The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating

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1 The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating Hanen Boujlida, Houssem Haddar, Moez Khenissi To cite this version: Hanen Boujlida, Houssem Haddar, Moez Khenissi. The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating. 27. <hal-6463> HAL Id: hal Submitted on 23 Nov 27 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 THE ASYMPTOTIC OF TRANSMISSION EIGENVALUES FOR A DOMAIN WITH A THIN COATING by H. Boujlida, H. Haddar and M.Khenissi Abstract. We consider the transmission eigenvalue problem for a medium surrounded by a thin layer of inhomogeneous material with different refractive index. We derive explicit asymptotic expansion for the transmission eigenvalues with respect to the thickness of the thin layer. We prove error estimate for the asymptotic expansion up to order for simple eigenvalues. This expansion can be used to obtain explicit expressions for constant index of refraction.. Introduction This work is a contribution to the study of transmission eigenvalues [, 4, 6] and their relation to the shape and material properties of scatterers. These special frequencies are associated with the existence of an incident field that does not scatter. They can be equivalently defined as the eigenvalues of a system of two coupled partial differential equations posed on the inclusion domain. One of these equations refers to the equation satisfied by the total field and the other one is satisfied by the incident field. The two equations are coupled on the boundary by imposing that the Cauchy data coincide. This eigenvalue problem can then be formulated as the eigenvalue problem a non-selfadjoint compact operator. Although non intuitive, it can be shown that this problem admits an infinite discrete set of real eigenvalues without finite accumulation points [7, 26]. These special frequencies can be identified from farfield data as proved in [5, 9, 4]. Since they carry information on the material properties of the scatterer, transmission eigenvalues would then be of interest for the inverse problem of retrieving qualitative information on the material properties from measured multistatic data [4, 5]. In this perspective, it appears important to study the dependence of these eigenfrequencies with respect to the material properties and the geometry. Several works in the literature have addressed this issue by considering asymptotic regimes and quantifying the dependence of the first leading Key words and phrases. transmission eigenvalue, thin layer, inverse scattering.

3 2 H.BOUJLIDA, H. HADDAR AND M.KHENISSI terms in the asymptotic expansion of the transmission eigenvalue with respect to the small parameter [, 8, 2, 6]. We here consider the case of a scatterer made of a thin coating which corresponds to frequently encountered configurations in the stealth technology for instance. The goal is to characterize the dependence of the first order term on the material properties and the thickness of the coating. A first work on this topic was done in [] where the case of coated perfect scatterer is considered. One proves in particular for the latter case that the first order term depends only on the thickness. We here address the more complicated configuration of a coated penetrable media. The analysis indicates that the first order asymptotic resembles to the shape derivative for the buckling plate equation [7] and contain non trivial dependence on the material properties. more importantly, this expansion allows us to obtain explicit approximate expressions for the thin layer index of refraction. This indeed can be useful for the solution of the inverse problem. Although the formal derivation follows the systematic procedure using the classical scaled expansion method as in [3, 2, 3] for instance, the rigorous justification is much more involved. For instance the arguments in [] are hard to extend to the present case since special uniform estimates have to be obtained for the transmission problem. We restrict ourselves here to the justification of the first two terms in the asymptotic expansion using the abstract theory developed in [23, 2]. We follow the procedure developed in [8] for the case of small obstacles asymptotic. The main technical point in the proof is to obtain the corrector for the main operator, which is here the biharmonic operator. Our main result provide explicit expansion for simple transmission eigenvalues. We analyze the problem where the contrast in material properties affect only the lower order term in the Helmholtz equation. We finally indicate that although the problem is considered only in dimension 2, the results of the main theorem including the expression of the first order asymptotic term remain true for three dimensions up to more complicated technicalities in the proof related to differential geometry. The paper is organized as follows. We first introduce the transmission eigenvalues and write them as the eigenvalues of a non selfadjoint operator. We then explain the outline of a classical formal procedure to obtain the expression of the asymptotic expansion. We give the expression till the second order term to emphasize for instance that the expression of the second order derivative is too complicated to be exploited in practice. We then proceed with the main result of the paper that provides explicit expressions and an error estimate for the first two terms in the asymptotic expansion. 2. Problem statement Let Ω R 2 be a bounded domain with a smooth boundary Γ. We denote by Ω = {x Ω, dx, Γ > } and its boundary

4 ASYMPTOTIC EXPANSION OF T.E 3 Γ = {x Ω, dx, Γ = } = Ω, for > a small enough parameter, where dx, Γ denotes the distance of a point x to the boundary Γ. Let Ω = Ω\Ω be the layer of thickness around Ω see Figure. Figure. Stretch of the geometry We consider the following transmission eigenvalue problem: w + k 2 n xw = in Ω, v + k 2 v = in Ω, w = v on Γ, w ν = v on Γ, ν where k denotes the unknown eigenfrequency and ν the unitary normal to Γ directed to the interior of Ω. The index of refraction n is defined as follows : { n x in Ω n x =, n x in Ω, where n and n are non negative real valued functions L R 2 that are independent from. For the sake of simplicity, we assume that the restriction of n and n to Ω are constant functions along the normal coordinate to Γ for sufficiently small. We finally assume that the function / n is either positive definite or negative definite on Ω. We remark that this assumption also implies that / n is either positive definite or negative definite on Ω and that / n x γ > for a.e. x Ω 2 with γ being independent from sufficiently small. The main goal of this paper is to find the asymptotic expansion of eigenfrequencies k with respect to. Assuming that L Ω, the transmission eigenvalue problem can n

5 4 H.BOUJLIDA, H. HADDAR AND M.KHENISSI be reformulated as the nonlinear eigenvalue problem for λ := k 2 R and u := w v HΩ 2 such that + λ n + λ u = in Ω, n which in variational form, after integration by parts, is formulated as finding λ R and non-trivial function u HΩ 2 such that u + λ u φ + λ n φdx =, φ H n Ω. 2 3 Ω The space H 2 Ω denotes the closure in H 2 Ω of the set of regular compactly supported functions in Ω. We shall work with the reformulation of 3 as a linear eigenvalue problem for a non selfadjoint compact operator [4]. First observe that 3 can be written as where A u + λ B u + λ 2 C u = in H 2 Ω 4 A : H 2 Ω H 2 Ω, B : H 2 Ω H 2 Ω, C : H 2 Ω H 2 Ω are defined by the Riesz representation theorem as A u, φ H 2 Ω := u φdx, 5 Ω n B u, φ H 2 Ω := u φ + n u φdx, 6 Ω n and n C u, φ H 2 Ω := u φdx. 7 Ω n Note that A : HΩ 2 HΩ 2 is a bounded, self-adjoint and invertible operator thanks to 2, B : HΩ 2 HΩ 2 is a bounded, compact and self-adjoint operator and C : HΩ 2 HΩ 2 is a non negative or non positive bounded, compact and self-adjoint operator. Observe that since A is invertible, λ. In order to avoid distinguishing the cases of n being positive or negative we shall abusively set C 2 C 2 in the case where n non positive. Setting U = u, λ C 2 u, the transmission eigenvalue problem 4 can be transformed into the linear eigenvalue problem, τ R, U H 2 Ω H 2 Ω such that T τ IU =, with τ = λ, 8 for the compact non-selfadjoint operator T : HΩ 2 HΩ 2 HΩ 2 HΩ 2 defined by A T = B A C 2. 9 C 2

6 ASYMPTOTIC EXPANSION OF T.E 5 We set T = A B A C 2 C 2 where A, B and C are defined by 5, 6 and 7 respectively for n = n in Ω. We state here the main result of this paper which will be proven in Section 4. In the following a transmission eigenvalue λ is called simple if the corresponding τ = /λ has an algebraic multiplicity equal to. Theorem 2.. Assume that n, n C 4 Ω. Let λ R be a simple transmission eigenvalue of 3 with n = n in Ω and let u HΩ 2 be an associated eigenfunction. This implies in particular that β := λ 2 n n u 2 u dx 2. Ω If we suppose in addition that u and A u are in C 6 Ω, then, for sufficiently small >, there exists a transmission eigenvalue λ of 3 such that λ = λ + λ + O 3 2 where λ is given by the following expression λ := λ n n β n u 2 dsx. 2 Γ This theorem is an immediate consequence of Theorem 4.8 that is stated and proven in the last section of this paper. From the practical point of view, this theorem implies in particular that λ gives a measure for the contrast n n. For instance, if n is constant and n is constant on Γ, one can approximate the value of n using the identity n Γ = n Γ λ λ λ n u 2 u dx 2 + O 2 α n λ with Ω α := Γ u 2 n 2 dsx. For the inverse problem where one would like to determine n from multistatic measurements of scattered waves, the value of λ can be approximated using sampling methods as in [5, 4] see also [9] for an alternative approach. The values of λ and u can be computed numerically if one has a priori knowledge of n and Ω see for instance [2, 8, 2] for numerical methods to approximate λ and u. We finally indicate that, although not carefully checked, we conjecture that the expression for λ remains true in three dimensions corrections due to the curvature of Γ only affect higher order terms.

7 6 H.BOUJLIDA, H. HADDAR AND M.KHENISSI 3. Formal asymptotic expansion In this section, we derive the formal asymptotic expansion for transmission eigenvalues and give explicit formulas for the terms up to order 2. The idea here is to provide a systematic formal way to quickly obtain the explicit expression of λ in Theorem 2. and also higher order terms. The latter turn out to have complicated expressions that would be of marginal interest for the solution of the inverse problem mentioned above. This formal stage will also be helpful in establishing the rigorous proof based of Osborn s theorem [23]. It allows one to have an intuition for the expression of the corrector in the asymptotic of the main operator A. We assume the following expansions for the transmission eigenvalues : λ = j λ j, 2 j= and then follow a classical technique for thin layers asymptotics based on rescaling and asymptotic expansion with respect to the thickness. We shall mainly follow the approach in []. 3.. Scaling. We assume that the boundary Γ is C -smooth, although much less regularity is needed if we restrict ourselves to only few terms in the expansion. The issue of optimal regularity assumptions for Γ is not discussed here. However, one can check that at least a C 2 regularity is needed to get the expression of λ. We parametrize Γ as Γ = {x Γ s, s [, L[}, with L being the length of Γ and s is the curvilinear abscissa. At the point x Γ s, the unit tangent vector is τs := dx Γs, the curvature κs is defined by: ds dτs = κsνs or dνs = κsτs. ds ds Within these notations, the boundary of Ω is parametrized as Γ = {x Γ s + νs, s [, L[}. For a function u defined in Ω, we consider ũ defined on [, L[ ], [ by ũs, η = uϕs, η where ϕs, η := x Γ s + ηνs. 3 Then, the gradient and Laplace operators are expressed in the local coordinates as: u = + ηκs s τs + η νs ũ, u = + ηκ s + ηκ s + κ + ηκ η + 2 ũ. 4 η 2 To make the formal calculations, we need to separate the thin layer and scaled it with respect to the thickness so that the equation are posed on a domain independent from.

8 ASYMPTOTIC EXPANSION OF T.E 7 We therefore rewrite the transmission eigenvalue problem in the following equivalent form w + + k 2 n w + = in Ω, w + k 2 n w = in Ω, v + k 2 v = in Ω, w + = w w +, ν = w on Γ, 5 ν w + = v on Γ, w + ν = v on Γ. ν We denote by ξ = η the stretched normal variable inside Ω and define ϕ : G = [, L[ ], [ Ω s, ξ ϕ s, ξ = x Γ s + ξνs. Then the expression of the Laplace operator in the scaled layer is: u = + ξκ s + ξκ s + κ + ξκ ξ + 2 û =: 2 ξ 2 s,ξ û 6 for ûs, ξ := uϕ s, ξ. The next step is to write the equation for w + in the scaled domain and solve for the asymptotic expansion of w + in terms of the boundary values on Γ. These boundary values are given by the asymptotic expansion of v. More specifically, setting ŵ s, ξ := w + ϕ s, ξ, we have that together with the boundary conditions We assume that ŵ s, ξ = s,ξ ŵ + λ n ŵ = in G 7 ŵ s, = v x Γ s ŵ ξ s, = v ν x Γs j ŵ j s, ξ, s, ξ G and v x = j= s [, L[, s [, L[. 8 j v j x, x Ω 9 for some functions ŵ j defined on G and v j defined on Ω that are independent from. Multiplying 7 by 2 + ξκ 3 and using 2, we obtain 5 k A k ŵ =, k= j=

9 8 H.BOUJLIDA, H. HADDAR AND M.KHENISSI where A k k=...5 are differential operators of order 2 at maximum with the following expressions for the first fourth terms: A = 2 ξ 2, A =3ξκ 2 ξ 2 + κ ξ, A 2 = 2 s 2 + 3ξ2 κ 2 2 ξ 2 + 2ξκ2 ξ + λ n, A 3 =ξ 3 κ 3 2 ξ + 2 ξ2 κ 3 ξ ξ κ s s + ξκ 2 s 2 + 3λ n ξκ + λ n. Inserting the ansatz 9 in 7 and 8 we obtain after equating the terms of same order in and using the convention ŵ j = v j = for j <, 2 5 ξ 2 ŵj = A k ŵ j k in G, k= ŵ j s, = v j x Γ s s [, L[, 2 ŵ j ξ s, = v j ν x Γs s [, L[. These equations can be solved inductively to get the expressions of ŵ j boundary values of v l, l j. One gets for j =,, 2 and 3 ŵ s, ξ = v x Γ s, in terms of the ŵ s, ξ = v ν x Γsξ + v x Γ s, ŵ 2 s, ξ = ξ2 2 and ŵ 3 s, ξ = ξ3 6 + ξ3 3 v 6 κ w ν x Γs + 2 w s x Γs + λ 2 n w x Γ s 2κ 2 w ν x Γs 3κ 2 w s 2 ν x Γs κ w s x Γs + v ν x Γsξ + v 2 x Γ s, 2 s 2 x Γs κλ n w x Γ s + λ n v ν x Γs + ξ2 κ v 2 ν x Γs + λ n v x Γ s + λ n w x Γ s Now, we also postulate the following expansion for w : w x = j w j x 23 j= + v 2 ν x Γsξ + v 3 x Γ s. 22

10 ASYMPTOTIC EXPANSION OF T.E 9 with w j : Ω R are functions independent of. Then w j, v j solves j w j + λ n w j = λ l n w j l in Ω, v j + λ v j = l= j λ l v j l in Ω. l= Note that the functions w j are defined in all Ω and not only Ω and therefore 23 gives a extension of w to all Ω. The continuity conditions at Γ can be written as w s, = ŵ s, and w η s, = ŵ s, ξ where w is defined from w using the local change of variables 3 in a neighborhood of Γ. Using Taylor s expansion up to the second order, which is sufficient to compute the first three terms in the asymptotic expansion we get 24 and w s, = w s, + w η s, w 2 η s, + 2 o2 = ŵ s, 25 w η s, = w η s, + w 2 η s, w 2 2 η 3 s, + o2 = ŵ s,. 26 ξ Injecting 9 and 23 into 25 and 26, we respectively obtain the following continuity conditions on Γ, w x Γ s = ŵ s,, w x Γ s + w ν x Γs = ŵ s,, 27 and w2 x Γ s + w ν x Γs + 2 w 2 ν x Γs = ŵ 2 2 s,, = ŵ s,, ξ w ν x Γs = ŵ s,, 28 ξ w ν x Γs + 2 w ν x Γs = ŵ 2 s,. 2 ξ System 24 coupled with the boundary conditions 28 and 27 provide an inductive way to determine w j, v j. We obtain the set of equations satisfied by these terms after substituting the expressions of ŵ j s, given by 2,22. We hereafter summarize the set of equations obtained for w j, v j and how to use them to get the expressions of λ j,

11 H.BOUJLIDA, H. HADDAR AND M.KHENISSI j =,, 2. We first obtain that the couple w, v solves w + λ n w = in Ω, v + λ v = in Ω, w v = on Γ, w ν v = ν on Γ. This means in particular that λ is a transmission eigenvalue for the limiting problem where the thin layer is removed. We then obtain that the couple w, v satisfies w + λ n w = λ n w in Ω, v + λ v = λ v in Ω, w v = on Γ, w ν v ν = λ n n w on Γ. Since λ is an eigenvalue of the associated homogeneous system, this problem is solvable only if a compatibility condition is satisfied by the right hand sides. This compatibility condition can be obtained by multiplying the first equation with w and the second equation with v, taking the difference then integrating by parts and using 29. One ends up with λ n n w 2 dsx Γ λ = n w 2 v 2 dx Ω which coincides with the expression of given in Theorem 2. expressed in terms of u = w v. Although not covered by the analysis of convergence, we also provide the expression of the third term in the asymptotic expression. One get that the couple w 2, v 2 solves where w2 + λ n w2 = λ n w λ 2 n w in Ω, v 2 + λ v 2 = λ v λ 2 v in Ω, w2 v 2 = h on Γ, w2 ν v 2 ν = h 2 on Γ, h = 2 w 2 ν 2 2 λ n n w

12 ASYMPTOTIC EXPANSION OF T.E and h 2 = κ 2 w 7κ ν w + 2κ 2 + λ s 2 n + n λ 2n n + λ κ n 2 n w 2 ν 3κ 2 w 2 w ν 2 + κ w ν. w s w 2 ν s 2 Writing the compatibility condition for 3, we obtain the following formula for λ 2 λ 2 u 2 λ n u 2 dx = λ 2 u ū + u 2 dx Ω n λ Ω λ n λ u ū + n u ū + 2n λ u ū dx n + Γ Ω h ν + λ ū n dsx Γ h 2 + λ ū dsx. 32 n This complicated expression shows in particular a nonlinear dependence of λ 2 in terms of n. It suggests that the use of λ 2 for solutions to the inverse problem of determining n may not be appropriate. 4. Convergence analysis The main goal of this section is to prove Theorem 2. that provides a rigorous mathematical justification of the formal asymptotic expansion for simple real transmission eigenvalues up to the first order. The proof is split into several steps. The first one is to establish the convergence in norm of the operator T to T. This ensures the convergence of λ to λ. In order to get to the term of order in, we shall apply the Osborn theorem which requires for instance a characterization of the pointwise asymptotic expansion of T U up to order in for some given function U HΩ 2 HΩ. 2 The latter can be obtained from the asymptotic expansions of A u, B u and C u for some u HΩ. 2 The difficult part to get the expansion of A u since for the two others, the first order terms are vanishing. This critical result is provided by Lemma 4.5. In all the following we use the notation f, g := f, g H 2 Ω = f gdx and g := g, g 2 H 2Ω. Ω For an operator A : V V, A denotes the operator norm. To simplify the writing, C will denote a generic constant whose value may change but remains independent from as. 4.. Pointwise convergence of the spectrum of T. In this first step, we prove pointwise convergence of the spectrum of the operator T to the spectrum of T. This is a direct consequence of the following convergence in norm [23, 8].

13 2 H.BOUJLIDA, H. HADDAR AND M.KHENISSI Theorem 4.. Assume that n C 2 Ω. Let T and T be defined by 9 and respectively. Then T converges to T in the operator norm. Proof. The proof follows from Lemma 4.2 and Lemma 4.4 below, using the definition of T and T. In the first lemma we prove norm convergence for B and C. Lemma 4.2. Let B, C, B and C be the operators defined by 6 and 7. Then, for sufficiently small, B B C 2 and C 2 C 2 C. 33 Proof. From the definitions of B and B, we have that for u, φ HΩ 2 B B u, φ = u φ + n uφ dx u φ + n uφ dx Ω n Ω n = u φ + uφ dx. n n Therefore, Ω B B u, φ C u L Ω φ L Ω + φ L Ω u L Ω. Using the Sobolev embedding theorem and the Cauchy Schwartz inequality, we get By choosing φ = B B u, we get B B u, φ C 2 u H 2 Ω φ H 2 Ω. Ω B B u H 2 Ω C 2 u H 2 Ω. The proof is similar for the second inequality. For u, φ HΩ, 2 we have C C u, φ = uφdx C Ω u L n n Ω φ L Ω From the Sobolev embedding theorem, we obtain C C u, φ C u H 2 Ω φ H 2 Ω. By choosing φ = C C u, we have C C u H 2 Ω C u H 2 Ω. 34 Using the square root Lemma in [24] and the fact that C n converges to C n at the same order O, we conclude that C 2 converges to C 2 at the same order O. Hence we have C 2 C 2 u H 2 Ω C u H 2 Ω. 35

14 ASYMPTOTIC EXPANSION OF T.E 3 Now we show convergence in the HΩ 2 norm for A f assuming smoothness of f. This will be useful in the proof of Lemma 4.4 since the operators B and C are regularizing. Lemma 4.3. Let A and A be defined by 5 for > and =, respectively and f H 2 Ω. If A f C 2 Ω, then for sufficiently small, A f A f C Proof. For a fixed f HΩ, 2 define z and z in HΩ 2 as z = A f and z = A f. Since A z = A z = f, we have that for φ HΩ 2 A z z, φ = A z A z, φ = z φdx. 37 n n If z C 2 Ω, we get z φdx C z φdx C 2 φ H 2 Ω n n Ω. Ω Thus, we have shown that Ω A z z, φ C 2 φ H 2 Ω. By plugging in φ = z z, we obtain the desired convergence using the coercivity of A. Lemma 4.4. Assume that n C 2 Ω. Let A, B, C, A, B and C be defined by 5, 6 and 7 for > and =, respectively. Then for sufficiently small, A B A B and A C 2 A C 2. Proof. From 37, we have that for f, φ HΩ 2 and with z = A f and z = A f Furthermore, A z z, φ C A f L 2 Ω φ H 2 Ω. A B f A B f H 2 Ω A A B f H 2 Ω + A B B f H 2 Ω C A B f L 2 Ω + A B B f H 2 Ω. 38 For estimating A B f L 2 Ω, observe that B u H 2 Ω is the weak solution B u = n n u + n u in Ω. Classical regularity results [22, 25] and the fact that n C 2 Ω imply that B u H 4 Ω H 2 Ω and therefore A B f H Ω C f H 2 Ω. By the Sobolev embedding theorem, this implies that

15 4 H.BOUJLIDA, H. HADDAR AND M.KHENISSI A B f L p Ω C f H 2 Ω, for p > 2. Let p = p 2 > and q such that p + q =. From 33 we obtain A B f 2 L 2 Ω A B f 2 L Ω Ω p q C 2q f H 2 Ω. 39 Using 38, 39 and 4 we have that A B B f H 2 Ω C 2 f H 2 Ω. 4 A B A B. The second convergence result follows from similar arguments. Now we would like to obtain explicit formula for the correction term in the asymptotic expansion for the operator T. More precisely, we define explicit formula for the corrector term associated with A A Corrector term for A A. In this subsection, we construct a corrector function and use it to estimate the convergence rate of z = A u for u HΩ. 2 Let z = A u HΩ, 2 i.e z HΩ 2 solution of z = u in Ω. 4 n Inspired by the formal calculations on the previous section, namely problem 3, we define z solution of z = in Ω, n z = on Γ, 42 z ν = n z on Γ. n We expect that z = z + z + O 2 in Ω. We extend z in Ω as z defined by { z z in Ω =, 43 z ψ in Ω where ψ is a polynomial of order 3 and satisfying the boundary conditions: ψ ψ =, ν = n z on Γ, n ψ = ψ 44 ν = on Γ. This gives the following expression of ψ that plays the role ŵ 2 in the formal calculations ψx = ψϕs, ξ = ˆψs, n ξ = z ϕs, ξ ξ 2. n

16 ASYMPTOTIC EXPANSION OF T.E 5 The choice of ψ ensures in particular that z HΩ. 2 To simplify the notation we set m :=. n n Now we have the following Lemma. Lemma 4.5. Assume that n and n are in C 4 Ω. Let u HΩ 2 then set z = A u and z = A u. We define z as in 43 and assume that z C 6 Ω. Then we have, for sufficiently small, Proof. For any φ H 2 Ω we have that We recall that Furthermore, we have that A z, φ = z z z H 2 Ω C 3 2. A z z z, φ = A z z, φ A z, φ. 45 A z z, φ = z φdx. Ω n n Ω z φdx + z ψ φdx. 46 n Ω n Using the fact that z = and the Green formula yield, n A z, φ φ = m z Γ ν dsx + z z φdsx Γ ν n ν n + ψ φ Γ n ν dsx ψ φdsx ψφdx. Γ ν n Ω n Using the expression of ψ we have n ψ Γ = n s,ξ ψs, = 2 m z ϕs,, ν ψ Γ = n η ψ Γ = n n ξ ψs, s,ξ = 6 m z ϕs,. 2 We then get after substitution of these expressions A z, φ = m z ϕs, + 2 φ Γ z ϕs, ν dsx m Γ ν z ϕs, + 6 z ϕs, φdsx ψφdx 2 Ω n = φ φ Γ ν dsx φ 2φdsx ψφdx Γ Ω n where we have set

17 6 H.BOUJLIDA, H. HADDAR AND M.KHENISSI φ s := m z ϕs, + 2 m z ϕs,, φ 2s := m ν z ϕs, m z ϕs, using the parametrization of the curve Γ, s ϕs, with ϕ defined by 3. Using this parametrization and setting φs, η := φϕs, η in Ω we have φ L ν dsx = φ φ L s, + κds = φ 2 φ s, η + κdsdη. η η2 Γ φ From the definition of φ we then get for φ HΩ, 2 φ ν dsx = 2 L m z ϕs, 2 φ η s, ηdsdη + O 2 2 φ H 2 Ω. Γ φ Here and in all the following O r denotes a function such that O r C r for a constant C independent from the test function φ but that may depend on z C 6 Ω. Using Taylor s expansion we also get for φ HΩ, 2 φ 2φdsx = L φ 2 φ 2 Γ 2 η s, η + κdsdη + O φ H 2 Ω = 3 L m z ϕs, 2 φ η 2 s, ηdsdη + O 2 φ H 2 Ω where the last equality is obtained after substituting the expression of φ 2. One ends up with φ φ L Γ ν φ 2φdsx = m z ϕs, 2 φ η s, ηdsdη + O φ H 2 Ω. Equation 45 then gives A z z z, φ = m z φdx A z, φ Ω L = m z ϕs, η φϕs, η + ηκdsdη Ω n ψφdx + O 3 2 φ H 2 Ω. L We use the expression of the Laplacien in local coordinates + ηκ φϕs, η = s m z ϕs, 2 φ ϕs, ηdsdη η2 φ + ηκ s s, η + κ φ η s, η + + φ ηκ 2 s, η η2

18 ASYMPTOTIC EXPANSION OF T.E 7 to make the decomposition L L + m z ϕs, η φϕs, η + ηκdsdη = L m z ϕs, η κ φ η s, η + φ ηκ 2 η s, η + 2 m z ϕs, η s L To estimate the first term, we integrate by parts on [, L[, we obtain L m z ϕs, η φ s + ηκ s s, η dηds = = = L L L = O 3 2 φ H 2 Ω. + ηκ s m z ϕs, η φ s, ηdsdη s m + ξκ s z ϕs, ξ φ s, ξdsdξ s m ξ s z ϕs, For the last term we proceed similarly to obtain L = m z ϕs, η φ s, ηdsdη = η ξ m z ϕs, L L φ + ηκ s s, η m z ϕs, η 2 φ s, η. η2 2 φ η s s, ηdη dsdξ + O 2 φ H 2 Ω m z ϕs, ξ φ s, ξdsdξ η 2 φ η s, ηdη dsdξ + O 2 φ 2 H 2 Ω = O 3 2 φ H 2 Ω Observing in addition that one ends up with L A z z z, φ = ηκm z ϕs, η 2 φ η 2 s, ηdsdη = O 3 2 φ H 2 Ω, L 2 φ m z ϕs, η z ϕs, s, ηdsdη η2 Ω n fψ φdx + O 3 2 φ H 2 Ω. To conclude we just observe that the two remaining terms are also of the form O 3 2 φ H 2 Ω. For the first term, we simply use a Taylor expansion for z while for the second one we just use that, due to the regularity of n and n, n ψ L Ω.

19 8 H.BOUJLIDA, H. HADDAR AND M.KHENISSI In conclusion, A z z z, φ C 3 2 φ H 2 Ω. Choosing φ = z z z, since the coercivity constant associated with A is independent from, we get which ends the proof. z z z H 2 Ω C 3 2 Lemma 4.6. Assume that n and n are in C 4 Ω. If u C 6 Ω H 2 Ω, then for sufficiently small, B A where C independent of. A u H 2 Ω C and C 2 A A u H 2 Ω C 47 Proof. From the estimate of Lemma 4.5 we have that Since z L 2 Ω = O, then z z z L 2 Ω C 3 2. z z L 2 Ω C. Since B is two orders smoothing, we have that The same proof holds for C 2. B z z H 2 Ω z z L 2 Ω C Now to derive the eigenvalue expansion, we will apply the Theorem of Osborn [23], which we state here for reader s convenience. Suppose X is a Hilbert space with inner product denoted by, and K n : X X is a sequence of compact linear operators such that K n converge in the operator norm to K. It then follows that the adjoint operators also converges in norm. Let µ be a nonzero eigenvalue of K of algebraic multiplicity m. It is well known that for n large enough, there exist m eigenvalues of K n : µ n, µ n 2,..., µ n m such that µ n j µ pour tout j =,..., m. Let E be the spectral projection onto the generalized j eigenspace of K corresponding to eigenvalue µ. The space X can be decomposed in terms of the range and null space of E as X = RE NE. Then form the proof of Theorem 3 in [23], one can state the following theorem.

20 ASYMPTOTIC EXPANSION OF T.E 9 Theorem 4.7. Let φ, φ 2,..., φ m be a normalized basis for RE, and let φ, φ 2,..., φ m be the dual basis of RE such that v, φj = for all v NE. Then there exists a constant C such that : µ m µ n j m K K n φ j, φ m m j C K K n RE K Kn RE 48 j= j= In order to apply Theorem 4.7 obtain explicit expression for the first order asymptotic m j= K K nφ j, φ j, one has to construct the basis φ j. In the case of selfadjoint operators, one take φ j = φ j. Once easily check that φ j are necessarily eigenvalues of K associated with µ. Reciprocally, such eigenvalues are admissible for dual basis if they belong to RE. We now turn our attention to application of this theorem with K n T and K T and X HΩ 2 HΩ. 2 We already showed that T converges to T in the operator norm in Theorem 4.. In order to simplify the calculations we define the inner product on H 2 Ω H 2 Ω by: u v w, z := A u, w H2 Ω + v, z H 2 Ω. Let τ be a simple real eigenvalue of T, then for small enough, some eigenvalue τ of T is such that τ τ. Let U = u λ C 2 u be an eigenvector of T associated with τ. Using the expression of u T one easily observes that Ũ = λ C is an eigenvector of T 2 associated with τ. u Then this eigenvector is proportional to the dual basis of U if and only if β := U, Ũ = A u, u λ 2 C u, u. 49 We remark that the fact that τ is assumed to a simple eigenvalue i.e. with algebraic multiplicity equals, guarantees that 49 holds. We then can define the dual vector as U = β Ũ and apply Theorem 4.7 to get that τ τ T T U, U C T T U H 2 Ω T T U H 2 Ω. 5 We are now in position to prove the main result of Theorem of this paper. Theorem 4.8. Assume that n and n are in C 4 Ω. Let λ be a simple real transmission eigenvalue corresponding to n and let u H 2 Ω be the corresponding eigenvector. This implies in particular that 49 holds. Further assume that u and A u are in C 6 Ω.

21 2 H.BOUJLIDA, H. HADDAR AND M.KHENISSI Then, for > small enough, there exists a transmission eigenvalue λ corresponding to n such that = n n λ λ β λ n u 2 dsx + O Proof. Using estimate 5 with λ = τ, we have Γ T T U, U C T T U λ λ H 2 Ω T T U H 2 Ω 52 From the definition of 9 of T, we have A T U = B u λ A C 2 C 2 u C 2 u A B u λ A C 2 C 2 u = + C 2 u A A B B u λ A + A A B u + λ C u A C 2 C 2 C 2 u Using the definition of T, we obtain A B T T U = B u λ A C 2 C 2 C 2 u C 2 C 2 u A A B + u + λ C u + From 33 and 36, we have On the other hand, we have T A A B B u λ A A C 2 C 2 C 2 u T T U H 2 Ω = O T U = B B A u B A A u λ C 2 C 2 C 2 u β C 2 C 2 A u C 2 A A u B B A β C 2 C 2 A From estimates 33 and 47, we obtain T A u A u T U H 2 Ω = O. 54

22 ASYMPTOTIC EXPANSION OF T.E 2 Next, 52 implies Using the expression of U we see that λ λ = T T U, U + O 3 2. T T U, U = β B B u, u + β λ C 2 C 2 C 2 u, u + β A A A B u + λ C u, u + β C 2 C 2 u, λ C 2 u + O 3 2 Recall that, by definition of u Au + λbu + λ 2 C u =. 55 Since C 2 is self-adjoint, we have T T U, U = β B B u, u + 2 β λ C 2 C 2 C 2 u, u A A A u, A u + O 3 2. β λ We then deduce = B B u, u + 2 λ C 2 C 2 λ λ β β C 2 u, u A A A u, A u + O β λ In order to conclude, we use the results of the two lemmas below that treat the asymptotic for each term in 56. Applying Lemma 4.9 with u = u and φ = u we have and with u = u and φ = C 2 u, we obtain B B u, u C C 2 C 2 u, C 2 u C Applying Lemma 4. with u = A u therefore z = u and φ = u, we get, using the fact that A is selfadjoint, A A A u, A u = L n n n 2 u ϕs, u ϕs, ds + O 3 2

23 22 H.BOUJLIDA, H. HADDAR AND M.KHENISSI We finally obtain L = n n λ λ β λ n u ϕs, 2 ds + O which corresponds with the formula announced in the theorem and concludes the proof. Lemma 4.9. Under the assumptions of Theorem 4.8 one has B B u, φ C 3 2 φ H 2 Ω and C 2 for some C independent of and φ. C 2 u, φ C 2 φ H 2 Ω Proof. Since B B u, φ = Ω u φ + uφdx, n n Using the local coordinates in Ω, we obtain L uφdx = m uϕs, ξ n n φs, ξ + ξκdsdξ Ω = L C 3 2 φ H 2 Ω ξ 2 φ m uϕs, η s, ηdη + ξκdsdξ 2 Hence B B u, φ C 3 2 φ H 2 Ω. Similarly, we compute the asymptotic formula of C C u, φ = = L L n C 2 φ H 2 Ω n uϕs, ξφϕs, ξ + ξκdsdξ n ξ n ũ s, ηdη η ξ φ η s, ηdη + ξκdsdξ Using the square root Lemma in [24] and the fact that C n converges to C n at the same order O 2, we conclude that C 2 converges to C 2 at the same order O 2. Thus we have C 2 C 2 u, φ C 2 φ H 2 Ω Lemma 4.. Under the assumptions of Theorem 4.8 one has for any φ HΩ 2 C 4 Ω L A A A n n u, φ = n z ϕs, φϕs, ds + O where z := A u.

24 ASYMPTOTIC EXPANSION OF T.E 23 Proof. With z := A u, z z φdx = z φdx = m z φdx Ω n Ω n n Ω = m z z z φdx + m z + z φdx Ω Ω Applying Lemma 4.5 we obtain z z φdx = m z + z Ω n φdx + O 3 2 Ω = m z φdx m ψ φdx + O 3 2. Ω Ω Making use of the local coordinates we show L m z φdx = m z ϕs, ξ φϕs, ξ + ξκdsdξ Ω = L m z ϕs, φϕs, ds + O 3 2. For the second term m ψ φdx = ψ m φdx m ψ Ω Ω Γ ν φdsx + mψ φ Γ ν dsx Or ψ Γ = and ψ ν n Γ = z ϕs,. Then we have n m ψ φdx = ψ m φdx m ψ Ω Ω Γ η φdsx = = L L L Consequently z z φdx = n Ω which implies A A and concludes the proof. ψs, ξ m φϕs, ξ + ξκdsdξ n m z ϕs, φϕs, ξds n n m z ϕs, φϕs, ds + O 3 2 n L n n n 2 z ϕs, φϕs, ds + O 3 2 L A n n u, φ = n z ϕs, φϕs, ds + O 3 2 2

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