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

Abstract

The Dirichlet problem for the Stokes system in a multiply connected domain of ℝ^𝑛(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.

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 of ℝ^𝑛 (𝑛≥2). Namely, we consider the simple layer potential approach for the Dirichlet problem in a domain Ω=Ω₀⧵^𝑚_𝑗=1Ω_𝑗, (1) where Ω_𝑗 (𝑗=0,…,𝑚) are suitable domains with connected boundaries in 𝐶^1,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 Section 2 we give an outlook of the method with a brief description of some previous results.

After the preliminary Section 3, in Section 4 we study in detail the case 𝑛=2, where some particular phenomena appear.

Section 5 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 Section 6 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, given 𝑓∈[𝑊^1,𝑝(𝜕Ω)]^𝑛, we can represent the solution of the Dirichlet problem 𝜇Δ𝑣=∇𝑟inΩ,div𝑣=0inΩ,𝑣=𝑓on𝜕Ω, (2) by means of a simple layer hydrodynamic potential if, and only if, the conditions _𝜕Ω𝑗𝑓⋅𝜈𝑑𝜎=0,𝑗=0,1,…,𝑚 (3) are satisfied (𝜈 being the outwards unit normal on 𝜕Ω). Moreover, if the data 𝑓 satisfies only the condition _𝜕Ω𝑓⋅𝜈𝑑𝜎=0 (4) (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 domain Ω⊂ℝ^𝑛, whose boundary we denote by Σ as follows: Δ𝑢=0inΩ,𝑢=𝑔onΣ. (5)

Suppose that 𝑔∈𝑊^1,𝑝(Σ), 1<𝑝<∞. If we want to find the solution in the form of a simple layer potential whose density belongs to 𝐿^𝑝(Σ), we have to solve an integral equation of the first kind on Σ as follows: _Σ𝜑𝑦𝑠𝑥,𝑦𝑑𝜎_𝑦=𝑔(𝑥),𝑥∈Σ, (6) where 𝑠(𝑥,𝑦) is the fundamental solution of Laplace equation 𝑠=−1𝑥,𝑦12𝜋log−1𝑥−𝑦,if𝑛=2,𝜔_𝑛()1𝑛−2𝑥−𝑦^𝑛−2,if𝑛≥3. (7)

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 _Σ𝜑𝑦𝑑_𝑥𝑠𝑥,𝑦𝑑𝜎_𝑦=𝑑𝑔(𝑥),𝑥∈Σ, (8) in which we look for a solution 𝜑∈𝐿^𝑝(Σ).

The integral on the left hand side is a singular integral and it can be considered as a linear and continuous operator from 𝐿^𝑝(Σ) to 𝐿^𝑝₁(Σ) (we denote by 𝐿^𝑝_ℎ(Σ) the space of the differential forms of degree ℎ whose coefficients belong to 𝐿^𝑝(Σ) in every local coordinate system).

It must be remarked that, if 𝑛≥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, if 𝐵 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 by 𝑆𝜑 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 _Σ𝑑𝑔∧ℎ=0 (9) are satisfied for any ℎ∈𝐿^𝑞_𝑛−2(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∈𝐿^𝑝(Σ) of (8).

A left reduction is said to be equivalent 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.

Since N(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 𝐶¹(Ω), where Ω is the multiple connected domain (1). There exists a 2-form 𝑣 conjugate to 𝑢 in Ω if, and only if,_𝜕Ω𝑗𝜕𝑢𝜕𝜈𝑑𝜎=0,𝑗=0,1,…,𝑚. (10)

An explicit integral expression for 𝑣 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 paper Ω denotes an (𝑚+1)-connected domain of ℝ^𝑛 (𝑛≥2), that is an open-connected set of the form (1), where each Ω_𝑗 (𝑗=0,…,𝑚) is a bounded domain of ℝ^𝑛 with connected boundaries Σ_𝑗∈𝐶^1,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 𝜇Δ𝑢=∇𝑝,div𝑢=0,inΩ, (11) where the unknowns 𝑢=(𝑢₁,…,𝑢_𝑛) 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 𝛾_𝑖𝑗=−1𝑥,𝑦𝛿4𝜋𝜇^𝑖𝑗1log+𝑥𝑥−𝑦_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦²−1,if𝑛=2,2𝜔_𝑛𝜇𝛿_𝑖𝑗1𝑛−2𝑥−𝑦^𝑛−2+𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦^𝑛𝜀,if𝑛≥3,_𝑗1𝑥,𝑦=−𝜔_𝑛𝑥_𝑗−𝑦_𝑗𝑥−𝑦^𝑛, (12)(13) (𝑖,𝑗=1,…,𝑛), 𝜔_𝑛 being the hypersurface measure of the unit sphere in ℝ^𝑛. For a solution (𝑢,𝑝) of (11) we consider the following classical boundary operators: 𝑇_𝑗𝑢=−𝛿_𝑖𝑗𝑝+𝜇𝜕_𝑗𝑢_𝑖+𝜕_𝑖𝑢_𝑗𝜈_𝑖,𝑇^′_𝑗𝑢=𝛿_𝑖𝑗𝑝+𝜇𝜕_𝑗𝑢_𝑖+𝜕_𝑖𝑢_𝑗𝜈_𝑖,𝑗=1,…,𝑛. (14)

Through this paper, 𝑝 indicates a real number such that 1<𝑝<+∞. We denote by [𝐿^𝑝(Σ)]^𝑛 the space of all measurable vector-valued functions 𝑢=(𝑢₁,…,𝑢_𝑛) such that |𝑢_𝑗|^𝑝 is integrable over Σ (𝑗=1,…,𝑛). If ℎ is any nonnegative integer, 𝐿^𝑝_ℎ(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 𝐿^𝑝(Σ) in a coordinate system of class 𝐶¹ and consequently in every coordinate system of class 𝐶¹. The space [𝐿^𝑝_ℎ(Σ)]^𝑛 is constituted by the vectors (𝑣₁,…,𝑣_𝑛) such that 𝑣_𝑗 is a differential form of 𝐿^𝑝_ℎ(Σ) (𝑗=1,…,𝑛). [𝑊^1,𝑝(Σ)]^𝑛 is the vector space of all measurable vector-valued functions 𝑢=(𝑢₁,…,𝑢_𝑛) such that 𝑢_𝑗 belongs to the Sobolev space 𝑊^1,𝑝(Σ) (𝑗=1,…,𝑛).

The pair (𝑣,𝑟) with components 𝑣_𝑖(𝑥)=−_Σ𝛾_𝑖𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦,𝑖=1,…,𝑛,𝑥∈ℝ^𝑛,𝑟(𝑥)=−_Σ𝜀_𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦,𝑥∈ℝ^𝑛 (15)(16) is the simple layer hydrodynamic potential with density 𝜑.

The pair (𝑤,𝑞) with components 𝑤_𝑖(𝑥)=_Σ𝑇^′_𝑗,𝑦𝛾^𝑖𝑥,𝑦𝜓_𝑗𝑦𝑑𝜎_𝑦,𝑖=1,…,𝑛,𝑥∈ℝ^𝑛,𝑞(𝑥)=2𝜇_Σ𝜕𝜕𝜈_𝑦𝜀_𝑗𝑥,𝑦𝜓_𝑗𝑦𝑑𝜎_𝑦,𝑥∈ℝ^𝑛 (17)(18) 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 _|𝑦|=1log𝑥−𝑦𝑑𝑠_𝑦=0,|𝑥|<1. (19)

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 domain Ω is exceptional if there exists some constant vector which cannot be represented in Ω by a simple layer potential.

Denoting by Σ_𝑅 the circle of radius 𝑅 centered at the origin, we have the following lemma.

Lemma 1. The circle Σ_𝑅 with 𝑅=exp(1/2) is exceptional for the Stokes system.

Proof. Keeping in mind that (see, e.g., [16, Section 4]) _Σ𝑅log𝑥−𝑦𝑑𝑠_𝑦=2𝜋𝑅log𝑅,_Σ𝑅𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦²𝑑𝑠_𝑦=𝛿_𝑖𝑗𝜋𝑅,|𝑥|<𝑅, (20) we find _Σ𝑅𝛾_𝑖𝑗𝑥,𝑦𝑑𝑠_𝑦=𝑅𝛿4𝜇^𝑖𝑗2log𝑅−1,|𝑥|<𝑅. (21)
Taking 𝑅=exp(1/2) we obtain the result.

Let us consider now the exceptional boundaries of not simply connected domains.

Proposition 2. Let Ω⊂ℝ² be an (𝑚+1)-connected domain. Denote by 𝒫 the eigenspace in [𝐿^𝑝(Σ)]² of the singular integral system _Σ𝜑_𝑗𝑦𝜕𝜕𝑠_𝑥𝛾_𝑖𝑗𝑥,𝑦𝑑𝑠_𝑦=0,𝑎.𝑒.𝑥∈Σ,𝑖=1,2. (22) Then dim𝒫=2(𝑚+1).

Proof. As in the proof of [16, Lemma 12], one can show that 𝜕𝜕𝑠_𝑥𝛾_𝑖𝑗=1𝑥,𝑦𝛿4𝜋𝜇^𝑖𝑗𝜕𝜕𝑠_𝑥log𝑥−𝑦+𝒪𝑦−𝑥^ℎ−1, (23) deduce that system (22) can be regularized to a Fredholm one, and see that its index is zero. Since the vectors 𝑒_𝑖𝜒_Σ𝑗  (by 𝜒_𝑋 we denote the characteristic function of the set 𝑋) (𝑖=1,2, 𝑗=0,1,…,𝑚) are the only eigensolutions of the adjoint system _Σ𝜑_𝑗𝑦𝜕𝜕𝑠_𝑦𝛾_𝑖𝑗𝑥,𝑦𝑑𝑠_𝑦=0,a.e.𝑥∈Σ,𝑖=1,2, (24) we have dim𝒫=2(𝑚+1).

Theorem 3. Let Ω⊂ℝ² be an (𝑚+1)-connected domain. The following conditions are equivalent(1)There exists a Hölder continuous vector function 𝜑≢0 such that _Σ𝛾𝑥,𝑦𝜑𝑦𝑑𝑠_𝑦=0,𝑥∈Σ. (25)(2)There exists a constant vector which cannot be represented in Ω by a simple layer potential;(3)Σ₀ is exceptional.(4)Let 𝜑₁,…,𝜑_2𝑚+2 be linearly independent vectors of 𝒫 (see Proposition 2), and let 𝑐_𝑗𝑘=(𝛼_𝑗𝑘,𝛽_𝑗𝑘)∈ℝ² be given by _Σ𝛾𝑥,𝑦𝜑_𝑗𝑦𝑑𝑠_𝑦=𝑐_𝑗𝑘,𝑥∈Σ_𝑘,𝑗=1,…,2𝑚+2,𝑘=0,1,…,𝑚. (26) Then det𝒞=0, where 𝛼𝒞=^1,0⋯𝛼_2𝑚+2,0𝛼⋯⋯⋯_1,𝑚⋯𝛼_2𝑚+2,𝑚𝛽_1,0⋯𝛽_2𝑚+2,0𝛽⋯⋯⋯_1,𝑚⋯𝛽_2𝑚+2,𝑚. (27)

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 by 𝒩_𝑝 the space of 𝜓∈[𝐿^𝑝_𝑛−2(Σ)]^𝑛 such that _Σ𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0,a.e.onΣ,𝑖=1,…,𝑛. (28)

We begin by proving the following result.

Lemma 4. Let 𝑢∈[𝐶^∞₀(ℝ^𝑛)]^𝑛. Then, for any 𝑥∈ℝ^𝑛, 𝑢_𝑖(𝑥)=𝜇_ℝ𝑛Δ𝑢_𝑗𝑦𝛾_𝑖𝑗𝑥,𝑦𝑑𝑦+_ℝ𝑛𝜕²𝑢_𝑗𝑦𝜕𝑦_𝑖𝜕𝑦_𝑗𝑠𝑥,𝑦𝑑𝑦, (29) where 𝛾(𝑥,𝑦) and 𝑠(𝑥,𝑦) are given by (12) and (7), respectively.

Proof. By the well-known Stokes identity we have 𝑢_𝑖(𝑥)=_ℝ𝑛Δ𝑢_𝑖𝑦𝑠𝑥,𝑦𝑑𝑦=_ℝ𝑛Δ𝑢_𝑗𝑦𝛿_𝑖𝑗𝑠𝑥,𝑦𝑑𝑦. (30) Since, for every 𝑛≠2,4, 𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦^𝑛=1()𝜕4−𝑛)(2−𝑛𝜕𝑦_𝑖𝜕²𝑦_𝑗𝑥−𝑦^4−𝑛−𝛿_𝑖𝑗𝜔_𝑛𝑠,𝛾𝑥,𝑦_𝑖𝑗=𝛿𝑥,𝑦^𝑖𝑗𝑠−12𝜇𝑥,𝑦2𝜇𝜔_𝑛𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦^𝑛,∀𝑛≥2, (31) we can rewrite 𝛾_𝑖𝑗=𝛿𝑥,𝑦^𝑖𝑗𝜇𝑠−1𝑥,𝑦2𝜇𝜔_𝑛1()𝜕4−𝑛)(2−𝑛𝜕𝑦_𝑖𝜕²𝑦_𝑗×𝑥−𝑦^4−𝑛. (32)
Then 𝛿_𝑖𝑗𝑠𝑥,𝑦=𝜇𝛾_𝑖𝑗+1𝑥,𝑦2𝜔_𝑛()𝜕4−𝑛)(2−𝑛𝜕𝑦_𝑖𝜕²𝑦_𝑗𝑥−𝑦^4−𝑛,𝑢_𝑖(𝑥)=𝜇_ℝ𝑛Δ𝑢_𝑗𝑦𝛾_𝑖𝑗+1𝑥,𝑦𝑑𝑦2𝜔_𝑛(4−𝑛)(2−𝑛)_ℝ^𝑛Δ𝑢_𝑖𝑦𝜕²𝜕𝑦_𝑖𝜕𝑦_𝑗𝑥−𝑦^4−𝑛𝑑𝑦. (33) Integrating by parts, it follows that the last integral is equal to 12𝜔_𝑛(4−𝑛)(2−𝑛)_ℝ^𝑛𝜕²𝑢_𝑗𝑦𝜕𝑦_𝑖𝜕𝑦_𝑗Δ_𝑦𝑥−𝑦^4−𝑛=𝑑𝑦_ℝ𝑛𝜕²𝑢_𝑗𝑦𝜕𝑦_𝑖𝜕𝑦_𝑗𝑠𝑥,𝑦𝑑𝑦, (34) since Δ_𝑦|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 𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦²=12𝜕²𝜕𝑦_𝑖𝜕𝑦_𝑗𝑥−𝑦²log𝑥−𝑦−𝛿_𝑖𝑗−1log𝑥−𝑦2𝛿_𝑖𝑗, (35)Δ_𝑦|𝑥−𝑦|²log|𝑥−𝑦|=4(log|𝑥−𝑦|+1), while, for 𝑛=4, 𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦⁴=𝛿_𝑖𝑗2𝑥−𝑦²+12𝜕²𝜕𝑦_𝑖𝜕𝑦_𝑗log𝑥−𝑦, (36)  Δ_𝑦log|𝑥−𝑦|=2/|𝑥−𝑦|².

Lemma 5. Let 𝜁₁,…,𝜁_𝑛 be differential forms in 𝐿^𝑝_𝑛−2(Σ) such that 𝑑𝜁_𝑗=(−1)^𝑛−1𝜈_𝑗𝑑𝜎 on Σ. One has 𝜓∈𝒩_𝑝 if, and only if, 𝜓_𝑗=^𝑚_ℎ=0𝑐_ℎ𝜒_Σℎ𝜁_𝑗+𝜂_𝑗,𝑗=1,…,𝑛, (37) where 𝑐₀,…,𝑐_𝑚∈ℝ and 𝜂₁…,𝜂_𝑛 are weakly closed forms belonging to 𝐿^𝑝_𝑛−2(Σ).

Proof. It is easy to construct the differential forms 𝜁₁,…,𝜁_𝑛. For example, one can take the restriction on Σ of the following forms: 𝜁₁=(−1)^𝑛−1𝑥₂𝑑𝑥³⋯𝑑𝑥^𝑛, 𝜁_𝑗=(−1)^𝑛−𝑗𝑥₁𝑑𝑥²⋯𝑗̂⋯𝑑𝑥^𝑛 (𝑗=2,…,𝑛). We remark that (37) holds if, and only if, the weak differentials 𝑑𝜓_𝑗 exist and 𝑑𝜓_𝑗=(−1)^𝑚𝑛−1_ℎ=0𝑐_ℎ𝜒_Σℎ𝜈_𝑗𝑑𝜎,𝑗=1,…,𝑛, (38) that is, _Σ𝜓_𝑗∧𝑑𝑢_𝑗=^𝑚_ℎ=0𝑐_ℎΣℎ𝑢⋅𝜈𝑑𝜎,∀𝑢∈𝐶^∞₀ℝ^𝑛𝑛. (39)
Let us prove that (39) holds if, and only if, _Σ𝑘𝜓_𝑗∧𝑑𝑢_𝑗=𝑐_𝑘Σ𝑘𝐶𝑢⋅𝜈𝑑𝜎,∀𝑢∈^∞₀ℝ^𝑛𝑛,𝑘=0,…,𝑚. (40)
It is obvious that (40) implies (39).
Conversely, suppose that (39) is true. Define 𝑈^𝜀_𝑘={𝑥∈ℝ^𝑛∣dist(𝑥,Σ_𝑘)<𝜀}, where 0<𝜀<min_{0≤ℎ<𝑘≤𝑚}dist(Σ_ℎ,Σ_𝑘). Let 𝑣_𝑘∈𝐶^∞₀(𝑈^𝜀_𝑘) be such that 𝑣_𝑘=1 in 𝑈_𝑘^𝜀/2. Since 𝑣_𝑘𝑢∈[𝐶^∞₀(ℝ^𝑛)]^𝑛, we may write _Σ𝜓_𝑗∧𝑑𝑣_𝑘𝑢_𝑗=^𝑚_ℎ=0𝑐_ℎΣℎ𝑣_𝑘𝑢⋅𝜈𝑑𝜎, (41) and (40) follows immediately.
Suppose now that (39) is true. From (40) it follows that _Σ𝑘𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=𝑐_𝑘Σ𝑘𝛾_𝑖𝑗𝑥,𝑦𝜈_𝑗𝑦𝑑𝜎_𝑦,∀𝑥∉Σ_𝑘. (42)
An integration by parts shows that _Σ𝑘𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0,∀𝑥∉Ω_𝑘. (43)
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 _Σ𝑘𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0 (44) a.e. on Σ_𝑘. Arguing as in [9, pages 189-190], this implies that _Σ𝑘𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0 (45) also in Ω_𝑘. Summing over 𝑘 we find _Σ𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0, (46) for every 𝑥∈ℝ^𝑛⧵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 _Σ𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦=0,𝑥∉Σ. (47) Since 𝜀_𝑗(𝑥,𝑦)=−𝜕_𝑥𝑗𝑠(𝑥,𝑦), system (11) implies that Δ_𝑥[𝛾_𝑖𝑗(𝑥,𝑦)]=−(1/𝜇)(𝜕²/𝜕𝑥_𝑖𝜕𝑥_𝑗)𝑠(𝑥,𝑦). Hence, 𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗Σ𝜓_𝑗𝑦∧𝑑_𝑦𝑠𝑥,𝑦=0,𝑥∉Σ. (48) Therefore, there exist some constants 𝑎₀,𝑎₁,…,𝑎_𝑚 such that 𝜕_𝑗Ψ_𝑗(𝑥)=−𝑎_ℎ𝑥∈Ω_ℎ,ℎ=1,…,𝑚,−𝑎₀𝑥∈Ω,0𝑥∈ℝ^𝑛⧵Ω, (49) where Ψ_𝑗(𝑥)=_Σ𝜓_𝑗𝑦∧𝑑_𝑦𝑠𝑥,𝑦. (50) Then, on account of Lemma 4, for every 𝑢∈[𝐶^∞₀(ℝ^𝑛)]^𝑛, _Σ𝜓_𝑗∧𝑑𝑢_𝑗=𝜇_ℝ𝑛Δ𝑢_𝑗(𝑥)𝑑𝑥_Σ𝜓_𝑗𝑦∧𝑑_𝑦𝛾_𝑖𝑗+𝑥,𝑦_ℝ𝑛𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)𝑑𝑥_Σ𝜓_𝑗𝑦∧𝑑_𝑦𝑠𝑥,𝑦. (51)
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 _ℝ𝑛𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)Ψ_𝑗(𝑥)=𝑑𝑥_{𝑚ℎ=1Ωℎ}𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)Ψ_𝑗(𝑥)𝑑𝑥+_Ω𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)Ψ_𝑗(𝑥)+𝑑𝑥_{ℝ𝑛⧵Ω0}𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)Ψ_𝑗(𝑥)𝑑𝑥=−^𝑚_ℎ=1Σℎ𝜕_𝑖𝑢_𝑖Ψ_𝑗𝜈_𝑗𝑑𝜎+^𝑚_ℎ=1Ωℎ𝜕_𝑖𝑢_𝑖𝜕_𝑗Ψ_𝑗+𝑑𝑥_Σ𝜕_𝑖𝑢_𝑖Ψ_𝑗𝜈_𝑗𝑑𝜎−_Ω𝜕_𝑖𝑢_𝑖𝜕_𝑗Ψ_𝑗−𝑑𝑥_Σ0𝜕_𝑖𝑢_𝑖Ψ_𝑗𝜈_𝑗𝑑𝜎+_{ℝ𝑛⧵Ω0}𝜕_𝑖𝑢_𝑖𝜕_𝑗Ψ_𝑗=𝑑𝑥_{𝑚ℎ=1Ωℎ}𝜕_𝑖𝑢_𝑖𝜕_𝑗Ψ_𝑗𝑑𝑥−_Ω𝜕_𝑖𝑢_𝑖𝜕_𝑗Ψ_𝑗𝑑𝑥. (52) Hence, by (49), _ℝ𝑛𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖(𝑥)Ψ_𝑗(𝑥)𝑑𝑥=−^𝑚_ℎ=1𝑎_ℎΩℎ𝜕_𝑖𝑢_𝑖𝑑𝑥+𝑎_0Ω𝜕_𝑖𝑢_𝑖=𝑑𝑥_𝑚ℎ=1𝑎_ℎΣℎ𝑢⋅𝜈𝑑𝜎+𝑎_0Σ𝑢⋅𝜈𝑑𝜎=𝑎_0Σ0𝑢⋅𝜈𝑑𝜎+^𝑚_ℎ=1𝑎₀+𝑎_ℎΣℎ𝑢⋅𝜈𝑑𝜎. (53) By setting 𝑐₀=𝑎₀ and 𝑐_ℎ=𝑎₀+𝑎_ℎ (ℎ=1,…,𝑚) we get the claim.

Remark 6. Lemma 5 shows that the dimension of the kernel 𝒩_𝑝 is infinite. However, if we consider the quotient space 𝒩_𝑝/Ξ_𝑝, Ξ_𝑝 being the space of weakly closed differential forms in 𝐿^𝑝_𝑛−2(Σ), we have dim(𝒩_𝑝/Ξ_𝑝)=𝑚+1.
We conclude this section by recalling some properties concerning the following eigenspaces: 𝒱_±=𝜑_𝑘∈𝐿^𝑝(Σ)1∶±2𝜑_𝑘(𝑥)+_Σ𝐹_𝑘𝑖𝑥,𝑦𝜑_𝑖𝑦𝑑𝜎_𝑦,𝒲=0,𝑘=1,…,𝑛_±=𝜑_𝑘∈𝐿^𝑝(Σ)1∶∓2𝜑_𝑘(𝑥)+_Σ𝐹_𝑖𝑘𝑦,𝑥𝜑_𝑖𝑦𝑑𝜎_𝑦=0,𝑘=1,…,𝑛, (54) where (see, e.g., [22]) 𝐹_𝑘𝑖𝑥,𝑦∶=𝑇^′_𝑖,𝑦𝛾^𝑘𝑛𝑥,𝑦=−𝜔_𝑛𝑥_𝑘−𝑦_𝑘𝑥_𝑖−𝑦_𝑖𝑥_𝑗−𝑦_𝑗𝑥−𝑦^𝑛+2𝜈_𝑗𝑦. (55) For the proofs of the following two results see [7, Lemma 3.3] and [8, Theorem 3.2], respectively.

Proposition 7. The sets 𝒱₊ and 𝒲₋ are linear subspaces of 𝐿¹(Σ) and dim𝒱₊=dim𝒲₋𝑛()𝑚=1+𝑛+12. (56)
A basis of 𝒲₋ is expressed by the fields {𝜓_𝑖ℎ,𝜈∶𝑖=1,…,𝑛(𝑛+1)/2,ℎ=1,…,𝑚}. The simple layer potentials 𝑣_𝑖ℎ whose densities are 𝜓_𝑖𝑘 such that: 𝑣_𝑖ℎ|_Ω𝑘=𝛿_ℎ𝑘𝜌_𝑖, 𝑖=1,…,𝑛(𝑛+1)/2,ℎ,𝑘=1,…,𝑚, where 𝜌_𝑖 are rigid displacement in ℝ^𝑛, specifically 𝜌_𝑖(𝑥)=𝑒_𝑖,𝑖=1,…,𝑛, and, for 𝑖=𝑛+1,…,𝑛(𝑛+1)/2, 𝜌_𝑖(𝑥)=(𝑒_ℎ∧𝑒_𝑘)𝑥, ℎ=1,…,𝑛−1,𝑘=ℎ+1,…,𝑛,ℎ[𝑛−(ℎ+1)/2]+ 𝑘=𝑖.
In addition, every 𝜓∈𝒲₋ has the property that 𝑣|_Σ0=0, where 𝑣 is the simple layer potential with density 𝜓.

Proposition 8. The sets 𝒱₋ and 𝒲₊ are linear subspaces of 𝐿¹(Σ) and dim𝒱₋=dim𝒲₊=𝑛(𝑛+1)2+𝑚. (57) A basis for 𝒲₊ is expressed by the fields {𝜓_𝑖,𝜈𝜒_Σℎ∶𝑖=1,…,𝑛(𝑛+1)/2,ℎ=1,…,𝑚}, where 𝜓_𝑖, 𝑖=1,…,𝑛(𝑛+1)/2 are zero on Σ⧵Σ₀, and such that the simple layer potentials with density 𝜓_𝑖 are 𝑛(𝑛+1)/2 rigid displacement in Ω₀ (linearly independent for 𝑛≥3).
Finally, every function 𝜑 which is the restriction to Σ of a rigid displacement belongs to 𝒱₋.

One recalls that if 𝜑∈[𝐿¹(Σ)]^𝑛 belongs to one of the eigenspaces 𝒱_±,𝒲_±, then 𝜑∈[𝐶^𝜆(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 Proposition 8 slightly more precise, saying that the simple layer potentials with density 𝜓_𝑖 are 𝑛(𝑛+1)/2 rigid displacement in Ω₀ linear independent for any 𝑛≥2, unless 𝑛=2 and Σ₀ is exceptional. Indeed, let us show that if 𝑛=2 and Σ₀ is not exceptional, such rigid displacements are linearly independent. Let 𝑐_𝑖 be such that ³_𝑖=1𝑐_𝑖Σ0𝜓_𝑖𝑦𝛾𝑥,𝑦𝑑𝜎_𝑦=0,inΩ₀. (58) We have also _Σ0³_𝑖=1𝑐_𝑖𝜓_𝑖𝑦𝛾𝑥,𝑦𝑑𝜎_𝑦=0,a.e.𝑥∈Σ₀. (59) Let 𝜑=∑³_𝑖=1𝑐_𝑖𝜓_𝑖. In view of the equivalence between (1) and (3) of Theorem 3, 𝜑 has to vanish. Therefore 𝑐_𝑖=0 (𝑖=1,2,3) because of the linearly independence of 𝜓_𝑖. On the other hand, if 𝑛=2 and Σ₀ is exceptional, Theorem 3 shows that the potentials with densities {𝜓_𝑖}_𝑖=1,2,3 are linearly dependent.

6. Reduction of a Certain Singular Integral Operator

For every 𝜓∈𝐿^𝑝₁(Σ), let Θ_ℎ be the operator defined by Θ_ℎ𝜓(𝑥)=∗_Σ𝑑_𝑥𝑠_𝑛−2𝑥,𝑦∧𝜓𝑦∧𝑑𝑥^ℎ,𝑥∈Ω, (60) where ∗ and 𝑑 denote the Hodge star operator and the exterior derivative, respectively, and 𝑠_ℎ(𝑥,𝑦) is the double ℎ-form introduced by Hodge in [23] as follows: 𝑠_ℎ𝑥,𝑦=_{𝑗1<⋯<𝑗ℎ}𝑠𝑥,𝑦𝑑𝑥^𝑗1⋯𝑑𝑥^𝑗ℎ𝑑𝑦^𝑗1⋯𝑑𝑦^𝑗ℎ. (61) Note that the operator Θ_ℎ satisfies the equation 𝜕_ℎΣ𝑢𝑦𝜕𝜕𝜈_𝑦𝑠𝑥,𝑦𝑑𝜎_𝑦=−Θ_ℎ(𝑑𝑢),𝑥∈Ω, (62) for each 𝑢∈𝑊^1,𝑝(Σ), since (see [9, page 187]) ∗𝑑_Σ𝑢𝑦𝜕𝜕𝜈_𝑦𝑠𝑥,𝑦𝑑𝜎_𝑦=𝑑_𝑥Σ𝑑𝑢𝑦∧𝑠_𝑛−2𝑥,𝑦,𝑥∈Ω. (63)

Moreover we introduce the operators ℋ_𝑗ℎ defined as ℋ_𝑗ℎ𝜓(𝑥)=Θ_ℎ𝜓_𝑗(𝑥)−𝛿^123⋯𝑛_{𝑙𝑖𝑗3⋯𝑗𝑛}()!×𝑛−2_Σ𝜕_𝑥ℎ𝐻_𝑙𝑗𝑥,𝑦∧𝜓_𝑖𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛, (64) for every 𝜓∈[𝐿^𝑝₁(Σ)]^𝑛, where 𝐻_𝑙𝑗=1𝑥,𝑦𝜔_𝑛𝑦_𝑙−𝑥_𝑙𝑦_𝑗−𝑥_𝑗𝑦−𝑥^𝑛. (65)

In the sequel 𝑑𝑢 denotes the vector (𝑑𝑢₁,…,𝑑𝑢_𝑛) whose elements are 1-forms, and 𝜓=(𝜓₁,…,𝜓_𝑛)∈[𝐿^𝑝₁(Σ)]^𝑛.

Lemma 10. Let (𝑤,𝑞) be the double layer hydrodynamic potential of (17)-(18) with density 𝑢∈[𝑊^1,𝑝(Σ)]^𝑛. Then, for 𝑥∉Σ, 𝜕_ℎ𝑤_𝑗(𝑥)=ℋ_𝑗ℎ(𝑥),𝑞(𝑥)𝑑𝑢)(=2𝜇Θ_ℎ𝑑𝑢_ℎ(𝑥), (66)(67) where ℋ_𝑗ℎ and Θ_ℎ 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, ⁽𝑤^𝑘)_𝑗(𝑥)=_Σ𝑢_𝑖𝑦𝐿^(𝑘)Γ_𝑖,𝑦^(𝑘)𝑗𝑥,𝑦𝑑𝜎_𝑦, (68) 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 𝜕𝑤_ℎ^(𝑘)_𝑗(𝑥)=ℋ^(𝑘)_𝑗ℎ(𝑑𝑢)(𝑥), (69) where ℋ^(𝑘)_𝑗ℎ𝜓(𝑥)=Θ_ℎ𝜓_𝑗(𝑥)−𝛿^123⋯𝑛_{𝑙𝑖𝑗3⋯𝑗𝑛}()!×𝑛−2_Σ𝜕𝐻_𝑥ℎ^(𝑘)_𝑙𝑗𝑥,𝑦∧𝜓_𝑖𝑦∧𝑑𝑦^𝑗3⋯𝑑𝑦^𝑗𝑛,_(𝑘)𝐻_𝑙𝑗=𝑘𝑥,𝑦𝜔_𝑛()𝑦𝑘+1^𝑙−𝑥_𝑙𝑦_𝑗−𝑥_𝑗𝑦−𝑥^𝑛−1𝛿𝑘+1^𝑙𝑗𝑠𝑥,𝑦, (70) and Θ_ℎ is given by (60).
From [16, formula (5)] (where we set 𝜉=1), letting 𝑘→+∞, we get 𝜕𝐿_𝑥ℎ^(𝑘)Γ_𝑖,𝑦^(𝑘)𝑗𝑛𝑥,𝑦⟶−𝜔_𝑛𝜕_𝑥ℎ𝑦_𝑖−𝑥_𝑖𝑦_𝑗−𝑥_𝑗𝑦_𝑘−𝑥_𝑘𝑥−𝑦^𝑛+2𝜈_𝑘𝑦=𝜕_𝑥ℎ𝑇^′_𝑖,𝑦𝛾^𝑗𝑥,𝑦, (71)𝑥∉Σ, from which 𝜕𝑤_ℎ^(𝑘)→𝜕_ℎ𝑤 as 𝑘→+∞. Therefore we obtain formula (66) by letting 𝑘→+∞ in (69). Formula (67) is an immediate consequence of (62) because 𝜀_𝑗(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].

Let 𝑓∈𝐿¹(Σ). If 𝜂∈Σ is a Lebesgue point for 𝑓, we get lim_{𝑥→𝜂Σ}𝑓𝑦𝜕_𝑥𝑠𝑦_𝑙−𝑥_𝑙𝑦_𝑗−𝑥_𝑗𝑥−𝑦^𝑛𝑑𝜎_𝑦=𝜔_𝑛2𝛿_𝑙𝑗−2𝜈_𝑗𝜂𝜈_𝑙𝜂𝜈_𝑠𝜂𝑓𝜂+_Σ𝑓𝑦𝜕_𝑥𝑠𝑦_𝑙−𝜂_𝑙𝑦_𝑗−𝜂_𝑗𝑥−𝑦^𝑛𝑑𝜎_𝑦, (72) 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, let 𝜓∈𝐿^𝑝₁(Σ) and write 𝜓 as 𝜓=𝜓_ℎ𝑑𝑥^ℎ with 𝜈_ℎ𝜓_ℎ=0. (73) Assumption (73) is not restrictive, because, given the 1-form 𝜓 on Σ, there exist scalar functions 𝜓_ℎ defined on Σ such that 𝜓=𝜓_ℎ𝑑𝑥^ℎ and (73) holds (see [24, page 41]). Then, for almost every 𝜂∈Σ, lim_𝑥→𝜂Θ_ℎ𝜓(𝑥)1=−2𝜓_ℎ𝜂+Θ_ℎ𝜓𝜂, (74) where Θ_ℎ is given by (60), and the limit has to be understood again as an internal angular boundary value.

Lemma 11. Let 𝜓∈𝐿^𝑝₁(Σ). Let one write 𝜓 as 𝜓=𝜓_ℎ𝑑𝑥^ℎ and suppose that (73) holds. Then, for almost every 𝜂∈Σ, lim_𝑥→𝜂1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}_Σ𝜕_𝑥𝑠𝐻_𝑙𝑗𝑥,𝑦∧𝜓𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛1=−2𝜈_𝑗𝜂𝜓_𝑖𝜂+𝜈_𝑖𝜂𝜓_𝑗𝜂𝜈_𝑠𝜂+1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}_Σ𝜕_𝑥𝑠𝐻_𝑙𝑗𝜂,𝑦∧𝜓𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛, (75) where 𝐻_𝑙𝑗 is defined by (65), and the limit has to be understood as an internal angular boundary value.

Proof. We have 1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}_Σ𝜕_𝑥𝑠𝐻_𝑙𝑗𝑥,𝑦∧𝜓𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛=1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}𝛿^123⋯𝑛_{𝑟ℎ𝑗3⋯𝑗𝑛Σ}𝜕_𝑥𝑠𝐻_𝑙𝑗𝑥,𝑦𝜓_ℎ𝑦𝜈_𝑟𝑦𝑑𝜎_𝑦=𝛿^𝑙𝑖_𝑟ℎΣ𝜕_𝑥𝑠𝐻_𝑙𝑗𝑥,𝑦𝜓_ℎ𝑦𝜈_𝑟𝑦𝑑𝜎_𝑦. (76) Hence, by (65) and (72), lim_𝑥→𝜂1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}_Σ𝜕_𝑥𝑠𝐻_𝑙𝑗𝑥,𝑦∧𝜓𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛=𝛿^𝑙𝑖_𝑟ℎ2𝛿_𝑙𝑗−2𝜈_𝑗𝜂𝜈_𝑙𝜂𝜈_𝑠𝜂𝜈_𝑟𝜂𝜓_ℎ𝜂+1()!𝛿𝑛−2^{123⋯𝑛𝑙𝑖𝑗3⋯𝑗𝑛}_Σ𝜕_𝑥𝑠𝐻_𝑙𝑗𝜂,𝑦∧𝜓𝑦∧𝑑𝑦^𝑗3⋯∧𝑑𝑦^𝑗𝑛. (77) Keeping in mind (73), we find 𝛿^𝑙𝑖_𝑟ℎ2𝛿_𝑙𝑗−2𝜈_𝑗𝜈_𝑙𝜈_𝑠𝜈_𝑟𝜓_ℎ=12𝛿_𝑙𝑗𝜈_𝑠−𝜈_𝑗𝜈_𝑙𝜈_𝑠𝜈_𝑙𝜓_𝑖−𝜈_𝑖𝜓_𝑙1=−2𝜈_𝑠𝜈_𝑗𝜓_𝑖−12𝜈_𝑠𝜈_𝑖𝜓_𝑗, (78) and the result follows.

Lemma 12. Let 𝜓=(𝜓₁,…,𝜓_𝑛)∈[𝐿^𝑝₁(Σ)]^𝑛. Then, for almost every 𝜂∈Σ, lim_𝑥→𝜂𝜇2𝛿_𝑖𝑗Θ_ℎ𝜓_ℎ(𝑥)+ℋ_𝑖𝑗𝜓(𝑥)+ℋ_𝑗𝑖𝜓(𝑥)𝜈_𝑖(𝑥)=𝜇2𝛿_𝑖𝑗Θ_ℎ𝜓_ℎ𝜂+ℋ_𝑖𝑗𝜓𝜂+ℋ_𝑗𝑖𝜓𝜂𝜈_𝑖𝜂, (79)Θ_ℎ 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 write 𝜓_𝑖 as 𝜓_𝑖=𝜓_𝑖ℎ𝑑𝑥^ℎ with 𝜈_ℎ𝜓_𝑖ℎ=0,𝑖=1,…,𝑛. (80) On account of (72) and (74), we infer lim_𝑥→𝜂𝜇2𝛿_𝑖𝑗Θ_ℎ𝜓_ℎ(𝑥)+ℋ_𝑖𝑗𝜓(𝑥)+ℋ_𝑗𝑖𝜓(𝑥)𝜈_𝑖(𝑥)=𝜇Ψ_𝑖𝑗𝜓𝜂𝜈_𝑖𝜂+𝜇2𝛿_𝑖𝑗Θ_ℎ𝜓_ℎ𝜂+ℋ_𝑖𝑗𝜓𝜂+ℋ_𝑗𝑖𝜓𝜂𝜈_𝑖𝜂, (81) where Ψ_𝑖𝑗𝜓=−𝛿_𝑖𝑗𝜓_ℎℎ−12𝜓_𝑖𝑗+12𝜈_𝑖𝜓_𝑠𝑠+𝜈_𝑠𝜓_𝑠𝑖𝜈_𝑗−12𝜓_𝑗𝑖+12𝜈_𝑗𝜓_𝑠𝑠+𝜈_𝑠𝜓_𝑠𝑗𝜈_𝑖. (82) By (80) we get Ψ_𝑖𝑗(𝜓)𝜈_𝑖=−𝜓_ℎℎ𝜈_𝑗−𝜓_𝑖𝑗𝜈_𝑖/2+𝜓_𝑠𝑠𝜈_𝑗+𝜈_𝑠𝜓_𝑠𝑗/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. Let 𝑤 be the double layer potential (17) with density 𝑢∈[𝑊^1,𝑝(Σ)]^𝑛. Then 𝑇_+,𝑗𝑤=𝑇_−,𝑗𝑤=𝜇[2𝛿_𝑖𝑗Θ_ℎ(𝑑𝑢_ℎ)+ℋ_𝑖𝑗(𝑑𝑢)+ℋ_𝑗𝑖(𝑑𝑢)]𝜈_𝑖 a.e. on Σ, where 𝑇₊𝑤 and 𝑇₋𝑤 denote the internal and the external angular boundary limits of 𝑇𝑤, respectively, and Θ_ℎ is given by (60) and ℋ by (64).

Proof. It is an immediate consequence of (66), (67), (79), and Remark 13.

Proposition 15. Let 𝑅∶[𝐿^𝑝(Σ)]^𝑛→[𝐿^𝑝₁(Σ)]^𝑛 be the following singular integral operator 𝑅𝜑(𝑥)=−_Σ𝑑_𝑥𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦. (83) Let one define 𝑅^′∶[𝐿^𝑝₁(Σ)]^𝑛→[𝐿^𝑝(Σ)]^𝑛 to be the singular integral operator 𝑅^′_𝑗𝜓(𝑥)=𝜇2𝛿_𝑖𝑗Θ_ℎ𝜓_ℎ(𝑥)+ℋ_𝑖𝑗𝜓(𝑥)+ℋ_𝑗𝑖𝜓(𝑥)𝜈_𝑖(𝑥). (84) Then 𝑅^′1𝑅𝜑=4𝜑−𝐾²𝜑, (85) where 𝐾𝜑(𝑥)=−_Σ𝑇_𝑥𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦. (86)

Proof. Let 𝑣 be the simple layer potential (15) with density 𝜑∈[𝐿^𝑝(Σ)]^𝑛. In view of Lemma 14, we have a.e. on Σ𝑅^′_𝑗𝑅𝜑=𝜇2𝛿_𝑖𝑗Θ_ℎ𝑑𝑣_ℎ+ℋ_𝑖𝑗(𝑑𝑣)+ℋ_𝑗𝑖(𝑑𝑣)𝜈_𝑖=𝑇_𝑗𝑤, (87) where 𝑤 is the double layer potential (17) with density 𝑣. Moreover, if 𝑥∈Ω, 𝑤_𝑘(𝑥)=_Σ𝑣_𝑖𝑦𝑇^′_𝑖,𝑦𝛾^𝑘𝑥,𝑦𝑑𝜎_𝑦=𝑣_𝑘(𝑥)+_Σ𝛾_𝑖𝑘𝑥,𝑦𝑇_𝑖𝑣𝑦𝑑𝜎_𝑦, (88) and then, on account of (86), 1𝑇𝑤=2()=1𝑇𝑣−𝐾𝑇𝑣2121𝜑+𝐾𝜑−𝐾2=1𝜑+𝐾𝜑4𝜑−𝐾²𝜑. (89)

7. The Dirichlet Problem

Let us consider the Dirichlet problem for the Stokes system 𝜇Δ𝑣=∇𝑟inΩ,div𝑣=0inΩ,𝑣=𝑓onΣ, (90) where the given data 𝑓∈[𝑊^1,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 symbol 𝒮^𝑝 we mean the class of the simple layer hydrodynamic potentials (15)-(16) with density in [𝐿^𝑝(Σ)]^𝑛. Whenever 𝑛=2 and Σ₀ is exceptional (see Section 4), we say that (𝑣,𝑟) belongs to 𝒮^𝑝 if, and only if, 𝑣(𝑥)=−_Σ𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦𝑟(𝑥)+𝑐,𝑥∈Ω,=−_Σ𝜀_𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦,𝑥∈Ω, (91) where 𝜑∈[𝐿^𝑝(Σ)]² and 𝑐∈ℝ².

We will see that condition (4) is not sufficient to prove the existence of the solution in the class 𝒮^𝑝, but it must be satisfied on each Σ_𝑗, 𝑗=0,1,…,𝑚.

We begin by proving the following result.

Theorem 16. Given 𝜔∈[𝐿^𝑝₁(Σ)]^𝑛, there exists a solution 𝜑∈[𝐿^𝑝(Σ)]^𝑛 of the singular integral system −_Σ𝑑_𝑥𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦=𝜔(𝑥),𝑎.𝑒.𝑥∈Σ, (92) if, and only if, _Σ𝜓_𝑖∧𝜔_𝑖=0, (93) for every 𝜓=(𝜓₁,…,𝜓_𝑛)∈[𝐿^𝑞_𝑛−2(Σ)]^𝑛  (𝑞=𝑝/(𝑝−1)) such that the weak differentials 𝑑𝜓_𝑗 exist and (38) holds for some real constants 𝑐₀,…,𝑐_𝑚.

Proof. Consider the adjoint of 𝑅 (see (83)), 𝑅^∗∶[𝐿^𝑞_𝑛−2(Σ)]^𝑛→[𝐿^𝑞(Σ)]^𝑛, that is, the operator whose components are given by 𝑅^∗_𝑖𝜓(𝑥)=−_Σ𝜓_𝑖𝑦∧𝑑_𝑦𝛾_𝑖𝑗𝑥,𝑦. (94) Proposition 15 implies that the integral system (92) has a solution 𝜑∈[𝐿^𝑝(Σ)]^𝑛 if, and only if, _Σ𝜓_𝑖∧𝜔_𝑖=0, (95) for each 𝜓=(𝜓₁,…,𝜓_𝑛)∈[𝐿^𝑞_𝑛−2(Σ)]^𝑛 such that 𝑅^∗𝜓=0. The result follows from Lemma 5.

Proposition 17. Given 𝑓∈[𝑊^1,𝑝(Σ)]^𝑛, there exists a solution of the BVP (𝑣,𝑟)̃∈𝒮^𝑝,̃𝑑𝜇Δ𝑣=∇𝑟𝑖𝑛Ω,div𝑣=0𝑖𝑛Ω,𝑣=𝑑𝑓𝑜𝑛Σ, (96) 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 solution 𝜑∈[𝐿^𝑝(Σ)]^𝑛 of the singular integral system −_Σ𝑑_𝑥𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦=𝑑𝑓(𝑥),a.e.𝑥∈Σ. (97)
In view of Theorem 16, there exists a solution 𝜑 of this system if, and only if, _Σ𝜓_𝑖∧𝑑𝑓_𝑖=0, (98) for every 𝜓=(𝜓₁,…,𝜓_𝑛)∈[𝐿^𝑞_𝑛−2(Σ)]^𝑛 satisfying 𝑅^∗𝜓=0, that is, such that the weak differentials 𝑑𝜓_𝑗 exist and (39) holds for some real constants 𝑐₀,…,𝑐_𝑚. Equation (39) being true for any 𝑢∈[𝑊^1,𝑝(Σ)]^𝑛, we can write _Σ𝜓_𝑖∧𝑑𝑓_𝑖=^𝑚_ℎ=0𝑐_ℎΣℎ𝑓⋅𝜈𝑑𝜎, (99) because of a density argument. In view of the arbitrariness of 𝑐₀,…,𝑐_𝑚, (98) is satisfied if, and only if, (3) holds.

Proposition 18. Let 𝑎_ℎ∈ℝ^𝑛  (ℎ=0,…,𝑚). Let 𝜓_𝑖𝑘, 𝑖=1,…,𝑛, 𝑘=1,…,𝑚, be the elements of the basis of 𝒲₋ given by Proposition 7. The pair 𝑣₀(𝑥)=^𝑚𝑛_𝑘=1𝑖=1𝑎^𝑖_𝑘−𝑎^𝑖_0Σ𝛾𝑥,𝑦𝜓_𝑖𝑘𝑦𝑑𝜎_𝑦+𝑎₀𝑟,𝑥∈Ω,₀(𝑥)=^𝑚𝑛_𝑘=1𝑖=1𝑎^𝑖_𝑘−𝑎^𝑖_0Σ𝜀𝑥,𝑦𝜓_𝑖𝑘𝑦𝑑𝜎_𝑦,𝑥∈Ω, (100) is the solution of the BVP 𝑣₀,𝑟₀∈𝒮^𝑝,𝜇Δ𝑣₀=∇𝑟₀𝑖𝑛Ω,div𝑣₀𝑣=0𝑖𝑛Ω,₀=𝑎_ℎ𝑜𝑛Σ_ℎ,ℎ=0,…,𝑚. (101)

Proof. The pair (𝑣₀,𝑟₀) belongs to 𝒮^𝑝 (for 𝑛=2, see Remark 9). Obviously it satisfies the Stokes system, and it satisfies the boundary conditions since, thanks to Proposition 7, 𝑣₀∣_Σ0=^𝑚𝑛_𝑘=1𝑖=1𝑎^𝑖_𝑘−𝑎^𝑖₀𝑣_𝑖𝑘∣_Σ0+𝑎₀=𝑎₀,𝑣₀∣_Σℎ=^𝑚𝑛_𝑘=1𝑖=1𝑎^𝑖_𝑘−𝑎^𝑖₀𝑣_𝑖𝑘∣_Σℎ+𝑎₀=^𝑚𝑛_𝑘=1𝑖=1𝑎^𝑖_𝑘−𝑎^𝑖₀𝛿_ℎ𝑘𝑒_𝑖+𝑎₀=𝑎_ℎ, (102) for any ℎ=1,…,𝑚.

Theorem 19. Given 𝑓∈[𝑊^1,𝑝(Σ)]^𝑛, the Dirichlet problem (𝑣,𝑟)∈𝒮^𝑝,𝜇Δ𝑣=∇𝑟𝑖𝑛Ω,div𝑣=0𝑖𝑛Ω,𝑣=𝑓𝑜𝑛Σ, (103) 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+𝑎_ℎ on Σ_ℎ (ℎ=0,…,𝑚) for some 𝑎_ℎ∈ℝ^𝑛. 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 Ω_𝑗. 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. If 𝑛≥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 −_Σ𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦=𝑓(𝑥) (104) on Σ. Therefore, Theorem 19 can be seen as an existence theorem for the integral system of the first kind (104) in 𝐿^𝑝(Σ).
If 𝑛=2 and Σ₀ is exceptional, we have the existence of a solution (𝜑,𝑐)∈[𝐿^𝑝(Σ)]²×ℝ² of the integral equation −_Σ𝛾𝑥,𝑦𝜑𝑦𝑑𝜎_𝑦=𝑓(𝑥)+𝑐onΣ. (105)

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.
By 𝒮^𝑝 we denote the space of all pairs (𝑣,𝑟) written as 𝑣_𝑖(𝑥)=−_Σ𝛾_𝑖𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦+_Σ𝑇^′_𝑗,𝑦𝛾^𝑖𝑥,𝑦𝜓_𝑗𝑦𝑑𝜎_𝑦𝑟(𝑥),𝑖=1,…,𝑛,𝑥∈Ω,=−_Σ𝜀_𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦+2𝜇_Σ𝜕𝜕𝜈_𝑦𝜀_𝑗𝑥,𝑦𝜓_𝑗𝑦𝑑𝜎_𝑦,𝑥∈Ω, (106) where 𝜑 and 𝜓 belong to [𝐿^𝑝(Σ)]^𝑛.

Theorem 23. Given 𝑓∈[𝑊^1,𝑝(Σ)]^𝑛 satisfying (4), the Dirichlet problem (𝑣,𝑟)∈𝒮^𝑝,𝜇Δ𝑣=∇𝑟𝑖𝑛Ω,div𝑣=0𝑖𝑛Ω,𝑣=𝑓𝑜𝑛Σ, (107) has one, and only one, solution (𝑣,𝑟) given by 𝑣_𝑖(𝑥)=−_Σ𝛾_𝑖𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦+_Σ𝑇^′_𝑗,𝑦𝛾^𝑖𝑥,𝑦𝑓_𝑗𝑦𝑑𝜎_𝑦𝑟(𝑥),𝑥∈Ω,=−_Σ𝜀_𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦+2𝜇_Σ𝜕𝜕𝜈_𝑦𝜀_𝑗𝑥,𝑦𝑓_𝑗𝑦𝑑𝜎_𝑦,𝑥∈Ω, (108)(109) where 𝜑∈[𝐿^𝑝(Σ)]^𝑛 is solution of the integral system of the first kind −_Σ𝛾_𝑖𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦=12𝑓_𝑖(𝑥)−_Σ𝑇^′_𝑗,𝑦𝛾^𝑖𝑥,𝑦𝑓_𝑗𝑦𝑑𝜎_𝑦,𝑎.𝑒.onΣ. (110)

Proof. Let 𝑣 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)−_Σ𝛾_𝑖𝑗𝑥,𝑦𝜑_𝑗𝑦𝑑𝜎_𝑦=𝑓_𝑖(𝑥)−𝑤_+,𝑖(𝑥),a.e.𝑥∈Σ. (111) In view of Remark 21 such a system is solvable if, and only if, _Σℎ𝑓_𝑖−𝑤_+,𝑖𝜈_𝑖𝑑𝜎=0,ℎ=0,…,𝑚. (112) On the other hand, because of the jump formulas, we have 𝑓_𝑖(𝑥)−𝑤_+,𝑖(𝑥)=𝑓_𝑖(𝑥)−12𝑓_𝑖(𝑥)+_Σ𝑇^′_𝑗,𝑦𝛾^𝑖𝑥,𝑦𝑓_𝑗𝑦𝑑𝜎_𝑦1∶=2𝑓_𝑖(𝑥)−𝑤_𝑖(𝑥)=−𝑤_−,𝑖(𝑥),a.e.𝑥∈Σ. (113) Therefore, conditions (112) become −∫_Σℎ𝑤₋⋅𝜈𝑑𝜎=0, ℎ=0,…,𝑚. Since 𝑤₋ can be considered as the datum of the interior Dirichlet problem in Ω_ℎ, for ℎ=1,…,𝑚, we have _Σℎ𝑤₋⋅𝜈𝑑𝜎=0,ℎ=1,…,𝑚. (114) As far as ℎ=0 is concerned, first we remark that (4) implies ∫_Σ𝑤⋅𝜈𝑑𝜎=0, because 0=_Σ𝑤₊⋅𝜈𝑑𝜎=_Σ12𝑓+𝑤⋅𝜈𝑑𝜎=_Σ𝑤⋅𝜈𝑑𝜎. (115) Keeping in mind (4) and (114), this leads to _Σ012=1𝑓−𝑤⋅𝜈𝑑𝜎2_Σ0𝑓⋅𝜈𝑑𝜎−_Σ01𝑤⋅𝜈𝑑𝜎=−2^𝑚_𝑗=1Σ𝑗𝑓⋅𝜈𝑑𝜎+^𝑚_𝑗=1Σ𝑗=𝑤⋅𝜈𝑑𝜎_{𝑚𝑗=1Σ𝑗}𝑤₋⋅𝜈𝑑𝜎=0. (116) Finally, assume that (𝑣,𝑟) is the solution of (107) with the data 𝑓=0. The integral representation (108) shows that (𝑣,𝑟)∈𝒮^𝑝, and then the uniqueness follows from Theorem 19.