=(ψ1,…,ψn)∈[Ln-2q(Σ)]n satisfying R*ψ=0, that is, such that the weak differentials dψj exist and (39) holds for some real constants c0,…,cm. Equation (39) being true for any u∈[W1,p(Σ)]n, we can write
(99)∫Σψi∧dfi=∑h=0mch∫Σhf·νdσ,
because of a density argument. In view of the arbitrariness of c0,…,cm, (98) is satisfied if, and only if, (3) holds.Proposition 18.
Letah∈ℝn  (h=0,…,m). Let ψik, i=1,…,n, k=1,…,m, be the elements of the basis of 𝒲- given by Proposition 7. The pair
(100)v0(x)=∑k=1m∑i=1n(aki-a0i)∫Σγ(x,y)ψik(y)dσy+a0,x∈Ω,r0(x)=∑k=1m∑i=1n(aki-a0i)∫Σε(x,y)ψik(y)dσy,x∈Ω,
is the solution of the BVP
(101)(v0,r0)∈𝒮p,μΔv0=∇r0inΩ,divv0=0inΩ,v0=ahonΣh,h=0,…,m.Proof.
The pair(v0,r0) belongs to 𝒮p (for n=2, see Remark 9). Obviously it satisfies the Stokes system, and it satisfies the boundary conditions since, thanks to Proposition 7,
(102)v0∣Σ0=∑k=1m∑i=1n(aki-a0i)vik∣Σ0+a0=a0,v0∣Σh=∑k=1m∑i=1n(aki-a0i)vik∣Σh+a0=∑k=1m∑i=1n(aki-a0i)δhkei+a0=ah,
for any h=1,…,m.Theorem 19.
Givenf∈[W1,p(Σ)]n, the Dirichlet problem
(103)(v,r)∈𝒮p,μΔv=∇rinΩ,divv=0inΩ,v=fonΣ,
is solvable if, and only if, conditions (3) are satisfied. Moreover the solution (v,r) is unique (r is unique up to an additive constant).Proof.
Suppose conditions (3) are satisfied. Let (v~,r~) be a solution of the problem (96). Since dv~=df on Σ, v~=f+ah on Σh (h=0,…,m) for some ah∈ℝn. The pair (v,r)=(v~,r~)- (v0,r0), where v0 and r0 are given by (100), solves the problem (103).
Conversely, if there exists a solution(v,r) of (103), the compatibility condition (4) has to be satisfied. Moreover, for any j=1,…,m, (v,r) is the solution of the Stokes system also in Ωj. Therefore conditions (3) are satisfied for j=1,…,m. These, together with (4), imply (3) also for j=0. The uniqueness is known [7, Theorem 5.5].Remark 20.
The density(φ,ε) of (v,r) can be written as (φ,ε)=(φ0+λ0,ε), where φ0 solves the singular integral system (97), and (λ0,ε) is the density of a simple layer potential which is constant on every connected component of Σ.Remark 21.
Ifn≥3 or n=2 and Σ0 is not exceptional, denoting by φ the density of the simple layer potential (15)-(16) obtained in Theorem 19, we have φ that solves the integral system of the first kind
(104)-∫Σγ(x,y)φ(y)dσy=f(x)
on Σ. Therefore, Theorem 19 can be seen as an existence theorem for the integral system of the first kind (104) in Lp(Σ).
Ifn=2 and Σ0 is exceptional, we have the existence of a solution (φ,c)∈[Lp(Σ)]2×ℝ2 of the integral equation
(105)-∫Σγ(x,y)φ(y)dσy=f(x)+conΣ.Remark 22.
Observe that the solvability of the Dirichlet problem (90) by means of a simple layer potential hinges on the singular integral system (97). Thanks to Proposition 15, the operator R′ provides a left reduction for such a system. This reduction is not an equivalent one, but, as in [25, pages 253-254], one can show that R′ is a weakly equivalent reduction (see definition in Section 3). Since the system Rφ=df is solvable, we have Rφ=df if, and only if, φ is solution of the Fredholm system R′Rφ=R′df. In this sense, such Fredholm system is equivalent to the problem (103).
In order to obtain a similar integral representation for the solution of the Dirichlet problem (90) when f satisfies the only condition (4), we need to modify the representation of the solution by adding an extra term.
By𝒮~p we denote the space of all pairs (v,r) written as
(106)vi(x)=-∫Σγij(x,y)φj(y)dσy+∫ΣTj,y′[γi(x,y)]ψj(y)dσy,i=1,…,n,x∈Ω,r(x)=-∫Σεj(x,y)φj(y)dσy+2μ∫Σ∂∂νy[εj(x,y)]ψj(y)dσy,x∈Ω,
where φ and ψ belong to [Lp(Σ)]n.Theorem 23.
Givenf∈[W1,p(Σ)]n satisfying (4), the Dirichlet problem
(107)(v,r)∈𝒮~p,μΔv=∇rinΩ,divv=0inΩ,v=fonΣ,
has one, and only one, solution (v,r) given by
(108)vi(x)=-∫Σγij(x,y)φj(y)dσy+∫ΣTj,y′[γi(x,y)]fj(y)dσy,x∈Ω,(109)r(x)=-∫Σεj(x,y)φj(y)dσy+2μ∫Σ∂∂νy[εj(x,y)]fj(y)dσy,x∈Ω,
where φ∈[Lp(Σ)]n is solution of the integral system of the first kind
(110)-∫Σγij(x,y)φj(y)dσy=12fi(x)-∫ΣTj,y′[γi(x,y)]fj(y)dσy,a.e.onΣ.Proof.
Letv be given by (108); imposing the boundary condition, we get (the symbol w+ (w-) stands for the interior (exterior) value of the double layer potential (17) on Σ)(111)-∫Σγij(x,y)φj(y)dσy=fi(x)-w+,i(x),a.e.x∈Σ.
In view of Remark 21 such a system is solvable if, and only if,
(112)∫Σh(fi-w+,i)νidσ=0,h=0,…,m.
On the other hand, because of the jump formulas, we have
(113)fi(x)-w+,i(x)=fi(x)-(12fi(x)+∫ΣTj,y′[γi(x,y)]fj(y)dσy)∶=12fi(x)-wi(x)=-w-,i(x),a.e.x∈Σ.
Therefore, conditions (112) become -∫Σh‍w-·νdσ=0, h=0,…,m. Since w- can be considered as the datum of the interior Dirichlet problem in Ωh, for h=1,…,m, we have
(114)∫Σhw-·νdσ=0,h=1,…,m.
As far as h=0 is concerned, first we remark that (4) implies ∫Σw·νdσ=0, because
(115)0=∫Σw+·νdσ=∫Σ(12f+w)·νdσ=∫Σw·νdσ.
Keeping in mind (4) and (114), this leads to
(116)∫Σ0(12f-w)·νdσ=12∫Σ0f·νdσ-∫Σ0w·νdσ=-12∑j=1m∫Σjf·νdσ+∑j=1m∫Σjw·νdσ=∑j=1m∫Σjw-·νdσ=0.
Finally, assume that (v,r) is the solution of (107) with the data f=0. The integral representation (108) shows that (v,r)∈𝒮p, and then the uniqueness follows from Theorem 19.



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