Steklov Problems in Perforated Domains with a Coefficient of Indefinite Sign

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American Institute of Mathematical Sciencesc

Volume7, Number1, March2012 pp.151–178

STEKLOV PROBLEMS IN PERFORATED DOMAINS WITH A COEFFICIENT OF INDEFINITE SIGN

Valeria Chiado Piat

Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi 24

10129 Torino, Italy

Sergey S. Nazarov

Institute for Problems in Mechanical Engineering RAS Bolshoi ave., 61, 199178, St-Petersburg, Russia

Andrey L. Piatnitski

Narvik University College, P.O.Box 385 Narvik 8505, Norway

and Lebedev Physical Institute RAS Leninski pr., 53, 119991, Moscow, Russia

(Communicated by Leonid Berlyand)

Abstract. We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in periodically perforated domain under the assumption that the spectral weight function changes sign. We show that the limit behaviour of the spectrum depends essentially on wether the average of the weight function over the boundary of holes is positive, or negative or equal to zero. In all these cases we construct the asymptotics of the eigenpairs.

1. Introduction. The paper studies Steklov spectral problem in a periodically perforated domain for the Laplace operator or for more general divergence form elliptic operator with periodic coefficients, under the assumptions that the Steklov condition is imposed on the perforation boundary and that the corresponding periodic weight function changes sign.

Previously, periodic homogenization of a bulk spectral problem with sign-changing density for an elliptic operator or an elliptic system was carried out in recent works [20], [19]. It was shown that the asymptotic behaviour of spectrum depends crucially on whether the mean value of the weight function is positive, or negative, or equal to zero.

The idea of studying Steklov and other spectral problems with sign-changing weight function arose during the conference “Differential Equations and Related Topics” in Moscow in 2007. It occurs after the talk “Homogenization in perforated domains with Fourier boundary conditions” that focused on homogenization of elliptic problems with Fourier boundary condition on the perforation surface under the assumption that the coefficient of the boundary operator changes sign. It turned out that the limit behaviour of solutions depend crucially on whether the average

2000Mathematics Subject Classification. Primary: 35B27.

Key words and phrases. Homogenization, Steklov problem.

151

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of this coefficient over the perforation surface is positive, or negative, or equal to zero (see [8] for further details).

Steklov spectral problem, although has many common features with the bulk problem, differs essentially from the bulk problem due to the facts that the surface volume of the perforation tends to infinity, as the period vanishes, and that the perforation geometry is asymptotically singular. This leads to a different scaling of the eigenpairs asymptotics. Also, the technique used here relies on a number of scaled versions of the trace and Poincare inequalities. There are also serious difficulties related to the non-trivial interaction of the perforation and the exterior boundary of the domain (see the last section of the paper).

In the existing literature the homogenization of Steklov spectral problem in periodically perforated domain with a constant positive weight function was studied for the first time in [24], where the effective spectral problem was constructed and the convergence of spectra was proven. The estimates for the rate of convergence have been obtained in [22].

In the case of inclusions situated along the exterior boundary the averaging of Steklov problem was considered in [23].

Boundary value problems in perforated domains with Robin and Dirichlet boundary condition have been considered in many works, see, for instance, [10], [11], [7].

In the case of sign-changing weight function the asymptotic behaviour of spectrum changes drastically. The detailed formulation of the Steklov problem studied in the present paper is







−∆uε= 0 in Ω_ε,

∂u_ε

∂ν_ε =λερεuε, on Γε, u_ε= 0, on∂Ω,

(1) here Ω is a smooth bounded domain, Ωεis the corresponding perforated domain, Γε

is the surface of a smooth periodic perforation consisting of disjoint inclusions,νε

is the exterior unit normal on Γ_ε, andεis a small positive parameter. We assume that the functionρis periodic and changes sign (see Section2 for further details).

By exactly the same methods one can study a slightly more general problem of the form







−div a x ε

∇uε) = 0 in Ωε,

∂uε

∂ν_ε^a =λερεuε, on Γε,

u_ε= 0, on∂Ω,

(2) with a periodic symmetric matrixa(y) that satisfies the uniform ellipticity conditions,ν_ε^a=a(x/ε)ν_ε.

We first prove that the spectrum of the considered Steklov problem (1) (or (2)) is discrete and, since the weight function ρ defines an indefinite metric on the perforation border (see [6])), the spectrum consists of two infinite sequences, one converges to +∞and another to−∞.

We show that the asymptotic behaviour of spectrum in (1) and (2), as ε →0, depends essentially on whether the average ofρover the surface of the hole is greater than zero, or less than zero, or equal to zero.

If the average ofρis positive (negative), then the positive (negative) part of the spectrum behaves in a regular way and admits homogenization like in the classical case whenρ >0. In particular, for anyk∈N, thek-th positive eigenvalue is of order ε, and the corresponding eigenfunction has a boundedH¹ norm. The convergence result in this case is presented in Theorem2.1.

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If ρ has zero average then both positive and negative eigenvalues have finite limits and the limit behaviour of the corresponding eigenpairs can be described in terms of the effective quadratic operator pencil. This operator pencil has a very simple structure and can be reduced to a standard eigenvalue problem for an elliptic operator in Ω. Notice that in this case thek-th negative and positive eigenfunctions are bounded inH¹-norm. The asymptotic behaviour of the spectrum in the case of zero averageρis described in Theorem2.3.

Finally, if the average of ρ is positive then the negative part of the spectrum of (1) (or (2)) shows a singular behaviour. Namely, for any k ∈ N the k-th negative eigenvalue is of order 1/ε and the corresponding eigenfunctions are rapidly oscillating.

We show that studying the negative part of the spectrum can be reduced to studying the negative part of the spectrum of an auxiliary problem that exhibits more regular behaviour. This reduction is done by means of factorization with the first negative eigenfunction of the corresponding cell periodic spectral problem.

Further details can be found in Theorem2.2and its proof.

The factorization principle has been widely used in the homogenization theory.

It applies efficiently when studying both spectral and boundary value problems for singularly perturbed operators or problems in perforated domains with Robin and Dirichlet boundary conditions. We refer to [15], [3], [4] for further details.

Independently of our work and at the same time closely related problem was considered in the recent preprint [13]. Several convergence results obtained there are similar to ours. However, the methods used there do not allow one to obtain estimates for the rate of convergence. This makes an essential difference with the present work. Also, we have serious doubts about the statement of Theorem 3.3 in [13]. Under the geometric assumptions made in [13], there might be additional eigenpairs supported in the vicinity of the exterior boundary of the perforated domain. Thus, the series of eigenpairs introduced in Theorem 3.3 need not be at the bottom of the negative part of spectrum.

2. Setting of the problem and main results. In this section we provide a detailed set up of the studied Steklov spectral problem, introduce necessary notation and auxiliary problems, and then formulate the main results of the paper.

Let Ω be a smooth bounded domain in Rⁿ. We denote by Y = (0,1)ⁿ the unit cube ofRⁿ, and byω =Y \B the perforated reference cell, for a given closed set B⊂Y with sufficiently smooth boundary∂B= Γ. We assume thatωis a connected set. Setting

Jε={z∈Zⁿ : ε(Y +z)⊂Ω}, (3) we define B_ε = S

z∈J_ε

ε(z+B), Γ_ε = S

z∈J_ε

ε(z+ Γ). Then a perforated domain is introduced as

Ω_ε= Ω\B_ε.

It should be noted that, under our assumptions, the perforationB_ε consists of an asymptotically large number of non-intersecting closed inclusions. Notice also that, according to (3),B_ε does not intersect the external boundary∂Ω.

Remark 1. Another possibility is not to remove the perforation in the vicinity of ∂Ω. Instead, we can keep this part of perforation and impose the homogeneous

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Dirichlet boundary condition on it. We denote Ωeε= Ω\ [

z∈Zⁿ

ε(z+B). (4)

If the perforation has an unbounded connected component, then the interface between Steklov and Dirichlet boundary conditions might be rather irregular. This leads to additional technical difficulties. We do not consider this case.

Throughout this paper we assume that the exterior boundary∂Ω has the regular- ityC^2,α. In many our statements this regularity can be replaced with just Lipschitz continuity of the boundary. However, in this case we obtain only convergence results without estimating the rate of convergence.

In what follows the symbol Γ#stands for the periodic extension of Γ inRⁿ. Also, the lower index # in the functional space notation indicates that the corresponding functions are periodic.

Given a functionρ∈L^∞_#(Γ), we study the asymptotic behaviour of the eigenvalue problems







−∆uε= 0 in Ωε,

∂u_ε

∂νε =λ_ερ_εu_ε, on Γ_ε, uε= 0, on∂Ω,

(5) asε→0. The corresponding weak formulation reads











u_ε∈H_ε, Z

Ωε

∇uε· ∇v dx=λε

Z

Γε

ρεuεvdσx ∀v∈Hε,

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where

Hε={v∈H¹(Ωε) :v= 0 on∂Ω}

is a Hilbert space equipped with the scalar product (u, v)H_ε=

Z

Ωε

∇u· ∇v dx,

andσ_xdenotes the (n−1)-dimensional surface Lebesgue measure.

We also consider a similar problem inΩe_ε







−∆uε= 0 in Ωf_ε,

∂u_ε

∂νε =λ_ερ_εu_ε, on Γ_ε, u_ε= 0, on∂Ωe_ε\Γ_ε.

(7) Every solutionu_εof problem (5) or (7) can be extended to the whole domain Ω as a function ˜u_ε∈H₀¹(Ω), with uniform estimates

Z

Ω

|∇˜uε|²dx≤c0

Z

Ω_ε

|∇uε|²dx, Z

Ω

|˜uε|²dx≤c0

Z

Ω_ε

|uε|²dx

for allε >0 and for somec0>0 that does not depend onε(see, for instance, [1]).

In the sequel, abusing slightly the notation, we still denote this extension byuε. Let us notice that, thanks to the above inequality, the usual Friedrichs inequality inHε

holds true with a constantc1 independent ofε, i.e., Z

Ω_ε

u²dx≤c₁ Z

Ω_ε

|∇u|²dx ∀u∈H_ε. (8)

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Throughout this paper we assume that the coefficient ρ satisfies the condition of indefinite sign

σy({y∈Γ :ρ(y)>0})>0 and σy({y∈Γ :ρ(y)<0})>0; (9) here and in the sequelσy stands for the (n−1)-dimensional surface Lebesgue measure. The limit behaviour of problems (5) appears to be different if the mean value ρofρ,

ρ= 1 σy(Γ)

Z

Γ

ρ(y)dσ_y(y), (10)

is zero or non zero.

We begin by considering problem (5) for a fixed positiveε.

Proposition 1. For each ε > 0 the spectrum of problem (5) consists of two sequences of eigenvalues

0< λ^ε₁≤λ^ε₂≤. . .≤λ^ε_j →+∞ (11) 0> λ^ε₋₁≥λ^ε₋₂≥. . .≥λ^ε_−j → −∞ asj→+∞ (12) Moreover, for all ε >0 there exists an orthonormal basis in Hε of eigenfunctions u^ε_j ∈ Hε which are solutions to problem (5) corresponding to λε=λ^ε_j, and for all i, j∈Z\ {0}

Z

Ω_ε

∇u^ε_i · ∇u^ε_jdx=δij. (13) Furthermore,

λ^ε₁ and λ^ε₋₁ are simple. (14)

The proof of this proposition will be given in Section 2.

Similar statement holds true for problem (7). Orthogonality condition in this case reads

Z

Ωeε

∇u^ε_i · ∇u^ε_jdx=δ_ij. (15) Ifρ >0, the asymptotic analysis of the positive eigenvalues (11) asε→0 involves the spectral properties of the Dirichlet problem

−div(a^eff∇u) =λρσy(Γ)u in Ω,

u= 0 on∂Ω. (16)

wherea^eff is a symmetric positive definite constant (n×n)-matrix whose associated quadratic form is defined by

a^effξ·ξ= inf Z

ω

|ξ+∇w(y)|²dy:w∈H_#¹(Y)

∀ξ∈Rⁿ, (17) andH_#¹(Y) denotes the space ofY-periodic functionsϕ(y) with finite norm

kϕk_H1

#(Y)= Z

Y

(|ϕ|²+|∇ϕ|²)dy ^1/2

.

The functionw_ξ that provides a minimum in (17) has the formw_ξ =ξ·χwith the vector-functionχ being a periodic solution to the classical cell problem

( ∆χ= 0 inω,

∇χ·ν =−ν(y) on Γ. (18)

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From the classical theory of elliptic operatotrs it follows that the spectrum of (16) is discrete and consists of a sequence{λj}j∈Nof positive eigenvalues,

0< λ₁≤λ₂≤. . .≤λ_j→+∞ asj→+∞, (19) and that the corresponding eigenfunctions {uj}_j∈_N ∈ H₀¹(Ω) form, under proper normalization, an orthonormal basis inL²(Ω). For our purposes it is convenient to normalizeuj,j∈N, as follows

Z

Ω

a^eff∇ui· ∇ujdx=δij. (20)

Then Z

Ω

u_iu_jdx= (ρλ_iσ_x(Γ))⁻¹δ_ij. (21) In what follows we use the notation

Λ ={λ_j:j∈N}.

The asymptotic analysis of negative eigenvalues in (12) as ε → 0 requires two more auxiliary spectral problems. The first one is stated in the periodicity cell with periodic boundary conditions:







−∆p= 0 inω,

∂p

∂ν =αρp, on Γ,

p isY-periodic.

(22) The corresponding weak formulation reads





 Z

ω

∇p· ∇w dy=α Z

Γ

ρpw dσy ∀w∈H_#¹(Y), p∈H_#¹(Y).

(23) Here,αis the spectral parameter. The statement below describes the behaviour of spectrum of problem (22). This statement will be proved in Section 2. The proof is more involved than that of Proposition 1 because the quadratic form related to (23) is not coercive.

Proposition 2. Let ρ >0. Then the spectrum of problem problem (22) is discrete and consists of two sequences of eigenvalues

0 =α1< α2≤. . .≤αj→+∞ asj→+∞, (24) 0> α₋₁> α₋₂≥. . .≥α_−j→ −∞ asj→+∞. (25) Moreover α1, α−1 are simple and the associated eigenfunctions p1, p−1∈H_#¹(Y)∩ L^∞(ω) can be normalized as follows

p_±1>0 in ω, Z

Γ

ρ(p_±1)²dσy=±1. (26) Finally, if ∂ω ∈ C^2,α andρ∈C^α(∂B), then p_± ∈ C²(ω), and 0< C₋ ≤p_± ≤C⁺ for some constants C₋ andC⁺.

Now, we introduce the second spectral problem, which is stated in the whole set Ω and involves a new weight functionρ^∗=ρ^∗(y) and its mean valueρ^∗:

ρ^∗=ρ p²₋₁, (27)

ρ^∗= 1 σ_y(Γ)

Z

Γ

ρ^∗(y)dσy. (28)

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Due to Proposition2, Z

ω

|∇p−1|²dy=α₋₁ Z

Γ

p²₋₁ρ dσ_y >0, and hence

ρ^∗ = 1 σy(Γ)

Z

Γ

ρ^∗dσ_y = Z

Γ

p²₋₁ρ dσ_y<0. (29) Define by ˜a^eff the constant positive definite (n×n)-matrix whose associated quadratic form is defined by

˜

a^effξ·ξ= inf Z

ω

|ξ+∇w(y)|²(p₋₁(y))²dy:w∈H_#¹(Y)

∀ξ∈Rⁿ, (30) Notice that a minimum in (30) is attained at the function ˜wξ = ξ·χ˜ with the vector-function ˜χ being a periodic solution to the following cell problem

( div (p₋₁)²(I+∇χ)˜

= 0 inω,

∇χ˜·ν=−ν(y) on Γ, (31)

hereIstands for the unit matrix.

We now introduce the effective spectral problem:

−div(˜a^eff∇v) =κρ^∗σ_y(Γ)v in Ω,

v= 0 on∂Ω, (32)

whereκ is a spectral parameter.

Problem (32) is classical. Sinceρ^∗<0, the spectrum of this problem consists of a sequence

0>κ−1>κ−2≥κ−3≥ · · · ≥κ−j −→ −∞, asj→ ∞. (33) The corresponding eigenfunctions {v_−j}_j∈_N, under proper normalization, form an orthonormal basis inL²(Ω). However, we normalize them in a different way. Namely, we assume that

Z

Ω

˜

a^eff∇v−i· ∇v−jdx=δij. (34) The following results concern the case of ρ > 0. It should be noted that, in this case, the positive and the negative parts of the spectrum show totally different behaviour. We first deal with the positive part of the spectrum.

Theorem 2.1. Let ρ > 0, and let (λ^ε_j, u^ε_j) be the j-th eigenpair of problem (5), (13), or problem (7) withj >0. Then

(i) For allj∈N

λ^ε_j

ε →λ_j asε→0, (35)

whereλj is thej-th eigenvalue of problem (16).

(ii) Under the additional assumption that Ωis aC^2,δ domain with someδ >0 the rate of convergence in (40) can be estimated as follows: for every j∈Nthere exist constants ε_j, C_j >0 such that

λ^ε_j ε −λ_j

≤C_j√

ε for allε∈(0, ε_j). (36)

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(iii) If, forj ∈N,λj is an eigenvalue of problem(16) of multiplicity mj, λ_j−1 <

λj =λj+1 = . . . =λj+m_j−1 < λj+m_j, then there exist orthogonal mj×mj

matricesU^ε and constantsεj >0and Cj >0 such that, for all ε∈(0, εj],

u^ε_j+l−1−

mj

X

k=1

U_lk^ε u_j+k−1 L²(Ω)

≤Cj

√ε, l= 1,· · · , mj, (37)

u^ε_j+l−1−

mj

X

k=1

U_lk^ε U_j+k−1^ε H^ε

≤Cj

√ε, l= 1,· · · , mj (38)

withU_j^ε(x) =uj(x) +εχ(x/ε)∇uj(x), here χis a solution of problem (18).

(iv) The function{U_j^ε} are almost orthogonal and normalized inH_ε that is

hU_k^ε, U_l^εiH_ε−δk,l

≤C√

ε. (39)

The same results hold true for problem (7)

We turn to the negative part of the spectrum. Here, in addition to the above assumptions, we suppose that the boundary ofB has regularityC^2,α and thatρis H¨older continuous, ρ∈C^α(∂B). Here we only consider problem (7).

Theorem 2.2. Let ρ > 0, and let (λ^ε_−j, u^ε_−j) be the j-th negative eigenpair of problem (7), (15). Then

(i) For allj∈N 1 ε

λ^ε_−j−α₋₁ ε

→κ−j asε→0, (40) where α−1 is defined in (25), and κ−j is the j-th (negative) eigenvalue of problem (32).

(ii) IfΩis aC^2,δdomain for someδ >0then for everyj∈Nthere exist constants ε_j, C_j>0 such that

1 ε

λ^ε_−j−α₋₁ ε

−κ−j

≤Cj

√ε for allε∈(0, εj). (41) (iii) If, forj∈N,κ−j is an eigenvalue of problem (32) of multiplicitym_−j,κ−j= κ−(j+1)=. . .=κ−(j+mj−1), then there exist orthogonalm_−j×m_−j matrices U^ε and constantsε_−j>0 andC_−j>0 such that, for allε∈(0, ε_−j],

u^ε_−(j+l−1) ku^ε_−(j+l−1)k_L₂_(Ω) −

m_j

X

k=1

U_lk^ε v_−(j+k−1)^ε L²(Ω)

≤C−j

√ε, l= 1,· · · , mj, (42)

with v^ε_−j(x) = (kv_−jk_L₂_(Ω))⁻¹v_−j(x)ˆp₋₁(x/ε); here pˆ₋₁ is the eigenfunction of problem (22) that corresponds toα₋₁ and is normalized by

Z

ω

(ˆp₋₁(y))²dy= 1.

(iv) The functions{U_−j^ε },U_−j^ε (x) =v−j(x) +ε˜χ(x/ε)∇v−j(x), are almost orthogonal and normalized inHε that is

hU_−k^ε , U_−l^ε iH_ε−δk,l

≤C√

ε. (43)

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Remark 2. In contrast with problem (7) we cannot assure that the interval ^α⁻¹_ε ,0 belongs to the resolvent set of spectral problem (5). If there are eigenvalues of problem (5) on this interval, then the corresponding eigenfunctions concentrate in the vicinity of∂Ω that is they are of boundary layer type.

In order to write down the limit problem in the case ρ = 0 we introduce one more cell problem:











−∆θ= 0 in ω,

∂θ

∂ν =ρ, on Γ,

θ isY-periodic,

(44)

Since ρ = 0, this problem is solvable, its solution is unique up to an additive constant. Denote

Ξ = Z

Γ

ρ(y)θ(y)dσy= Z

ω

∇θ(y)· ∇θ(y)dy >0, and consider the following operator pencil

( −div(a^eff∇u) =λ²Ξu in Ω,

u= 0 on∂Ω. (45)

and a spectral problem

( −div(a^eff∇u) =νΞu in Ω,

u= 0 on∂Ω. (46)

witha^eff defined in (17).

Since (46) has a discrete spectrum 0 < ν₁ < ν₂ ≤ν₃ ≤ · · · ≤ν_j → ∞, and all the eigenvaluesν_j are positive, the spectrum of (45) is discrete, real and consists of two series

λ⁺_j =√

ν_j, λ⁻_j =−√

ν_j, j = 1,2, . . . (47) Here, for the corresponding eigenfunctions, we impose the following normalization conditions

Z

Ω

a^eff∇ui· ∇ujdx+ Ξ√ ν_iν_j

Z

Ω

u_iu_jdx=δ_ij. (48) Theorem 2.3. Let ρ = 0, and let (λ^ε_j, u^ε_j), j ∈ Z\ {0}, be the j-th eigenpair of problem (5), (13). Then

(i) For allj∈N

λ^ε_±j→λ^±_j, asε→0, (49)

whereλ^±_j are defined in (47).

(ii) Under the additional assumption that Ω is a C^2,δ domain with some δ > 0, for everyj ∈Nthere exist constants εj, Cj >0 such that

λ^ε_±j−λ^±_j ≤Cj

√ε for allε∈(0, εj). (50) (iii) If, for j ∈ N, ν_j is an eigenvalue of problem(46) of multiplicity m_j, ν_j−1 <

ν_j = ν_j+1 = . . . =ν_j+m_j₋₁ < ν_j+m_j, then there exist orthogonal m_j ×m_j matricesU^ε and constantsεj >0and Cj >0 such that, for all ε∈(0, εj],

u^ε_±(j+l−1)−

m_j

X

k=1

U_lk^ε uj+k−1

L²(Ω)

≤Cj

√ε, l= 1,· · ·, mj, (51)

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u^ε_±(j+l−1)−

m_j

X

k=1

U_lk^ε U_±(j+k−1)^ε _H_ε_(Ω)

≤C_j√

ε, l= 1,· · ·, m_j (52) with U_±j^ε (x) = uj(x) +εχ(x/ε)∇uj(x) +λ^±_jθ(x/ε)uj(x), here χ and θ are solutions of problems (18) and (44), respectively.

(iv) The function{U_j^ε} are almost orthogonal and normalized inHε that is

hU_k^ε, U_l^εiH_ε−δk,l

≤C√

ε, k, j∈Z\ {0}. (53) 3. Preliminary statements. We begin this section by recalling some inequalities valid inH_ε. In what follows we denote

ω_εⁱ =ε(ω+i), Γⁱ_ε=ε(Γ +i), i∈Zⁿ.

Poincar´e-Wirtinger inequality. Under our assumptions on Ωεand Γε, there exist a positive constantksuch that for eachu∈Hεthe following inequality holds:

Z

Γε

|u−uε|²dσx≤k ε Z

Ωε

|∇u|²dx, (54)

where we denote byuε(·) the piece-wise constant function obtained by taking the mean value ofuover each perforated cellωⁱ_ε, i.e.,

uε(x) = 1

|ω_εⁱ| Z

ωⁱ_ε

u(y)dy, ifx∈ωⁱ_ε; (55) here|ω_εⁱ|stands for the Lebesgue measure ofω_εⁱ. The above inequality remains valid ifuεis replaced with the piece-wise constant function being equal in eachω_εⁱ to the surface average ofuover Γⁱ_ε.

Trace inequality Z

Γ_ε

|u|²dσ_x≤k_t ε⁻¹

Z

Ω_ε

|u|²dx+ε Z

Ω_ε

|∇u|²)dx

, (56)

Both inequalities can be easily obtained from the standard Poincar´e-Wirtinger and trace inequalities, (see [2], [25]) by means of scaling arguments.

Given g ∈ L²(Γ_ε), consider the following boundary value problem with non- homogeneous Neumann boundary conditions on Γ_ε







∂uε

∂nε =g, on Γε, uε= 0, on∂Ω.

(57) The corresponding weak formulation reads







uε∈Hε, Z

Ωε

∇uε· ∇v dx= Z

Γε

gvdσx ∀v∈Hε, (58)

where

H_ε={v∈H¹(Ω_ε) :v= 0 on∂Ω}

is a Hilbert space equipped with the scalar product (u, v)_H_ε=

Z

Ω_ε

∇u· ∇v dx.

(11)

Proposition 3. For every g ∈L²(Γε) there exists a unique solution uε ∈ Hε to problem (57). Moreover uεsatisfies the following a-priori estimate

||u_ε||_H_ε≤cε^−1/2||g||_L2(Γ_ε), (59) where the constant c >0 is independent ofε.

Proof. The existence and uniqueness ofuε is a straightforward consequence of the Reisz representation theorem for the problem

a(u, v) =F(v) ∀v∈H, where

a(u, v) = Z

Ωε

∇u· ∇v dx, F(v) = Z

Γε

gv dσx, H=Hε.

Moreover, replacingv=uε in the weak formulation (58), and using Friedrichs and trace inequalities (8), (56),we obtain that

||uε||²_H_ε = Z

Ωε

|∇ue|²dx= Z

Γε

guεdσx≤ ||g||_L2(Γε)||uε||_L2(Γε)≤

≤ ||g||_L2(Γ_ε)

k_t

ε⁻¹ Z

Ω_ε

|u|²dx+ε Z

Ω_ε

|∇u|²)dx^1/2

≤

≤ cε^−1/2||g||_L2(Γ_ε)||u_ε||_H_ε.

Dividing by||uε||Hε we obtain the desired inequality (59).

We introduce the operatorKε:Hε→Hεin the following way. For everyu∈Hε, we defineKεuas the unique solution to the problem

Z

Ω_ε

∇(Kεu)· ∇v dx= Z

Γ_ε

ρ_εuv dσ_x, ∀v∈H_ε. (60) The existence and uniqueness ofKεufollows directly from Proposition 3.

Proposition 4. The operatorKε:Hε→Hε is linear, compact and self-adjoint.

Proof. The linearity and self-adjointness of K_ε follows directly from its definition (see (60)). In order to prove the compactness ofK_εnotice that formula (60) defines a bounded linear operator ˜K_εthat mapsL²(Γ_ε) inH_ε. SinceK_εis the composition of the trace operator H_ε 7→L²(Γ_ε) and ˜K_ε, the desired compactness follows from the compactness of the mentioned trace operator (see, for instance, [16]).

Assume thatµε6= 0 is an eigenvalue of the operatorKεanduεis a corresponding eigenfunction. It means that

K_εu_ε=µ_εu_ε i.e.







∂u_ε

∂nε =_µ¹

ερ_εu_ε, on Γ_ε, uε= 0, on∂Ω, Thus,λε= _µ¹

ε is an eigenvalue of problem (5). Now, we recall the spectral properties ofKε.

From general spectral theory, the spectrum of the operatorKεis at most count- able, it consists of two sequences (possibly finite or empty) of positive and negative eigenvalues, and of zero. The latter implies the essential spectrum of Kε. Every non-zero eigenvalue has finite multiplicity. We denote by µ^ε_j, µ^ε_−j the positive and

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negative eigenvalues, for every j ∈N\ {0}, with the convention that the positive eigenvalues are enumerated in decreasing order, the negative ones in increasing order, and each eigenvalue is repeated a number of times equal to its multiplicity.

Moreover, we denote by u^ε_j, and u^ε_−j a sequence of corresponding Hε-normalized eigenfunctions. The following variational characterizations hold true

µ^ε₁= max u∈Hε,

u6= 0 Z

Γε

u²ρεdσx

Z

Ω_ε

|∇u|²dx

, (61)

µ^ε₋₁= min u∈Hε,

u6= 0 Z

Γ_ε

u²ρεdσx

Z

Ω_ε

|∇u|²dx

. (62)

For eachj ∈N,j≥2 one has also

µ^ε_j= max

(u,u^ε_i)_Hε=0, i=1,...,j−1

Z

Γ_ε

u²ρ_εdσ_x Z

Ωε

|∇u|²dx

= min

dimV=j−1 max

u∈V^⊥

Z

Γ_ε

u²ρ_ε, dσ_x Z

Ωε

|∇u|²dx

, (63)

µ^ε_−j= min

(u,u^ε_−i)_Hε=0, i=1,...,j−1

Z

Γ_ε

u²ρ_εdσ_x Z

Ωε

|∇u|²dx

= max

dimV=j−1 min

u∈V^⊥

Z

Γ_ε

u²ρ_εdσ_x Z

Ωε

|∇u|²dx

, (64)

whereV^⊥ stands for the orthogonal complement ofV inHε.

Remark 3. From (56) and the fact thatρ∈L^∞(Γ), it follows that there exists a positive constantk0 such that

εµ^ε_j≤k0 ε²+ 1 β_j^ε

!

for allε >0, j∈N, (65) where β_j^ε is the j-th eigenvalue of the Laplacian with homogeneous Neumann boundary conditions at the boundary of the perforation. More precisely, {β_j^ε}^∞_j=1, 0< β₁^ε≤β₂^ε≤. . ., is the spectrum of the problem







−∆v_j^ε=β_j^εv^ε_j in Ω_ε,

∂v^ε_je

∂ν_ε = 0, on Γε, vε= 0, on∂Ω,

(66) It is known (see, for instance, [24]) that for allj∈N

β^ε_j →β_j asε→0, (67)

withβj eigenvalue of the corresponding homogenized problem −div(a^eff∇vj) =βj|ω|vj in Ω,

vj = 0, on∂Ω, (68)

and

β_j→+∞, as j→+∞. (69)

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Proposition 5. Ifρ satisfies condition (9), then for eachε >0 the sets {j∈N : µ^ε_j>0} and {j∈N : µ^ε_−j <0}

have infinitely many elements.

Proof.

Step 1.We first prove that

µ^ε₋₁<0< µ^ε₁. Letting

ρ⁺_ε = max{ρ_ε,0} ρ⁻_ε = min{ρ_ε,0}, under our assumption (9) onρwe have

Z

Γ_ε

ρ_ερ⁺_ε dσ_x>0.

Denote by{uη}η>0a family of functionsuη∈Hεsuch thatkp

ρ⁺ε−uηk_L₂_(Γ

ε)−→0, asη→0. Such functionsu_η can be easily constructed by means of smoothingp

ρ⁺ε

on Γ_ε. Since

Z

Γε

ρεu²_ηdσx−→

Z

Γε

ρερ⁺_ε dσx, asη→0, then for all sufficiently smallη >0 it holds

Z

Γε

ρ_εu²_ηdσ_x>0. (70)

It remains to combine the last inequality with (61) in order to conclude thatµ^ε₁>0.

In a similar way, one can prove thatµ^ε₋₁<0.

Step 2.Our next goal is to show that for any j∈Nthe inequalitiesµ^ε_−j <0 and µ^ε_j >0 hold.

Assume that µ^ε₁ > 0, . . . , µ^ε_j−1 > 0, and let u^ε₁, . . . , u^ε_j−1 be the corresponding normalized eigenfunctions,hu^ε_i, u^ε_kiH_ε =δik withi, k= 1,2, . . . , j−1.

Consider a collection of sets{S^ε_i}^j_i=1withS_i^ε⊂ {x∈Γε : ρ(x)>0},σx(S_i^ε)>0, S_i^ε∩S_k^ε=∅,i6=k, and denote χ^ε_i the characteristic functions of these sets.

Letχ^δ,ε₁ , . . . , χ^δ,e_j be elements ofHεsuch thatkχ^ε_i−χ^δ,ε_i k_L2(Γε)≤δ,i= 1, . . . , j.

It is clear that for sufficiently small δ > 0 the functions χ^δ,ε₁ , . . . , χ^δ,ε_j are linearly independent. Therefore, there is a non-trivial linear combination Ξ = β^δ,ε₁ χ^δ,ε₁ +

· · ·+β_j^δ,εχ^δ,ε_j such thathΞ, u^ε_iiH_ε= 0, i= 1, . . . , j−1.

It is also clear that for sufficiently smallδ >0 we have Z

Γ_ε

Ξ²ρ⁺dσx>0.

Using Ξ as a test function in (63) we conclude that µ^ε_j >0. In the same way one can show thatµ^ε_−j <0.

It remains to use the induction.

Proof of Proposition1. All the statements of Proposition1 except for (14) follow from the spectral properties of the operator Kε, the fact that λ^ε_j = (µ^ε_j)⁻¹, and from Proposition5.

It remains to prove (14): we will do it forλ^ε₁, the proof forλ^ε₋₁ being analogous.

We first show that each eigenfunctionurelated toλ^ε₁does not change sign in Ω_ε.

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Assume the contrary. Then there is an eigenfunction urelated to λ^ε₁ such that u⁺= max{u,0}andu⁻= min{u,0} are non-trivial functions. Clearly,

Z

Γ_ε

ρ_ε(u⁺)²dσ_x>0 and Z

Γ_ε

ρ_ε(u⁻)²dσ_x>0.

Indeed, if R

Γερ_ε(u⁺)²dσ_x ≤ 0, then R

Γερ_ε(u⁻)²dσ_x ≥ 1. Since R

Ωε|∇u⁻|²dx <

R

Ω_ε|∇u|²dx, this contradicts the variational principle (63). Therefore, Z

Γ_ε

ρ_ε(u⁺)²dσ_x>0.

Similarly,R

Γερ_ε(u⁻)²dσ_x>0.

By (63) we have Z

Ω_ε

|∇u⁻|²dx≤λ^ε₁ Z

Γ_ε

ρ_ε(u⁻)²dσ_x, Z

Ω_ε

|∇u⁺|²dx≤λ^ε₁ Z

Γ_ε

ρ_ε(u⁺)²dσ_x. Summing up these inequalities and considering the relation

Z

Ω_ε

|∇u|²dx=λ^ε₁ Z

Γ_ε

ρ_ε(u)²dσ_x we conclude that

Z

Ωε

|∇u⁺|²dx=λ^ε₁ Z

Γε

ρε(u⁺)²dσx.

Thus, u⁺ is an eigenfunction related toλ^ε₁. Then u⁺ is a non-negative solution of the equation ∆u⁺= 0 in Ω_ε, and the fact thatu⁺is equal to zero at interior points of Ω_εcontradicts the maximum principle.

If we assume that there are two linearly independent positive eigenfunctions u, v∈H_εrelated toλ^ε₁, then

Z

Ωε

(u−cv)dx= 0, forc=Z

Ωε

vdx−1Z

Ωε

udx.

Therefore, u−cv is an eigenfunction that changes sign. This contradiction shows thatλ^ε₁ is simple.

Proof of Proposition2. Our goal is to show that for sufficiently small δ > 0 the quadratic form

J(u) = Z

ω

|∇u(y)|²dy+δ Z

Γ

ρ(y)(u(y))²dσy

is coercive that is

J(u)≥C(δ)kuk²_H₁_(ω) for allu∈H_#¹(Y) (71) withC(δ)>0. The spectral problem for the operator associated withJ reads





 Z

ω

∇p· ∇w dy+δ Z

Γ

ρpw dσy= ˜α Z

Γ

ρpw dσy ∀w∈H_#¹(Y), p∈H_#¹(Y).

(72) The spectrum of this problem coincides with the spectrum of problem (23) shifted byδ. Exploiting (71) by the same arguments as in the proof of Proposition 1 one can deduce that the spectrum of (72), and thus of (23), is discrete and consists of two infinite sequences of eigenvalues, one of these sequences tends to−∞, another to +∞.

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Other statements of Proposition 2 can be justified in the same way as in the proof of Proposition1.

To prove (71) we represent ρas ρ= ρ+ ˆρ with ρ > 0 defined in (10). For an arbitrary functionu∈H_#¹(Y) denoteu= (σy(Γ))⁻¹R

Γu(y)dσy, ˆu=u−u. Then Z

Γ

ρu²dσy = Z

Γ

ρu²+ ˆρ u+ ˆu² dσy=

Z

Γ

ρu²+ 2 ˆρuˆu+ ˆρˆu² dσy

≥ Z

Γ

ρu²−C_ρ(|uˆu|+ ˆu²) dσ_y

with Cρ = 2kρkL^∞. Using the trace and Poincare inequalities we deduce that for anyδ1>0

Z

Γ

Cρ(|u˜u|+ ˜u²)dσy≤ Z

Γ

Cρ

δ1u²+1 δ1

+ 1 ˆ u²

dσy

≤ Z

Γ

Cρδ1u²dσy+C1

1 δ₁ + 1Z

ω

|∇u|²dy.

Combining the last two inequalities and choosingδ1 in such a way that Cρδ1= ¹₂ρ we obtain

Z

Γ

ρu²dσy ≥ Z

Γ

1

2ρu²dσy−C1

1 δ1

+ 1Z

ω

|∇u|²dy.

This yields J(u)≥

Z

ω

|∇u|²dy+δ 2

Z

Γ

ρu²dσ_y−C₁δ1 δ₁ + 1Z

ω

|∇u|²dy.

Finally, takingδsuch thatC₁δ (1/δ₁) + 1

≤1/2, we get J(u)≥1

2 Z

ω

|∇u|²dy+δ 2

Z

Γ

ρu²dσy≥C(δ)kuk²_H1(ω).

4. The case ρ > 0. The aim of this section is to prove Theorem2.1. We begin with an auxiliary statement.

Lemma 4.1. Letu_ε, u∈H₀¹(Ω),||uε||_H¹

0 ≤c,u_ε→ustrongly inL²(Ω)andρ >0.

Then

ε Z

Γ_ε

ρεu²_εdσx→ρσy(Γ) Z

Ω

u²dx, asε→0. (73)

Moreover, for all v∈H₀¹(Ω) ε

Z

Γε

ρεuεv dσx→ρσy(Γ) Z

Ω

uvdx, asε→0. (74)

Proof. Let us denote by ˆuεthe piece-wise constant function that takes the value of the average ofuεin eachε-cell that is

ˆ

uε(x) = ˆu^ε_j ifx∈Y_εⁱ, uˆ^ε_j = 1

|ω_j^ε| Z

ω_j^ε

uεdx.

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Note that, by our assumptions and Poincar´e inequality, it follows that ˆuε → u strongly inL²(Ω) asε→0. In fact

Z

Ω

|uε−uˆε|²dx=X

j

Z

Y_εⁱ

|uε−uˆ^ε_j|²dx≤cε²X

j

Z

Y_εⁱ

|∇uε|²dx≤cε². (75) In order to prove (73), we write

ε Z

Γ_ε

ρεu²_εdσx=ε Z

Γ_ε

ρεˆu²_εdσx+ε Z

Γ_ε

ρε(u²_ε−uˆ²_ε)dσx. (76) The first term can be rearranged as follows

ε Z

Γε

ρεuˆ²_εdσx=εX

j

Z

Γ^ε_j

ρε( ˆu^ε_j)²δσx=

=εX

j

( ˆu^ε_j)²εⁿ⁻¹ Z

Γ

ρ(y)dσ_y=ρσ_y(Γ) Z

Ω

(ˆu_ε)²dx+o(1)

.

Hence, by (75), we can conclude that ε

Z

Γ_ε

ρ_εuˆ²_εdσ_x→ρσ_y(Γ) Z

Ω

u²dx asε→0. The second term in (76) is negligible, since

ε

Z

Γ_ε

ρε(u²_ε−uˆ²_ε)dσx

≤ε Z

Γ_ε

|ρε| |uε−uˆε| |uε+ ˆuε|dσx≤

≤ε Z

Γ_ε

|ρ_ε| |u_ε−uˆ_ε|²dσ_x

^1/2Z

Γ_ε

|ρ_ε| |u_ε+ ˆu_ε|²dσ_x ^1/2

.

The first term on the right hand side can be estimated by means of Poicar´e inequality. We have

ε Z

Γε

|ρε| |uε−uˆε|²dσx

^1/2

≤εkρk^1/2_L_∞

ε Z

Ωε

|∇uε|²dx ^1/2

≤cε^3/2. The second term can be estimated by means of the trace inequality:

Z

Γ_ε

|ρε| |uε+ ˆu_ε|²dσ_x

≤

2kρk_L∞

Z

Γ_ε

u²_εdσ_x+ 2kρk_L∞σ_y(Γ) Z

Ω_ε

u²_εdx

≤c

ε⁻¹ Z

Ωε

u²_εdx+ε Z

Ωε

|∇uε|²dx

. Hence, combining the last two inequalities, we finally have

ε

Z

Γ_ε

ρ_ε(u²_ε−uˆ²_ε)dσ_x

≤cε^1/2,

and (73) follows.

To prove (74) it suffices to notice that uεv= 1

2(uε+v)²−1 2u²_ε−1

2v², then (73) applies.