### Abstract

The present paper is concerned with an indirect method to solve the Dirichlet and
the traction problems for Lamé system in a multiply connected bounded domain of ℝ* ^{n}*,

*n*≥ 2. It hinges on the theory of reducible operators and on the theory of differential forms. Differently from the more usual approach, the solutions are sought in the form of a simple layer potential for the Dirichlet problem and a double layer potential for the traction problem.

**2000 Mathematics Subject Classification**. 74B05; 35C15; 31A10; 31B10; 35J57.

##### Keywords:

Lamé system; boundary integral equations; potential theory; differential forms; multiply connected domains### 1 Introduction

In this paper we consider the Dirichlet and the traction problems for the linearized
*n*-dimensional elastostatics. The classical indirect methods for solving them consist
in looking for the solution in the form of a double layer potential and a simple layer
potential respectively. It is well-known that, if the boundary is sufficiently smooth,
in both cases we are led to a singular integral system which can be reduced to a Fredholm
one (see, e.g., [1]).

Recently this approach was considered in multiply connected domains for several partial differential equations (see, e.g., [2-7]).

However these are not the only integral representations that are of importance. Another
one consists in looking for the solution of the Dirichlet problem in the form of a
simple layer potential. This approach leads to an integral equation of the first kind
on the boundary which can be treated in different ways. For *n *= 2 and Ω simply connected see [8]. A method hinging on the theory of reducible operators (see [9,10]) and the theory of differential forms (see, e.g., [11,12]) was introduced in [13] for the *n*-dimensional Laplace equation and later extended to the three-dimensional elasticity
in [14]. This method can be considered as an extension of the one given by Muskhelishvili
[15] in the complex plane. The double layer potential ansatz for the traction problem
can be treated in a similar way, as shown in [16].

In the present paper we are going to consider these two last approaches in a multiply
connected bounded domain of ℝ* ^{n }*(

*n*≥ 2). Similar results for Laplace equation have been recently obtained in [17]. We remark that we do not require the use of pseudo-differential operators nor the use of hypersingular integrals, differently from other methods (see, e.g., [[18], Chapter 4] for the study of the Neumann problem for Laplace equation by means of a double layer potential).

After giving some notations and definitions in Section 2, we prove some preliminary results in Section 3. They concern the study of the first derivatives of a double layer potential. This leads to the construction of a reducing operator, which will be useful in the study of the integral system of the first kind arising in the Dirichlet problem.

Section 4 is devoted to the case *n *= 2, where there exist some exceptional boundaries in which we need to add a constant
vector to the simple layer potential. In particular, after giving an explicit example
of such boundaries, we prove that in a multiply connected domain the boundary is exceptional
if, and only if, the external boundary is exceptional.

In Section 5 we find the solution of the Dirichlet problem in a multiply connected domain by means of a simple layer potential. We show how to reduce the problem to an equivalent Fredholm equation (see Remark 5.5).

Section 6 is devoted to the traction problem. It turns out that the solution of this problem does exist in the form of a double layer potential if, and only if, the given forces are balanced on each connected component of the boundary. While in a simply connected domain the solution of the traction problem can be always represented by means of a double layer potential (provided that, of course, the given forces are balanced on the boundary), this is not true in a multiply connected domain. Therefore the presence or absence of "holes" makes a difference.

We mention that lately we have applied the same method to the study of the Stokes
system [19]. Moreover the results obtained for other integral representations for several partial
differential equations on domains with lower regularity (see, e.g., the references
of [20] for *C*^{1 }or Lipschitz boundaries and [21] for "worse" domains) lead one to hope that our approach could be extended to more
general domains.

### 2 Notations and definitions

Throughout this paper we consider a domain (open connected set) Ω ⊂ ℝ* ^{n}*,

*n*≥ 2, of the form

*(*

_{j }*j*= 0, ...,

*m*) are

*m*+ 1 bounded domains of ℝ

*with connected boundaries Σ*

^{n }*∈*

_{j }*C*

^{1, λ }(

*λ*∈ (0, 1]) and such that

*m*+ 1)

*-connected domain*. We denote by

*ν*the outwards unit normal on Σ = ∂Ω.

Let *E *be the partial differential operator

where *u *= (*u*_{1}, ..., *u _{n}*) is a vector-valued function and

*k >*(

*n*- 2)/

*n*is a real constant. A fundamental solution of the operator -

*E*is given by Kelvin's matrix whose entries are

*i*, *j *= 1, ..., *n*, *ω _{n }*being the hypersurface measure of the unit sphere in ℝ

*.*

^{n}As usual, we denote by

where *ε _{ih }*(

*u*) and

*σ*(

_{ih }*u*) are the linearized strain components and the stress components respectively, i.e.

Let us consider the boundary operator *L ^{ξ }*whose components are

*ξ *being a real parameter. We remark that the operator *L*^{1 }is just the stress operator 2*σ _{ih}ν_{h}*, which we shall simply denote by

*L*, while

*L*

^{k/(k+2) }is the so-called pseudo-stress operator.

By the symbol
*n*. It is well-known that the dimension of this space is *n*(*n *- 1)/2. From now on *a *+ *Bx *stands for a rigid displacement, i.e. *a *is a constant vector and
*n*(*n *+ 1)/2. As usual {*e*_{1}, ..., *e _{n}*} is the canonical basis for ℝ

*.*

^{n}For any 1 *< p < *+∞ we denote by [*L ^{p}*(Σ)]

*the space of all measurable vector-valued functions*

^{n }*u*= (

*u*

_{1}, ...,

*u*) such that |

_{n}*u*|

_{j}*is integrable over Σ (*

^{p }*j*= 1, ...,

*n*). If

*h*is any non-negative integer,

*h*defined on Σ such that their components are integrable functions belonging to

*L*(Σ) in a coordinate system of class

^{p}*C*

^{1 }and consequently in every coordinate system of class

*C*

^{1}. The space

*v*

_{1}, ...,

*v*) such that

_{n}*v*is a differential form of

_{j }*W*

^{1, p}(Σ)]

*is the vector space of all measurable vector-valued functions*

^{n }*u*= (

*u*

_{1}, ...,

*u*) such that

_{n}*u*belongs to the Sobolev space

_{j }*W*

^{1,p}(Σ) (

*j*= 1, ...,

*n*).

If *B *and *B*' are two Banach spaces and *S *: *B *→ *B*' is a continuous linear operator, we say that *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 of *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., [9,10]).

We end this section by defining the spaces in which we look for the solutions of the BVPs we are going to consider.

**Definition 2.1**. *The vector-valued function u belongs to *
*if, and only if, there exists φ *∈ [*L ^{p}*(Σ)]

^{n }such that u can be represented by a simple layer potential

**Definition 2.2**. *The vector-valued function w belongs to *
*if, and only if, there exists ψ *∈ [*W*^{1,p}(Σ)]^{n }such that w can be represented by a double layer potential

*where *[*L _{y}*Γ(

*x*,

*y*)]'

*denotes the transposed matrix of L*[Γ(

_{y}*x*,

*y*)].

### 3 Preliminary results

#### 3.1 On the first derivatives of a double layer potential

Let us consider the boundary operator *L ^{ξ }*defined by (2). Denoting by Γ

*(*

^{j}*x*,

*y*) the vector whose components are Γ

*(*

_{ij}*x*,

*y*), we have

We recall that an immediate consequence of (5) is that, when *ξ *= *k/*(2 + *k*) we have

while for *ξ *≠ *k/*(2 + *k*) the kernels

Let us denote by *w ^{ξ }*the double layer potential

It is known that the first derivatives of a harmonic double layer potential with density
*φ *belonging to *W*^{1,p}(Σ) can be written by means of the formula proved in [[13], p. 187]

Here * and *d *denote the Hodge star operator and the exterior derivative respectively, *s*(*x*, *y*) is the fundamental solution of Laplace equation

and *s _{h}*(

*x*,

*y*) is the double

*h*-form introduced by Hodge in [22]

Since, for a scalar function *f *and for a fixed *h*, we have **df *∧ *dx ^{h }*= (

*-*1)

^{n-1}∂

*, denoting by*

_{h}f dx*w*the harmonic double layer potential with density

*φ*∈

*W*

^{1,p}(Σ), (8) implies

where, for every

The following lemma can be considered as an extension of formula (9) to elasticity.
Here *du *denotes the vector (*du*_{1}, ..., *du _{n}*) and

*ψ*= (

*ψ*

_{1}, ...,

*ψ*) is an element of

_{n}**Lemma 3.1**. *Let w ^{ξ }be the double layer potential *(7)

*with density u*∈ [

*W*

^{1,p}(Σ)]

^{n}. Then

*where*

*and *Θ* _{h }is given by *(10),

*h*= 1, ...,

*n*.

*Proof*. Let *n *≥ 3. Denote by *M ^{hi }*the tangential operators

*M*=

^{hi }*ν*∂

_{h}*-*

_{i }*ν*∂

_{i}*,*

_{h}*h*,

*i*= 1, ...,

*n*. By observing that

we find in Ω

An integration by parts on Σ leads to

Therefore, by recalling (9),

If *f *is a scalar function, we may write

This identity is established by observing that on Σ we have

Then we can rewrite (14) as

Similar arguments prove the result if *n *= 2. We omit the details. □

#### 3.2 Some jump formulas

**Lemma 3.2**. *Let f *∈ *L*^{1}(Σ). *If η *∈ Σ *is a Lebesgue point for f, we have*

*where the limit has to be understood as an internal angular boundary value*^{1}.

*Proof*. Let *h _{pj}*(

*x*) =

*x*|

_{p}x_{j}*x*|

^{-n}. Since

*h*∈

*C*

^{∞}(ℝ

*{0}) is even and homogeneous of degree 2 -*

^{n}\*n*, due to the results proved in [23], we have

where

(see also [24] and note that in [23,24]*ν *is the inner normal). On the other hand

and, since

(see, e.g., [[25], p. 156]), we find

Finally, keeping in mind that *ω _{n }*=

*n π*

^{n/2}/Γ(

*n*/2 + 1) and Γ(

*n*/2 + 1) =

*n*(

*n*- 2)Γ(

*n*/2 - 1)/4, we obtain

Combining this formula with (16) we get (15). □

**Lemma 3.3**. *Let *
*Let us write ψ as ψ *= *ψ _{h}dx^{h }with*

*Then, for almost every η *∈ Σ,

*where *Θ* _{s }is given by *(10)

*and the limit has to be understood as an internal angular boundary value*.

*Proof*. First we note that the assumption (17) is not restrictive, because, given the 1-form
*ψ *on Σ, there exist scalar functions *ψ _{h }*defined on Σ such that

*ψ*=

*ψ*and (17) holds (see [[26], p. 41]). We have

_{h}dx^{h }

and then

a.e. on Σ. From (17) it follows that

**Lemma 3.4**. *Let *
*Let us write ψ as ψ *= *ψ _{h}dx^{h }and suppose that *(17)

*holds. Then, for almost every η*∈ Σ,

*where K ^{ξ }is defined by *(13)

*and the limit has to be understood as an internal angular boundary value*.

*Proof*. We have

Keeping in mind (13), formula (15) leads to

On the other hand

and the result follows. □

**Lemma 3.5**. *Let *
*Then, for almost every η *∈ Σ,

*being as in *(12) *and the limit has to be understood as an internal angular boundary value*.

*Proof*. Let us write *ψ _{i }*as

*ψ*=

_{i }*ψ*with

_{ih}dx^{h }

In view of Lemmas 3.3 and 3.4 we have

Therefore

where

Conditions (21) lead to

The bracketed expression vanishing, Φ = 0 and the result is proved. □

**Remark 3.6**. In Lemmas 3.2, 3.3, 3.4 and 3.5 we have considered internal angular boundary values.
It is clear that similar formulas hold for external angular boundary values. We have
just to change the sign in the first term on the right hand sides in (15), (18) and
(19), while (20) remains unchanged.

#### 3.3 Reduction of a certain singular integral operator

The results of the previous subsection imply the following lemmas.

**Lemma 3.7**. *Let w ^{ξ }be the double layer potential *(7)

*with density u*∈ [

*W*

^{1,p}(Σ)]

^{n}. Then

*a.e. on *Σ, *where *
*and *
*denote the internal and the external angular boundary limit of L ^{ξ}*(

*w*)

^{ξ}*respectively and*

*is given by*(12).

*Proof*. It is an immediate consequence of (11), (20) and Remark 3.6. □

**Remark 3.8**. The previous result is connected to [[1], Theorem 8.4, p. 320].

**Lemma 3.9**. *Let R : *
*be the following singular integral operator*

*Let us define *
*to be the singular integral operator*

*Then*

*where*

*Proof*. Let *u *be the simple layer potential with density *φ *∈ [*L ^{p}*(Σ)]

*. In view of Lemma 3.7, we have a.e. on Σ*

^{n}

where *w ^{ξ }*is the double layer potential (7) with density

*u*. Moreover, if

*x*∈ Ω,

and then, on account of (26),

□

**Corollary 3.10**. *The operator R defined by *(23) *can be reduced on the left. A reducing operator is given by R*'* ^{ξ }with ξ *=

*k*/(2 +

*k*).

*Proof*. This follows immediately from (25), because of the weak singularity of the kernel
in (26) when *ξ *= *k*/(2 + *k*) (see (6)). □

#### 3.4 The dimension of some eigenspaces

Let *T *be the operator defined by (26) with *ξ *= 1, i.e.

and denote by *T** its adjoint.

In this subsection we determine the dimension of the following eigenspaces

We first observe that the (total) indices of singular integral systems in (28)-(29)
vanish. This can be proved as in [[1], pp. 235-238]. Moreover, by standard techniques, one can prove that all the eigenfunctions
are hölder-continuous and then these eigenspaces do not depend on *p*. This implies that

The next two lemmas determine such dimensions. Similar results for Laplace equation can be found in [[27], Chapter 3].

**Lemma 3.11**. *The spaces *
*and *
*have dimension n*(*n *+ 1)*m*/2. *Moreover*

*where *{*v _{h }*:

*h*= 1 ...,

*n*(

*n*+ 1)/2}

*is an orthonormal basis of the space*

*and*

*is the characteristic function of*Σ

*.*

_{j}*Proof*. We define the vector-valued functions *α _{j}*,

*j*= 1, ...,

*m*as

*x*∈ Σ. For a fixed

*j*= 1, ...,

*m*, the function

*α*(

_{j}*x*) belongs to

because of

Now we prove that the following *n*(*n *+ 1)*m*/2 eigensolutions of

are linearly independent. Indeed, if

Then, by applying a classical uniqueness theorem to the domain Ω* _{j}*,

from which it easily follows that

Thus,
*u *be the simple layer potential with density *φ*. Since *E _{u }*= 0 in Ω

*and*

_{j }*L*= 0 on Σ

_{-}u*,*

_{j}*u*=

*a*+

^{j }*B*on each connected component Ω

^{j}x*,*

_{j}*j*= 1, ...,

*m*, and

*u*= 0 in

*n*= 2, because

*τ*as follows

If *τ*(*φ*) = 0, from a classical uniqueness theorem, we have that *φ *≡ 0 in ℝ* ^{n}*. Thus,

*τ*is an injective map and

**Lemma 3.12**. *The spaces *
*and *
*have dimension n*(*n *+ 1)/2. *Moreover *
*is constituted by the restrictions to *Σ *of the rigid displacements*.

*Proof*. Let
*x *∈ Σ, we have

thanks to

This shows that the restriction to Σ of *α *belongs to
*u *be the simple layer potential with density *ϕ*. Since *Eu *= 0 in Ω and *L*_{+}*u *= 0 on Σ, *u *= *a *+ *Bx *in Ω. Let *σ *be the linear map

If *n *≥ 3, we have that *σ*(*ϕ*) = 0 implies *u *≡ 0 in ℝ* ^{n }*and then

*ϕ*≡ 0 on Σ, in view of classical uniqueness theorems.

If *n *= 2, define
^{2}. Therefore

### 4 The bidimensional case

The case *n *= 2 requires some additional considerations. It is well-known that there are some
domains in which no every harmonic function can be represented by means of a harmonic
simple layer potential. For instance, on the unit disk we have

Similar domains occur also in elasticity. In order to give explicitly such an example, let us prove the following lemma.

**Lemma 4.1**. *Let *Σ_{R }be the circle of radius R centered at the origin. We have

*Proof*. Denote by *u*(*x*) the function on the left hand side of (32) and by Ω* _{R }*the ball of radius

*R*centered at the origin. Let us fix

*x*

_{0 }∈ Σ

*. For any*

_{R}*x*∈ Σ

*we have*

_{R }

and then *u *is constant on Σ* _{R}*. Moreover

and then also Δ*u *is constant on Σ* _{R}*. Since Δ

*u*is harmonic in Ω

*and continuous on*

_{R }*and then*

_{R }

The function *u*(*x*) *- *2*πR*(1 + log *R*) |*x*|^{2 }is continuous on
* _{R }*and constant on Σ

*. Then it is constant in Ω*

_{R}*and*

_{R }

□

**Corollary 4.2**. *Let *Σ_{R }be the circle of radius R centered at the origin. We have

*Proof*. Since

formula (32) implies

In a similar way

From (32) we have also

Keeping in mind the expression (1), (33) follows. □

This corollary shows that, if *R *= exp[*k*/(2(*k *+ 2))], we have

This implies that in Ω* _{R}*, for such a value of

*R*, we cannot represent any smooth solution of the system

*E*= 0 by means of a simple layer potential.

_{u }If there exists some constant vector which cannot be represented in the simply connected
domain Ω by a simple layer potential, we say that the boundary of Ω is *exceptional*. We have proved that

**Lemma 4.3**. *The circle *Σ* _{R }with R *= exp[

*k*/(2(

*k*+ 2))]

*is exceptional for the operator*Δ +

*k*∇

*div*.

Due to the results in [28], one can scale the domain in such a way that its boundary is not exceptional.

Here we show that also in some (*m *+ 1)-connected domains one cannot represent any constant vectors by a simple layer
potential and that this happens if, and only if, the exterior boundary Σ_{0 }(considered as the boundary of the simply connected domain Ω_{0}) is exceptional.

We note that, if any constant vector *c *can be represented by a simple layer potential, then any sufficiently smooth solution
of the system *Eu *= 0 can be represented by a simple layer potential as well (see Section 5 below).

We first prove a property of the singular integral system

**Lemma 4.4**. *Let *Ω ⊂ ℝ^{2 }*be an *(*m *+ 1)*-connected domain. Denote by *
*the eigenspace in *[*L ^{p}*(Σ)]

^{2 }

*of the system*(34).

*Then*

*Proof*. We have

and, since

(the dot denotes the derivative with respect to the arc length on Σ), we find^{3}

We have proved that^{4}

and then the system (34) is of regular type (see [15,29]). From the general theory we know that such a system can be regularized to a Fredholm one. Let us consider now the adjoint system

It is not difficult to see that the index is zero and then systems (34) and (35) have the same number of eigensolutions.

The vectors
*ψ *satisfies the system (35) then

for any *f *∈ [*C*^{∞ }(ℝ^{2})]^{2}. This can be siproved by the same method in [[13], pp. 189-190]. Therefore *ψ *has to be constant on each curve Σ* _{j }*(

*j*= 0, ...,

*m*), i.e.

*ψ*is a linear combination of

**Theorem 4.5**. *Let *Ω ⊂ ℝ^{2 }*be an *(*m *+ 1)*-connected domain. The following conditions are equivalent:*

*I. there exists a Hölder continuous vector function *
*such that*

*II. there exists a constant vector which cannot be represented in *Ω *by a simple layer potential (i.e., there exists c *∈ ℝ^{2 }*such that *
*;*

*III*. Σ_{0 }*is exceptional;*

*IV. let φ*_{1}, ..., *φ*_{2m+2 }*be linearly independent functions of *
*and let c _{jk }*= (

*α*,

_{jk}*β*) ∈ ℝ

_{jk}^{2 }

*be given by*

*Then*

*where*

*Proof*. I ⇒ II. Let *u *be the simple layer potential (3) with density *φ*.

Since *u *= 0 in Ω, and then on Σ* _{k}*, we find that

*u*= 0 also in Ω

*(*

_{k }*k*= 1, ...,

*m*) in view of a known uniqueness theorem.

On the other hand *L*_{+}*u *- *L*_{-}*u *= *φ *on Σ and *φ *= 0 on Σ* _{k}*,

*k*= 1, ...,

*m*. This means that

If II is not true, we can find two linear independent vector functions *ψ*_{1 }and *ψ*_{2 }such that

Arguing as before, we find *ψ _{j }*= 0 on Σ

*,*

_{k}*k*= 1, ...,

*m*,

*j*= 1, 2, and then

Since *φ*, *ψ*_{1}, *ψ*_{2 }belong to the kernel of the system

Lemma 4.4 shows that they are linearly dependent. Let *λ*, *μ*_{1}, *μ*_{2 }∈ ℝ such that (*λ*, *μ*_{1}, *μ*_{2}) ≠ (0, 0, 0) and

This implies

i.e. *μ*_{1}*e*_{1 }+ *μ*_{2}*e*_{2 }= 0, and then *μ*_{1 }= *μ*_{2 }= 0. Now (38) leads to *λφ *= 0 and thus *λ *= 0, which is absurd.

II ⇒ III. If Σ_{0 }is not exceptional, for any *c *∈ ℝ^{2 }there exists ϱ ∈ [*C ^{λ}*(Σ

_{0})]

^{2 }such that

Setting

we can write

and this contradicts II.

III ⇒ IV. Let us suppose
*c *= (*α*, *β*) ∈ ℝ^{2 }there exists *λ *= (*λ*_{1}, ..., *λ*_{2m+2}) solution of the system

i.e.

Therefore

Arguing as before, this leads to
* _{k }*for

*k*= 1, ...,

*m*. Then Σ

_{0 }is not exceptional.

IV ⇒ I. From (37) it follows that there exists an eigensolution *λ *= (*λ*_{1}, ..., *λ*_{2m+2}) of the homogeneous system

Set

In view of the linear independence of *φ*_{1}, ..., *φ*_{2m+2}, the vector function *φ *does not identically vanish and it is such that (36) holds. □

**Definition 4.6**. *Whenever n *= 2 *and *Σ_{0 }*is exceptional, we say that u belongs to *
*if, and only if*,

*where φ *∈ [*L ^{p}*(Σ)]

^{2 }

*and c*∈ ℝ

^{2}.

### 5 The Dirichlet problem

The purpose of this section is to represent the solution of the Dirichlet problem
in an (*m *+ 1)-connected domain by means of a simple layer potential. Precisely we give an existence
and uniqueness theorem for the problem

where *f *∈ [*W*^{1,p}(Σ)]* ^{n}*.

We establish some preliminary results.

**Theorem 5.1**. *Given *
*there exists a solution of the singular integral system*

*if, and only if*,

*for every *