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Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

Alberto Cialdea*, Vita Leonessa and Angelica Malaspina

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Department of Mathematics and Computer Science, University of Basilicata, V.le dell'Ateneo Lucano, 10, Campus of Macchia Romana, 85100 Potenza, Italy

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Citation and License

Boundary Value Problems 2011, 2011:53  doi:10.1186/1687-2770-2011-53


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/53


Received:21 May 2011
Accepted:12 December 2011
Published:12 December 2011

© 2011 Cialdea et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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.

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 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.

2 Notations and definitions

Throughout this paper we consider a domain (open connected set) Ω ⊂ ℝn, n ≥ 2, of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M1">View MathML</a>, where Ωj (j = 0, ..., m) are m + 1 bounded domains of ℝn with connected boundaries Σj C1, λ (λ ∈ (0, 1]) and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M3">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M4">View MathML</a>

where u = (u1, ..., un) 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M5">View MathML</a>

(1)

i, j = 1, ..., n, ωn being the hypersurface measure of the unit sphere in ℝn.

As usual, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M6">View MathML</a> the bilinear form defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M7">View MathML</a>

where εih (u) and σih (u) are the linearized strain components and the stress components respectively, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M8">View MathML</a>

Let us consider the boundary operator Lξ whose components are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M9">View MathML</a>

(2)

ξ being a real parameter. We remark that the operator L1 is just the stress operator 2σihνh, which we shall simply denote by L, while Lk/(k+2) is the so-called pseudo-stress operator.

By the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M267">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M10">View MathML</a> the space of all rigid displacements whose dimension is n(n + 1)/2. As usual {e1, ..., en} is the canonical basis for ℝn.

For any 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 non-negative integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M11">View MathML</a> is the vector space of all differential forms of degree h 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M12">View MathML</a> is constituted by the vectors (v1, ..., vn) such that vj is a differential form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M13">View MathML</a>. [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).

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 = 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M14">View MathML</a>if, and only if, there exists φ ∈ [Lp(Σ)]n such that u can be represented by a simple layer potential

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M15">View MathML</a>

(3)

Definition 2.2. The vector-valued function w belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M16">View MathML</a>if, and only if, there exists ψ ∈ [W1,p(Σ)]n such that w can be represented by a double layer potential

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M17">View MathML</a>

(4)

where [LyΓ(x, y)]' denotes the transposed matrix of Ly[Γ(x, y)].

3 Preliminary results

3.1 On the first derivatives of a double layer potential

Let us consider the boundary operator Lξ defined by (2). Denoting by Γj(x, y) the vector whose components are Γij(x, y), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M18">View MathML</a>

(5)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M19">View MathML</a>

(6)

while for ξ k/(2 + k) the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M20">View MathML</a> have a strong singularity on Σ.

Let us denote by wξ the double layer potential

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M21">View MathML</a>

(7)

It is known that the first derivatives of a harmonic double layer potential with density φ belonging to W1,p(Σ) can be written by means of the formula proved in [[13], p. 187]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M22">View MathML</a>

(8)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M23">View MathML</a>

and sh(x, y) is the double h-form introduced by Hodge in [22]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M24">View MathML</a>

Since, for a scalar function f and for a fixed h, we have *df dxh = (-1)n-1hf dx, denoting by w the harmonic double layer potential with density φ W1,p(Σ), (8) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M25">View MathML</a>

(9)

where, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M26">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M27">View MathML</a>

(10)

The following lemma can be considered as an extension of formula (9) to elasticity. Here du denotes the vector (du1, ..., dun) and ψ = (ψ1, ..., ψn) is an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M28">View MathML</a>.

Lemma 3.1. Let wξ be the double layer potential (7) with density u ∈ [W1,p(Σ)]n. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M29">View MathML</a>

(11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M30">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M31">View MathML</a>

(13)

and Θh is given by (10), h = 1, ..., n.

Proof. Let n ≥ 3. Denote by Mhi the tangential operators Mhi = νhi - νih, h, i = 1, ..., n. By observing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M32">View MathML</a>

we find in Ω

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M33">View MathML</a>

An integration by parts on Σ leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M34">View MathML</a>

Therefore, by recalling (9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M35">View MathML</a>

(14)

If f is a scalar function, we may write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M36">View MathML</a>

This identity is established by observing that on Σ we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M37">View MathML</a>

Then we can rewrite (14) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M38">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M39">View MathML</a>

(15)

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

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M40">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M42">View MathML</a> being the Fourier transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M43">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M44">View MathML</a>

and, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M45">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M46">View MathML</a>

Finally, keeping in mind that ωn = n πn/2/Γ(n/2 + 1) and Γ(n/2 + 1) = n(n - 2)Γ(n/2 - 1)/4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M47">View MathML</a>

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

Lemma 3.3. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M48">View MathML</a>. Let us write ψ as ψ = ψhdxh with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M49">View MathML</a>

(17)

Then, for almost every η ∈ Σ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M50">View MathML</a>

(18)

where Θs 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 ψh defined on Σ such that ψ = ψhdxh and (17) holds (see [[26], p. 41]). We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M51">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M52">View MathML</a>

a.e. on Σ. From (17) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M53">View MathML</a> and (18) is proved.   □

Lemma 3.4. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M54">View MathML</a>. Let us write ψ as ψ = ψhdxh and suppose that (17) holds. Then, for almost every η ∈ Σ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M55">View MathML</a>

(19)

where Kξ is defined by (13) and the limit has to be understood as an internal angular boundary value.

Proof. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M56">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M57">View MathML</a>

On the other hand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M58">View MathML</a>

and the result follows.   □

Lemma 3.5. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M59">View MathML</a>. Then, for almost every η ∈ Σ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M60">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M61">View MathML</a>being as in (12) and the limit has to be understood as an internal angular boundary value.

Proof. Let us write ψi as ψi = ψihdxh with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M62">View MathML</a>

(21)

In view of Lemmas 3.3 and 3.4 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M63">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M64">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M65">View MathML</a>

Conditions (21) lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M66">View MathML</a>

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 wξ be the double layer potential (7) with density u ∈ [W1,p(Σ)]n. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M67">View MathML</a>

(22)

a.e. on Σ, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M68">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M69">View MathML</a>denote the internal and the external angular boundary limit of Lξ(wξ) respectively and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M70">View MathML</a>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 : <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M71">View MathML</a>be the following singular integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M72">View MathML</a>

(23)

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M73">View MathML</a>to be the singular integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M74">View MathML</a>

(24)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M75">View MathML</a>

(25)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M76">View MathML</a>

(26)

Proof. Let u be the simple layer potential with density φ ∈ [Lp(Σ)]n. In view of Lemma 3.7, we have a.e. on Σ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M77">View MathML</a>

where wξ is the double layer potential (7) with density u. Moreover, if x ∈ Ω,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M78">View MathML</a>

and then, on account of (26),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M79">View MathML</a>

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.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M80">View MathML</a>

(27)

and denote by T* its adjoint.

In this subsection we determine the dimension of the following eigenspaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M81">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M82">View MathML</a>

(29)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M83">View MathML</a>

(30)

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

Lemma 3.11. The spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M84">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M85">View MathML</a>have dimension n(n + 1)m/2. Moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M86">View MathML</a>

where {vh : h = 1 ..., n(n + 1)/2} is an orthonormal basis of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M87">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M88">View MathML</a>is the characteristic function of Σj.

Proof. We define the vector-valued functions αj, j = 1, ..., m as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M89">View MathML</a>, x ∈ Σ. For a fixed j = 1, ..., m, the function αj(x) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M90">View MathML</a>; indeed

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M91">View MathML</a>

because of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M92">View MathML</a>

(31)

Now we prove that the following n(n + 1)m/2 eigensolutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M93">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M94">View MathML</a>

are linearly independent. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M95">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M96">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M97">View MathML</a>

from which it easily follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M98">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M99">View MathML</a>. On the other hand, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M100">View MathML</a> and let u be the simple layer potential with density φ. Since Eu = 0 in Ωj and L-u = 0 on Σj, u = aj + Bjx on each connected component Ωj, j = 1, ..., m, and u = 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M101">View MathML</a>. Note that this is true also for n = 2, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M102">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M103">View MathML</a>. We can define a linear map τ as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M104">View MathML</a>

If τ(φ) = 0, from a classical uniqueness theorem, we have that φ ≡ 0 in ℝn. Thus, τ is an injective map and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M105">View MathML</a>. The assertion follows from (30).   □

Lemma 3.12. The spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M106">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M107">View MathML</a>have dimension n(n + 1)/2. Moreover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M108">View MathML</a>is constituted by the restrictions to Σ of the rigid displacements.

Proof. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M109">View MathML</a>. If x ∈ Σ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M110">View MathML</a>

thanks to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M111">View MathML</a>

This shows that the restriction to Σ of α belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M112">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M113">View MathML</a>. On the other hand, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M114">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M115">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M116">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M117">View MathML</a> is injective and its range does not contain the vectors ((1, 0), 0) and ((0, 1), 0)2. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M118">View MathML</a>. On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M119">View MathML</a>and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M120">View MathML</a>. In any case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M121">View MathML</a> and the result follows from (30).   □

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M122">View MathML</a>

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

Lemma 4.1. Let ΣR be the circle of radius R centered at the origin. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M123">View MathML</a>

(32)

Proof. Denote by u(x) the function on the left hand side of (32) and by ΩR the ball of radius R centered at the origin. Let us fix x0 ∈ ΣR. For any x ∈ ΣR we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M124">View MathML</a>

and then u is constant on ΣR. Moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M125">View MathML</a>

and then also Δu is constant on ΣR. Since Δu is harmonic in ΩR and continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M126">View MathML</a>, it is constant in ΩR and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M127">View MathML</a>

The function u(x) - 2πR(1 + log R) |x|2 is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M128">View MathML</a>, harmonic in ΩR and constant on ΣR. Then it is constant in ΩR and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M129">View MathML</a>

Corollary 4.2. Let ΣR be the circle of radius R centered at the origin. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M130">View MathML</a>

(33)

Proof. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M131">View MathML</a>

formula (32) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M132">View MathML</a>

In a similar way

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M133">View MathML</a>

From (32) we have also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M134">View MathML</a>

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

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M135">View MathML</a>

This implies that in ΩR, for such a value of R, we cannot represent any smooth solution of the system Eu = 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 ΣR with R = exp[k/(2(k + 2))] is exceptional for the operator Δ + kdiv.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M136">View MathML</a>

(34)

Lemma 4.4. Let Ω ⊂ ℝ2 be an (m + 1)-connected domain. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M137">View MathML</a>the eigenspace in [Lp(Σ)]2 of the system (34). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M138">View MathML</a>.

Proof. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M139">View MathML</a>

and, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M140">View MathML</a>

(the dot denotes the derivative with respect to the arc length on Σ), we find3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M141">View MathML</a>

We have proved that4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M142">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M143">View MathML</a>

(35)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M144">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M145">View MathML</a>

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 Σj (j = 0, ..., m), i.e. ψ is a linear combination of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M146">View MathML</a>.   □

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M147">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M148">View MathML</a>

(36)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M149">View MathML</a>);

III. Σ0 is exceptional;

IV. let φ1, ..., φ2m+2 be linearly independent functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M150">View MathML</a>and let cjk = (αjk, βjk) ∈ ℝ2 be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M151">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M152">View MathML</a>

(37)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M153">View MathML</a>

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

Since u = 0 in Ω, and then on Σk, we find that u = 0 also in Ωk (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 Σk, k = 1, ..., m. This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M154">View MathML</a>

If II is not true, we can find two linear independent vector functions ψ1 and ψ2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M155">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M156">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M157">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M158">View MathML</a>

(38)

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M159">View MathML</a>

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 ∈ [Cλ0)]2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M160">View MathML</a>

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M161">View MathML</a>

we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M162">View MathML</a>

and this contradicts II.

III ⇒ IV. Let us suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M163">View MathML</a>. For any c = (α, β) ∈ ℝ2 there exists λ = (λ1, ..., λ2m+2) solution of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M164">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M165">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M166">View MathML</a>

Arguing as before, this leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M167">View MathML</a> on Σk for k = 1, ..., m. Then Σ0 is not exceptional.

IV ⇒ I. From (37) it follows that there exists an eigensolution λ = (λ1, ..., λ2m+2) of the homogeneous system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M168">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M169">View MathML</a>

In view of the linear independence of φ1, ..., φ2m+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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M170">View MathML</a>if, and only if,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M171">View MathML</a>

(39)

where φ ∈ [Lp(Σ)]2 and c ∈ ℝ2.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M172">View MathML</a>

(40)

where f ∈ [W1,p(Σ)]n.

We establish some preliminary results.

Theorem 5.1. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M173">View MathML</a>, there exists a solution of the singular integral system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M174">View MathML</a>

(41)

if, and only if,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M175">View MathML</a>

(42)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M176">View MathML</a>such that γ is a weakly closed (n - 2)-form.

Proof. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M177">View MathML</a> the adjoint of R (see (23)), i.e. the operator whose components are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M178">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M179">View MathML</a>

(43)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M180">View MathML</a> 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 ∈ [W1,p(Σ)]n there exists a solution of the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M181">View MathML</a>

(44)

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

Proof. Consider the following singular integral system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M182">View MathML</a>

(45)

in which the unknown is φ ∈ [Lp(Σ)]n and the datum is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M183">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M184">View MathML</a> of the Fredholm integral system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M185">View MathML</a>

The dimension of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M186">View MathML</a> is nm. This can be proved as in [[30], p. 63], where the case n = 3 is considered.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M187">View MathML</a>

(46)

It is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M188">View MathML</a>

(47)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M189">View MathML</a>satisfy the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M190">View MathML</a>

Proof. Let ψ1, ..., ψnm be nm linearly independent eigensolutions of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M191">View MathML</a>. For a fixed j = 1, ..., nm we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M192">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M193">View MathML</a> on Σ. As in [[30], Theorem III, p. 45], this implies that Vj is constant on each connected component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M194">View MathML</a>. Then Vj = 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M196">View MathML</a> in Ω k (k = 1, ..., m). For every k = 1, ..., m, consider the n × nm matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M197">View MathML</a> defined as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M198">View MathML</a>

The nm × nm matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M199">View MathML</a> has a not vanishing determinant. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M200">View MathML</a>, the linear system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M201">View MathML</a> admits an eigensolution λ = (λ1, ..., λnm) ∈ ℝnm. Hence the potential

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M202">View MathML</a>

vanishes not only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M203">View MathML</a>, but also on Ωk (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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M204">View MathML</a> and this is absurd.

For each h = 1, ..., m and i = 1, ..., n, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M205">View MathML</a> be the solution of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M206">View MathML</a>

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M207">View MathML</a>

we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M209">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M210">View MathML</a>

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M211">View MathML</a>

The potential v belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M212">View MathML</a>, thanks to the isomorphism σ introduced in the proof of Lemma 3.12 (for n = 2 see Definition 4.6). Moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M213">View MathML</a>

i.e. v = ck on Σk (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 ∈ [W1,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 + ch on Σh (h = 0, ..., m) for some ch ∈ 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M214">View MathML</a>

(48)

where Tk/(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M215">View MathML</a> and we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M216">View MathML</a>

from which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M217">View MathML</a>

(49)

The solution of (49) is u(x) = a + Bx, where a ∈ ℝn and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268">View MathML</a> are arbitrary. Finally, u = 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M218">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M220">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M221">View MathML</a>. 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) = R'k/(k+2)ψ.

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

6 The traction problem

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 Σj of the boundary.

More precisely, we consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M222">View MathML</a>

(50)

where f ∈ [Lp(Σ)]n is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M223">View MathML</a>

(51)

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 Σj, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M224">View MathML</a>be a double layer potential with density ψ ∈ [W1,2(Σ)]n. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M225">View MathML</a>

(52)

Proof. Let (ψk)k≥1 be a sequence of functions in [C1,h(Σ)]n (0 < h < λ) such that ψk ψ in [W1,2(Σ)]n.

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M226">View MathML</a>

we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M227">View MathML</a>, Ewk = 0 and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M228">View MathML</a>

(53)

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

On the other hand we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M229">View MathML</a> in L2(Ω). By applying formula (11), we see that ∇wk → ∇w in [L2(Ω)]n. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M230">View MathML</a> also in L2(Σ), (22) shows that Lwk Lw in [L2(Σ)]n. We get the claim by letting k → +∞ in (53).   □

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M231">View MathML</a>

(54)

for every j = 0, 1, ..., m, a ∈ ℝn and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M268">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M232">View MathML</a>

(55)

ϕ being a solution of the singular integral system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M233">View MathML</a>

(56)

where T is given by (27).

Proof. Assume that conditions (54) hold. If u is the double layer potential with density ψ ∈ [W1,p(Σ)]n, in view of (22) the boundary condition Lu = f turns into the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M234">View MathML</a>

(57)

where R' is given by (24) with ξ = 1.

On account of Theorem 5.4, if n = 2 and Σ0 is exceptional, any ψ ∈ [W1,p(Σ)]2 can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M235">View MathML</a>

with ϕ ∈ [Lp(Σ)]2, c ∈ ℝ2. In all other cases, ψ can be written as (55) with ϕ ∈ [Lp(Σ)]n. In any case, since = (R being defined by (23)), we infer R'() = R'. 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 ∈ [Lp(Σ)]n of the singular integral system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M236">View MathML</a>

(58)

if, and only if, f is orthogonal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M237">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M238">View MathML</a>

(59)

From Lemma 3.11 the dimension of the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M239">View MathML</a> is n(n + 1)m/2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M240">View MathML</a> is a basis of it. The equation (59) has a solution if, and only if,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M241">View MathML</a>

(60)

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 Σj we find (see (31))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M242">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M243">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M244">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M245">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M246">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M247">View MathML</a>, ϕ being as in (55), which can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M248">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M249">View MathML</a>

(61)

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 W1,2(Σ), i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M250">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M251">View MathML</a>as the space of all the functions w written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M252">View MathML</a>

where ψ ∈ [W1,p(Σ)]n, {vh : h = 1, ..., n(n + 1)/2} is an orthonormal basis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M253">View MathML</a>and cjh ∈ ℝ.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M254">View MathML</a>

(62)

admits a solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M255">View MathML</a>

(63)

where ψ ∈ [W1,p(Σ)]n is solution of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M256">View MathML</a>

(64)

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

Proof. First observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M257">View MathML</a>

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'() = g has a solution if, and only if, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M258">View MathML</a> for any k = 0, 1, ..., m, l = 1, ..., n(n + 1)/2. By integrating on Σk (k = 1, ..., m), for every l we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M259">View MathML</a>

On the other hand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M260">View MathML</a>

Finally, assume that w is solution of (62) with f = 0. From (63) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M261">View MathML</a> and then w is a rigid displacement in Ω by virtue of the uniqueness proved in Theorem 6.2.   □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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

Endnotes

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M262">View MathML</a>, is such that u(x) = c in Ω, then u(x) = c in Ω0. Since u(∞) = 0, we find u(x) = 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M263">View MathML</a> and this leads to u = 0 in ℝ2, c = 0.

3It is not difficult to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M264">View MathML</a>

4We remark that for n ≥ 3 the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M265">View MathML</a>

is false.

5This is true also for n = 2 because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/53/mathml/M266">View MathML</a>.

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