L I E C

 M

Keywords: 2000 Mathematics Subject Classification. Primary: 35B40, 35C20, 49Q10, 74P15 Secondary Spectral problem, singular perturbation, eigenvalues of Laplacian, shape sensitivity analysis, topology optimization

SHAPE SENSITIVITY ANALYSIS OF EIGENVALUES REVISITED

S.A. NAZAROV AND J. SOKOLOWSKI A. The paper can be considered as a complement to previous papers of the authors. An insight into applied asymptotic analysis of boundary value problems in singularly perturbed domains is presented. As a result, the asymptotic expansions of eigenvalues are obtained and discussed in terms of integral attributes of the geometrical perturbations including the virtual mass tensor, polarization tensor etc. The results are presented in such a way that can be easily employed in numerical methods for shape optimization and inverse problems.

I

Shape optimization problems for eigenvalues are among the most popular subject of extended studies in applied PDE's, we refer the reader e.g., to [START_REF] B | B Variational methods in shape optimisation problems Progress in Nonlinear Differential Equations and their Applications[END_REF][START_REF] H | Extremum problems for eigenvalues of elliptic operators Frontiers in Mathematics[END_REF][START_REF] S | Z Introduction to shape optimization[END_REF][START_REF] Z | Semiderivatives of repeated eigenvalues[END_REF] for a review of known results, and to [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] for a list of references from the field of asymptotic analysis.

Recently, the asymptotic analysis in singularly perturbed geometrical domains [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] is applied to shape optimization [START_REF] S | Ż On topological derivative in shape optimization[END_REF] and the topological derivatives of shape functionals are obtained for elliptic boundary value problems with singularly perturbed boundaries. In the paper we present certain results on topological derivatives for the spectral problems with the Laplace operator. Namely, the asymptotic analysis of eigenvalues is performed with respect to singular perturbations of domains (see Fig. 1, a, b, andc). The results can be directly used in some applications, in particular, in the shape and topology sensitivity analysis of the Helmholtz equation. Compared to the existing results in the literature, the technical difficulties of the asymptotic procedures concern the variable coefficients of differential operators in limit problems that particularly arise from the curved boundaries. The known results are mainly given for singular perturbations of isolated points of the boundary (small holes in the domain, see [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF], [START_REF] K | N Spectral problems in singular perturbed domains and self adjoint extensions of differential operators Trudy St.-Petersburg Mat. Obshch[END_REF], [START_REF] C | N Asymptotics of eigenvalues of a plate with small clamped zone[END_REF], [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF], [START_REF] O | S Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains-the Neumann condition[END_REF] and others), perturbations of straight boundaries including perturbations by changing the type of boundary conditions (cf. [START_REF] G |  Asymptotic form of the eigenvalue of a singularly perturbed elliptic problem with a small parameter in the boundary condition[END_REF] and others), and the dependence on the curvature has been clarified only in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF], where it was shown that the first order correction term for an eigenvalue is independent of the curvature, even if the appropriate change of curvilinear variables leads to differential expressions depending explicitly on the curvature. We revisit our results in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] with two goals. First, we correct all misprints which, unfortunately have appeared in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] (cf. the end of Section 2).

Second, we elucidate and explicate here the integral characteristics of geometrical perturbations which form the asymptotic expansions for eigenvalues, and, therefore, the topological derivative of the eigenvalues as the main correction term.

The description of shape optimisation problems for eigenvalues can be found e.g., in monographs [START_REF] B | B Variational methods in shape optimisation problems Progress in Nonlinear Differential Equations and their Applications[END_REF], [START_REF] H | Extremum problems for eigenvalues of elliptic operators Frontiers in Mathematics[END_REF], [START_REF] S | Z Introduction to shape optimization[END_REF], [START_REF] Z | Semiderivatives of repeated eigenvalues[END_REF], and we recall that the method of boundary variations goes back to Hadamard so the structure of the shape gradient of an differentiable shape functional is called the Hadamard formula [START_REF] S | Z Introduction to shape optimization[END_REF]. There is a natural gap between the regularity of boundaries, from one side for the results on the existence of optimal domains, and the necessary optimality conditions where stronger assumptions on the regularity of boundaries of admissible domains are necessary to compute the directional derivatives of eigenvalues with respect to domain perturbations.

We provide the analysis of non-smooth perturbations of boundaries which uses the same tools [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] as the derivation of topological derivatives of shape functionals. In this way we extend the notion of shape gradient to the case of singular boundary perturbations. The obtained formulae can be employed to obtain informations from optimality conditions about the decreasing or increasing of eigenvalues for the specific boundary perturbations in the form of caverns and knops. Such an information is interesting on its own for the analysis of optimal solutions to shape optimisation problems for eigenvalues.

The outline of the paper is the following. In Section 2 asymptotics of solutions to spectral problems are introduced. In Section 3 integral characteristics of small domains which serves as perturbations are defined by certain solutions to boundary value problems in unbounded domains with specific data. In Section 4 the minmax principle for eigenvalues is recalled and discussed for asymptotics in specific boundary value problems. In Section 5 the case of multiple eigenvalues is focused on. In Sections 6 a control on eigenvalues increments and a simple example of singular boundary perturbations are presented.

A      

Let Ω, ω ⊂ R 2 be domains with the boundaries ∂Ω, ∂ω and the compact closures Ω, ω, respectively. ∂Ω is assumed to be of class C ∞ for simplicity. Given a small parameter ε > 0, we introduce the sets

Ω(ε) = Ω ω ε , ω ε = {ξ ∈ R 2 : ξ := ε -1 x ∈ ω} . (1)
We further have to distinguish between several situations drawn in Fig. 1 where a bullet perces a pillow. If the coordinate origin O is located on ∂Ω and in the interior of ω, we come across the boundary perturbation by the cavity θ ε = Ω ∩ ω ε (Fig. 1,a). Otherwise, we find a small hole (opening) θ ε = ω ε which is situated near the boundary ∂Ω (Fig. 1,b) or far from the boundary in the interior of the domain Ω (Fig. 1,c). We emphasize that the analysis of the first two geometrical situations is performed in the same way, while for the third one is performed in a slightly different way. We proceed with the Neumann spectral problem

-∆ x u ε (x) = λ ε u ε (x), x ∈ Ω(ε), ∂ n u ε (x) = 0, x ∈ ∂Ω(ε), (2)
where ∂ n stands for the outward normal derivative defined almost everywhere on the Lipschitz (by the assumption) boundary ∂Ω(ε). The problem (2) admits the eigenvalue sequence

λ ε 1 < λ ε 2 ≤ λ ε 3 ≤ • • • ≤ λ ε j ≤ • • • → ∞ , (3)
where the eigenvalues are listed according to multiplicity and the first eigenvalue λ ε 1 = 0 is simple. The first limit (ε = 0) problem in the entire domain

Ω -∆ x u 0 (x) = λ 0 u 0 (x), x ∈ Ω, ∂ n u 0 (x) = 0, x ∈ ∂Ω, (4)
has the eigenvalue sequence

λ 0 1 < λ 0 2 ≤ λ 0 3 ≤ • • • ≤ λ 0 j ≤ • • • → ∞ , (5)
with the same properties while the corresponding eigenfunctions u 0 1 , u 0 2 , . . . , u 0 n , . . . are subject to the normalization and orthogonality conditions

(u 0 j , u 0 k ) Ω = δ j,k , j, k ∈ N := {1, 2, . . . }, (6)
where (•, •) Ω stands for the scalar product in the Lebesque space L 2 (Ω) and δ j,k is the Kronecker symbol. In particular, the first eigenfunction is constant and the first eigenvalue λ 0 1 stays unperturbed. The remaining eigenvalues in (5) get certain perturbations in (3) and, we refer for the proof to [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF][START_REF] K | N Spectral problems in singular perturbed domains and self adjoint extensions of differential operators Trudy St.-Petersburg Mat. Obshch[END_REF][START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] and [ [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]; Ch. 9] that the eigenvalues take the asymptotic form

λ ε j = λ 0 j + ε 2 ∇ x u 0 j (O) ⊤ M(θ)∇ x u 0 j (O) + λ 0 j |u 0 j (O)| 2 mes 2 θ + O ε 5/2 (7)
in the case of a simple eigenvalue λ 0 j (see Section 5 for the multiple case). In ( 7), the gradient

∇ x u 0 j (O) is a column vector in R 2 , ∇ x u 0 j (O) ⊤ is the transposed line vector and M(θ) is a matrix of size 2 × 2.
We emphasize that θ = ω in the case of O ∈ Ω but θ must be reconstructed by dilatation from θ ε in the case of O ∈ ∂Ω.

Remark 2.1. Actually, the majorant for the asymptotic remainder in [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] is Cε 3 and, furthermore, the whole asymptotic expansions in powers of ε are available although coefficients in the expansions may become polynomial in | ln ε| (cf. [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF][START_REF] K | N Spectral problems in singular perturbed domains and self adjoint extensions of differential operators Trudy St.-Petersburg Mat. Obshch[END_REF] and [ [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]; Ch. 9]). In the paper we formulate the relation [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] in the same way as the new results given in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] for the perturbations of spectral problem (2) in Fig. 1, a andb. The main result in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] reads: For the spectral problem (4) the first correction term ε 2 λ ′ j is independent of | ln ε| and of the curvature of the contour ∂Ω at the point O.

In the case O ∈ ∂Ω, the asymptotic formula [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] keeps its validity for the mixed boundary value problem

-∆ x u ε (x) = λ ε u ε (x), x ∈ Ω(ε), u ε (x) = 0, x ∈ ∂Ω ω ε , ∂ n u ε (x) = 0, x ∈ ∂ω ε ∩ Ω, (8)
and for the Dirichlet problem

-∆ x u ε (x) = λ ε u ε (x), x ∈ Ω(ε), u ε (x) = 0, x ∈ ∂Ω(ε) = ∂Ω ω ε ∪ ∂ω ε ∩ Ω. (9)
We point out, that the matrix M(θ) depends on the particular problem, i.e. as it can be expected, on the shape of the void and on the boundary conditions on the void as well as on the unperturbed boundary closeby the void. Moreover, the first limit problem in Ω provides ∂ n u 0 (O) = 0 for (8) and u 0 j (O) = ∂ s u 0 j (O) = 0 for [START_REF] M | V M Mathematical modelling of solids with nonregular boundaries[END_REF]. In other words, the asymptotic formula [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] simplifies and involves only scalar characteristics for the boundary perturbations (Fig. 1, a andb).

Remark 2.2. For some specific cases, e.g., the Dirichlet problem with O ∈ Ω and the mixed boundary value problem with O ∈ ∂Ω and the Dirichlet and Neumann conditions on ∂ω ε ∩ Ω and ∂Ω ω ε , respectively, the asymptotic expansions of eigenvalues [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF][START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] are much more elaborated. In particular, the main correction term is of order | ln ε| -1 with the unsatisfactory remainder O(| ln ε| -2 ) while the main term with the remainder O(ε) becomes a holomorphic function in | ln ε| -1 . A serious complication of the asymptotic procedure for systems of differential equations, e.g., in elasticity, provokes for mistakes(cf. [START_REF] M | V M Mathematical modelling of solids with nonregular boundaries[END_REF] and the requisite correction in [START_REF] C | N Asymptotics of eigenvalues of a plate with small clamped zone[END_REF]).

In Fig. 2 and Fig. 3 we outline disposition of the Dirichlet and Neumann boundary conditions which lead to the eigenvalue perturbation of order ε 2 and | ln ε| -1 , respectively.

The asymptotic formula (7) first of all, needs an appropriate description of the matrix M(θ) as an integral characteristics of the perturbation set θ. Unfortunately, the authors had chosen in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] a lame way to introduce M(θ) due to the wrong sign of the Poisson kernel (cf. formula (18) below) that has distorted the final asymptotic formulae in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] although after returning the sign minus to the kernel all calculations get the validity. Our immediate objective is to introduce M(θ) properly.

I 

Let us assume for simplicity that λ 0 j is a simple eigenvalue (see Section 5 for the multiple eigenvalues). According to the asymptotic procedures developed in [ [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]; Ch. 9, 10], the asymptotic ansatz for the eigenfunction u ε j reads

u ε j (x) = u 0 j (x) + εw j (ε -1 x) + ε 2 u ′ j (x) + . . . . ( 10
)
Here w j is the boundary layer term in the form [START_REF] O | S Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains-the Neumann condition[END_REF] and u ′ implies the main regular correction. The function [START_REF] O | S Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains-the Neumann condition[END_REF] is written in the streched coordinates ξ (see formula [START_REF] B | Spectral theory of selfadjoint operators in Hilbert space[END_REF]) and it is a solution of a boundary value problem in an unbounded domain with a proper decay as |ξ| → ∞.

w j (ξ) = 2 p=1 W p (ξ) ∂u 0 j ∂y p (O)
First, we consider the Neumann problem [START_REF] B | Spectral theory of selfadjoint operators in Hilbert space[END_REF] in the case of O ∈ ω (Fig. 1,c). The second limit problem, obtained by streching the coordinates and setting ε = 0, is but the exterior Neumann problem

-∆ ξ W(ξ) = 0, ξ ∈ R 2 ω, ∂ n W(ξ) = G(ξ), ξ ∈ ∂ω . ( 12
)
Due to the Taylor formula

u 0 j (x) = u 0 j (O) + x ⊤ ∇ x u 0 j (O) + O(|x| 2 ) = u 0 j (O) + εξ ⊤ ∇ x u 0 j (O) + O(ε 2 ), ξ ∈ ∂ω ε , (13)
the main discrepancy of the eigenfunction u 0 j of the problem (4) in the Neumann boundary condition on ∂ω ε

G(ξ) = -n(ξ) ⊤ ∇ x u 0 j (O) (14)
is compensated by the linear combination [START_REF] O | S Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains-the Neumann condition[END_REF], where W p is the decaying solution of ( 12) with the specific right-hand side G p (ξ) = -n p (ξ). Since components of the unit normal n(ξ) = (n 1 (ξ), n 2 (ξ)) ⊤ are of mean zero value over the contour ∂ω, the solutions exist and take the form

W p (ξ) = 2 q=1 M pq (ω) ∂Φ ∂ξ q (ξ) + O(|ξ| -2 ) = (15) = - 2 q=1 M pq (ω) ξ q 2π|ξ| 2 + O(|ξ| -2 ), |ξ| → ∞ .
Here Φ(ξ) = -(2π) -1 ln |ξ| is the fundamental solution of the operator -∆ ξ in R 2 .

The matrix M(ω) composed from the coefficients in ( 15) is called [ [START_REF] P | S Isoperimetric inequalties in mathematical physics[END_REF]; Appendix G] the matrix associated with the virtual mass form of the set ω. The representation

M pq (ω) = - R 2 ω ∇ ξ W p (ξ) ⊤ ∇ ξ W q (ξ)dξ -δ p,q mes 2 ω (16)
is known (see [START_REF] P | S Isoperimetric inequalties in mathematical physics[END_REF]). Thus M(ω) is a symmetric and negative definite matrix if the area mes 2 ω of ω is positive.

Remark 3.1. If ω = {ξ : |ξ 1 | ≤ ℓ, ξ 2 = 0}
is a crack of length 2ℓ > 0, the function G 1 and, therefore, the solution W 1 vanish so that the matrix M(ω) is degenerate. However, all asymptotic formulae remain valid (see [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF]).

Finally, we refer to paper [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF] and book [ [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]; Ch. 9] for the asymptotic procedure to compose the Neumann problem in the punctured domain Ω O in order to find out the correction term u ′ in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF]. We emphasize that the compability condition in this problem provides the explicit formula for the correction term ε 2 λ ′ in [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF].

In the case of O ∈ ∂Ω we assume that Ω is located on the right of the x 2 -axis, and that x 2 -axis is tangent to the contour ∂Ω at the point O (see Fig. 1,a). Then 1 F 4. Sets Θ obtained as a union of θ with its miror reflection.

the second limit problem reads:

-∆ ξ W(ξ) = 0, ξ ∈ R 2 + ω, ∂ n W(ξ) = G(ξ), ξ ∈ ∂ω ∩ R 2 + , (17) ∂W ∂ξ 1 (ξ) = 0, ξ ∈ ∂R 2 + ω,
where R 2 + = {ξ : ξ 1 > 0} is the half-plane. Owing to the Neumann condition ∂ n u 0 j (O) = 0, the discrepancy ( 14) takes the form

G(ξ) = -n 2 (ξ) ∂u 0 j ∂x 2 (O).
Hence, W 1 = 0 in (11) while W 2 solves problem (17) with G(ξ) = -n 2 (ξ) and admits the asymptotic form

W 2 (ξ) = -M N (θ) ξ 2 π|ξ| 2 + O(|ξ| -2 ), ξ → +∞ , (18)
where θ = ω ∩ R 2 + . We point out that the factor of M 22 (θ) implies the Poisson kernel and differs by 1/2 compared to the derivative of the fundamental solution Φ in [START_REF] Z | Semiderivatives of repeated eigenvalues[END_REF].

Let Θ be the union of the set θ and its miror reflection (cf. Fig. 4, a and b, with Fig. 1, a andb, respectively), that is

Θ = θ ∪ {ξ : (-ξ 1 , ξ 2 ) ∈ θ} . ( 19
)
We observe that there is a simple relation between the virtual mass matrix of the set Θ and the matrix M(θ) in the eigenvalue asymptotics. To this end, let us note that the restriction to R 2 + θ of the decaying solution of the exterior problem (12) in the domain R 2 + Θ and with the right-hand side G 2 (ξ) = -n 2 (ξ) coincides with the solution W 2 (ξ) of the problem (17). Hence,

M N (θ) = 1 2 M 22 (Θ) . ( 20
)
In other words, the matrix M(θ) in the eigenvalue asymptotics for the Neumann problem [START_REF] B | B Variational methods in shape optimisation problems Progress in Nonlinear Differential Equations and their Applications[END_REF] takes the form

M(θ) = 0 0 0 M N (θ) , ( 21
)
with the nontrivial entry (20).

For the mixed boundary value problem ( 8), a similar argument can be used. Namely, the decaying solution of the corresponding second limit problem

-∆ ξ W 1 (ξ) = 0, ξ ∈ R 2 + ω, W 1 (ξ) = 0, ξ ∈ ∂R 2 + ω, (22) ∂ n W 1 (ξ) = -n 1 (ξ), ξ ∈ ∂ω ∩ R 2 + , (23)
is but the restriction to R 2

+ ω of the odd in the variable x 1 solution of the exterior problem ( 12) in R 2 Θ with the same right-hand side -n 1 (ξ) as in (23). Therefore, formulae (18), ( 20) and ( 21) can be replaced by

W 1 (ξ) = - 1 2 M 11 (Θ) ξ 1 π|ξ| 2 + O(|ξ| -2 ), M(θ) = 1 2 M 11 (Θ) 0 0 0 . ( 24
)
The Dirichlet problem (9) gives rise to the second limit problem

-∆ ξ W(ξ) = 0, ξ ∈ R 2 + ω, W(ξ) = 0, ξ ∈ ∂R 2 + ω, W(ξ) = -ξ 1 , ξ ∈ ∂ω ∩ R 2 + . (25)
Let us consider again the symmetrized set (19) and replace ( 22), (25) by the exterior Dirichlet problem

-∆ ξ W(ξ) = 0, ξ ∈ R 2 + Θ, W(ξ) = G(ξ), ξ ∈ ∂Θ (26)
Let W p be a bounded solution of (26) for G(ξ) = -ξ p , p = 1, 2. Such a solution is unique and admits the asymptotic expansion

W p (ξ) = c p - 2 q=1 P pq (Θ) ξ q 2π|ξ| 2 + O(|ξ| -2 ), |ξ| → ∞ ,
where c p is a constant, c 1 = 0 by the symmetry, and the coefficients P pq (Θ) form the matrix P(Θ) associated with the polarization tensor of Θ (see [ [START_REF] P | S Isoperimetric inequalties in mathematical physics[END_REF]; Appendix G]). It is known that 16)) and, therefore, P(Θ) is a symmetric positive definite matrix 2 × 2-matrix. The restriction of W 1 onto R 2 ω solves the problem (22), (25) and it follows that

P pq (Θ) = R 2 Θ ∇ ξ W p (ξ) ⊤ ∇ ξ W q (ξ)dξ + δ p,q mes 2 Θ (27) (cf. (
M(θ) = 1 2 P 11 (Θ) 0 0 0 , . (28)
We refer to [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] for the arguments completing the asymptotic ansatz [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] and the derivation of an expression for the correction term ε 2 λ ′ j in the eigenvalue asymptotics [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF].

M-   

The operator theory in Hilbert spaces furnishes the representation of eigenvalues for the Dirichlet problem [START_REF] M | V M Mathematical modelling of solids with nonregular boundaries[END_REF],

λ ε j = max E j inf v∈E j {0} ∇ x v; L 2 (Ω(ε)) 2 v; L 2 (Ω(ε)) 2 , j ∈ N, (29)
(cf. [[1]; Section 10.2]
) where E j is an arbitrary subspace in

H 1 0 (Ω(ε); ∂Ω(ε)) of codimension j -1, i.e., E 1 = H 1 0 (Ω(ε); ∂Ω(ε)
) is a subspace of the Sobolev space H 1 (Ω(ε)) of functions which vanish on the boundary ∂Ω(ε).

Since by construction Ω(ε) = Ω ω ε ⊂ Ω, it follows that H 1 0 (Ω(ε); ∂Ω(ε)) ⊂ H 1 0 (Ω; ∂Ω) and, thus, ( 29) and ( 29) with ε = 0 provide the relationship

λ ε j > λ 0 j , j ∈ N, (30)
which is in accord with the asymptotic expansion [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] taking, in view of (28), the form

λ ε j = λ 0 j + ε 2 2 P 11 (Θ) ∂u 0 j ∂x 1 (O) 2 + O(ε 5/2 ) . (31)
We emphasize that P 11 (Θ) > 0 by ( 27) and the equalities u 0 j (O) = 0, ∂u 0 j ∂x 2 (O) = 0, which simplify [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF], follow from the Dirichlet condition in the first limit problem

-∆ x u 0 (x) = λ 0 u 0 (x), x ∈ Ω, u 0 (x) = 0, x ∈ Ω . (32)
Note that the spectral problem (32) admits the eigenvalues (5) where λ 0 1 > 0 is simple by the maximum principle.

If O ∈ ∂Ω one may consider the domain Ω(ε) = Ω ∪ ω ε perturbed by a knoll. All asymptotic formulae are preserved, however, by the same argument as above the inequality (30) changes for λ ε j < λ 0 j while, simultaneously the factor P 11 (Θ) becomes negative (see [ [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF]; Lemma 5.1]).

For the Neumann problem, the max-min principle (29) applies in the same manner but for a crack ω only (cf. Remark 3.1). Clearly, H 1 (Ω) ⊂ H 1 (Ω(ε)) because functions in the domain Ω(ε) with the cut ω can have a jump over the crack lips. Thus the relation λ ε j ≤ λ 0 j is valid, which in the case of a selvage microcrack is consistent with the asymptotic formula

λ ε j = λ 0 j + ε 2          1 2 M 11 (Θ) ∂u 0 j ∂x 1 (O) 2 + λ 0 j |u 0 j (O)| 2 mes 2 θ          + O(ε 3/2 ) , (33)
with the simple observations: M 11 (Θ) < 0 and mes 2 θ = 0. The above examination of asymptotic formulae for eigenvalues is an obvious indirect way to check the signs of the second terms of the asymptotic ansatz [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]. Sadly enough, this simple step was not taken into account in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF].

P    

Let us consider the Neumann spectral problem [START_REF] B | B Variational methods in shape optimisation problems Progress in Nonlinear Differential Equations and their Applications[END_REF] in the particular case of Fig. 1, a, we refer to [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] for the justification of our asymptotic procedure. Assume, that λ 0 j is an eigenvalue of the multiplicity κ j > 1, i.e., (34)

λ 0 j-1 < λ 0 j = • • • = λ 0 j+κ j -1 < λ 0 j+κ j .
In such a case the asymptotic ansätze [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] and ( 35)

λ ε p = λ 0 j + ε 2 λ ′ p + O(ε 5/2
) are still valid for p = j, . . . , j + κ j -1, however, the principal term takes the form of the linear combinations (36)

u p0 = a p 1 u 0 j + • • • + a p κ j u 0 j+κ j -1
of eigenfunctions corresponding to the eigenvalue λ 0 j . Coefficients of the columns a p = (a p 1 , . . . , a p κ j ) in (36) are to be determined such that (37) a p • a q = δ p,q , p, q = j, . . . , j + κ j -1 .

Since λ 0 j is an eigenvalue of multiplicity κ j , each of the problem for the regular correction terms u ′ j , . . . , u ′ j+κ j -1 in (10) gets κ j compability conditions, which can be written in the form of the following linear system of κ j algebraic equations Formula (39) is derived in exactly the same way as it is for the term ε 2 λ ′ j in [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] (see [START_REF] M | P Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes[END_REF][START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] and [ [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF]; Ch. 9] for details).

The matrix M is symmetric, and its real eigenvalues λ j′ , . . . , λ j+κ j -1′ correspond to the eigenvectors a j , . . . , a j+κ j -1 , satisfying the orthogonality and normalization conditions (37). Actually, just these attributes of the matrix M with the elements (39) are included in the asymptotic ansätze [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] and (35) for the eigenvalues λ ε p and the eigenfunctions u ε p of the problem (2) for p = j, . . . , j + κ j -1 in case (34). An estimate of the asymptotic remainder in the eigenvalue expansion (35) is obtained in [START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF].

C  

The asymptotic expansion [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] for the first eigenvalue λ ε 1 of the Dirichlet problem (9) in the domain Ω(ε) with the small cavity θ ε (Fig. 1, a andb) takes the form (31) where the coefficient P 11 (Θ) is positive (see ( 27), (28)). Thus, the eigenvalue increment △λ ε

1 = λ ε 1 -λ 0 1 > 0 (cf. ( 30
)) becomes maximal (is maximized) provided that the absolute maximum of the function ∂Ω ∋ x → ∂ n u 0 1 (x) is attained at the point O ∈ ∂Ω.

For the Neumann problem (2), the first eigenvalue λ ε 1 = 0 is stable and in the case of the simple eigenvalue λ 0 1 the increment △ λ ε j is given by (33) with the negative coefficient M 11 (Θ) while △ λ ε j can be of any sign. Indeed, if O constitutes a local maximum of the function ∂Ω ∋ x → |u 0 j (x)|, then ∇ x u 0 j (O) = 0 and △λ ε j ≥ 0, however, in the case u 0 j (O) = 0, ∇ x u 0 j (O) 0 we have △λ ε j < 0 because the coefficient M 22 (Θ) is negative.

If θ ε is a selvage micro-crack, i.e., a cut of length ε on the boundary ∂Ω (cf. Remark 2.2 and the end of Section 4 then mes 2 θ = 0 and, therefore, △ λ ε j ≤ 0. The asymptotic expansion can be also employed for solving one more shape optimization problem, namely to maximize the difference λ ε 3λ ε 2 in the case of simple eigenvalues λ ε 3 > λ ε 2 > 0. From formulae [START_REF] M | P Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten[END_REF] and ( 20

F 1 .

 1 A bullet pierces a pillow.

F 2 .

 2 The boundary conditions provide the eigenvalue perturbation of order ε 2 . F 3. The boundary conditions provide the eigenvalue perturbation of order | ln ε| -1 .

  = Ma p with the matrix M = (M mk ) κ j -1 m,k=0 of the size κ j × κ j , (39) M mk = M(θ)∂ s u 0 j+k (O)∂ s u 0 j+m (O) + λ 0 j u 0 j+k (O)u 0 j+m (O)mes 2 (ω).

= 2 π sin px 1 sin qx 2 2 π 2 π

 222 ), (21) it follows that the difference becomes maximal provided at the point O the absolute maximum of the function∂Ω ∋ x → |∇ x u 0 3 (x)| 2 -|∇ x u 0 2 (x)| 2 is attained.Example: Dirichlet problem with Neumann hole We consider Ω = (0, π) 2 and the Dirichlet spectral problem in Ω. In such a case we can determine all eigenvalues and eigenfunctions, namelyλ n = p 2 + q 2 , p, q = 1, 2, . . . , u n = 2 π sin px 1 sin qx 2and therefore, ought to follow the formulae in Section 5. For a simple eigenvalue, e.g., for the case of p = q, we have the following formula for the topological derivative at a point O ∈ Ω,λ ε nλ n = ε 2 [-2π|∇u n (O)| 2 + πΛ n |u n (O)| 2 ] + . . .When we can exchange p q, we have a double eigenvalue λ n = λ n+1 = p 2 +q 2 , with the eigenfunctions of the form u n also in such a case, namely we construct the 2 × 2-matrix M = (M jk ), and the coefficients of M are given byM jk = -2π∇u j (O) ⊤ ∇u k (O) + πλ n u j (O)u k (O) ,(42)where we denote u j = sin px 1 sin qx 2 , u k = sin qx 1 sin px 2 . The eigenvalues of matrix M are denoted by γ 1 , γ 2 , respectively, and determined from the problem Mz = γz, and the formula for the topological derivative of the double eigenvalue λ n takes the form λ ε nλ n = ε 2 γ 1 + . . . (43) λ ε n+1λ n+1 = ε 2 γ 2 + . . . (44) R

According to the calculation applied in[START_REF] N | S Spectral problems in the shape optimisation. Singular boundary perturbations[END_REF] to the Laplacian in curvilinear coordinates the same second limit problem (17) occurs in the case of curved boundary ∂Ω.