2. Two-Scale Convergence on Periodic Surfaces

Volume 2010, Article ID 853717,15pages doi:10.1155/2010/853717

Research Article

Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains

Hermann Douanla

Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenburg, Sweden

Correspondence should be addressed to Hermann Douanla,[email protected] Received 31 July 2010; Accepted 11 November 2010

Academic Editor: Gary Lieberman

Copyrightq2010 Hermann Douanla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloffeigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem.

1. Introduction

We are interested in the spectral asymptoticsas ε → 0 of the linear elliptic eigenvalue problem

−^N

i,j1

∂

∂xi

aij

x,x

ε ∂uε

∂xj

0, inΩ^ε,

N i,j1

aij

x,x

ε ∂uε

∂xjνiλεuε, on∂T^ε, uε0, on∂Ω,

ε

S^ε

|uε|²dσεx 1,

1.1

where Ω is a bounded open set in ^N_x the numerical space of variables x x1, . . . , xN, with integerN ≥ 2with Lipschitz boundary∂Ω,aij ∈ CΩ;L^{∞ N}_y 1 ≤ i, j ≤ N, with

the symmetry conditionaji aij, the periodicity hypothesis: for eachx ∈ Ωand for every k∈^None hasaijx, yk aijx, yalmost everywhere iny∈ ^N_y , and finally the ellipticity condition: there existsα >0 such that for anyx∈Ω

Re N i,j1

aij

x, y ξjξi≥α|ξ|² 1.2

for allξ∈^N and for almost ally∈ ^N_y , where|ξ|² |ξ1|²· · ·|ξN|².

The setΩ^ε ε >0is a domain perforated as follows. LetT ⊂Y 0,1^Nbe a compact subset in ^N_y with smooth boundary∂T≡Sand nonempty interior. Forε >0, we define

t^ε

k∈^N:εkT⊂Ω , T^ε

k∈t^ε

εkT, Ω^ε Ω\T^ε.

1.3

In this setup,Tis the reference hole, whereasεkTis a hole of sizeεandT^εis the collection of the holes of the perforated domainΩ^ε. The familyT^εis made up with a finite number of holes sinceΩis bounded. Finally,ν νidenotes the outer unit normal vector to∂T^ε≡S^ε with respect toΩ^ε.

The asymptotics of eigenvalue problems has been widely explored. Homogenization of eigenvalue problems in a fixed domain goes back to Kesavan1,2 . In perforated domains it was first considered by Rauch3 and Rauch and Taylor4 , but the first homogenization results on this topic pertains to Vanninathan5 , where he considered eigenvalue problems for the laplace operator aij δij Kronecker symbol in perforated domains, and combined asymptotic expansion with Tartar’s energy method to prove homogenization results. Concerning homogenization of eigenvalue problems in perforated domains, we also mention the work of Conca et al.6 , Douanla and Svanstedt7 , Kaizu 8 , Ozawa and Roppongi 9 , Roppongi 10 , and Pastukhova 11 and the references therein. In this paper, we deal with the spectral asymptotics of Stekloffeigenvalue problems for an elliptic linear differential operator of order two in divergence form with variable coefficients depending on the macroscopic variable and one microscopic variable. We obtain a very accurate, precise, and concise homogenization resultTheorem3.7by using the two-scale convergence method12–16 introduced by Nguetseng15 and further developed by Allaire 12 . A convergence result for gradients of eigenfunctions is provided, which improves the understanding of the asymptotic behavior of eigenfunctions. We also justify that the natural local problem is not an eigenvalue problem.

Unless otherwise specified, vector spaces throughout are considered over the complex field,, and scalar functions are assumed to take complex values. Let us recall some basic notations. LetY 0,1^N, and letF ^Nbe a given function space. We denote byFperYthe space of functions inFloc ^Nthat areY-periodic and byF#Ythe space of those functions u ∈FperYwith

Yuydy 0. Finally, the letterEdenotes throughout a family of strictly positive real numbers0< ε≤1admitting 0 as accumulation point. The numerical space ^N and its open sets are provided with the Lebesgue measure denoted bydxdx1· · ·dxN.

The rest of the paper is organized as follows. In Section2, we recall some results about the two-scale convergence method, and the homogenization process is consider in Section3.

2. Two-Scale Convergence on Periodic Surfaces

We first recall the definition and the main compactness theorems of the two-scale convergence method. LetΩbe an open bounded set in ^N_x integerN≥2andY 0,1^N, the unit cube.

Definition 2.1. A sequenceuε_ε∈E ⊂ L²Ω is said to two-scale converge inL²Ωto someu0 ∈L²Ω×Yif, asE ε → 0,

Ωuεxφ

x,x ε

dx−→

Ω×Yu0

x, y φ

x, y dx dy 2.1

for allφ∈L²Ω;CperY.

Notation 1. We express this by writinguε−−→2s u0inL²Ω.

The following theorem is the backbone of the two-scale convergence method.

Theorem 2.2. Letuε_ε∈Ebe a bounded sequence inL²Ω. Then, a subsequenceEcan be extracted fromEsuch that, asE ε → 0, the sequenceuε_ε∈E two-scale converges inL²Ωto someu0 ∈ L²Ω×Y.

Here follows the cornerstone of two-scale convergence.

Theorem 2.3. Letuε_ε∈Ebe a bounded sequence inH¹Ω. Then, a subsequenceEcan be extracted fromEsuch that, asE ε → 0,

uε−→u0, in H¹Ω-weak, uε−→u0, inL²Ω,

∂uε

∂xj

−−→2s ∂u0

∂xj ∂u1

∂yj, in L²Ω

1≤j≤N ,

2.2

whereu0∈H¹Ωandu1∈L²Ω;H_#¹Y.

In the sequel, we denote bydσy y∈Y,dσεx x∈Ω, ε∈E, the surface measures onSandS^ε, respectively. The surface measure ofSis denoted by|S|. The space of squared integrable functions, with respect to the previous measures onSandS^εare denoted byL²S andL²S^ε, respectively. Since the volume ofS^ε grows proportionally to 1/εasε → 0, we endowL²S^εwith the scaled scalar product17

u, v_L2S^εε

S^ε

uxvxdσεx

u, v∈L²S^ε

. 2.3

Definition2.1then generalizes as.

Definition 2.4. A sequenceuε_ε∈E⊂L²S^εis said to two-scale converge to someu0∈L²Ω× Sif as follows.E ε → 0,

ε

S^ε

uεxφ

x,x ε

dσεx−→

Ω×Su0

x, y φ

x, y dx dσ

y 2.4

for allφ∈ CΩ;CperY.

The following result paves the way of the general version of Theorem2.2.

Lemma 2.5. Letφ∈ CΩ;CperY. Then, we have

ε

S^ε

φ

x,x ε

²dσεx≤Cφ²_∞ 2.5

for some constantCindependent ofεand, asE ε → 0,

ε

S^ε

φ

x,x ε

²dσεx−→

Ω×S

φ

x, y ²dx dσ

y . 2.6

Proof. The first part is left to the reader. Letϕ∈ CΩandψ∈ CperY. We have

ε

S^ε

ϕxψ x

ε

²dσεx ε

k∈t^ε

εkS

ϕxψ x

ε

²dσεx. 2.7

Using the second mean value theorem, for anyk∈t^ε, we have

εkS

ϕxψ x

ε

²dσεx ϕxk²

εkS

ψ x

ε

²dσεx 2.8

for somexk∈εkS⊂εkY. Hence,

ε

S^ε

ϕxψ x

ε

²dσεx ε

k∈t^ε

εkS

ϕxψ x

ε

²dσεx

ε

k∈t^ε

ϕ x^k2

εkS

ψ x

ε

²dσεx

ε

k∈t^ε

ϕ

x^k2ε^N−1

kS

ψ

y ²dσ y

S

ψ

y ²dσ y

k∈t^ε

ε^Nϕ x^k2.

2.9

But, asE ε → 0,

k∈t^ε

ε^Nϕ

x^k2−→

Ω

ϕx²dx, 2.10

and the proof is completed due to the density ofCΩ⊗ CperYinCΩ;CperY.

Remark 2.6. Even if often usedsee, e.g.,13,17 , this is the first time Lemma2.5is rigorously proved. It is worth noticing that because of a trace issue one cannot replace therein the spaceCΩ;CperY byL²Ω;CperY.

Theorem2.2generalizes as follows.

Theorem 2.7. Letuε_ε∈Ebe a sequence inL²S^εsuch that

ε

S^ε

|uεx|²dσεx≤C, 2.11

whereCis a positive constant independent ofε. There exists a subsequenceEofEsuch thatuε_ε∈E

two-scale converges to someu0∈L²Ω;L²Sin the sense of Definition2.4.

Proof. PutFεφ ε

S^εuεxφx,x/εdσεxforφ∈ CΩ;CperY. We have Fε

φ ≤C

ε

S^ε

φ

x,x ε

²dσεx 1/2

≤Cφ_∞, 2.12

which allows us to viewFεas a continuous linear form onCΩ;CperY. Hence, there exists a bounded sequence of measuresμε_ε∈Esuch thatFεφ με, φ. Due to the separability of CΩ;CperYthere exists a subsequenceEofEsuch that in the weak^∗topology of the dual ofCΩ;CperYwe haveμε → μ0 asE ε → 0. A limit passageE ε → 0in2.12 yields

μ0, φ≤C

Ω×S

φ

x, y ²dx dσ y

1/2

. 2.13

Butμ0is a continuous form onL²Ω;L²Sby density ofCΩ;CperYin the later space, and there existsu0 ∈L²Ω;L²Ssuch that

μ0, φ

Ω×Su0

x, y φ

x, y dx dσ

y 2.14

for allφ∈ CΩ;CperY, which completes the proof.

In the case whenuε_ε∈Eis the sequence of traces onS^εof functions inH¹Ω, a link can be established between its usual and surface two-scale limits. The following proposition whose proof’s outlines can be found in13 clarifies this.

Proposition 2.8. Letuε_ε∈E⊂H¹Ωbe such that

uε_L²_ΩεDuε_L2Ω^N ≤C, 2.15 whereCis a positive constant independent ofεandD denotes the usual gradient. The sequence of traces ofuε_ε∈EonS^εsatisfies

ε

S^ε

|uεx|²dσεx≤C ε∈E, 2.16

and up to a subsequence E of E, it two-scale converges in the sense of Definition 2.4 to some u0 ∈ L²Ω;L²Swhich is nothing but the trace on Sof the usual two-scale limit, a function in L²Ω;H_#¹Y. More precisely, asE ε → 0,

ε

S^ε

uεxφ

x,x ε

dσεx−→

Ω×Su0

x, y φ

x, y dx dσ y ,

Ωuεxφ

x,x ε

dx dy−→

Ω×Yu0

x, y φ

x, y dx dy

2.17

for allφ∈ CΩ;CperY.

3. Homogenization Procedure

We make use of the notations introduced earlier in Section1. Before we proceed we need a few details.

3.1. Preliminaries

We introduce the characteristic functionχGof

G ^N_y \Θ 3.1

with

Θ

k∈ ^N

kT. 3.2

It follows from the closeness ofT thatΘis closed in ^N_y so thatGis an open subset of ^N_y . Next, letε∈Ebe arbitrarily fixed, and define

Vε

u∈H¹Ω^ε:u0 on∂Ω

. 3.3

We equipVεwith theH¹Ω^ε-norm which makes it a Hilbert space. We recall the following classical result18 .

Proposition 3.1. For eachε ∈ Ethere exists an operatorPε ofVε intoH₀¹Ω with the following properties:

iPεsends continuously and linearlyVεintoH₀¹Ω;

ii Pεv|_Ω^ε vfor allv∈Vε;

iiiDPεv_L2Ω^N≤cDv_L2Ω^εNfor allv∈Vε, wherecis a constant independent ofεand Ddenotes the usual gradient operator.

It is also a well-known fact that, under the hypotheses mentioned earlier in the introduction, the spectral problem1.1has an increasing sequence of eigenvalues{λ^k_ε}^∞_k1,

0< λ¹_ε≤λ²_ε≤λ³_ε≤ · · · ≤λⁿ_ε,

λⁿ_ε −→∞, asn−→∞. 3.4

It is to be noted that if the coefficients a^ε_ij are real valued then the first eigenvalue λ^ε₁ is isolated. Moreover, to each eigenvalue,λ^k_ε is attached to an eigenvectoru^k_ε ∈Vεand{u^k_ε}^∞_k1 is an orthonormal basis in L²S^ε. In the sequel, the couple λ^k_ε, u^kε will be referred to as eigencouple without further ado.

We finally recall the Courant-Fisher minimax principle which gives a usefulas will be seen latercharacterization of the eigenvalues to problem1.1. To this end, we introduce the Rayleigh quotient defined, for eachv∈Vε\ {0}, by

R^εv

Ω^εA^εDv, Dvdx

S^ε|v|²dσεx , 3.5

whereA^εis theN²-square matrixa^ε_{ij1≤i,j≤N}andDdenotes the usual gradient. Denoting by E^kk≥0the collection of all subspaces of dimensionkofVε, the minimax principle is stated as follows: for anyk≥1, thekth eigenvalue to1.1is given by

λ^k_ε min

W∈E^k

v∈Wmax\{0}R^εv

max

W∈E^k−1

v∈Wmin^⊥\{0}R^εv

. 3.6

In particular, the first eigenvalue satisfies λ¹_ε min

v∈Vε\{0}R^εv, 3.7

and every minimum in3.6is an eigenvector associated withλ¹_ε.

Now, letQ^ε Ω\εΘ. This is an open set in ^N, andΩ^ε\Q^εis the intersection of Ωwith the collection of the holes crossing the boundary∂Ω. We have the following result which implies, as will be seen later, that the holes crossing the boundary∂Ωare of no effects as regards the homogenization process since they are in arbitrary narrow stripe along the boundary.

Lemma 3.2see19 . LetK⊂Ωbe a compact set independent ofε. There is someε0 >0 such that Ω^ε\Q^ε⊂Ω\Kfor any 0< ε≤ε0.

Next, we introduce the space

1

0 H₀¹Ω×L²

Ω;H_#¹Y

. 3.8

Endowed with the following norm

v¹

0 Dxv0Dyv1

L²Ω×Y

v v0, v1∈¹₀

, 3.9

1

0 is a Hilbert space admittingF₀^∞DΩ×DΩ⊗ C^∞_# Y as a dense subspace. This being so, for u,v∈¹₀×¹₀, let

aΩu,v ^N

i,j1

Ω×Y^∗aij

x, y ∂u0

∂xj ∂u1

∂yj

∂v0

∂xj ∂v1

∂yj

dx dy. 3.10

This defines a hermitian, continuous sesquilinear form on¹₀×¹₀. We will need the following results.

Lemma 3.3. FixΦ ψ0, ψ1∈F₀^∞, and defineΦε:Ω → (ε >0) by

Φεx ψ0x εψ1

x,x

ε

x∈Ω. 3.11

Ifuε_ε∈E⊂H₀¹Ωis such that

∂uε

∂xi

−−→2s ∂u0

∂xi ∂u1

∂yi, inL²Ω 1≤i≤N 3.12

asE ε → 0, whereu u0, u1∈¹₀, then

a^εuε,Φε−→aΩu,Φ 3.13

asE ε → 0, where

a^εuε,Φε

N i,j1

Ω^εa^ε_ij∂uε

∂xj

∂Φε

∂xi dx. 3.14

Proof. Forε >0,Φε∈ DΩand all the functionsΦεε >0have their supports contained in a fixed compact setK⊂Ω. Thanks to Lemma3.3, there is someε0>0 such that

Φε0, inΩ^ε\Q^ε E ε≤ε0. 3.15

Using the decomposition Ω^ε Q^ε∪Ω^ε\Q^ε and the equalityQ^ε Ω∩εG, we get for E ε≤ε0

a^εuε,Φε

N i,j1

Ω^εaij

x,x

ε ∂uε

∂xj

∂Φε

∂xidx ^N

i,j1

Q^ε

aij

x,x

ε ∂uε

∂xj

∂Φε

∂xidx ^N

i,j1

Ω∩εGaij

x,x

ε ∂uε

∂xj

∂Φε

∂xidx ^N

i,j1

Ωaij

x,x

ε

χεGx∂uε

∂xj

∂Φε

∂xi dx ^N

i,j1

Ωaij

x,x

ε

χG

x ε

∂uε

∂xj

∂Φε

∂xidx.

3.16

Bear in mind that, asE ε → 0, we havesee, e.g.,19, Lemma 2.4

N i,j1

∂uε

∂xj

∂Φε

∂xi

−−→2s ^N

i,j1

∂u0

∂xj ∂u1

∂yj

∂ψ0

∂xj ∂ψ1

∂yj

, inL²Ω. 3.17

We also recall that aijx, yχGy ∈ CΩ;L²_perY1 ≤ i, j ≤ Nand that Property2.1 in Definition2.1still holds forf inCΩ;L²_perYinstead ofL²Ω;CperYwhenever the two- scale convergence therein is ensuredsee, e.g.,14, Theorem 15 . Thus, asE ε → 0,

a^εuε,Φε

N i,j1

Ωaij

x,x

ε

χG

x ε

∂uε

∂xj

∂Φε

∂xidx

−→^N

i,j1

Ω×Yaij

x, y χG

y ∂u0

∂xj ∂u1

∂yj

∂ψ0

∂xj ∂ψ1

∂yj

dx dy

^N

i,j1

Ω×Y^∗aij

x, y ∂u0

∂xj

∂u1

∂yj

∂ψ0

∂xj

∂ψ1

∂yj

dx dy aΩu,Φ,

3.18

which completes the proof.

We now construct and point out the main properties of the so-called homogenized coefficients. Let 1≤j≤N, and fixx∈Ω. Put

ax;u, v ^N

i,j1

Y^∗

aij

x, y ∂u

∂yj

∂v

∂yidy,

ljx, v ^N

k1

Y^∗

akj

x, y ∂v

∂ykdy

3.19

foru, v∈H_#¹Y. Equipped with the seminorm Nu Dyu_L2Y^∗N

u∈H_#¹Y

, 3.20

H_#¹Y is a pre-Hilbert space that is nonseparate and noncomplete. Let H_#¹Y^∗ be its separated completion with respect to the seminorm N· and i the canonical mapping of H_#¹YintoH_#¹Y^∗. We recall that

iH_#¹Y^∗is a Hilbert space;

iii is linear;

iiiiH_#¹Yis dense inH_#¹Y^∗; iviu_H¹

#Y^∗Nufor everyuinH_#¹Y;

vifFis a Banach space andla continuous linear mapping ofH_#¹YintoF, then there exists a unique continuous linear mappingL:H_#¹Y^∗ → Fsuch thatlL◦i.

Proposition 3.4. Letj 1, . . . , N, and fixxinΩ. The noncoercive local variational problem u∈H_#¹Y, ax;u, v ljx, v, ∀v∈H_#¹Y 3.21

admits at least one solution. Moreover, ifχ^jxandθ^jxare two solutions,

Dyχ^jx Dyθ^jx a.e.inY^∗. 3.22 Proof. Proceeding as in the proof of19, Lemma 2.5 , we can prove that there exists a unique hermitian, coercive, continuous sesquilinear form Ax;·,· on H_#¹Y^∗×H_#¹Y^∗ such that Ax;iu,iv ax;u, vfor allu, v∈H_#¹Y. Based onvabove, we consider the antilinear formljx,·onH_#¹Y^∗such thatljx,iu ljx, ufor anyu∈H_#¹Y. Then,χ^jx∈H_#¹Y satisfies3.21if and only ifiχ^jxsatisfies

i χ^jx

∈H_#¹Y^∗, A x;i

χ^jx , V

ljx, V, ∀V ∈H_#¹Y^∗. 3.23

Butiχ^jxis uniquely determined by3.23 see, e.g.,20, page 216 . We deduce that3.21 admits at least one solution, and ifχ^jxandθ^jxare two solutions, theniχ^jx iθ^jx,

which means thatχ^jxandθ^jxhave the same neighborhoods inH_#¹Yor equivalently Nχ^jx−θ^jx 0. Hence,3.22.

Corollary 3.5. Let 1≤ i, j ≤N, andxfixed inΩ. Letχ^jx∈H_#¹Ybe a solution to3.21. The following homogenized coefficients

qijx

Y^∗

aij

x, y dy−^N

l1

Y^∗

ail

x, y ∂χ^j

∂yl

x, y dy 3.24

are well defined in the sense that they do not depend on the solution to3.21.

Lemma 3.6. The following assertions are true:

iqij∈ CΩ, iiqjiq_ij,

iiithere exists a constantα0>0 such that

Re N i,j1

qijxξjξ_i≥α0|ξ|² 3.25

for allx∈Ωand allξ∈^N. Proof. See for example,21 .

We are now in a position to state the main result of this paper.

3.2. Homogenization Result

Theorem 3.7. For eachk≥1 and eachε∈E, letλ^k_ε, u^k_εbe the k’th eigencouple to1.1. Then, there exists a subsequenceEofEsuch that

1

ελ^k_ε −→λ^k₀, in asE ε−→0, 3.26

Pεu^k_ε −→u^k₀, inH₀¹Ω-weak asE ε−→0, 3.27

Pεu^k_ε −→u^k₀, inL²ΩasE ε−→0, 3.28

∂Pεu^k_ε

∂xj

−−→2s ∂u^k₀

∂xj ∂u^k₁

∂yj, inL²Ω, asE ε−→0

1≤j ≤N , 3.29

whereλ^k₀, u^k₀∈ ×H₀¹Ωis the k’th eigencouple to the spectral problem

−^N

i,j1

∂

∂xi

1

|S|qijx∂u0

∂xj

λ0u0, in Ω,

u00, on∂Ω,

Ω|u0|²dx 1

|S|,

3.30

whereu^k₁ ∈L²Ω;H_#¹Y. Moreover, for almost everyx∈Ω, the following hold true:

iu^k₁xis a solution to the noncoercive variational problem u^k₁x∈H_#¹Y,

a

x;u^k₁x, v −^N

i,j1

∂u^k₀

∂xj

Y^∗

aij

x, y ∂v

∂yidy,

∀v∈H_#¹Y.

3.31

iiWe have

i u^k₁x

^N

j1

∂u^k₀

∂xjxi χ^jx

, 3.32

whereχ^jis any function inH_#¹Ydefined by the cell problem3.21.

Proof. Let us first recall that, according to the properties of the coefficientsqijLemma3.6, the spectral problem3.30admits a sequence of eigencouples with similar properties to those of problem1.1. However, this is also proved by our homogenization process.

Now, fixk≥1. There exists a constant 0< c1<∞independent ofεsuch that

0< λ^k_ε ≤c1μ^k_ε, 3.33 where

μ^kε min

W∈E^k

v∈W\{0}max

Ω^ε|Dv|²dx

S^ε|uε|²dσεx

, 3.34

E^k still being the collection of subspaces of dimension k of Vε. But it is proved in 5, Proposition 12.1 that 0 < μ^k_ε < c2ε, c2 being a constant independent ofε. Hence the sequence1/ελ^k_εε∈Eis bounded in.

Clearly, for fixedE ε >0,u^k_εlies inVεand N

i,j1

Ω^εa^ε_ij∂u^k_ε

∂xj

∂v

∂xidx 1

ελ^k_ε

ε

S^ε

u^k_εv dσεx 3.35

for anyv ∈ Vε. Bear in mind thatε

S^ε|u^k_ε|²dσεx 1, and chose v u^k_ε in 3.35. The boundedness of the sequence1/ελ^k_εε∈Eand the ellipticity assumption1.2implies at once by means of Proposition3.1that the sequencePεu^k_εε∈Eis bounded inH₀¹Ω. Theorem2.3 and Proposition 2.8 apply simultaneously and give us u^k u^k₀, u^k₁ ∈ ¹₀ such that for someλ^k₀ ∈ and some subsequenceE ⊂ Ewe have 3.26–3.29, where3.28is a direct consequence of3.27by the Rellich-Kondrachov theorem. For fixedε ∈ E, letΦε be as in Lemma3.3. Multiplying both sides of the first equality in1.1byΦεand integrating overΩ leads us to the variationalε-problem

N i,j1

Ω^εa^ε_ij∂Pεu^k_ε

∂xj

∂Φε

∂xi dx 1

ελ^k_ε

ε

S^ε

Pεu^k_ε

Φεdσεx. 3.36

Sendingε∈Eto 0, keeping3.26–3.29and Lemma3.3in mind, we obtain N

i,j1

Ω×Y^∗aij

∂u0

∂xj ∂u1

∂yj

∂ψ0

∂xj ∂ψ1

∂yj

dx dyλ^k₀

Ω×Su^k₀ψ0dx dσ

y . 3.37

The right-hand side follows by means of Proposition2.8as explained:

ε

S^ε

Pεu^kε

Φεdσεx ε

S^ε

Pεu^kε

ψ₀dσεx ε

ε

S^ε

Pεu^kε

ψ₁

x,x

ε

dσεx

−→

Ω×Su^k₀ψ0dx dσ

y 0, asE ε−→0.

3.38

Therefore,λ^k₀,u^k∈ ×¹₀ solves the following global homogenized spectral problem:

findλ,u∈ ×¹₀ such that N

i,j1

Ω×Y^∗aij

∂u0

∂xj ∂u1

∂yj

∂ψ0

∂xi ∂ψ1

∂yi

dx dyλ|S|

Ωu0ψ₀dx, ∀Φ∈¹₀. 3.39 To provei, chooseΦ ψ0, ψ1in3.39such thatψ0 0 andψ1 ϕ⊗v1, whereϕ∈ DΩ andv1∈H_#¹Yto get

Ωϕx

⎡

⎣^N

i,j1

Y^∗

aij

∂u^k₀

∂xj ∂u^k₁

∂yj

∂v1

∂yidy

⎤

⎦dx0. 3.40

Hence, by the arbitrariness ofϕ, we have a.e. inΩ N

i,j1

Y^∗

aij

∂u^k₀

∂xj ∂u^k₁

∂yj

∂v1

∂yidy0 3.41

for anyv1inH_#¹Y, which is nothing but3.31.

Regarding ii, pick any χ^jx solution to the cell problem 3.21, and put zx _N

j1∂u^k₀/∂xjxχ^jx.

By multiplying both sides of3.21by−∂u^k₀/∂xjxand then summing over 1≤j ≤ N, we see thatzxsatisfies3.31. Hence,izx iu^kxby uniqueness of the solution to the coercive variational problem inH_#¹Y^∗corresponding to the noncoercive variational problem3.31 see the proof of Proposition3.4. Thus,3.32follows sincei is linear.

Now, by consideringΦ ψ0, ψ1in3.39such thatψ10 andψ0∈ DΩ, we get N

i,j1

Ω×Y^∗aij

∂u^k₀

∂xj ∂u^k₁

∂yj

∂ψ₀

∂xi dx dy|S|λ^k₀

Ωu^k₀ψ₀dx. 3.42 As3.32is equivalentsee the proof of Proposition3.4to

Dyu^k₁x ^N

j1

∂u^k₀

∂xjxDyχ^jx, a.e.inY^∗, 3.43

we arrive at N i,j1

Ω

Y^∗

aijdy−^N

l1

Y^∗

ail∂χ^j

∂yldy ∂u^k₀

∂xj

∂ψ₀

∂xi dx|S|λ^k₀

Ωu^k₀ψ₀dx, 3.44

that is,see3.24

N i,j1

Ω

1

|S|qijx∂u^k₀

∂xj

∂ψ₀

∂xidxλ^k₀

Ωu^k₀ψ₀dx. 3.45 Thanks to the arbitrariness ofψ0and the weak derivative formula, we conclude thatλ^k₀, u^k₀ is thekth eigencouple to3.30and the whole sequence1/ελ^k_εε∈Econverges.

Finally, by using3.28and a similar line of reasoning as in the proof of Lemma2.5, we arrive at

Elim ε→0ε

S^ε

P^εu^k_εP^εu^l_εdσεx |S|

Ω

u^k₀u^l₀dx. 3.46

The normalization condition in3.30follows thereby, and moreover{u^k₀}^∞_k1is an orthogonal basis inL²Ω.

References

1 S. Kesavan, “Homogenization of elliptic eigenvalue problems. I,” Applied Mathematics and Optimiza- tion, vol. 5, no. 2, pp. 153–167, 1979.

2 S. Kesavan, “Homogenization of elliptic eigenvalue problems. II,” Applied Mathematics and Optimiza- tion, vol. 5, no. 3, pp. 197–216, 1979.

3 J. Rauch, “The mathematical theory of crushed ice,” in Partial Differential Equations and Related Topics, vol. 446, pp. Lecture Notes in Math.370–379, Springer, Berlin, Germany, 1975.

4 J. Rauch and M. Taylor, “Potential and scattering theory on wildly perturbed domains,” Journal of Functional Analysis, vol. 18, pp. 27–59, 1975.

5 M. Vanninathan, “Homogenization of eigenvalue problems in perforated domains,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 90, no. 3, pp. 239–271, 1981.

6 C. Conca, J. Planchard, and M. Vanninathan, “Existences and location of eigenvalues for fluid-solid structures,” Publications del departemento de matematicas y ciencias de la computation informes tecnicos, Facultad de ciencias fisicas y matematicas, Universitad de Chile, Informe Interno, No. MA- 88-8-352.

7 H. Douanla and N. Svanstedt, “Reiterated homogenization of linear eigenvalue problems in multiscale perforated domains beyond the periodic setting,” Communications in Mathematical Analysis, vol. 11, no. 1, pp. 61–93, 2011.

8 S. Kaizu, “Homogenization of eigenvalue problems for the Laplace operator with nonlinear terms in domains in many tiny holes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 28, no. 2, pp.

377–391, 1997.

9 S. Ozawa and S. Roppongi, “Singular variation of domain and spectra of the Laplacian with small Robin conditional boundary. II,” Kodai Mathematical Journal, vol. 15, no. 3, pp. 403–429, 1992.

10 S. Roppongi, “Asymptotics of eigenvalues of the Laplacian with small spherical Robin boundary,”

Osaka Journal of Mathematics, vol. 30, no. 4, pp. 783–811, 1993.

11 S. E. Pastukhova, “On the error of averaging for the Steklov problem in a punctured domain,”

Differential Equations, vol. 31, no. 6, pp. 975–986, 1995.

12 G. Allaire, “Homogenization and two-scale convergence,” SIAM Journal on Mathematical Analysis, vol.

23, no. 6, pp. 1482–1518, 1992.

13 G. Allaire, A. Damlamian, and U. Hornung, “Two-scale convergence on periodic surfaces and applications,” in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995), A. Bourgeat et al., Ed., pp. 15–25, World Scientific, Singapore, 1996.

14 D. Lukkassen, G. Nguetseng, and P. Wall, “Two-scale convergence,” International Journal of Pure and Applied Mathematics, vol. 2, no. 1, pp. 35–86, 2002.

15 G. Nguetseng, “A general convergence result for a functional related to the theory of homogeniza- tion,” SIAM Journal on Mathematical Analysis, vol. 20, no. 3, pp. 608–623, 1989.

16 V. V. Zhikov, “On two-scale convergence,” Trudy Seminara imeni I. G. Petrovskogo, no. 23, pp. 149–187, 2003, translation in Journal of Mathematical Sciences, vol. 120, no. 3, pp. 1328–1352, 2004.

17 M. Radu, Homogenization techniques, Diplomarbeit, University of Heidelberg, Faculty of Mathematics, July 1992.

18 D. Cioranescu and J. Saint Jean Paulin, “Homogenization in open sets with holes,” Journal of Mathematical Analysis and Applications, vol. 71, no. 2, pp. 590–607, 1979.

19 G. Nguetseng, “Homogenization in perforated domains beyond the periodic setting,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 608–628, 2004.

20 J.-L. Lions and E. Magenes, Probl`emes aux limites non homog`enes et applications. Vol. 1, Travaux et Recherches Math´ematiques, No. 17, Dunod, Paris, France, 1968.

21 A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1978.