THE INTEGRAL EQUATION METHODS FOR THE
PERTURBED HELMHOLTZ EIGENVALUE PROBLEMS
ABDESSATAR KHELIFIReceived 10 May 2004 and in revised form 31 January 2005
It is well known that the main difficulty in solving eigenvalue problems under shape de-formation relates to the continuation of multiple eigenvalues of the unperturbed config-uration. 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 op-erator under boundary variations of the domain. Using surface potentials, we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions.
1. 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 expan-sions 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 multi-ple 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 Rouch´e’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.
Copyright©2005 Hindawi Publishing Corporation
Our analysis and uniform asymptotic formulas of the eigenfunctions, which are rep-resented 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]. Power-ful techniques from the theory of meromorphic operator-valued functions and carePower-ful asymptotic analysis of integral kernels are combined for solving this problem. Similar results can be obtained when considering the homogeneous Neumann boundary condi-tion with only minor modificacondi-tions 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.
2. Problem formulation
Letγ(t) : [0, 1]→R2andβ(t) : [0, 1]→R2be two analytic, 1-periodic functions satisfying |γ(t)|> Cfor allt∈[0, 1], whereCis a positive constant. We introduce
γ(t)=γ(t) +β(t), ∈R. (2.1)
With this definition, (t,)→γ(t) is an analytic function on [0, 1]×R, 1-periodic in the variablet, and satisfies
γ
(t)>C2 (2.2)
for any such that 2|||β(t)|Ꮿ0([0,1])< C. We consider the bounded domainΩ inR2
with boundary∂Ωparameterized by the functionγ(t):
∂Ω=γ(t),t∈[0, 1]. (2.3)
The outward unit normal to∂Ωis denoted byν.
In this paper, we deal with the asymptotics of eigenvalues and eigenfunctions associ-ated to the following eigenvalue problem:
∆u() +λ2()u()=0 inΩ, u()=0 on∂Ω. (2.4)
It is well known that the operator−∆onL2(Ω
) with domainH2(Ω)∩H01(Ω) is selfadjoint with compact resolvent. Consequently, its spectrum only consists of isolated, real, and positive eigenvalues with finite multiplicity, and there are corresponding eigen-functions which build an orthonormal basis ofL2(Ω
).
Letλ20>0 denote an eigenvalue of the eigenvalue problem (2.4) for=0 with geo-metric multiplicitym≥1. There exists a small constantr0>0 such thatλ20is the unique eigenvalue of (2.4) for=0 in the set{λ2,λ∈D
r0(λ0)}, whereDr0(λ0) is a disc of center λ0and radiusr0. We define theλ0-group as the whole perturbed eigenvalues of (2.4) for
analytic functions with respect tosatisfyingλi(0)=λ0,i=1,. . .,m.The normalized eigen-functions associated to theλ0-group of eigenvalues are analytic with respect to and their values at0aremlinearly independent solutions of the unperturbed eigenvalue problem.
Classical regularity results and the previous theorem imply that the eigenfunctions associated to theλ0-group of eigenvalues are separately analytic in the small parameter and the spatial variablex. 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.
3. 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.
Letube the solution to the Helmholtz equation:
∆u+λ2u=0, inR2. (3.1)
We begin by defining the outgoing Green’s functionG(x,y) as the solution of ∆xG(x,y) +λ2G(x,y)= −δ
y(x), inR2, (3.2)
with the radiation condition as|x| →+∞:
∂∂G|x|−iλG=O
1 |x|
. (3.3)
In fact,Gis explicitly given as
G(x,y)= i 4H
(1) 0
λ|x−y|, (3.4)
whereH0(1)(z) is the Hankel function of the first kind and of order zero [1]. Its singularity has the form
G(x,y)∼ 1
2πlog|x−y|+··· asx−→y. (3.5) 3.1. Preliminary results. Consider (3.1) for the functionuin the exterior ofΩ, multiply by the Green’s functionGand integrate by parts, we get that forx∈R2\Ω
,
u(x)=
∂Ω
∂u
∂ν
+
(y)G(x,y)−u(y) ∂G
∂ν(y)
+ (x,y)
dσ(y). (3.6)
The jump condition for∂u/∂νon∂Ωyields
u(x)= −
∂Ω
∂G ∂ν(y)
+
(x,y)u(y)dσ(y) +
∂Ω
G(x,y)∂u
∂ν
Of course, the above equations does not hold up to the boundary ofΩ, but if we take the limit asx→∂Ω, we get from, for instance, [5,12] that
1 2u
∂Ω (x)= −
∂Ω
∂G ∂ν(y)
+(
x,y)u(y)dσ(y)
+
∂Ω
G(x,y)∂u
∂ν
−(y)dσ(y) forx∈∂Ω.
(3.8)
We introduce the following operators, called single- and double-layer potentials, re-spectively:
Sl(λ)g(x)=
∂Ω
G(x,y)g(y)dσ(y), x∈R2\∂Ω ,
Dl(λ)g(x)=
∂Ω
∂ν(y)G(x,y)g(y)dσ(y), x∈R2\∂Ω.
(3.9)
Forg∈Ꮿ∞(∂Ω
), or eveng∈L1(∂Ω), the functions Sl(λ)gand Dl(λ)gare well de-fined and smooth forx∈R2\∂Ω
. Now define the following operators:
S(λ) :H−1/2∂Ω
−→H1/2∂Ω, D(λ) :H1/2∂Ω−→H1/2∂Ω, (3.10) where
S(λ) :g−→
∂Ω
G(·,y)g(y)dσ(y),
D(λ) :g−→
∂Ω
∂G ∂ν(y)
+(·,y)g(y)dσ(y).
(3.11)
For suchg and everyx∈∂Ω, we denote byg+(x) andg−(x) the limits of g(y) as
y→x, from y∈Ω and y∈R2\Ω, respectively, when these limits exist. It is a well-known classical result that, forx∈∂Ω,
Sl(λ)g+(x)=Sl(λ)g−(x)=S(λ)g(x),
Dl(λ)g±(x)= ±1
2g(x) +D(λ)g(x),
(3.12)
whereS(λ) is pseudodifferential operator of order−1.
Throughout this paper, we use for simplicity the notationHs(]0, 1[)=Hs(R/]0, 1[), for s∈R, where Hs(R/]0, 1[) denotes the classical Sobolev Hs-space on the quotient R/]0, 1[.
Using a change of variables and integral equations, the following important result im-mediately holds from Taylor [16, page 184].
Proposition3.1. LetL(λ) :H−1/2(]0, 1[)→H1/2(]0, 1[)be defined as follows:
L(λ)f(t)=S(λ)fγ−1γ(t)
= i 4
1
0H (1) 0
λγ(t)−γ(s)γ(s)f(s)ds for f ∈H−1/2
Then the operator-valued functionL(λ)is Fredholm analytic with index0inC\iR−. More-over,L−1
(λ)is a meromorphic function and its poles are in
(z)≤0. (z) means the imaginary part ofzand(z)is the real part.
From now on we will focus our attention on solving the eigenvalue problem (2.4).
3.2. 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 Sections4and5.
Proposition3.2. There exist a constantη >0and a complex neighborhoodᐂof0such that for every functionϕ(t,)∈H−1/2(]0, 1[)analytic in(t,)∈| (t)| ≤η×ᐂ, the function
L(λ)ϕ(t,)∈H1/2(]0, 1[)is analytic with respect to(t,,λ)in| (t)| ≤η×ᐂ×Dr0(λ0),
whereDr0(λ0)is a disc of centerλ0and radiusr0.
Proof. The proof of this proposition heavily relies on the work of Bruno and Reitich [3] where they expressed the kernel ofL(λ) in the following form:
i
4H (1) 0
λγ(t)−γ(s)γ(s)
= − 1
p=−1
log|t−s−p|hλ2γ(t)−γ
(s)2γ(s)+k(t,s,,λ),
(3.14)
for (t,s,,λ)∈∩ { (t)=0} ∩ { (s)=0} ∩ { ()=0},s /∈ {t,t+ 1,t−1}, wherehis an analytic function inC, andkis analytic in. Here= {| (t)| ≤η;| (s)| ≤η;|| ≤
δ;λ∈Dr0(λ0);−ρ≤ (t)≤1 +ρ;−ρ≤ (s)≤1 +ρ}, for the positive numbers ρ,r0,η,
andδ.
The central difficulty to prove the analytic property of the operatorLcomes from the spatial regularity of its kernel. We show that logarithmic singularity of the kernel ofL(λ) is independent of the parameter.
We introduce
Φ(t,,λ)= −
1
0 1
p=−1
log|t−s−p|hλ2γ(t)−γ(s)2
ϕ(s,)γ(s)ds. (3.15)
Using a change of variables, this function can be rewritten as follows:
Φ(t,,λ)= −
2
−1log|t−s|h
λ2γ
(t)−γ(s)2
ϕ(s,)γ(s)ds. (3.16)
An integration by parts leads to
Φ(t,,λ)=log|t−2|ψ(t, 2,,λ)
−log|t+ 1|ψ(t,−1,,λ)−
2
−1
ψ(t,s,,λ) |t−s| ds,
where
ψ(t,s,,λ)= −
s
th
λ2γ
(t)−γ(z)2
ϕ(z,)γ(z)dz. (3.18)
Clearly,Φ(t,,λ) can be extended to a complex analytic function inC×ᐂ×Dr0(λ0).
Furthermore, from the following identity
L(λ)ϕ(t,)=Φ(t,,λ) +
1
0k(t,s,,λ)ϕ(s,)ds, (3.19)
we deduce the desired result applying (3.14).
4. Reduction to a characteristic value problem
In this section, we first recall some definitions and notations from [6,10]. Using bound-ary 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.
4.1. Notations and definitions. LetᏱandᏲbe two Banach spaces and letᏸ(Ᏹ,Ᏺ) be the algebra of all bounded-valued function acting fromᏱintoᏲ.ᏱandᏲdenote the dual spaces ofᏱandᏲ, respectively.
Letλ0be a fixed complex value inC. We denote byᏭ(λ) an operator-valued function acting fromDr0(λ0) intoᏸ(Ᏹ,Ᏺ), whereDr0(λ0) is a disc of centerλ0and radiusr0>0. λ0is called acharacteristicvalue ofᏭ(λ) if
(i)Ꮽ(λ) is holomorphic in some neighborhood ofλ0, except possibly atλ0, (ii) there exists a vector-valued functionφ(λ):Dr0(λ0)→Ᏼholomorphic atλ0and
verifyingφ(λ0)=0, such thatᏭ(λ)φ(λ) is holomorphic atλ0and vanishes at this point.
φ(λ) is called aroot function ofᏭ(λ) associated to λ0 and the vectorφ0=φ(λ0) is called aneigenvector. The closure of the linear set of eigenvectors corresponding toλ0 is denoted by KerᏭ(λ0).
Suppose thatλ0is a characteristic value of the functionᏭ(λ) andφ(λ) is a root func-tion satisfying (ii). Then there exists a numberm(φ)≥1 and a vector-valued function
ψ(λ):Dr0(λ0)→Ᏼholomorphic such that
Ꮽ(λ)φ(λ)=λ−λ0
m(φ)
ψ(λ), ψ(λ0)=0. (4.1)
The numberm(φ) is called themultiplicityof the root functionφ(λ). Letφ0be an eigen-vector corresponding toλ0and let
φ0
=m(φ);φ(λ) is a root function suchφλ0
=φ0
. (4.2)
Then byrankofφ0we mean rank(φ0)=max(φ0). Suppose thatn=dim KerᏭ(λ0)< +∞and that the ranks of all vectors in KerᏭ(λ0) are finite. A system of eigenvectorsφ0j,
in some direct complement in dim KerᏭ(λ0) of the linear span of the vectorsφ01,. . .,φ j−1 0 . Letrj=rank(φ0j). Then (rj)juniquely determines the functionᏭ(λ). We call
NᏭλ0
= n
j=1
rj (4.3)
thenull multiplicity of the characteristicvalueλ0ofᏭ(λ).
Ifλ0is not a characteristic value ofᏭ(λ), we putN(Ꮽ(λ0))=0.
Suppose thatᏭ−1(λ) exists and is holomorphic in some neighborhood ofλ
0, except possibly atλ0. Then the number
MᏭλ0
=NᏭλ0
−NᏭ−1λ 0
(4.4)
is called themultiplicityof the characteristic valueλ0ofᏭ(λ). Suppose thatλ1is a pole of the operator-valued function. The Laurent expansion ofᏭ(λ) inλ1is given by
Ꮽ(λ)= j≥−s
λ−λ1
j
Aj. (4.5)
If in the last expression the operatorsA−j, j=1,. . .,s, are finite dimensional, thenᏭ(λ) is calledfinitely meromorphicatλ1.
The operator-valued functionᏭ(λ) is said to be of Fredholm type at the pointλ1if the operatorA0in the last expansion is a Fredholm operator.
IfᏭ(λ) is holomorphic at the pointλ0and the operatorᏭ(λ0) is invertible, thenλ0is called aregular pointofᏭ(λ).
We set
(y⊗v)(w) :=(w,v)y (w∈Ᏺ),
(v⊗y)(u) :=(y,u)v u∈Ᏹ (4.6)
for y∈Ᏹand v∈Ᏺ. Note that y⊗v∈ᏸ(Ᏺ,Ᏹ),v⊗y∈ᏸ(Ᏹ,Ᏺ), and (y⊗v)∗=
v⊗y. An ordered set{y0,y1,. . .,yh} ⊂Ᏹis called a chain of eigenvectors and associated vectors (CEAV) ofᏭatλif
Y=: h
l=0
(· −λ)ly
l (4.7)
is a root function ofᏭatλwithm(Y)≥h+ 1. Conversely, ifY is a root function ofᏭat
λandm(Y)≥h+ 1, then the function
Y(µ)=: ∞
l=0
(µ−λ)lyl, (4.8)
A system{yl(j): 1≤ j≤h, 0≤l≤mj}is called a canonical system of eigenvectors and associated vectors (CSEAV) ofᏭatλ if{y(0j): 1≤j≤h}is a basis of Ker(Ꮽ(λ)), {y0(j),y
(j) 1 ,. . .,y
(j)
m j}is a CEAV ofᏭatλ(j=1,. . .,h),m j=sup{m(y) :y∈Ker(Ꮽ(λ))\ Span{y(k)0 :k < j}}; (1≤j≤h). Obviouslym j=m(y
(j) 0 ). We recall the generalization of Steinberg’s theorem [15].
Theorem4.1. Suppose thatᏭis an operator-valued function which is finitely meromorphic and of Fredholm type in the domainDr0(λ0). If the operatorᏭ(λ)is invertible at one point
ofDr0, thenᏭ(λ)has a bounded inverse for allλ∈Dr0, except possibly for certain isolated
points.
We will reduce, as mentioned in the introduction, the eigenvalue problem to some characteristic problem. FromProposition 3.1, we know that ifλ2
0is an eigenvalue of (2.4), thenλ0 is a characteristic value ofL0(λ). Moreover, for r0 small enough, the function
L−01(λ) is meromorphic inDr0(λ0), whereDr0(λ0) the disc of centerλ0and radiusr0, and λ0is its unique pole inDr0.
We begin by establishing the following lemma which characterises the eigenvalues of (2.4) if=0.
Lemma4.2. Any eigenvalue of the problem (2.4) is a simple pole ofL−01(λ).
Proof. Letλ20be an eigenvalue of (2.4). Letu0be an associated eigenfunction normalized inL2(Ω0). Using an integration by parts, we know that∂
ν0u0 is in fact a characteristic
function of the operator-valued functionλ→L0(λ) corresponding to the characteristic valueλ0, that is,L0(λ0)∂ν0u0=0 on∂Ω0. We defineφ(λ) as a root function ofL0(λ)
corre-sponding to (λ0;∂ν0u0); it is holomorphic inDr0(λ0),φ(λ0)=∂ν0u0and satisfies the
iden-tityL0(λ)φ(λ)|λ=λ0=0. The multiplicity ofφ(λ) is the order ofλ0as a zero ofL0(λ)φ(λ).
But it is well known that the order ofλ0as a pole ofL−01(λ) is precisely the maximum of the ranks of the eigenvectors in KerL0(λ0). Then it suffices to show that the rank of an arbitrary eigenvector is equal to one.
We writeL0(λ)φ(λ)=(λ2−λ20)ψ(λ), where ψ(λ) is a holomorphic function inH1/2 (∂Ω0). Forλ∈Dr0(λ0), we denote byu(λ) the unique solution of (∆+λ
2)u(λ)=0 inΩ0 with the boundary conditionu(λ)=ψ(λ)γ−1on∂Ω0. Using a trivial integration by parts overΩ0, we find that
Ω0
u(λ)u0dx=
λ2−λ2
0
−1 ∂Ω0
u(λ)∂ν0u0dσ(x)=
∂Ω0
ψ(λ)∂ν0u0dσ(x), (4.9)
which immediately implies that
∂Ω0 ψλ0
∂ν0u0dσ(x)=
Ω0 u02
dx=1, (4.10)
sinceΩ0u(λ)u0dxis holomorphic inDr0(λ0). Therefore,|ψ(λ0)|
2 L2(∂Ω
0)=0 and thus the
functionψ(λ0) is nontrivial.
Theorem 4.3. There exists a positive constant1=1(r0,0)such that 1≤0 and for ||<1, the operator-valued functionλ→L(λ)has exactlymcharacteristic values(λi())i (counted according to their multiplicity) inDr0(λ0). These characteristic values build theλ0
-group associated to the perturbed eigenvalue problem (2.4), and are analytic with respect to
in]−1,1[. They satisfyλi(0)=λ0, fori=1,. . .,m. Moreover, if(λi())τi=1denotes the set of distinct values of(λi())mi=1, then the following assertions hold:
τ
i=1
MLλi()=m,
L−1 (λ)=
τ
i=1
λ−λi()−1i() +(λ),
(4.11)
wherei() : Ker(L(λi()))→Ker(L(λi()))and(λ)is a holomorphic function with respect to(,λ)∈]−0,0[×Dr0(λ0).
Proof. We first recall thatmis the geometric multiplicity ofλ2
0as an eigenvalue of the eigenvalue problem (2.4).Proposition 3.2implies thatL(λ) is an analytic operator-val-ued function with respect to (,λ)∈R×Dr0(λ0). Then there exists a constant1(r0,0)>
0 such that for anylying in ]−1,1[, the following holds:
L(λ)−L0(λ) L−1
0 (λ)ᏸ(H1/2,H1/2)<1, ∀λ∈∂Dr0
λ0. (4.12)
From the generalized Rouch´e’s theorem and the results of Gohberg and Sigal [6], we deduce thatL(λ) is invertible on∂Dr0and hasτcharacteristic valuesλi()
iinDr0(λ0)
which are (obviously) the poles of the functionL−1
(λ) in the discDr0(λ0). Thus, with the
definitions introduced earlier, the following holds:
τ
i=1
MLλi()=ML0
λ0
. (4.13)
Using Theorem 2.1andProposition 3.2, it can now be easily seen that the set of these characteristic values build precisely theλ0-group of eigenvalues introduced in the last section, that is, (λi())i≡(λi())i. Hence, they are analytic in the variable. Notice that in general we haveM(L0(λ0))≥dim KerL0(λ0)=m. ButLemma 4.2impliesM(L0(λ0))=
m. Furthermore, we have the Laurent expansion
L−1 0 (λ)=
λ−λ0
−1
0+0(λ), (4.14)
where0=
τ0
i=1i(0) and0(λ) is a holomorphic function. From the decomposition (4.14), we obtain
0L0
λ0
=0 inᏸH−1/2]0, 1[,H−1/2]0, 1[,
L0
λ0
0=0 inᏸ
It follows that0: KerL∗0(λ0)→KerL0(λ0). But from the properties of the Green’s func-tionG(x,y), we know that
KerL∗0
λ0=KerL0λ0. (4.16)
Note that using similar arguments, we can prove that (λi())i are also simple poles of
L−1
(λ) andM(L(λi()))=dim KerL(λi()), fori=1,. . .,m. Moreover, we have
L−1(λ)= τ
i=1
λ−λi()
−1
i() +(λ),
i() : Ker
Lλi()
−→KerLλi()
,
(4.17)
where(λ) is a holomorphic function which completes the proof of the theorem. Letλ0be a characteristic value ofL0(λ). From Keldys’s theorem which is simplified in [10, page 462], there exist{φ0i: 1≤i≤m}CSEAV ofL0atλ0and{ψ0i: 1≤i≤m}CSEAV ofL∗0 such that the operator
A0= 1 2iπ
|λ−λ0|=ρ
L0(λ)
−1
dλ= m
i=1
φi0⊗ψ0i (4.18)
is well defined.
Analogously, by the result of Reinhard and M¨oller which is due to Keldyˇs[8], for each characteristic valueλi() (1≤i≤m), there exist{φi,j() : 1≤i≤m, 1≤j≤mi}CSEAV ofLatλi() and{ψi,j() : 1≤i≤m, 1≤j≤mi}CSEAV ofL∗ such that the operator
Ai()=21iπ
|λ−λi()|=ρ
L(λ)−1dλ= mi
j=1
φi,j()⊗ψi,j() (4.19)
is well defined. Introduce the operator
A()= m
i=1
Ai(), for||<1. (4.20)
Based onTheorem 4.3, [15] and on relation (4.20), one can see that the operatorA() is selfadjoint and holomorphic function with respect to∈]−1,1[. It is quite easy to see thatA0=A(=0) and the following results hold.
Proposition4.4. For||<1and fori∈ {1,. . .,τ}, (1)the operatorAi()satisfies
Ai()(Ker(L(λi())))⊥≡0; (4.21)
(2)ifλi()andλj()are two characteristic values ofL(λ)withi=j, then
KerLλi()
⊥KerLλj()
Proof. (1) Letφ∈H−1/2such thatAi()φ=0, then
mi
j=1
φ,ψi,j()
φi,j()=0. (4.23)
The fact that (φi j)i j and (ψi j)i j are CSEAV implies thatφ=0 and, therefore, yields the desired result.
(2) It suffices to prove the result only fori=1 and j=2. From relation (4.16), we deduce that
Lλ1()
KerLλ2()
⊂KerLλ2()
. (4.24)
Then,
Lλ1()
KerLλ2()
⊂KerLλ2()
∩KerLλ1()
⊥
. (4.25)
Using the last relation, the relation
KerLλ2()
=Lλ1()
KerLλ2()
+I−Lλ1()
KerLλ2()
(4.26)
becomes
I−Lλ1()KerLλ2()=0. (4.27)
The operator-valued function L(λ1()) is Fredholm of index 0 which completes the
proof.
Our strategy now is to investigate the properties of the eigenelements corresponding to the operatorsA0 andA. Let (µ0j)1≤j≤h be the family of eigenvalues of the operator
A0 with multiplicity mj each. Using the generalisation ofTheorem 2.1, see [7,13], we know that there exists2=2(1)>0 such that for||<2and forj∈ {1,. . .,h}, theµ0j -group consists ofmjeigenvalues ofA(),µj,l(),l=1,. . .,mj(repeated according to their multiplicity).
Let3=inf(1,2). For||<3, the following projector is well defined:
Pj()=21iπ
|µ−µj,i()|=ρ1
µ−A()−1dµ= mj
l=1 mj,l
s=1
q(l,sj)()⊗q(j)l,s(), (4.28)
where for 1≤j≤hand for 1≤l≤mj, the family (q (j)
l,s())1≤s≤mjldenotes the orthogonal
family of eigenfunctions corresponding to the eigenvaluesµj,l(). For=0, we have
Pj(0)=21iπ
|µ−µ0j|=ρ2
µ−A(0)−1dµ= mj
l=1
where the family (q(lj)(0))1≤l≤mj is the orthogonal family of eigenfunctions
correspond-ing to the eigenvalueµ0j. Now it seems natural, from the previous results, that for all
j=1,. . .,h, the family (q(j)l (0))1≤l≤mj ismj-characteristic functions ofL0(λ0) and for all l=1,. . .,mj, the family (q
(j)
l,s())1≤s≤mj,l ismj,l-characteristic functions ofL(λi()) and
h
j=1mj=m. We set
P()= h
j=1
Pj(), for||<3, P(0)= h
j=1
Pj(0). (4.30)
5. 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 eigenvalueλ2
0of−∆is not simple. Our strategy, for deriving asymptotic expansions of the perturbations in a multiple eigenvalue
λ0with multiplicitymthat are due to boundary deformations, relies on finding the ana-lyticity and complete asymptotic expansions of the eigenelements ofA(). The following holds.
Proposition5.1. For||<3,
(1)the operatorP()is holomorphic for∈]−3,3[andP()=P(0) +R(), where
R()is holomorphic with respect to, (2)P()=m
j=1q(j)()⊗q(j)(), where (q(j)())1≤j≤m denotes an orthonormal basis ofKerL(λi()). Also,P(0)=
m
j=1q(j)(0)⊗q(j)(0), where(q(j)(0))1≤j≤m is an or-thonormal basis ofKerL0(λ0).
Proof. (1) This property is clear by recalling that the operatorA() is holomorphic and has the expansion A()=A(0) +A(), where the operator A() is holomorphic with respect toand goes to 0 as→0 and by considering, for∈]−3,3[, the Neumann series
µ−A()−1=µ−A(0)−1+ ∞
p=1
µ−A(0)−1A()µ−A(0)−1p, (5.1)
which converges uniformly with respect toµin a neighborhood ofµj,i.
(2) As said at the end ofSection 4, the elements of the family (q(j)l (0))1≤j≤h,1≤l≤mjare
characteristic functions ofL0(λ0) and the idea here is to organize this family as follows:
q(1)1 (0)=q0(1),q(1)2 (0)=q(2)0 ,. . .,q(h)mh(0)=q
(m)
0 . (5.2)
Then, the family (q(j)0 )1≤j≤m defines an orthogonal basis in KerL0(λ0). Using similar ar-guments, the family (q(l,sj)())1≤l≤mj,1≤s≤mjl,1≤j≤h, given by (4.28), can be organized:
q(1)1,1()=q(1)(),q (1)
1,2(0)=q(2)(),. . .,q(h)mh,mhmh()=q
where for all j=1,. . .,m,q(j)()→q(j)
0 as→0. The family (q(j)())1≤j≤m defines an orthogonal basis inmi=1Ker(L(λi())), and so the order of organization of its elements directly depends on the order of organization of the basis (q(j)0 )1≤j≤m.
LetB=[als] be the (m×m) matrix, where forl=1,. . .,mand fors=1,. . .,m,
als=
q0(l),q(s)
. (5.4)
Then,B0=Im, whereImdenotes the identity matrix. FromProposition 5.1, we have m
j=1
·,q(j)()q(j)()= m
j=1
·,q0(j)
q(j)0 +R(). (5.5)
Relation (5.5) implies that
Bφl=q (l)
0 +R()q(l)0 , (5.6)
where
q0=
q(1)0 ,. . .,q0(m)
T
, φ=
q(1)(),. . .,q(m)()T. (5.7)
Proposition5.2. There exists some constant4=4(3)>0,(4≤3), such that for j∈ {1,. . .,m},
(1)the functionsq(j)()(t)are holomorphic in(t,)and satisfy the following uniform expansions: fort∈[0, 1],
q(j)()(t)=q(j)0 (t) + n≥1
q(j)n (t)n, (5.8)
where the first coefficient satisfies
q1(j)=R1q(j)0 , (5.9)
and forn≥2, the coefficientsqn(j)are given by
q(j)n =Rnq(j)0 − n−1
k=1 m
i=1
q(0j),q(i)k
q(i)n−k, (5.10)
withR0=0andRn(n≥1) being the Taylor coefficients ofR(), (2)the characteristic valuesλj()=λ(j)()satisfy
λ(j)()=λ 0+
n≥1
λ(nj)n. (5.11)
The first coefficient satisfies
λ(j)1 = −1−
L0
λ0
q(j)1 ,l
(0) 1 q
(j) 0
l1(0)q
(j) 0
and forn≥2,
λ(j)n = − 1
l(0)1 q (j) 0
2
n
i=2
s1+···+si=n λ(j)s1 λ
(j) s2 ···λ
(j) si
l(0)i q(j)0 ,l
(0) 1 q
(j) 0
+ n−1
l=0 l
k=0
Fk,l−kq(j)n−l,l (0) 1 q
(j) 0
+ n
k=1
Fk,n−kq(j)0 ,l (0) 1 q
(j) 0
.
(5.13)
Here for all integerskands, the expressionsl(s)k andFk,sare two operator-valued functions with simple forms.
Proof. (1) Define the matrix:Ᏸ=(dsp)sp; the coefficientsdspare given by
dsp =δsp+
R()q(s)0 ,q (p) 0
, δsp=1 ifp=sandδsp=0 ifp=s. (5.14)
The analyticity of the operator-valued functionR() with respect to∈]−3,3[ guar-antees the analyticity ofᏰ. The inner product of (5.5) byq(p)0 gives
m
l=1
q0(s),q(l)()
q(l)(),q(p)0
=δsp+
R()q(s)0 ,q (p) 0
(5.15)
which implies that
B2
=Ᏸ. (5.16)
Relation (5.5) implies
Bφ=φ0+R()φ0. (5.17)
We now verify that the functionφ is jointly analytic in (t,). Through relation (5.17), we deduce that
φ=B−1φ0+R()φ0
. (5.18)
The analyticity ofφ0(t) intis a classical result. Then we deduce the result by using the analyticity of the matrixB−1
in , which is obvious from (5.16), and the fact that the functionR()φ0(t) is jointly analytic in (t,). From relation (5.7), we then deduce the analyticity of the functionsq(j)() follows for all j=1,. . .,m.
The analyticity of the matrix operatorBallows writing, in a neighborhood of 0, the expansion
B=B0+B1+2B2+···. (5.19)
By expanding
φ= k≥0
kq
and (5.7), relation (5.17) implies
+∞
n=0
nB n
+∞
s=0
sq s
=q0+ +∞
n=1
nR
nq0, (5.21)
where we have consideredφ0=q0andR()=
n≥1nRn. Then, n
k=0
Bkqn−k=q0+Rnq0. (5.22)
Thejth component of the vectorBkqn−kis given by
Bkqn−kj= m
i=1
q(j)0 ,q(i)k
qn(i)−k (5.23)
and then relation (5.22) becomes
B0qnj+ n
k=1 m
i=1
q(j)0 ,q(i)k
qn(i)−k=q0(j)+Rnq0(j). (5.24)
The relationB0=Imimplies
q(j)n = − n
k=1 m
i=1
q(j)0 ,qk(i)
q(i)n−k+q0(j)+Rnq(j)0 . (5.25)
Therefore,
q(j)n = − m
i=1
q(j)0 ,q(i)n
q0(i)−
n−1
k=1 m
i=1
q(j)0 ,qk(i)
q(i)n−k+q0(j)+Rnq0(j). (5.26)
Taking the inner product withq(s)0 ,s=1,. . .,m, we have
q(s)0 ,q
(j) n
= −q(0j),q(s)n
− n−1
k=1 m
i=1
q(j)0 ,q(i)k
q(s)0 ,qn(i)−k
+δs j+
q(s)0 ,Rnq(0j)
. (5.27)
Then,
q(0j),q(s)n
= −q(s)0 ,qn(j)
− n−1
k=1 m
i=1
q(j)0 ,q(i)k
q(s)0 ,qn(i)−k
+δs j+
q(s)0 ,Rnq(0j)
. (5.28)
If we replace this equality in (5.26), we find
q(j)n = m
i=1
q(i)0 ,q
(j) n
q(i)0 +
m
s=1
n−1
k=1 m
i=1
q(j)0 ,q(i)k
q(s)0 ,q(i)n−k
−δs j−
q(s)0 ,Rnq
(j) 0
q(s)0
− n−1
k=1 m
i=1
q(0j),q(i)k
q(i)n−k+q(j)0 +Rnq(j)0 .
We recall that
P(0)q(j)n = m
i=1
q(i)0 ,q
(j) n
q(i)0 . (5.30)
Then relation (5.29) becomes
I−P0
q(j)n =
m
s=1
n−1
k=1 m
i=1
q(j)0 ,q(i)k
q(s)0 ,qn(i)−k
−δs j−
q(s)0 ,Rnq(0j)
q(s)0
− n−1
k=1 m
i=1
q0(j),q(i)k
q(i)n−k+q0(j)+Rnq(0j).
(5.31)
From the previous relation and the properties of the operatorI−P0, we deduce
I−P0
q(j)n + n−1
k=1 m
i=1
q(j)0 ,q(i)k
qn(i)−k−Rnq(j)0
=0. (5.32)
In other words, it is obvious thatRnq(j)0 ∈/ Ker(L0(λ0)) for allj=1,. . .,m. Then
qn(j)+
n−1
k=1 m
i=1
q0(j),q(i)k
q(i)n−k−Rnq(0j)
/
∈KerL0
λ0
. (5.33)
Thus, relation (5.32) means that
q(nj)+ n−1
k=1 m
i=1
q(0j),qk(i)
q(i)n−k−Rnq(j)0 =0. (5.34)
(2) In order to find out the coefficients in (5.11), our method is based on expanding the expressionL(λ(j)()) for near zero. To handle this, we have to expand, first, the operator-valued functionL(λ) around=0 and so the resulting expression aroundλ=
λ0.
We first recall the formula (see [1] and [3, page 332])
H0(1)(z)=J0(z) +iY0(z), (5.35)
whereJ0andY0are the Bessel functions
J0(z)= +∞
n=0 (−1)n
n!2
z
2
2n
,
Y0(z)= 2
π
Γ0+ log
z
2
J0(z)−2
π
+∞
n=1 (−1)n
(n!)2
z
2
2n
1 +1 2+···
1
n
.
Thus, in particular, we can write
H0(1)(z)=h1
z2+h
2
z2log(z)=h 1
z2+Γ0+z2h 3
z2log(z), (5.37)
for some entire functionsh1,h2, andh3, withΓ0being a real constant.
Combining the last result and the kernel of the operatorL(λ) (seeProposition 3.1), we formally obtain the following uniform expansion: for all (λ,t,s)∈Dr0(λ0)×[0, 1]×[0, 1],
G(λ,t,s)= i 4H
(1) 0
λγ(t)−γ(s)= n≥0
Gn(λ,t,s)n, (5.38)
with the Taylor coefficients
Gn(λ,t,s)=4ni! n
k=0
λk β(t)−β(s),γ(t)−γ(s)
γ(t)−γ(s) k β(s),γ(s) γ(s)
n−k
×dkH (1) 0
dzk
λγ(t)−γ(s).
(5.39)
As an immediate consequence, the following holds:
L(λ)= +∞
n=0
Ln
λn, forsmall enough. (5.40)
By consideringProposition 3.1, it seems clear that each operatorLn(λ) has kernelGn(λ,
t,s). One also gets from (5.39) the following uniform expansion: for (t,s)∈[0, 1]×[0, 1],
Gn(λ,t,s)= +∞
k=0
λ−λ0kG(n)k (t,s), (5.41)
whereλis in a neighborhood ofλ0, with the coefficients
G(n)k (t,s)= i 4n!
n
i=0 k
l=0 1
l!(k−l)!
β(t)−β(s),γ(t)−γ(s)i
γ(t)−γ(s)i+l−k
× β(s),γ(s)
γ(s)
n−idlλi
dλl
λ=λ0 dk+i−l
dzk+i−lH (1) 0
λ0γ(t)−γ(s). (5.42)
Now, for (,λ) in a neighborhood of (0,λ0), we have the following expansions:
L(λ)= +∞
n=0 +∞
k=0
nλ−λ 0
k
where the operatorlk(n)has kernelG(n)k (t,s). We know thatλ(j)()→λ
0as→0. Then for
small enough, we can replaceλ(j)() given by (5.11) in relation (5.43), and so we obtain
Lλ(j)()= +∞
n=0
n
k=0
Fk,n−k
n, (5.44)
where
Fn,0=l0(n),
Fn,k= k
i=1
s1+···+si=k λ(s1j)λ
(j) s2 ···λ
(j) si
l(n)i . (5.45)
Remember that
Lλ(j)()q(j)()=0, ∀j=1,. . .,m. (5.46)
Then by using (5.10) and (5.44) at ordern≥1, we can easily write
n
l=0 l
k=0
Fk,l−kq(j)n−l=0. (5.47)
Thus, from (5.45) and by simple calculus, we find forn≥2,
λ(j)n = − 1
l(0)1 q (j) 0
2
n
i=2
s1+···+si=n λ(j)s1 λ
(j) s2 ···λ
(j) si
li(0)q(j)0 ,l
(0) 1 q
(j) 0
+ n−1
l=0 l
k=0
Fk,l−kq(j)n−l,l (0) 1 q
(j) 0
+ n
k=1
Fk,n−kq(j)0 ,l (0) 1 q
(j) 0
.
(5.48)
We now give the following lemma which seems useful to prove the fundamental result in this section.
Lemma5.3. Let 0be a bounded neighborhood of Ω0inR2. The functionsui,j()(x)=
S(λj())q(i)()(γ−1)are jointly analytic in the variables(x,)∈0×]−4,4[.
Proof. The function ui,j()(x)=S(λj())q(i)()(γ−1) satisfies the Helmholtz equation in Ω with the boundary conditions ui,j()|∂Ω=0 and ∂νui,j()(γ(t))=q(j)()(t), which are jointly analytic with respect to the variables (t,)∈[0, 1]×]−4,4[. The out-ward unit normal ν to∂Ω is given byγ(t)/|γ(t)|, as a function oft=γ−1(x). The symbol of the operator∆x=∂2
x1+∂
2
x2 is(ξ1,ξ2,)=ξ
2
1+ξ22. Thus(ν)=1>0. Since the surface∂Ωis noncharacteristic for∆x, the Cauchy-Kowaleski theorem implies that
ui,j()(x) is jointly analytic with respect to (x,) in{|x−γ(t)| ≤α0}×]−4,4[, where
α0is a positive constant.
Theorem5.4. Let 0be a bounded neighborhood of Ω0 inR2. Then there exists a con-stant5>0smaller than4such that an orthonormal basis of eigenfunctions(uj())j corre-sponding to theλ0-group,(λj())j, inH01(Ω)can be chosen to depend holomorphically on (x,)∈0×]−5,5[. Moreover, these eigenfunctions satisfy the following uniform expan-sion: forx∈0,
uj()=u(0j)+ n≥1
u(j)n n, (5.49)
where the familyu(0j)builds a basis of eigenfunctions of (2.4) associated toλ20and normalized inL2(Ω0). The termsu(j)
n are computed from the Taylor coefficients ofq(j)().
Proof. Propositions 5.1 and 5.2 imply that there exists an orthonormal basis (q(i)())
1≤i≤m j(t,)∈H−1/2(]0, 1[) of Ker(L(λ(j)())), which is analytic in (t,)∈[0, 1]× ]−4,4[. We know thatS(λj())q(i)()(γ−1) builds a basis of eigenfunctions of the eigen-value problem (2.4) associated toλ2
j(). Using the Schmidt orthogonalization process once again, we construct the desired orthonormal basis. Clearly, the functions (ui,j())i j, introduced inLemma 5.3, build a basis of the eigenspaces corresponding to theλ0-group, (λj())jinH01(Ω). We will now give the asymptotic expansion of these functions when
tends to 0. To simplify notations, we drop the subscriptsiand j. Integral equations give
u()(x)=
1
0G
λ()x−γ(t)q()(t)γ(t)dt, x∈0. (5.50)
The perturbed eigenvalueλ() lies in a small neighborhood ofλ0for small values of. Then, there exists5>0 (5≤4), such that we have the following Taylor expansion:
Gλ()x−γ(t)γ(t)=G
λ0x−γ(t)γ(t)+ k≥1
kG
k(x,t), (5.51)
which holds uniformly inx∈0andt∈[0, 1]. We useProposition 5.2to write
q()(t)=q0(t) + k≥1
kq
k(t), (5.52)
uniformly int∈[0, 1]. Substituting the last two expansions into (5.50), we find
u()=u(0) + k≥1
k
k
n=1
1
0qk−n(t)Gn(x,t)dt
. (5.53)
Now we use the Schmidt orthogonalization process to construct from the eigenfunctions (uj())jan orthonormal basis (uj())jof the direct sum of eigenspaces associated to the
λ0-group. This method allows us to compute the asymptotic expansion of these functions.
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Abdessatar Khelifi: Centre de Math´ematiques Appliqu´ees (CMAP), ´Ecole Polytechnique, 91128 Palaiseau cedex, France
