On the singular behavior of solutions of a transmission problem in a Dihedral

In this paper, we study the singular behavior of solutions of a boundary value problem with mixed conditions in a neighborhood of an edge. The considered problem is defined in a nonhomogeneous body of R, this is done in the general framework of weighted Sobolev spaces. Using the results of Benseridi-Dilmi, Grisvard and Aksentian, we show that the study of solutions’ singularities in the spatial case becomes a study of two problems: a problem of plane deformation and the other is of normal plane deformation.


Introduction
Many research papers have been written recently, both on the singular behavior of solutions for elasticity system in a homogeneous polygon or a polyhedron, see for example [2,6,7,11] and the references cited therein.In the homogeneous domain, in [14] it is introduced a unified and general approach to the asymptotic analysis of elliptic boundary value problems in singularly perturbed domains.The construction of this method capitalizes on the theory of elliptic boundary value problems with nonsmooth boundary.On the other hand, in [15] the authors developed an asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities.
The case of a nonhomogeneous polygon was already considered in [3].The regularity of the solutions of transmission problem for the Laplace operator in R 3 was studied in [4].
The aim of this paper, is to study the regularity of solutions for the following transmission problem: where σ i , (i = 1, 2) designate the stress tensor with σ i = (σ ijk ), j, k = 1, 2, 3 and i = 1, 2. The σ ijk elements are given by the Hooke's law and Ω 1 , Ω 2 are two homogeneouse elastic and isotropic bodies occupying a domain of R 3 with a polyhedral boundary.We suppose that the lateral surface Γ 2 forms an arbitrary angle ω 2 (0 < ω 2 ≤ 2π) to the surface Γ 1 .In addition we suppose that Ω is an nonhomogeneous body constituted by two bodies (Ω 1 ∪ Ω 2 ) rigidly joined along the cylindrical surface Λ × R, which passes through the edge A. The generator of this surface is inclined at an angle ω 1 (0 < ω 1 ≤ 2π) to the surface of the first body.For a function u, defined on Ω, we designate by u 1 (resp.u 2 ) its restriction on Ω 1 (resp.Ω 2 ).Let µ i and be, respectively, the shear modulus and Poisson's ratio for the material of the body Ω i , bounded by the surfaces Γ i and Λ × R, i = 1, 2.
The vector N (resp.τ ) denotes the normal (resp.the tangent) on Λ toward the interior of Ω 1 .B i is the infinite subset of R 3 defined by: The paper is organised as follows: In section 1 we recall some definitions and properties of Sobolev spaces with double weights introduced by Pham The Lai [13].In section 2 we transform the problem (P 1 ) using the partial complex Fourier transform with respect to the first variable, we obtain then a new problem.In section 3 we prove a result of existence and uniqueness of the η− solutions according to boundary conditions and we find transcendental equations which govern the singular behavior of solution, then we compare these η− solutions.This comparison will be very useful because it allows us to find a sufficient condition for the existence and the uniqueness of the solution of our initial problem.Finaly, we state our main result on the regularity for the problem (P 1 ).

Preliminary results and lemma
In this section we give some basic tools and properties of the weighted Sobolev spaces used in the next.Definition 2.1.For s ∈ N, we define the spaces equiped with the scalar product D α e θ0 t (1 + e t ) θ∞−θ0 u D α e θ0 t (1 + e t ) θ∞−θ0 v dtdθdx 3 .
Proof.Use cylindrical coordinates together with the change of variable r = e −t .Definition 2.2.The application is an isomorphism.

Use cylindrical coordinates
We put x 1 = r cos θ , x 2 = r sin θ and x 3 = x 3 with r = e −t .Let us write the equations of the Lamé' system in this coordinates, the problem (P 1 ) becames where g i (t, θ, x 3 ) = e 2t f i ( e −t cos θ, e −t sin θ, x 3 ), u ir , u iθ and u ix3 are the components of the displacement vector, taken in the directions of the introduced coordinates.Property 3.1.
which gives the result for s = 2. Property 3.2.The problems (P 1 ) and (P 2 ) are equivalents.Proof.It follows from property 3.1.Remark 3.1 1-To express the behavior of the solution of the boundary value problem far away from the vertex, noting that the neighborhood of A is sufficiently small so that terms containing the factor e −t may be neglected.2-According to the mixed condition it is shown that the surface Γ 2 is free of stresses while the surface Γ 1 is rigidly clamped.Since Γ 1 , Λ × R and Γ 2 are coordinate surfaces corresponding to θ = 0, θ = ω 1 and θ = ω 2 respectively.3-The boundary conditions are 4-The indicated stresses, in terms of displacements in the above coordinate system, are given by: where, τ irθ and σ iθθ , are the tangential stress tensor and the normal stress tensor respectively.

Fourier transform of (P 2 )
With the condition f i ∈ L 2 θ0,θ∞ (Ω i ) 3 the function g i (t, θ, x 3 ) admits a Fourier transform g i (ξ, θ, x 3 ) for any ξ in the strip C η0,η∞ defined by This strip is not empty since it was assumed that θ 0 ≤ θ ∞ .On the other hand u i (x 1 , x 2 , x 3 ) ∈ H 2 θ0,θ∞ (Ω i ) 3 , u i and its derivatives of order ≤ 2 admit a Fourier transform in the same strip.
Applying the Fourier transform on (P 2 ) and taking into account the smallness of the neighborhood, we obtain the following problem where u i and σ i are the Fourier transforms of u i and σ i respectively.More exactly we have: Remark 3.2 1-From equations of (P 3 ) it can be seen that the problem (P 1 ) can be divided into two problems: The first is a plane deformation to which correspond the two first equations (I) and (II), while the second is a normal plane deformation, expressed by the third equation (III).
2-Finally, we get the following problem: for a fixed ξ in the strip C η0,η∞ , we look for a possible solution The study of the homogeneous problem corresponding to (P 3 ) gives the following results.Proposition 3.1.The transcendental equations governing the singular behavior of the problem (P 3 ) given by: Problem of plane deformation

Problem of normal plane deformation
Proof.Using the boundary conditions on Γ 1 , Γ 2 and Λ × R, we obtain a system of homogeneous equations.The condition of the vanishing of the system's determinant gives the transcendental equations with respect to ξ. Proposition 3.2.Let F and G be the zeros of (3.1) and (3.2) repectively, then the homogeneous problem (P 3 ) admits a unique solution, if and only if, ξ / ∈ (F ∪ G).

The main result
In this section, we are going to prove a result of existence and uniqueness of the η− solutions and then, we compare them η− solutions.This comparison will be very useful because it allows us to find a sufficient condition for the existence and the uniqueness of the solution of our initial problem (P 1 ).It is important to introduce the following definition.Defnition 4.1.Let η ∈ [η 0 , η ∞ ] , we call η−solutions for the problem (P 1 ), all elements u = (u 1 , u 2 ) of H 2 η+1,η+1 (Ω 1 ) 3 × H 2 η+1,η+1 (Ω 2 ) 3 , verifying (P 1 ).The following property is a straightforward consequence of lemma 2.1.Property 4.1.u is a solution for the problem (P 1 ), iff, u is a η 0 −solutions and η ∞ −solutions of (P 1 ).Proof.Let u be a solution of (P 1 ), then and from lemma 2.1, we have Then u is a η 0 −solution and η ∞ −solution of the (P 1 ).Property 4.2.If the transcendental equations (3.k), k = 1, 2 have no zeros of imaginary part η, the problem (P 1 ) has a unique η− solutions, in addition there exists a positive constant c such that The proof of this property is based on the following lemmas.Lemma 4.1.K is a compact containing no zeros of (3.k), k = 1, 2, then there exist a constant c depending on K such that for all u and all ξ ∈ K: where EJQTDE, 2010 No. 13, p. 6 Lemma 4.2.Let R > 0, there exists α > 0 and c > 0 such that for any ξ verifying |Reξ| ≥ α, |Imξ| ≤ R and for all u i of H 2 (]0, ω i [) 3 , we have Remark 4.1.For the proof of the two first lemmas we refer the reader to [10] .Lemme 4.3.For a given η 1 , η 2 ∈ R such that, , then It suffies to show that e η t g ≤ e η1 t 1 + e t η2−η1 g .(4.1) Indeed, for t ∈ R + , we have (1 + e t ) η2−η1 ≥ e (η2−η1) t and e η2 t ≥ e η t , as e η t g ≤ e η t 1 + e t η2−η1 g , and for t ≤ 0 (1 + e t ) η2−η1 ≥ 1 and e η1 t ≥ e η t .

Hence the inequality (4.1).
Therefore, Proof.(property 4.2).This amounts to showing that the problem (P 2 ) admits a unique η− solution, i.e. that there exists one and only one u = (u 1 , u 2 ) in H 2 η,η (B 1 ) 3 × H 2 η,η (B 2 ) 3 verifying (P 2 ).Existence.The hypothesis that (3.k) has no zeros on the half plane R + iη ensures that the problem (P 3 ) admits a solution We set where T −1 is the inverse Fourier transform with respect to ρ.One can easily verify that u is a solution of (P 2 ) and Uniqueness.Let u 1 and u 2 two solutions of the problem (P 1 ), then u 1 and u 2 are two solutions of (P 3 ).It follows from the proposition 3.3, that u 1 = u 2 , now applying the inverse Fourier transform to both sides of this equality, we obtain u 1 = u 2 , hence the uniqueness.
We show now that For this, it suffies to show that First recall that the application is an isomorphism, this allows us to write Let R = |η| and α as defined in lemma 4.2, then for all ρ, |ρ| ≥ α Set K = {ξ = ρ + iη : |ρ| ≤ α} , which is a compact set containing no zeros of (3.k).It comes from lemma 4.1 that , we deduce that (4.2) is valid for ρ such that |ρ| ≤ α, so it is also valid for any ρ ∈ R. By integrating both members of (4.2) with respect to ρ, we find Moreover, from lemma 4.3 EJQTDE, 2010 No. 13, p. 8 Hence η 0 ,η∞ (B2) 3 .Finally, from the proposition 2.1, we deduce that The following proposition is devoted to the decomposition of the solution of the problem (P 1 ) to a singular and a regular parts.Proposition 4.1.
This shows that u η0 = u η∞ .We put now u = u η0 , it is clear that Thus u is a solution of (P 1 ) by construction.
(3) Continuity with respect to the data.We deduce from property 4.2, that Ω2) 3 , and from lemma 2.1, we get Which proves the theorem.

Singularity solutions of the homogeneous elasticity system
Let us now examine the case of a homogeneous plate, the side surface of which makes an angle ω with the plane of the face.This case may be obtained by setting: ν = ν 1 = ν 2 , µ = µ 1 = µ 2 and ω = ω 1 = ω 2 in the relations previously derived.Proposition 5.1.The transcendental equations governing the singular behavior of the problem (P 1 ) take the form if ξ is a simple root of (5.1), where By setting θ = 0 and θ = ω in the boundary conditions (BC), we obtain a system of homogeneous equations.The condition of the vanishing of the system's determinant gives the transcendental equations (5.1) with respect to ξ.So for any ξ a complex solution of (5.1), the solutions of this system give the singular solution ℑ(r, θ, x 3 ) for ω ∈ ]0, π[ ∪ ]π, 2π[.
In the same way setting θ = 2π in (BC), we obtain the component of the singular solution for ω = 2π.
This ends the proof.
EJQTDE, 2010 No. 13, p. 11 The purpose of this paper is to study the singular behavior of solutions of a boundary value problem with mixed conditions in a neighborhood of an edge in the general framework of weighted Sobolev spaces.This work is an extension to similary ones in Sobolev spaces with null and single weight.In the non homogeneous case, it's not easy to solve the transcendental equations defined in the proposition 3.1, this does not permit us to find the singular solutions.
We will devote a further paper for the generalization of the results obtained here for the non-homogeneous case with presence of discontinuity of the boundary value on the intersection surface.