THE INTEGRAL EQUATION METHODS FOR THE PERTURBED HELMHOLTZ EIGENVALUE PROBLEMS

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues. In this paper, we address the integral equation method in the evaluation of eigenfunctions and the corresponding eigenvalues of the two-dimensional Laplacian operator under boundary variations of the domain. Using surface potentials, we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions.


Introduction
The properties of eigenvalue problems under shape deformation have been the subject of comprehensive studies [7,13] and the area continues to carry great importance up to now [14].A substantial portion of these investigations is related to smoothness properties of eigenfunctions with respect to boundary perturbations.Recently, Bruno and Reitich have presented in [4] explicit constructions of high-order boundary perturbation expansions for eigenelements.Their algorithm is based on certain properties of joint analytic dependence on the boundary perturbations and spatial variables of the eigenfunctions.The main difficulty in solving eigenvalue problems relates to the continuation of multiple eigenvalues of the unperturbed configuration.These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and this splitting may only become apparent at high orders in their Taylor expansions.
In this paper, we use the technique of integral equation to evaluate the analyticity properties and asymptotic expansions of the eigenfunctions and the eigenvalues of the Laplacian operator under boundary variations of the domain of definition.Using surface potentials, we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions which are of Fredholm type with index 0. We then proceed using the generalized Rouché's theorem [6] and the result found in [10] to construct their complete asymptotic expressions.Our approach concerning the question of analytic dependence and asymptotic expansion is based on a holomorphic formulation of the boundary integral equations and its characteristic problem version.
Our analysis and uniform asymptotic formulas of the eigenfunctions, which are represented as sum of a single-layer potential and of a double-layer potential involving the Green's function, are considerably different from that in [9].We would here like to find an efficient and accurate method, different from what we have presented in [2].Powerful techniques from the theory of meromorphic operator-valued functions and careful asymptotic analysis of integral kernels are combined for solving this problem.Similar results can be obtained when considering the homogeneous Neumann boundary condition with only minor modifications of the arguments and techniques presented in this work.We just replace the Dirichlet Green's function by the Neumann Green's function in the integral representation and in a way completely similar, we establish our asymptotic formulas.
In this paper, we deal with the asymptotics of eigenvalues and eigenfunctions associated to the following eigenvalue problem: It is well known that the operator −∆ on L 2 (Ω ) with domain H 2 (Ω ) ∩ H 1 0 (Ω ) is selfadjoint with compact resolvent.Consequently, its spectrum only consists of isolated, real, and positive eigenvalues with finite multiplicity, and there are corresponding eigenfunctions which build an orthonormal basis of L 2 (Ω ).
Classical regularity results and the previous theorem imply that the eigenfunctions associated to the λ 0 -group of eigenvalues are separately analytic in the small parameter and the spatial variable x.Using an integral equation technique, we will also establish the joint analytic dependence of these functions with respect to (x, ).As it is well known, joint analyticity does not follow from separate real analyticity properties.

Integral equation method
The use of integral equations is a convenient tool for a number of investigations [5].We now develop a boundary integral formulation for solving the eigenvalue problem (2.4).We use this method to characterize the eigenvalue and the normal derivative of the eigenfunction as characteristic value and root function of some operator-valued function.This characterization is the key point in deriving our results and asymptotic formulae.
Let u be the solution to the Helmholtz equation: We begin by defining the outgoing Green's function G(x, y) as the solution of with the radiation condition as |x| → +∞: In fact, G is explicitly given as where H (1)  0 (z) is the Hankel function of the first kind and of order zero [1].Its singularity has the form Of course, the above equations does not hold up to the boundary of Ω , but if we take the limit as x → ∂Ω , we get from, for instance, [5,12]  We introduce the following operators, called single-and double-layer potentials, respectively: For g ∈ Ꮿ ∞ (∂Ω ), or even g ∈ L 1 (∂Ω ), the functions Sl(λ)g and Dl(λ)g are well defined and smooth for x ∈ R 2 \ ∂Ω .Now define the following operators:  For such g and every x ∈ ∂Ω , we denote by g + (x) and g − (x) the limits of g(y) as y → x, from y ∈ Ω and y ∈ R 2 \ Ω , respectively, when these limits exist.It is a wellknown classical result that, for x ∈ ∂Ω , where S(λ) is pseudodifferential operator of order −1.Throughout this paper, we use for simplicity the notation Using a change of variables and integral equations, the following important result immediately holds from Taylor [16, page 184].
]0,1[) be defined as follows: .13)Abdessatar Khelifi 1205 Then the operator-valued function L (λ) is Fredholm analytic with index 0 in C \ iR − .Moreover, L −1 (λ) is a meromorphic function and its poles are in (z) ≤ 0 .(z) means the imaginary part of z and (z) is the real part.
From now on we will focus our attention on solving the eigenvalue problem (2.4).

Joint analyticity of kernel.
Based on the result found in [3] and on the argument of Millar [11], we will now prove the following proposition.The following result will be useful in Sections 4 and 5.
Proposition 3.2.There exist a constant η > 0 and a complex neighborhood ᐂ of 0 such that for every function ϕ(t, where D r0 (λ 0 ) is a disc of center λ 0 and radius r 0 .
Proof.The proof of this proposition heavily relies on the work of Bruno and Reitich [3] where they expressed the kernel of L (λ) in the following form: for (t,s, ,λ , where h is an analytic function in C, and k is analytic in .
The central difficulty to prove the analytic property of the operator L comes from the spatial regularity of its kernel.We show that logarithmic singularity of the kernel of L (λ) is independent of the parameter .
We introduce Using a change of variables, this function can be rewritten as follows: An integration by parts leads to where Clearly, Φ(t, ,λ) can be extended to a complex analytic function in C × ᐂ × D r0 (λ 0 ).Furthermore, from the following identity we deduce the desired result applying (3.14).

Reduction to a characteristic value problem
In this section, we first recall some definitions and notations from [6,10].Using boundary integral equations, we reduce the problem (2.4) to the existence and the distribution of the characteristic values of selfadjoint integral operators in the complex plane.
Suppose that λ 0 is a characteristic value of the function Ꮽ(λ) and φ(λ) is a root function satisfying (ii).Then there exists a number m(φ) ≥ 1 and a vector-valued function The number m(φ) is called the multiplicity of the root function φ(λ).Let φ 0 be an eigenvector corresponding to λ 0 and let Then by rank of φ 0 we mean rank(φ 0 ) = max (φ 0 ).Suppose that n = dimKer Ꮽ(λ 0 ) < +∞ and that the ranks of all vectors in KerᏭ(λ 0 ) are finite.A system of eigenvectors φ j 0 , j = 1,...,n, is called a canonical system of eigenvectors of Ꮽ(λ) associated to λ 0 if the ranks possess the following property: rank(φ j 0 ) is the maximum of the ranks of all eigenvectors Abdessatar Khelifi 1207 in some direct complement in dimKerᏭ(λ 0 ) of the linear span of the vectors φ 1 0 ,...,φ j−1 0 .Let r j = rank(φ j 0 ).Then (r j ) j uniquely determines the function Ꮽ(λ).We call the null multiplicity of the characteristic value λ 0 of Ꮽ(λ).If λ 0 is not a characteristic value of Ꮽ(λ), we put N(Ꮽ(λ 0 )) = 0. Suppose that Ꮽ −1 (λ) exists and is holomorphic in some neighborhood of λ 0 , except possibly at λ 0 .Then the number is called the multiplicity of the characteristic value λ 0 of Ꮽ(λ).Suppose that λ 1 is a pole of the operator-valued function.The Laurent expansion of Ꮽ(λ) in λ 1 is given by If in the last expression the operators A − j , j = 1,...,s, are finite dimensional, then Ꮽ(λ) is called finitely meromorphic at λ 1 .
The operator-valued function Ꮽ(λ) is said to be of Fredholm type at the point λ 1 if the operator A 0 in the last expansion is a Fredholm operator.
A system {y ( j) l : 1 ≤ j ≤ h, 0 ≤ l ≤ m j } is called a canonical system of eigenvectors and associated vectors (CSEAV) of Ꮽ at λ if {y ( j) 0 ).We recall the generalization of Steinberg's theorem [15].Theorem 4.1.Suppose that Ꮽ is an operator-valued function which is finitely meromorphic and of Fredholm type in the domain D r0 (λ 0 ).If the operator Ꮽ(λ) is invertible at one point of D r0 , then Ꮽ(λ) has a bounded inverse for all λ ∈ D r0 , except possibly for certain isolated points.
We will reduce, as mentioned in the introduction, the eigenvalue problem to some characteristic problem.From Proposition 3.1, we know that if λ 2 0 is an eigenvalue of (2.4), then λ 0 is a characteristic value of L 0 (λ).Moreover, for r 0 small enough, the function , where D r0 (λ 0 ) the disc of center λ 0 and radius r 0 , and λ 0 is its unique pole in D r0 .
Our main results in this section are summarized in the following theorem.
Proof.We first recall that m is the geometric multiplicity of λ 2 0 as an eigenvalue of the eigenvalue problem (2.4).Proposition 3.2 implies that L (λ) is an analytic operator-valued function with respect to ( ,λ) ∈ R × D r0 (λ 0 ).Then there exists a constant 1 (r 0 , 0 ) > 0 such that for any lying in ] − 1 , 1 [, the following holds: From the generalized Rouché's theorem and the results of Gohberg and Sigal [6], we deduce that L (λ) is invertible on ∂D r0 and has τ characteristic values λ i ( ) i in D r0 (λ 0 ) which are (obviously) the poles of the function L −1 (λ) in the disc D r0 (λ 0 ).Thus, with the definitions introduced earlier, the following holds: Using Theorem 2.1 and Proposition 3.2, it can now be easily seen that the set of these characteristic values build precisely the λ 0 -group of eigenvalues introduced in the last section, that is, ( λ i ( )) i ≡ (λ i ( )) i .Hence, they are analytic in the variable .Notice that in general we have M(L 0 (λ 0 )) ≥ dimKer L 0 (λ 0 ) = m.But Lemma 4.2 implies M(L 0 (λ 0 )) = m.Furthermore, we have the Laurent expansion where 0 = τ0 i=1 i (0) and 0 (λ) is a holomorphic function.From the decomposition (4.14), we obtain It follows that 0 : KerL * 0 (λ 0 ) → Ker L 0 (λ 0 ).But from the properties of the Green's function G(x, y), we know that Ker L * 0 λ 0 = Ker L 0 λ 0 .(4.16) Note that using similar arguments, we can prove that (λ i ( )) i are also simple poles of L −1 (λ) and M(L (λ i ( ))) = dimKer L (λ i ( )), for i = 1,...,m.Moreover, we have where (λ) is a holomorphic function which completes the proof of the theorem.
Let λ 0 be a characteristic value of L 0 (λ).From Keldys's theorem which is simplified in [10, page 462], there exist {φ i 0 : 1 ≤ i ≤ m} CSEAV of L 0 at λ 0 and {ψ i 0 : 0 such that the operator is well defined.Analogously, by the result of Reinhard and Möller which is due to Keldyš [8], for each characteristic value λ i ( ) (1 ≤ i ≤ m), there exist {φ i, j ( ) : is well defined.Introduce the operator The fact that (φ i j ) i j and (ψ i j ) i j are CSEAV implies that φ = 0 and, therefore, yields the desired result.
(2) It suffices to prove the result only for i = 1 and j = 2. From relation (4.16), we deduce that Then, Using the last relation, the relation becomes The operator-valued function L (λ 1 ( )) is Fredholm of index 0 which completes the proof.
Our strategy now is to investigate the properties of the eigenelements corresponding to the operators A 0 and A .Let (µ j 0 ) 1≤ j≤h be the family of eigenvalues of the operator A 0 with multiplicity m j each.Using the generalisation of Theorem 2.1, see [7,13], we know that there exists 2 = 2 ( 1 ) > 0 such that for | | < 2 and for j ∈ {1, ...,h}, the µ j 0group consists of m j eigenvalues of A( ), µ j,l ( ), l = 1,...,m j (repeated according to their multiplicity).
Let 3 = inf( 1 , 2 ).For | | < 3 , the following projector is well defined: where for 1 ≤ j ≤ h and for 1 ≤ l ≤ m j , the family (q ( j) l,s ( )) 1≤s≤mjl denotes the orthogonal family of eigenfunctions corresponding to the eigenvalues µ j,l ( ).For = 0, we have 1212 The integral equation methods for eigenvalue problems where the family (q ( j) l (0)) 1≤l≤mj is the orthogonal family of eigenfunctions corresponding to the eigenvalue µ j 0 .Now it seems natural, from the previous results, that for all j = 1,...,h, the family (q ( j) l (0)) 1≤l≤mj is m j -characteristic functions of L 0 (λ 0 ) and for all l = 1,...,m j , the family (q ( j) l,s ( )) 1≤s≤mj,l is m j,l -characteristic functions of L (λ i ( )) and h j=1 m j = m.We set

Analyticity and asymptotic expansion
This section is devoted to the study of the asymptotics of the characteristic elements and, therefore, the asymptotics of the eigenelements of (2.4) when the parameter goes to zero.We will give a method in order to calculate the coefficients of the expansions of the eigenelements in a neighborhood of zero when the eigenvalue λ 2 0 of −∆ is not simple.Our strategy, for deriving asymptotic expansions of the perturbations in a multiple eigenvalue λ 0 with multiplicity m that are due to boundary deformations, relies on finding the analyticity and complete asymptotic expansions of the eigenelements of A( ).The following holds.
Proof.(1) This property is clear by recalling that the operator A( ) is holomorphic and has the expansion A( ) = A(0) + A( ), where the operator A( ) is holomorphic with respect to and goes to 0 as → 0 and by considering, for ∈] − 3 , 3 [, the Neumann series which converges uniformly with respect to µ in a neighborhood of µ j,i .
Proof.(1) Define the matrix: Ᏸ = (d sp ) sp ; the coefficients d sp are given by The analyticity of the operator-valued function R( ) with respect to ∈] − 3 , 3 [ guarantees the analyticity of Ᏸ .The inner product of (5.5) by q which implies that (5.17) We now verify that the function φ is jointly analytic in (t, ).Through relation (5.17), we deduce that The analyticity of φ 0 (t) in t is a classical result.Then we deduce the result by using the analyticity of the matrix B −1 in , which is obvious from (5.16), and the fact that the function R( )φ 0 (t) is jointly analytic in (t, ).From relation (5.7), we then deduce the analyticity of the functions q ( j) ( ) follows for all j = 1,...,m.The analyticity of the matrix operator B allows writing, in a neighborhood of 0, the expansion where we have considered φ 0 = q 0 and R( The jth component of the vector B k q n−k is given by and then relation (5.22) becomes (5.24) The relation (5.26) Taking the inner product with q (s) 0 , s = 1,...,m, we have If we replace this equality in (5.26), we find (5.29) 1216 The integral equation methods for eigenvalue problems We recall that n q (i) 0 . (5.30) Then relation (5.29) becomes 0 , q (i) k q (s) 0 , q (i) n−k − δ s j − q (s) 0 ,R n q 0 , q (i) k q (i) n−k + q ( j) 0 + R n q ( j) 0 . (5.31) From the previous relation and the properties of the operator I − P 0 , we deduce 0 , q (i) k q (i) n−k − R n q ( j) 0 = 0. (5.32) In other words, it is obvious that R n q ( j) 0 / ∈ Ker(L 0 (λ 0 )) for all j = 1,...,m.Then 0 , q (i) k q (i) n−k − R n q ( j) 0 / ∈ Ker L 0 λ 0 .
(5.33) Thus, relation (5.32) means that 0 , q (i) k q (i) n−k − R n q ( j) 0 = 0. (5.34) (2) In order to find out the coefficients in (5.11), our method is based on expanding the expression L (λ ( j) ( )) for near zero.To handle this, we have to expand, first, the operator-valued function L (λ) around = 0 and so the resulting expression around λ = λ 0 .