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 ℝⁿ, 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.

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 ℝⁿ (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 C¹ or Lipschitz boundaries and [21] for "worse" domains) lead one to hope that our approach could be extended to more general domains.

Throughout this paper we consider a domain (open connected set) Ω ⊂ ℝⁿ, n ≥ 2, of the form , where Ω_j (j = 0, ..., m) are m + 1 bounded domains of ℝⁿ with connected boundaries Σ_j ∈ C^{1, λ} (λ ∈ (0, 1]) and such that and . For brevity, we shall call such a domain an (m + 1)-connected domain. We denote by ν the outwards unit normal on Σ = ∂Ω.

Let E be the partial differential operator

where u = (u₁, ..., 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 ℝⁿ.

As usual, we denote by the bilinear form defined as

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 L¹ is just the stress operator 2σ_ihν_h, which we shall simply denote by L, while L^k/(k+2) is the so-called pseudo-stress operator.

By the symbol we denote the space of all constant skew-symmetric matrices of order 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 . We denote by the space of all rigid displacements whose dimension is n(n + 1)/2. As usual {e₁, ..., e_n} is the canonical basis for ℝⁿ.

For any 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 non-negative integer, is the vector space of all differential forms of degree h 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 is constituted by the vectors (v₁, ..., v_n) such that v_j is a differential form of . [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).

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_yΓ(x, y)]' denotes the transposed matrix of L_y[Γ(x, y)].

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 have a strong singularity on Σ.

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∂_hf dx, denoting by 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 (du₁, ..., du_n) and ψ = (ψ₁, ..., ψ_n) is an element of .

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 ∈ L¹(Σ). If η ∈ Σ is a Lebesgue point for f, we have

where the limit has to be understood as an internal angular boundary value¹.

Proof. Let h_pj(x) = x_px_j|x|^-n. Since h ∈ C^∞(ℝⁿ\{0}) is even and homogeneous of degree 2 - n, due to the results proved in [23], we have

where , being the Fourier transform

(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_hdx^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_hdx^h and (17) holds (see [[26], p. 41]). We have

and then

a.e. on Σ. From (17) it follows that and (18) is proved.   □

Lemma 3.4. Let . Let us write ψ as ψ = ψ_hdx^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_ihdx^h with

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)]ⁿ. In view of Lemma 3.7, we have a.e. on Σ

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 ; indeed

because of

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

are linearly independent. Indeed, if , we have

Then, by applying a classical uniqueness theorem to the domain Ω_j,

from which it easily follows that

Thus, . On the other hand, suppose and let u be the simple layer potential with density φ. Since E_u = 0 in Ω_j and L_-u = 0 on Σ_j, u = a^j + B^jx on each connected component Ω_j, j = 1, ..., m, and u = 0 in . Note that this is true also for n = 2, because implies . We can define a linear map τ as follows

If τ(φ) = 0, from a classical uniqueness theorem, we have that φ ≡ 0 in ℝⁿ. Thus, τ is an injective map and . The assertion follows from (30).   □

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 . If x ∈ Σ, we have

thanks to

This shows that the restriction to Σ of α belongs to and then . On the other hand, suppose and let 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 ℝⁿ and then ϕ ≡ 0 on Σ, in view of classical uniqueness theorems.

If n = 2, define . We have is injective and its range does not contain the vectors ((1, 0), 0) and ((0, 1), 0)². Therefore . On the other hand, and then . In any case, and the result follows from (30).   □

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 x₀ ∈ Σ_R. For any x ∈ Σ_R we have

and then u is constant on Σ_R. Moreover

and then also Δu is constant on Σ_R. Since Δu is harmonic in Ω_R and continuous on , it is constant in Ω_R and then

The function u(x) - 2πR(1 + log R) |x|² is continuous on , harmonic in Ω_R and constant on Σ_R. Then it is constant in Ω_R and

□

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_u = 0 by means of a simple layer potential.

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 Σ₀ (considered as the boundary of the simply connected domain Ω₀) 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 ⊂ ℝ² 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 find³

We have proved that⁴

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 are the only linearly independent eigensolutions of (35). Indeed it is obvious that such vectors satisfy the system (35). On the other hand, if ψ satisfies the system (35) then

for any f ∈ [C^∞ (ℝ²)]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 ⊂ ℝ² 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 ∈ ℝ² such that );

III. Σ₀ is exceptional;

IV. let φ₁, ..., φ_2m+2 be linearly independent functions of and let c_jk = (α_jk, β_jk) ∈ ℝ² 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 ψ₁ and ψ₂ such that

Arguing as before, we find ψ_j = 0 on Σ_k, k = 1, ..., m, j = 1, 2, and then

Since φ, ψ₁, ψ₂ belong to the kernel of the system

Lemma 4.4 shows that they are linearly dependent. Let λ, μ₁, μ₂ ∈ ℝ such that (λ, μ₁, μ₂) ≠ (0, 0, 0) and

This implies

i.e. μ₁e₁ + μ₂e₂ = 0, and then μ₁ = μ₂ = 0. Now (38) leads to λφ = 0 and thus λ = 0, which is absurd.

II ⇒ III. If Σ₀ is not exceptional, for any c ∈ ℝ² there exists ϱ ∈ [C^λ(Σ₀)]² such that

Setting

we can write

and this contradicts II.

III ⇒ IV. Let us suppose . For any c = (α, β) ∈ ℝ² there exists λ = (λ₁, ..., λ_2m+2) solution of the system

i.e.

Therefore

Arguing as before, this leads to on Σ_k for k = 1, ..., m. Then Σ₀ is not exceptional.

IV ⇒ I. From (37) it follows that there exists an eigensolution λ = (λ₁, ..., λ_2m+2) of the homogeneous system

Set

In view of the linear independence of φ₁, ..., 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 Σ₀ is exceptional, we say that u belongs to if, and only if,

where φ ∈ [L^p(Σ)]² and c ∈ ℝ².

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 such that γ is a weakly closed (n - 2)-form.

Proof. Denote by the adjoint of R (see (23)), i.e. the operator whose components are given by

Thanks to Corollary 3.10, the integral system (41) admits a solution φ ∈ [L^p(Σ)]ⁿ if, and only if,

for any such that R*ψ = 0. Arguing as in [13], R*ψ = 0 if, and only if, all the components of ψ are weakly closed (n - 2)-forms. It is clear that (43) is equivalent to conditions (42).   □

Lemma 5.2. For any f ∈ [W^1,p(Σ)]ⁿ there exists a solution of the BVP

It is given by (3), where the density φ ∈ [L^p(Σ)]ⁿ solves the singular integral system Rφ = df with R as in (23).

Proof. Consider the following singular integral system:

in which the unknown is φ ∈ [L^p(Σ)]ⁿ and the datum is . In view of Theorem 5.1, there exists a solution φ of system (45) because conditions (42) are satisfied.   □

In the next result we consider the eigenspace of the Fredholm integral system

The dimension of is nm. This can be proved as in [[30], p. 63], where the case n = 3 is considered.

Theorem 5.3. Given c₀, c₁, ..., c_m ∈ ℝⁿ, there exists a solution of the BVP

It is given by

where satisfy the following conditions

Proof. Let ψ₁, ..., ψ_nm be nm linearly independent eigensolutions of the space . For a fixed j = 1, ..., nm we set

Then on Σ. As in [[30], Theorem III, p. 45], this implies that V_j is constant on each connected component of . Then V_j = 0 in and in Ω _k (k = 1, ..., m). For every k = 1, ..., m, consider the n × nm matrix defined as follows

The nm × nm matrix has a not vanishing determinant. Indeed, if , the linear system admits an eigensolution λ = (λ₁, ..., λ_nm) ∈ ℝ^nm. Hence the potential

vanishes not only on , but also on Ω_k (k = 1, ..., m). Since this implies W = 0 on Σ, we find W = 0 in Ω, thanks to the classical uniqueness theorem for the Dirichlet problem. Accordingly, W = 0 all over ℝⁿ, from which and this is absurd.

For each h = 1, ..., m and i = 1, ..., n, let be the solution of the system

Setting

we get , and

Put

The potential v belongs to , thanks to the isomorphism σ introduced in the proof of Lemma 3.12 (for n = 2 see Definition 4.6). Moreover

i.e. v = ck on Σ_k (k = 0, 1, ..., m). This shows that v is solution of (46).   □

We are now in a position to establish the main result of this section.

Theorem 5.4. The Dirichlet problem (40) has a unique solution u for every f ∈ [W^1,p(Σ)]ⁿ. If n ≥ 3 or n = 2 with Σ₀ is not exceptional, u is given by (3). If n = 2 and Σ₀ is exceptional, it is given by (39). In any case, the density φ solves the singular system (45).

Proof. Let w be a solution of the problem (44). Since dw = df on Σ, w = f + c_h on Σ_h (h = 0, ..., m) for some c_h ∈ ℝⁿ. The function u = w - v, where v is given by (47), solves the problem (40).

In order to show the uniqueness, suppose that (3) is solution of (40) with f = 0. From Corollary 3.10 it follows that the condition u = 0 on Σ implies that

where T^k/(k+2) is the compact operator given by (26). By bootstrap techniques, (48) implies that φ is a Hölder function on Σ. Then u belongs to and we get that

from which

The solution of (49) is u(x) = a + Bx, where a ∈ ℝⁿ and are arbitrary. Finally, u = 0 in by virtue of the classical uniqueness theorem for the Dirichlet problem.   □

Remark 5.5. In order to solve the Dirichlet problem (40), we need to solve the singular integral system (45). We know that this system can be reduced to a Fredholm one by means of the operator R'^k/(k+2). This reduction is not an equivalent reduction in the usual sense (for this definition see, e.g., [[10], p. 19]), because , being the kernel of the operator R'^k/(k+2).

However R'^k/(k+2) still provides a kind of equivalence. In fact, as in [[31], pp. 253-254], one can prove that . This implies that if ψ is such that there exists at least a solution of the equation Rφ = ψ, then Rφ = ψ if, and only if, R'^k/(k+2) Rφ = R'^k/(k+2)ψ.

Since we know that the system Rφ = df is solvable, we have that Rφ = df if, and only if, φ is solution of the Fredholm system R'^k/(k+2)Rφ = R'^k/(k+2)df.

Therefore, even if we do not have an equivalent reduction in the usual sense, such Fredholm system is equivalent to the Dirichlet problem (40).

The aim of this section is to study the possibility of representing the solution of the traction problem by means of a double layer potential. As we shall see, in an (m + 1)-connected domain this is possible if, and only if, the given forces are balanced on each connected component Σ_j of the boundary.

More precisely, we consider the problem

where f ∈ [L^p(Σ)]ⁿ is such that

We shall prove that, in order to have the existence of a solution of such a problem, condition (51) is not sufficient, but it must be satisfied on each Σ_j, j = 0, 1, ..., m (see Theorem 6.2 below).

If f satisfies the only condition (51), we need to modify the representation of the solution by adding some extra terms (see Theorem 6.4 below).

Lemma 6.1. Let be a double layer potential with density ψ ∈ [W^1,2(Σ)]ⁿ. Then

Proof. Let (ψ_k)_k≥1 be a sequence of functions in [C^1,h(Σ)]ⁿ (0 < h < λ) such that ψ_k → ψ in [W^1,2(Σ)]ⁿ.

Setting

we have that , Ew_k = 0 and then

From ψ_k → ψ in [L²(Σ)]ⁿ, it follows that w_k → w in [L²(Σ)]ⁿ because of well-known properties of singular integral operators.

On the other hand we have that in L²(Ω). By applying formula (11), we see that ∇w_k → ∇w in [L²(Ω)]ⁿ. Moreover, since also in L²(Σ), (22) shows that Lw_k → Lw in [L²(Σ)]ⁿ. We get the claim by letting k → +∞ in (53).   □

Theorem 6.2. Given f ∈ [L^p(Σ)]ⁿ, the traction problem (50) admits a solution if, and only if,

for every j = 0, 1, ..., m, a ∈ ℝⁿ and . The solution is determined up to an additive rigid displacement.

Moreover, (4) is a solution of (50) if, and only if, its density ψ is given by

ϕ being a solution of the singular integral system

where T is given by (27).

Proof. Assume that conditions (54) hold. If u is the double layer potential with density ψ ∈ [W^1,p(Σ)]ⁿ, in view of (22) the boundary condition Lu = f turns into the equation

where R' is given by (24) with ξ = 1.

On account of Theorem 5.4, if n = 2 and Σ₀ is exceptional, any ψ ∈ [W^1,p(Σ)]² can be written as

with ϕ ∈ [L^p(Σ)]², c ∈ ℝ². In all other cases, ψ can be written as (55) with ϕ ∈ [L^p(Σ)]ⁿ. In any case, since dψ = Rϕ (R being defined by (23)), we infer R'(dψ) = R'Rϕ. Keeping in mind Lemma 3.9, we find that equation (57) is equivalent to (56), with ψ given by (55).

Therefore there exists a solution of the traction problem (50) if, and only if, the singular integral system (56) is solvable.

On the other hand, there exists a solution γ ∈ [L^p(Σ)]ⁿ of the singular integral system

if, and only if, f is orthogonal to . In view of Lemma 3.12, this occurs if, and only if, (51) is satisfied. Then conditions (54) imply the existence of a solution of (58).

Consider now the singular integral system

From Lemma 3.11 the dimension of the kernel is n(n + 1)m/2 and is a basis of it. The equation (59) has a solution if, and only if,

Since γ is solution of (58), conditions (60) are fulfilled. Indeed, picking j = 1, ..., m and h = 1, ..., n(n+1)/2, by integrating (58) on Σ_j we find (see (31))

Conditions (60) follow from (54) since the last ones are equivalent to

Let ϕ be a solution of (59); taking (58) into account, we have that ϕ solves (56) and then the traction problem (50) admits a solution.

Conversely, if u is a solution of (50), from Lemma 3.7, we have that

By Lemma 6.1, for any fixed j = 1, ..., m we have

since .

Now we pass to discuss the uniqueness. Let u be a solution of (50) with datum f = 0. As we know, the condition L₊u = 0 is equivalent to the singular integral system , ϕ being as in (55), which can be written as

Set

Since -Ξ/2+TΞ = 0 and the operator -I/2+T can be reduced to Fredholm one, as shown by Kupradze [[1], Chapter IV, §7], Ξ has to be Hölder continuous. By a similar argument, the vector-valued function ϕ, being solution of the singular integral system (61), is Hölder continuous. Therefore the relevant simple layer potential ψ belongs to W^1,2(Σ), i.e. . By applying formula (52), we get that u is a rigid displacement in Ω.   □

We remark that, by Theorem 6.2, a solution of the traction problem (50) can be written as a double layer potential if, and only if, conditions (54) are satisfied.

In order to consider the problem (50) under the only condition (51), we introduce the following space.

Definition 6.3. We define as the space of all the functions w written as

where ψ ∈ [W^1,p(Σ)]ⁿ, {v_h : h = 1, ..., n(n + 1)/2} is an orthonormal basis for and c_jh ∈ ℝ.

Theorem 6.4. Given f ∈ [L^p(Σ)]ⁿ satisfying (51), the traction problem

admits a solution given by

where ψ ∈ [W^1,p(Σ)]ⁿ is solution of the system

The solution is uniquely determined up to an additive rigid displacement.

Proof. First observe that

for h = 1, ..., n(n + 1)/2 and j = 1, ..., m. If w is given by (63), taking into account (57), we find that Lw = f if, and only if, is (64) satisfied.

Denote by g the right hand side of (64). In view of Theorem 6.2, R'(dψ) = g has a solution if, and only if, for any k = 0, 1, ..., m, l = 1, ..., n(n + 1)/2. By integrating on Σ_k (k = 1, ..., m), for every l we get

On the other hand

Finally, assume that w is solution of (62) with f = 0. From (63) it follows that and then w is a rigid displacement in Ω by virtue of the uniqueness proved in Theorem 6.2.   □

The authors declare that they have no competing interests.

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

¹For the definition of internal (external) angular boundary values see, e.g., [[23], p. 53].

²If a simple layer potential u, whose density belongs to , is such that u(x) = c in Ω, then u(x) = c in Ω₀. Since u(∞) = 0, we find u(x) = 0 in and this leads to u = 0 in ℝ², c = 0.

³It is not difficult to see that

⁴We remark that for n ≥ 3 the formula

is false.

⁵This is true also for n = 2 because .

1. Kupradze, VD, Gegelia, TG, Basheleĭshvili, MO, Burchuladze, TV: In: Kupradze VD (ed.) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, Volume 25 of North-Holland Series in Applied Mathematics and Mechanics, Amsterdam: North-Holland Publishing Co (1979)

2. Maremonti, P, Russo, R, Starita, G: On the Stokes equations: the boundary value problem. Advances in fluid dynamics, Volume 4 of Quad Mat, Dept Math, Seconda Univ Napoli, Caserta. 69–140 (1999)

3. Starita, G, Tartaglione, A: On the traction problem for the Stokes system. Math Models Methods Appl Sci. 12(6), 813–834 (2002). Publisher Full Text

4. Kohr, M: A mixed boundary value problem for the unsteady Stokes system in a bounded domain in ℝⁿ. Eng Anal Bound Elem. 29(10), 936–943 (2005). Publisher Full Text

5. Kohr, M: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ℝⁿ. Math Nachr. 280(5-6), 534–559 (2007). Publisher Full Text

6. Kohr, M: The interior Neumann problem for the Stokes resolvent system in a bounded domain in ℝⁿ. Arch Mech (Arch Mech Stos). 59(3), 283–304 (2007)

7. Kohr, M: Boundary value problems for a compressible Stokes system in bounded domains in ℝⁿ. J Comput Appl Math. 201, 128–145 (2007). Publisher Full Text

8. Constanda, C: Direct and indirect boundary integral equation methods, Volume 107 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL (2000)

9. Fichera, G: Una introduzione alla teoria delle equazioni integrali singolari. Rend Mat e Appl (5). 17, 82–191 (1958)

10. Mikhlin, SG, Prössdorf, S: Singular integral operators, Berlin: Springer-Verlag (1986) [Translated from the German by Albrecht Böttcher and Reinhard Lehmann]

11. Fichera, G: Spazi lineari di k-misure e di forme differenziali. Proc Internat Sympos Linear Spaces (Jerusalem, 1960), pp. 175–226. Jerusalem: Jerusalem Academic Press (1961)

12. Flanders, H: Differential forms with applications to the physical sciences, New York: Academic Press (1963)

13. Cialdea, A: On oblique derivative problem for Laplace equation and connected topics. Rend Accad Naz Sci XL Mem Mat (5). 12, 181–200 (1988)

14. Cialdea, A, Hsiao, GC: Regularization for some boundary integral equations of the first kind in mechanics. Rend Accad Naz Sci XL Mem Mat Appl (5). 19, 25–42 (1995)

15. Muskhelishvili, NI: Singular integral equations, Wolters-Noordhoff Publishing, Groningen (1972) [Boundary problems of functions theory and their applications to mathematical physics, Revised translation from the Russian, edited by J. R. M. Radok, Reprinted]

16. Malaspina, A: On the traction problem in mechanics. Arch Mech (Arch Mech Stos). 57(6), 479–491 (2005)

17. Cialdea, A, Leonessa, V, Malaspina, A: On the Dirichlet and the Neumann problems for Laplace equation in multiply connected domains. In: Complex Variables and Elliptic Equations to appear [http://dx.doi.org/10.1080/17476933.2010.534156] webcite

18. Lifanov, IK, Poltavskii, LN, Vainikko, GM: Hypersingular integral equations and their applications, Volume 4 of Differential and Integral Equations and Their Applications, Chapman & Hall/CRC, Boca Raton, FL (2004)

19. Cialdea, A, Leonessa, V, Malaspina, A: On the Dirichlet problem for the Stokes system in multiply connected domains. [To appear]

20. Kohr, M, Pintea, C, Wendland, WL: Brinkman-type operators on Riemannian manifolds: transmission problems in Lipschitz and C¹-domains. Potential Anal. 32(3), 229–273 (2010). Publisher Full Text

21. Medková, D: The integral equation method and the Neumann problem for the Poisson equation on NTA domains. Integral Equations Operator Theory. 63(2), 227–247 (2009). Publisher Full Text

22. Hodge, W: A Dirichlet problem for harmonic functionals, with applications to analytic varieties. Proc Lond Math Soc, II Ser. 36, 257–303 (1933)

23. Cialdea, A: A general theory of hypersurface potentials. Ann Mat Pura Appl (4). 168, 37–61 (1995). Publisher Full Text

24. Cialdea, A, Malaspina, A: Completeness theorems for the Dirichlet problem for the polyharmonic equation. Rend Accad Naz Sci XL Mem Mat Appl (5). 29, 153–173 (2005)

25. Neri, U: Lecture Notes in Mathematics. Singular integrals, Berlin: Springer-Verlag (1971) [Notes for a course given at the University of Maryland, College Park, Md., 1967]

26. Cialdea, A: On the finiteness of the energy integral in elastostatics with non-absolutely continuous data. Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl. 4, 35–42 (1993)

27. Folland, GB: Introduction to partial differential equations, Princeton, NJ: Princeton University Press (1995)

28. Steinbach, O: A note on the ellipticity of the single layer potential in two-dimensional linear elasto-statics. J Math Anal Appl. 294, 1–6 (2004). Publisher Full Text

29. Vekua, NP: Systems of singular integral equations, Groningen: P. Noordhoff Ltd (1967) [Translated from the Russian by A. G. Gibbs and G. M. Simmons. Edited by J. H. Ferziger]

30. Fichera, G: Sull'esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all'equilibrio di un corpo elastico. Ann Scuola Norm Super Pisa (3). 4(248), 35–99 Consiglio Naz Ricerche Pubbl Ist Appl Calcolo (1950) (1950)

31. Cialdea, A: The multiple layer potential for the biharmonic equation in n variables. Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl. 3(4), 241–259 (1992)