bloch wave homogenization of a non-homogeneous neumann problem

Z. angew. Math. Phys. 58 (2007) 969–9930044-2275/07/060969-25DOI 10.1007/s00033-007-6142-7c© 2007 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Bloch wave homogenization of a non-homogeneous

Neumann problem

Jaime Ortega, Jorge San Martın and Loredana Smaranda∗

Abstract. In this paper, we use the Bloch wave method to study the asymptotic behavior of the

solution of the Laplace equation in a periodically perforated domain, under a non-homogeneousNeumann condition on the boundary of the holes, as the size of the holes goes to zero morerapidly than the domain period. This method allows to prove that, when the hole size exceeds a

given threshold, the non-homogeneous boundary condition generates an additional term in thehomogenized problem, commonly referred to as “the strange term” in the literature.

Mathematics Subject Classification (2000). 35B27, 35A25, 42C30.

Keywords. Homogenization, perforated domains, Bloch waves.

1. Introduction

The general question that forms the focus of this paper is the Bloch wave homog-enization of a Neumann-type problem in a periodically perforated domain.

For any ε > 0, we consider the domain Ωε obtained by removing a periodicnetwork of holes from an open bounded domain Ω ⊂ RN (N ≥ 2). The periodicityof the medium is 2πε and the hole size is r(ε). In this domain, we study thehomogenization of the Laplace equation with homogeneous Dirichlet condition onthe exterior boundary and a non-homogeneous Neumann condition on the bound-ary of the holes. In this paper, we are going to consider that the hole size dependson the micro-structure size ε such that r(ε) goes to zero more rapidly than themicro-structure size.

The homogenization process in this same problem, using the classical methodof Tartar’s test functions, was studied in Conca and Donato [5]. Depending onthe hole size, the non-homogeneous boundary condition generates an additionalterm in the homogenized problem, which is referred to as ”the strange term” inthe literature. This terminology has been introduced by Cioranescu and Murat [3]in the study of a similar homogenization problem, with a Dirichlet condition onthe boundary of the holes.

∗Corresponding author

970 J. Ortega, J. San Martın and L. Smaranda ZAMP

Since the seventies, several authors have been driving forces behind the mathe-matical formulation of homogenization. Among others, we may mention I. Babuska,E. De Giorgi, J.-L. Lions, O. Oleinik, G. Papanicolaou, E. Sanchez-Palencia,S. Spagnolo, F. Murat and L. Tartar. Of particular note, Sanchez-Palencia [18]introduced the small parameter in order to describe the micro-structure size andthus, to find the homogenized equation, by studying the asymptotic behavior ofthe problem when this parameter goes to zero. For more details on homogenizationtheory, the reader is referred to the book of Bensoussan, Lions and Papanicolaou[1].

The study of the homogenization process in domains with small holes has beenwidely studied in the literature, using different methods. We briefly mention justa few references: Cioranescu and Saint Jean Paulin [4], Oleinik and Shaposh-nikova [16], Damlamian and Donato [11], Briane [2], Corbo Esposito, D’Apice andGaudiello [10], Conca, Murat and Timofte [8] and the references therein.

Even though there exists plenty of results in homogenization theory whichuse classical methods, there is still a lack of analogous results using the spectralmethod of Bloch waves in the literature. In this paper, we are concerned withthe latter approach to homogenization (for instance, see Bensoussan, Lions andPapanicolaou [1], Morgan and Babuska [14], Santosa and Symes [19], Conca andVanninathan [9]).

The Bloch waves are a family of eigenvalues and eigenvectors associated withthe partial differential equations, providing the spectral resolution of elliptic op-erators. These waves transform the system of partial differential equations into afamily of algebraic equations. The idea of this method is to show that the limit ofthis family, as the micro-structure size tends to zero, represents the homogenizedequation written in the Fourier space. However, this limit process must be studiedon a case-by-case basis. Each such case produces its own difficulties which justi-fies the development of special techniques adapted to each situation. This couldexplain why there are more results using classical methods than the Bloch waveapproach in the literature.

In the sequel, we briefly recall some references to studying homogenizationproblems by using the Bloch wave decomposition. Conca and Vanninathan [9]used this method, in order to homogenize the classical problem of elliptic operatorsin arbitrary domains with periodically and symmetrically oscillating coefficients.Recently, Ganesh and Vanninathan [12] introduced the so-called dominant Blochmode to recover the homogenization result of Murat and Tartar [15] in the generalperiodic case for non-selfadjoint operators.

In terms of using the Bloch wave decomposition to solve problems on periodi-cally perforated domains, we indicate the works of Conca, Gomez, Lobo and Perez[6], [7]. There, the authors studied the asymptotic behavior of the solution of anelliptic boundary value problem with strongly oscillating coefficients in the casewhere the structure periodicity and the hole size are of the same order as ε andwith homogeneous Neumann condition on the boundary of the holes. In contrast,

Vol. 58 (2007) Bloch wave homogenization of a non-homogeneous Neumann problem 971

we consider what happens as the hole size goes to zero faster than the structureperiodicity ε. Furthermore, we shall employ a non-homogeneous Neumann bound-ary condition on the holes. This condition not only gives rise to the strange term,but also generates new difficulties in the analysis.

The main novelty of this paper is to show that the strange term is amenableto analysis using an approach based on the spectral method of Bloch waves. Tothe best of our knowledge, this is the first article to deal with the strange term inthe homogenization theory by using the Bloch wave decomposition.

The paper is organized as follows. In the next Section we state the problemand present our main result. Section 3 is devoted to a brief discussion about Blochwaves related to our problem, their existence and regularity. Then, we introducethe Bloch waves at ε-scale and we give some useful preliminary results about theirconvergence. Finally, in Section 4 we prove our main result.

2. Setting of the problem and main result

Let Ω ⊂ RN , N ≥ 2 be a smooth open bounded set and let ε be a positive realnumber. Assume that r : R+ −→ R+ is a continuous map verifying r(ε) < πε and

limε→0

r(ε)

ε= 0. (2.1)

Let T ⊂ RN be a non-empty open bounded set with a smooth boundary.We suppose that 0 belongs to T and T is star-shaped with respect to 0 such

that d(0, ∂T ) > 0 and maxs∈∂T

|s| = 1. Then, we have that( r(ε)

ε

)T ⊂ Y and

d(( r(ε)

ε

)T , ∂Y

)> 0, where Y = [−π, π[

Nis the reference cell in RN (see Fig-

ure 1).For each ε and for any integer vector p ∈ ZN , we shall denote by T ε

p thetranslated image of r(ε)T by the vector 2πεp, i.e.,

T εp = 2πεp + r(ε)T.

Also, let us denote by T ε the set of all holes contained in Ω, that is,

T ε =⋃

T εp | T ε

p ⊂ Ω, p ∈ ZN

and we set

Ωε = Ω \ T ε.

Hence, Ωε is the periodically perforated domain with holes of size r(ε). All ofthem have the same shape, the distance between two adjacent holes is of order εand from the condition (2.1) they do not overlap. Also, let us remark that theholes do not intersect the boundary ∂Ω (see Figure 1).

Let γ ∈ R be a constant different of zero. We are interested in studying theasymptotic behavior, as ε goes to zero, of the solution vε of the following non-


Figure 1. Periodically perforated domain and reference cell

homogeneous Neumann boundary-value problem:

−∆vε = 0 in Ωε,

∂vε

∂n= γ on ∂T ε,

vε = 0 on ∂Ω.

(2.2)

Let us observe that the variational formulation of the problem (2.2) is written asfollows:

Find vε ∈ V ε such that∫

Ωε

∇vε · ∇Ψ dx = γ

∫

∂T ε

Ψ ds ∀Ψ ∈ V ε,(2.3)

where

V ε =Ψ ∈ H1(Ωε) | Ψ = 0 on ∂Ω

is a Hilbert space endowed with the inner product in H1(Ωε).Since the functions vε are only defined in Ωε, we shall introduce a family of

linear continuous extension operators P ε ∈ L(V ε,H1

0 (Ω))

such that

P εϕ|Ωε = ϕ and ‖∇P εϕ‖L2(Ω) ≤ C‖∇ϕ‖L2(Ωε) ∀ϕ ∈ V ε, (2.4)

where C is a real positive constant independent of ε. The proof of the existenceof one such family can be found in [5] and makes use of a similar extension resultwhich has been proved in [4].

The asymptotic behavior of the solution vε depends on the hole size r(ε),

and there exist two critical sizes. The first one is r(ε) = αεN

N−1 , where α is areal positive constant, and it separates the cases when the measure of the holesboundary tends to zero or infinity, as ε goes to zero.

When the size r(ε) is strictly less than this first critical size, there still existsa second critical size, which depends on the dimension of the space and which


gives us different asymptotic behavior for the solution vε. This second critical size

corresponds to the case when r(ε)ε goes to zero such that the limit

limε→0

( r(ε)ε )N−2G( r(ε)

ε )

ε2(2.5)

is a strictly positive real number. Here G denotes the real function, dependent onthe dimension of the space, given by:

G(r) =

− (ln r)

−1if N = 2,

1 if N ≥ 3.(2.6)

The case when the limit (2.5) is equal to +∞ (respectively 0) corresponds tor(ε) greater (respectively less) than this second critical size.

Now we state the main homogenization result that we are going to prove usingthe spectral method of Bloch waves.

Theorem 2.1. Assume that the function r(ε) verifies (2.1) and γ 6= 0. Let vε in

V ε be the sequence of the unique solutions of the problem (2.2). For any family of

linear continuous extension operators P ε verifying (2.4), we have:

(i) If limε→0

r(ε)N−2ε−NG( r(ε)ε ) = ℓ ∈ (0,+∞], then

r(ε)−(N−1)εNP εvε v weakly in H10 (Ω), as ε → 0,

where v is the unique solution of the following homogenized problem:

−∆v =|∂T |γ

(2π)Nin Ω,

v = 0 on ∂Ω.(2.7)

(ii) If limε→0

r(ε)N−2ε−NG( r(ε)ε ) = 0, then

(r(ε)

ε

)−N/2

G( r(ε)

ε

)1/2P εvε 0 weakly in H1

0 (Ω), as ε → 0.

Remark 2.2. Let us notice that if r(ε) is of order of the first critical size, that is

r(ε) = αεN

N−1 , with α > 0, then we have

limε→0

r(ε)N−2ε−NG( r(ε)ε ) = +∞.

Therefore, the above theorem states that P εvε, without any multiplicative term,weakly converges in H1

0 (Ω). For this reason, in this particular case, the proof ofthe theorem can be slightly modified in order to obtain that the extension to Ω ofthe solution of the non-homogeneous boundary value problem

−∆uε = f in Ωε,

∂uε

∂n= γ on ∂T ε,

uε = 0 on ∂Ω,

(2.8)


weakly converges in H10 (Ω) to the unique solution u of the homogenized problem

−∆u = f +|∂T |γ

αN−1(2π)Nin Ω,

u = 0 on ∂Ω,(2.9)

where f is a given function in L2(Ω), non identically zero.

3. Bloch waves

In order to study the problem (2.2), in this section, we are going to introduce theBloch waves associated to the homogeneous Neumann problem. In subsection 3.1,we consider a periodically perforated domain obtained from RN by removing afixed periodic network of holes. We state some theorems related to the existenceand the spectral decomposition of Bloch waves, and some regularity properties.These results are either easy to prove or classical and, for this reason, we shallomit several proofs. Then, in subsection 3.2, we introduce the Bloch waves atε-scale and we prove some convergence results, as ε → 0.

3.1. Existence and regularity of Bloch waves

Let us begin by introducing some notations. For a small positive real parameterr such that r T ⊂ Y and d(r T , ∂Y ) > 0, we set

Y ⋆ = Y \ r T , S =⋃

p∈ZN

2πp + r T.

The Bloch waves related to the problem (2.2) are the eigenvalues and eigen-vectors of the following family of spectral problems, parameterized by η ∈ RN ,associated with the Laplace operator in the periodically perforated domain RN \S,with homogeneous Neumann condition on the boundary of the holes:

Find λ(r; η) ∈ R and ψ(r; y; η) (not identically zero) such that

−∆ψ(r; ·; η) = λ(r; η)ψ(r; ·; η) in RN \ S,

∂ψ

∂n(r; ·; η) = 0 on ∂S,

ψ(r; ·; η) is (η;Y )-periodic,

(3.1)

where the condition “ψ(r; ·; η) is (η;Y )-periodic” means

ψ(r; y + 2πm; η) = e2πim·ηψ(r; y; η) ∀m ∈ ZN , y ∈ RN , η ∈ RN .

This (η;Y )-periodicity condition is a natural generalization to deal with peri-odic structures, and it is satisfied, for instance, by the plane waves.

Since the (η;Y )-periodicity condition is invariant under translations by ele-

ments of ZN in the η-variable, η may be restricted to the set Y ′ =[− 1

2 , 12

[Nwhich


is referred to as the dual cell or Brillouin zone. In the sequel, without loss ofgenerality, we shall take η ∈ Y ′.

In order to get the existence of the solutions of the above spectral problem, letus introduce the following change of variable:

ψ(r; y; η) = eiη·yφ(r; y; η),

φ(r; ·; η) is Y -periodic.(3.2)

Therefore, the above spectral problem is equivalent to the following one:

Find λ(r; η) ∈ R and φ(r; y; η) (not identically zero) such that

A(η)φ(r; ·; η) = λ(r; η)φ(r; ·; η) in RN \ S,

∂φ

∂n(r; ·; η) + iη · nφ(r; ·; η) = 0 on ∂S,

φ(r; ·; η) is Y -periodic,

(3.3)

where the operator A(η) is the so-called shifted-operator, and is given by

A(η) = −(∆ + 2iη · ∇ − |η|2). (3.4)

It is well known that for each fixed η ∈ Y ′, the above spectral problem (3.3)admits a discrete sequence of eigenvalues, each of them having finite multiplicity.As usual, we arrange them in increasing order repeating each values as many timesas its multiplicity:

0 ≤ λ1(r; η) ≤ λ2(r; η) ≤ . . . ≤ λm(r; η) ≤ . . . → ∞.

Besides, the corresponding eigenvectors denoted by φm(r; ·; η)m≥1 form an or-

thonormal basis of the space L2#(Y ⋆). It is worthwhile to remark that these eigen-

vectors in fact belong to the space H1#(Y ⋆), where

L2#(Y ⋆) = φ ∈ L2

loc(RN \ S) | φ is Y -periodic,

H1#(Y ⋆) =

φ ∈ L2

#(Y ⋆) | ∇φ ∈ L2#(Y ⋆)N

.

Due to the above Bloch waves, we can completely describe the spectral resolu-tion of the Laplace operator as an unbounded self-adjoint operator in L2(RN \S).More precisely, we have the following result: (for the proof see, for instance, [1,pp. 614])

Theorem 3.1. Let g ∈ L2(RN \ S) be arbitrary. For any m ∈ N∗, the mth Bloch

coefficient of g is defined by

(Bmg) (η) =

∫

RN\S

g(y)e−iη·yφm(r; y; η)dy ∀η ∈ Y ′.

Then the following inverse formula holds:

g(y) =

∫

Y ′

∞∑

m=1

(Bmg) (η)eiη·yφm(r; y; η)dη.


Further, we have Plancherel’s identity: ∀g, h ∈ L2(RN \ S),

∫

RN\S

g(y)h(y)dy =

∫

Y ′

∞∑

m=1

(Bmg) (η)(Bmh) (η)dη,

and, in particular, Parseval’s identity:

∫

RN\S

|g(y)|2dy =

∫

Y ′

∞∑

m=1

| (Bmg) (η)|2dη.

In the sequel, we are going to study the Bloch waves regularity with respectto the parameter η. First of all, let us observe that in spite of the fact that theoperator A(η) depends polynomially on η, it is well known (see, for instance,[13]) that, since η is a multiparameter, the eigenvalues are not necessarily smoothfunctions of η ∈ Y ′ because of the possible change of the multiplicity.

We begin by proving that all Bloch eigenvalues are Lipschitz functions withrespect to the variable η and that the Lipschitz constant is independent of thehole size r.

Proposition 3.2. For all m ≥ 1 and r > 0 small enough, λm(r; η) is a Lipschitz

function of η ∈ Y ′, with the Lipschitz constant independent of r.

Proof. Let observe that the quadratic form associated to the operator A(η),

ar(η;φ;φ) =

∫

Y ⋆

[|∇φ|2 − 2i(η · ∇φ)φ + |η|2|φ|2

]dy ∀φ ∈ H1

#(Y ⋆),

can be decomposed as

ar(η;φ;φ) = ar(η′;φ;φ) + Rr(η, η′;φ, φ),

where

Rr(η, η′;φ, φ) =

∫

Y ⋆

[2i ((η − η′) · ∇φ) φ +

(|η|2 − |η′|2

)|φ|2

]dy.

This term can be easily estimated by using the Cauchy–Schwarz inequality, thenthere exists a positive constant C, independent of r, such that

|Rr(η, η′;φ, φ)| ≤ C|η − η′|‖φ‖2H1

#(Y ⋆).

Due to the min-max characterization of eigenvalues, i.e.,

λm(r; η) = minW⊂H1

#(Y ⋆)

dimW=m

maxφ∈W

ar(η;φ, φ)

‖φ‖2L2(Y ⋆)

,

we deduce that

λm(r; η) ≤ λm(r; η′) + C |η − η′| µm(r),


where µm(r) is the mth eigenvalue of the following spectral problem

−∆ϕ(r; ·) + ϕ(r; ·) = µ(r) ϕ(r; ·) in RN \ S,

∂ϕ

∂n(r; ·) = 0 on ∂S,

ϕ(r; ·) is Y -periodic.

We observe that the sequence µm(r)r is bounded because it converges to themth eigenvalue µm of the spectral problem associated to the Laplace operator plusidentity in Y with periodicity condition, as r → 0. (see [17])

Interchanging η and η′, we finally get that

|λm(r; η) − λm(r; η′)| ≤ C |η − η′|,

where C is a positive constant independent of r.

The previous regularity of the Bloch eigenvalues is necessary, but is not enoughfor studying the homogenization process by using the spectral method of Blochwaves. For this reason, we state an analyticity result in a neighborhood of η = 0,independent of the hole size. We begin by the following Proposition which givesus this regularity, but the neighborhood could be dependent of the hole size.

Proposition 3.3. For any r > 0 small enough, there exists δ(r) > 0 such that

the first Bloch eigenvalue λ1(r; η) is an analytic function on Bδ(r) = η ∈ Y ′||η| <δ(r), and there is a choice of the first Bloch eigenvector φ1(r; y; η) satisfying

η−→ φ1(r; ·; η) ∈ H1

#(Y ⋆) is analytic on Bδ(r),

φ1(r; y; 0) = |Y ⋆|−1/2.

The proof of this result can be found, for instance, in [6].In order to prove that the neighborhood can be chosen independent of the

hole size, let us first state the following lemma, which gives a lower bound of thesuperior Bloch eigenvalues (m ≥ 2), independent of r.

Lemma 3.4. For any sufficiently small r > 0, we have

λm(r; η) ≥1

2λ

(N)2 > 0 ∀m ≥ 2, ∀η ∈ Y ′, (3.5)

where λ(N)2 is the second eigenvalue of the spectral problem for −∆ in the cell Y

with Neumann boundary conditions on ∂Y .

Proof. Firstly, we note that

λm(r; η) ≥ λ2(r; η) ∀m ≥ 2, ∀η ∈ Y ′.


Using the min-max principle it follows that: for any η ∈ Y ′,

λ2(r; η) = minW⊂H1

#(Y ⋆)

dimW=2

maxφ∈W

∫

Y ⋆

∣∣∇(eiη·yφ

)∣∣2 dy∫

Y ⋆

|φ|2dy

≥ minW⊂H1(Y ⋆)

dimW=2

maxψ∈W

∫

Y ⋆

|∇ψ|2dy∫

Y ⋆

|ψ|2dy

= λ(N)2 (r),

where λ(N)2 (r) is the second eigenvalue of the following spectral problem:

−∆Ψ(r; ·) = λ(N)(r)Ψ(r; ·) in Y ⋆,

∂Ψ

∂n(r; ·) = 0 on ∂Y ⋆.

Since λ(N)2 (r) −→ λ

(N)2 > 0, as r goes to zero (see [17]), then for any r suffi-

ciently small we obtain

λ(N)2 (r) ≥

1

2λ

(N)2 .

Now we are able to prove the existence of a neighborhood of the origin, inde-pendent of r, where the first eigenvalue is an analytic function with respect to thevariable η.

Proposition 3.5. There exists a neighborhood Bδ of η = 0, independent of r,such that λ1(r; η) is analytic on Bδ, for all small enough r > 0.

Proof. It is enough to prove that there exists a neighborhood of η = 0, independentof r, where λ1(r; η) is a simple eigenvalue, for all r small enough.

Since λ1(r; 0) = 0, we get

|λ1(r; η) − λ2(r; η)| ≥ |λ2(r; 0)| − |λ1(r; η) − λ1(r; 0)| − |λ2(r; η) − λ2(r; 0)|.

Due to the fact that λ1(r; η) and λ2(r; η) are Lipschitz functions on η, with theLipschitz constant C > 0, independent of r, and using Lemma 3.4 it follows that

|λ1(r; η) − λ2(r; η)| ≥1

2λ

(N)2 − 2C|η|.

If δ < λ(N)2 /(4C), then for all r > 0 small enough, we have that

|λ1(r; η) − λ2(r; η)| > 0 ∀|η| < δ.

Due to the previous regularity, in the following proposition we state high orderderivatives of the first Bloch eigenvalue and eigenvector at η = 0.


Proposition 3.6. For any sufficiently small r > 0, the first Bloch eigenvalue and

eigenvector satisfy the following properties:

(i) λ1(r; 0) = 0.(ii) The origin is a critical point of λ1(r; η), that is,

∂λ1

∂ηk(r; 0) = 0 ∀k = 1, . . . , N. (3.6)

(iii) The derivative of the first Bloch eigenvector satisfies

∂φ1

∂ηk(r; y; 0) = i |Y ⋆|

−1/2χk(y) ∀k = 1, . . . , N, (3.7)

where χk ∈ H1#(Y ⋆) solves the following cell problem:

−∆χk = 0 in Y ⋆,

∂χk

∂n= −nk on ∂(rT ),∫

Y ⋆

χk(y) dy = 0,

χk is Y -periodic.

(3.8)

(iv) The second order derivatives of the first Bloch eigenvalue have the following

asymptotic behavior:

1

2

∂2λ1

∂ηk∂ηl(r; 0) = δkl + O

(rN/2

)if N ≥ 3, (3.9)

1

2

∂2λ1

∂ηk∂ηl(r; 0) = δkl + O

(r |ln r|

1/2)

if N = 2, (3.10)

for all k, l = 1, . . . , N.

Proof. The proof of properties (i)–(iii) are classical and we refer the reader, forinstance, to [9].

Let us prove (iv). By an easy computation, we obtain that

1

2

∂2λ1

∂ηk∂ηl(r; 0) = δkl +

1

|Y ⋆|

∫

Y ⋆

(δks

∂χl

∂ys+ δls

∂χk

∂ys

)dy ∀k, l = 1, . . . , N,

where χk is the solution of the cell problem (3.8).Multiplying the first equation of (3.8) by χk and integrating by parts, one

obtains that ∫

Y ⋆

|∇χk(y)|2dy = −

∫

∂(rT )

nk χk(s)ds.

We have that the measure of each hole boundary is |∂ (rT )| = |∂T |rN−1, thenusing the Cauchy–Schwarz inequality we get

‖∇χk‖2L2(Y ⋆) ≤ Cr

N−12 ‖χk‖L2(∂(rT )). (3.11)


In order to bound this last term, we use the estimate (2.29) from [5], written forour case, that is,

‖χk‖2L2(∂(rT )) ≤ CrN−1

(‖χk‖

2L2(Y ⋆) + τ1r‖∇χk‖

2L2(Y ⋆)

), (3.12)

where

τ1r =

∫ π

br

t−(N−1)dt and b = d(0, ∂T ).

By hypothesis, we have that b > 0, then the term τ1r can be bounded as follows:

τ1r ≤ Cr−(N−2) if N ≥ 3,

τ1r ≤ C |ln r| if N = 2.

Due to the Poincare inequality and the previous two estimates, (3.12) becomes

‖χk‖2L2(∂(rT )) ≤ Cr‖∇χk‖

2L2(Y ⋆) if N ≥ 3,

‖χk‖2L2(∂(rT )) ≤ Cr |ln r| ‖∇χk‖

2L2(Y ⋆) if N = 2.

Therefore, the estimate (3.11) can be written as follows:

‖∇χk‖L2(Y ⋆) ≤ CrN2 if N ≥ 3,

‖∇χk‖L2(Y ⋆) ≤ Cr |ln r|1/2

if N = 2.

Since for all k, l = 1, ..., N, we have that∣∣∣∣

1

|Y ⋆|

∫

Y ⋆

(δks

∂χl

∂ys+ δls

∂χk

∂ys

)dy

∣∣∣∣ ≤1

|Y ⋆|1/2

[‖∇χl‖L2(Y ⋆) + ‖∇χk‖L2(Y ⋆)

],

we conclude the result.

3.2. Bloch waves at ε-scale and some properties

In this subsection, we prove some technical results which are useful for the proofof our main result. First, let us denote

Sε =⋃

p∈ZN

T εp ,

Y ⋆ε,p = εYp \ T ε

p = ε(2πp + y) \ T εp ∀p ∈ ZN .

Now, we introduce the Bloch eigenvalues λεm (r(ε); ξ)m≥1 and eigenvectors

φεm (r(ε);x; ξ)m≥1 at the ε-scale, which will diagonalize the Laplace operator

with homogeneous Neumann condition on the boundary of the holes ∂Sε in thespace L2

(RN \ Sε

). By homothecy, for all m ∈ N∗, we have the following relations:

λεm (r(ε); ξ) = ε−2λm

(r(ε)

ε ; η)

and φεm (r(ε);x; ξ) = φm

(r(ε)

ε ; y; η)

,


where λm and φm are the Bloch eigenvalues and eigenvectors that have alreadyintroduced in Subsection 3.1. The variables (x; ξ) and (y; η) are related by

y =x

ε, η = εξ.

We recall that y ∈ Y ⋆ε = Y \( r(ε)

ε )T and η ∈ Y ′, then x ∈ εY \r(ε)T and ξ ∈ ε−1Y ′.Using the Bloch waves at ε-scale, we can describe the spectral resolution of the

Laplace operator in L2(RN \ Sε). More precisely, we have the following result,which is similar to Theorem 3.1.

Theorem 3.7. Let g ∈ L2(RN \Sε) be arbitrary. For each m ∈ N∗, the mth Bloch

coefficient of g at ε-scale is defined by

(Bεmg) (ξ) =

∫

RN\Sε

g(x)e−iξ·xφεm (r(ε);x; ξ) dx ∀ξ ∈ ε−1Y ′.

Then the following inverse formula holds:

g(x) =

∫

ε−1Y ′

∞∑

m=1

(Bεmg) (ξ)eiξ·xφε

m (r(ε);x; ξ) dξ.

Further, we have Plancherel’s identity: ∀g, h ∈ L2(RN \ Sε),∫

RN\Sε

g(x)h(x)dx =

∫

ε−1Y ′

∞∑

m=1

(Bεmg) (ξ)(Bε

mh) (ξ)dξ,

and, in particular, Parseval’s identity:

∫

RN\Sε

|g(x)|2dy =

∫

ε−1Y ′

∞∑

m=1

| (Bεmg) (ξ)|2dξ.

In the sequel, we prove that in the homogenization process, one can neglect allsuperior Bloch modes. Precisely, we have the following result:

Proposition 3.8. Let consider ϕε the solution of the following problem:

−∆ϕε = 0 in RN \ Sε,∂ϕε

∂n= γ on ∂Sε.

(3.13)

We define

wε(x) =

∫

ε−1Y ′

∞∑

m=2

(Bεmϕε) (ξ)eiξ·xφε

m(r(ε);x; ξ)dξ,

then

‖wε‖L2(RN\Sε) ≤ Cε‖∇ϕε‖L2(RN\Sε),

with C a real positive constant independent of ε.


Proof. Since ϕε verifies (3.13), we have∫

RN\Sε

−∆ϕεϕεdx = 0,

then, by Plancherel’s identity, we deduce that∫

ε−1Y ′

∞∑

m=1

λεm(r(ε); ξ) |(Bε

mϕε) (ξ)|2

= γ

∫

∂Sε

ϕεds. (3.14)

By Parseval’s identity we have

‖wε‖2L2(RN\Sε) =

∫

ε−1Y ′

∞∑

m=2

|(Bεmϕε) (ξ)|

2dξ

≤(

supm≥2,

ξ∈ε−1Y ′

1

λεm(r(ε); ξ)

)∫

ε−1Y ′

∞∑

m=2

λεm(r(ε); ξ) |(Bε

mϕε) (ξ)|2dξ.

Since λεm(r(ε); ξ) = ε−2λm

(r(ε)

ε ; η), then due to Lemma 3.4 we get

supm≥2,

ξ∈ε−1Y ′

1

λεm(r(ε); ξ)

≤2ε2

λ(N)2

.

Finally, combining the last two estimates and (3.14) we conclude that

‖wε‖2L2(RN\Sε) ≤

2ε2

λ(N)2

γ

∫

∂Sε

ϕεds ≤ Cε2‖∇ϕε‖2L2(RN\Sε).

Now, we focus our study on the first Bloch mode, as ε goes to zero. We shallprove that the first Bloch transform converges to the classical Fourier transform ina suitable topology. For this purpose, let us start by stating the following technicalLemma.

Lemma 3.9. Let consider g ∈ L2(RN \ Sε

), with compact support K ⊆ RN ,

independent of ε, and ρ ∈ L∞(Y ′;L2#(Y ⋆

ε )). Then there exists a positive constant

C, independent of ε, such that∣∣∣∣∣

∫

K\Sε

g(x)e−iξ·xρ(x

ε; εξ

)dx

∣∣∣∣∣ ≤ C‖g‖L2(K\Sε) ‖ρ(·; εξ)‖L2#(Y ⋆

ε ) ∀ξ ∈ ε−1Y ′.

(3.15)

Proof. We start by decomposing the integral from (3.15) as follows:∣∣∣∣∣

∫

K\Sε

g(x)e−iξ·xρ(x

ε; εξ

)dx

∣∣∣∣∣ ≤∑

p∈ZN

Y ⋆ε,p∩K 6=∅

∣∣∣∣∣

∫

Y ⋆ε,p∩K

g(x)e−iξ·xρ(x

ε; εξ

)dx

∣∣∣∣∣ .


For each integral from the previous sum, we apply the Cauchy–Schwarz in-equality and we get

∣∣∣∣∣

∫

K\Sε

g(x)e−iξ·xρ(x

ε; εξ

)dx

∣∣∣∣∣ ≤∑

p∈ZN

Y ⋆ε,p∩K 6=∅

‖g‖L2(Y ⋆ε,p∩K)

∥∥∥ρ( ·

ε; εξ

)∥∥∥L2(εY ⋆

ε )

≤ ‖g‖L2(K\Sε)N1/2ε

∥∥∥ρ( ·

ε; εξ

)∥∥∥L2(εY ⋆

ε ),

where Nε denotes the number of cells contained in the compact set K. Since theorder of Nε is ε−N |K| and

∥∥∥ρ( ·

ε; εξ

)∥∥∥L2(εY ⋆

ε )= εN/2 ‖ρ (·; εξ)‖L2(Y ⋆

ε ) ,

we conclude the result.

Using the previous lemma, we are able to prove the Proposition below, thatshows how the first Bloch transform approximates the Fourier transform.

Proposition 3.10. Let hε be a sequence in L2(RN \ Sε) and h ∈ L2(RN ). We

denote by hε the extension by zero of hε on Sε and by h the usual Fourier transform

of h. If hε h weakly in L2(RN ), then

Bε1h

ε h weakly in L2loc(R

N ),

provided there is a fixed compact set K such that supp hε ⊆ K, ∀ε.

Proof. It is understood that Bε1h

ε, which is a priori defined for ξ ∈ ε−1Y ′, isextended by zero outside of ε−1Y ′. Using the definition of the first Bloch mode,we can write:

(Bε1h

ε) (ξ) =

∫

RN\Sε

hε(x)e−iξ·xφε1(r(ε);x; 0)dx

+

∫

K\Sε

hε(x)e−iξ·x[φε

1(r(ε);x; ξ) − φε1(r(ε);x; 0)

]dx.

Since φε1 (r(ε);x; 0) = |Y ⋆

ε |−1/2

and |Y ⋆ε |

−1/2 −→ (2π)−N/2, as ε → 0, we observethat the first integral from the right-hand side of the previous equality weaklyconverges to h in L2(RN ).

On the other hand, since the second integral of the previous identity is on acompact set, we use Lemma 3.9 in order to bound it from above by the followingterm:

C‖hε‖L2(RN )

∥∥∥φ1

(r(ε)

ε ; ·; εξ)− φ1

(r(ε)

ε ; ·; 0)∥∥∥

L2(Y ⋆ε )

,

where the positive constant C is independent of ε.

Here is where the regularity of the first Bloch mode η 7→ φ1

(r(ε)

ε ; ·; η)

∈

L2#(Y ⋆

ε ) is required when η is around zero. Analyticity of this map is established


in Subsection 3.1. Here we use simply the fact that the above map is a Lipschitzfunction (we recall that the Lipschitz constant is independent of ε). Thus, if|ξ| ≤ M, we see that it is bounded from above by CMε and so, in particular, itconverges to zero in L∞

loc(RN ). This completes the proof.

We finish this subsection by stating the following technical Lemma, based ongeometric properties of the periodically perforated domain. For the proof, we referthe reader to [5].

Lemma 3.11. There exists a positive constant C, independent of ε, such that:

If limε→0

r(ε)N−2ε−NG( r(ε)ε ) = ℓ ∈ (0,+∞], then

‖Ψ‖L2(∂T ε) ≤ Cr(ε)(N−1)/2ε−N/2‖Ψ‖H1(Ωε) ∀Ψ ∈ V ε. (3.16)

On the other hand, if limε→0

r(ε)N−2ε−NG( r(ε)ε ) = 0, then

‖Ψ‖L2(∂T ε) ≤ Cr(ε)1/2G( r(ε)

ε

)−1/2‖Ψ‖H1(Ωε) ∀Ψ ∈ V ε. (3.17)

4. Proof of the main result

In this section, using the previous properties on Bloch waves, we prove the mainresult of this paper according to different asymptotic behavior of r(ε), and for thisreason, we shall divide our proof into two steps. The first one corresponds to holeswhich order size is greater or equal than the second critical size. In the secondstep, the size of the holes is strictly less than the second critical size.

4.1. First step of the proof

Let us assume that limε→0

r(ε)N−2ε−NG( r(ε)ε ) = ℓ ∈ (0,+∞]. Due to Lemma 3.11

and the Poincare inequality in Ω, it is an easy matter to see that the sequencer(ε)−(N−1)εNP εvε

remains bounded in H1

0 (Ω). Therefore, we can extract a

subsequence, still denoted byr(ε)−(N−1)εNP εvε

, weakly convergent in H1

0 (Ω),that is,

r(ε)−(N−1)εNP εvε v weakly in H10 (Ω), as ε → 0. (4.1)

Our goal in what follows is to prove, using the Bloch waves method, that the weaklimit v is the unique solution of the problem (2.7).

For this purpose, we use the cut-off function ϕ ∈ D(Ω). Since vε is solution ofthe problem (2.2), we deduce that the function ϕvε, defined in all RN \Sε, satisfies


the following problem:

−∆(ϕvε) = −2∇ϕ · ∇vε − ∆ϕvε in RN \ Sε,

∂(ϕvε)

∂n= γϕ + vε ∂ϕ

∂non ∂Sε.

(4.2)

Now, we can apply the Bloch transform to (4.2), then for all m ≥ 1 and ξ ∈ ε−1Y ′,we get

λεm(r(ε); ξ) (Bε

m (ϕvε)) (ξ) −

∫

∂T ε

∂(ϕvε)

∂n(s)e−iξ·sφε

m (r(ε); s; ξ) ds

= −2Bεm (∇ϕ · ∇vε) (ξ) − Bε

m ((∆ϕ)vε) (ξ).

Before passing to the limit in the previous identity, as ε goes to zero, we multiplyit by r(ε)−(N−1)εN . Since

ϕr(ε)−(N−1)εNvε

is bounded in H1(RN \Sε), we can

neglect all the harmonics corresponding to m ≥ 2 (see Proposition 3.8). For thisreason, we shall pass to the limit, as ε → 0, only in the component correspondingto m = 1:

λε1(r(ε); ξ)

(Bε

1

(ϕr(ε)−(N−1)εNvε

))(ξ)

− r(ε)−(N−1)εN

∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds

= −2Bε1

(∇ϕ · r(ε)−(N−1)εN∇vε

)(ξ) − Bε

1

(∆ϕr(ε)−(N−1)εNvε

)(ξ),

which reads

λε1(r(ε); ξ)

(Bε

1

(ϕ|Ωε r(ε)−(N−1)εNP εvε

))(ξ)

− r(ε)−(N−1)εN

∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds

= −2Bε1

(˜(∇ϕ)|Ωε · r(ε)−(N−1)εN∇(P εvε)

)(ξ)

− Bε1

(˜(∆ϕ)|Ωε r(ε)−(N−1)εNP εvε

)(ξ). (4.3)

Let us note that since (2.1) holds then, for any function χ ∈ D(Ω) we have that

χ|Ωε −→ χ strongly in L2(RN ). (4.4)

Hence, combining this result with Proposition 3.10, we obtain the following weaklyconvergences in L2

loc(RN ):

Bε1

(ϕ|Ωε r(ε)−(N−1)εNP εvε

) ϕv, (4.5)

Bε1

(˜(∇ϕ)|Ωε · r(ε)−(N−1)εN∇(P εvε)

) ∇ϕ · ∇v, (4.6)

Bε1

(˜(∆ϕ)|Ωε r(ε)−(N−1)εNP εvε

) (∆ϕ)v. (4.7)


Due to Proposition 3.5 we can expand λ1(r(ε)

ε ; εξ) by the Taylor formula aroundξ = 0. Using Proposition 3.6 and convergences (4.5)–(4.7), we pass to the limit inidentity (4.3) and we get

δklξkξl(ϕv)(ξ) − limε→0

r(ε)−(N−1)εN

∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds

= −2∇ϕ · ∇v(ξ) − (∆ϕ)v(ξ). (4.8)

Now, we proceed to pass to the limit in the above boundary integral. To dothis, we decompose it into the sum of the following four integrals:

I1 = r(ε)−(N−1)εN

∫

∂T ε

γϕ(s)e−iξ·sφε1 (r(ε); s; 0) ds, (4.9)

I2 = r(ε)−(N−1)εN

∫

∂T ε

∂ϕ

∂n(s)vε(s)e−iξ·sφε

1 (r(ε); s; 0) ds, (4.10)

I3 = r(ε)−(N−1)εN

∫

∂T ε

γϕ(s)e−iξ·s[φε

1 (r(ε); s; ξ) − φε1 (r(ε); s; 0)

]ds, (4.11)

I4 = r(ε)−(N−1)εN

∫

∂T ε

∂ϕ

∂n(s)vε(s)e−iξ·s

[φε

1 (r(ε); s; ξ) − φε1 (r(ε); s; 0)

]ds.(4.12)

Let us pass to the limit in the first integral I1, using similar arguments as in[5]. Precisely, we introduce the sequence νε of positive Radon measure definedin C0

0 (Ω) by

〈νε, ψ〉 = r(ε)−(N−1)εN

∫

∂T ε

ψds ∀ψ ∈ C00 (Ω), (4.13)

which is bounded as follows

|〈νε, ψ〉| ≤ r(ε)−(N−1)εN |∂Tε|‖ψ‖C00 (Ω).

Since the number of holes in Ω is of order of |Ω|ε−N , then

limε→0

r(ε)−(N−1)εN |∂T ε| =|∂T ||Ω|

(2π)N. (4.14)

This implies that the sequence νε is bounded. Therefore, we can extract asubsequence, still denoted by νε , such that weakly* converges to a positivemeasure ν in the space of Radon measures on Ω, that is,

〈νε, ψ〉 → 〈ν, ψ〉 ∀ψ ∈ C00 (Ω). (4.15)

In order to identify the measure ν, we use the Riesz Representation Theorem andwe have that ν can be identified with a positive measure on Ω, denoted by ν, suchthat

〈ν, ψ〉 =

∫

Ω

ψ dν ∀ψ ∈ C00 (Ω),

where ν is uniquely defined by ν(A) = sup〈ν, ψ〉

∣∣ ψ ∈ C00 (A), 0 ≤ ψ ≤ 1

, for

all open subset A of Ω.


Using (4.14), for an arbitrary open subset A of Ω, it follows that

ν(A) =|∂T ||A|

(2π)N,

which implies that

limε→0

r(ε)−(N−1)εN

∫

∂T ε

ψds =|∂T |

(2π)N

∫

Ω

ψdx ∀ψ ∈ C00 (Ω). (4.16)

Using (4.16), we pass to the limit in the first integral I1 as follows:

limε→0

I1 = limε→0

γ|Y ⋆ε |

−1/2r(ε)−(N−1)εN

∫

∂T ε

ϕ(s)e−iξ·s ds

=γ

(2π)N/2

|∂T |

(2π)N

∫

Ω

ϕ(x)e−ix·ξ dx =γ|∂T |

(2π)Nϕ(ξ). (4.17)

Now, let us pass to limit in the integral I2. We are going to prove that thissecond integral converges to zero in L∞

loc(RN ), as ε goes to zero. For this purpose,

let us remark that I2 can be written as follows

I2 = |Y ⋆ε |

−1/2r(ε)−(N−1)εN

∫

∂T ε

(∇ϕ(s)e−iξ·sP εvε(s)

)· ~nds.

Due to Green’s formula written inside of the holes and the Cauchy–Schwarzinequality, we get

|I2| ≤ |Y ⋆ε |

−1/2r(ε)−(N−1)εN∥∥div

(∇ϕe−iξ·xP εvε

)∥∥L2(Ω)

|T ε|1/2

≤ C(1 + |ξ|)|Y ⋆ε |−1/2r(ε)−(N−1)εN‖P εvε‖H1(Ω)

(r(ε)

ε

)N/2

.

Since the sequence r(ε)−(N−1)εNP εvε is bounded in H1(Ω), we deduce that

if |ξ| ≤ M , the integral I2 is bounded by C(1 + M)(

r(ε)ε

)N/2

, then we conclude.

In the sequel, we study the convergence of the boundary integral I3, defined in(4.11). We begin by using the variational formulation (2.3) in order to obtain

I3 = r(ε)−(N−1)εNγ

∫

Ωε

∇vε(x) · ∇(ϕ(x)e−iξ·x

[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

])dx,

then we get

|I3|≤

∣∣∣∣r(ε)−(N−1)εNγ

∫

Ωε

∇vε(x) · (∇ϕ(x)) e−iξ·x[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

]dx

∣∣∣∣

+

∣∣∣∣r(ε)−(N−1)εNγ

∫

Ωε

∇vε(x) · ξϕ(x)e−iξ·x[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

]dx

∣∣∣∣

+

∣∣∣∣r(ε)−(N−1)εNγ

∫

Ωε

∇vε(x) · ϕ(x)e−iξ·x∇([

φε1 (r(ε);x; ξ) − φε

1 (r(ε);x; 0)])

dx

∣∣∣∣ .


In the last three integrals we use Lemma 3.9 and the fact r(ε)−(N−1)εNvε isbounded in H1(Ωε), to obtain

|I3| ≤ C(1 + |ξ|)

∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)− φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

,

where the constant C is independent of ε.

We recall that the application η 7→ φ1

(r(ε)

ε ; ·; η)

∈ H1#(Y ⋆

ε ) is a Lipschitz

function, then

|I3| ≤ C(1 + |ξ|)|ξ|ε.

Therefore, if |ξ| ≤ M, we obtain that the integral I3 is bounded by C(1 + M)Mε,and in particular, converges to zero in L∞

loc(RN ).

In order to obtain the limit of the boundary integral in (4.3), we need toestimate the term I4. For this purpose, we begin by introducing the auxiliaryfunctions vε

j , j = 1, . . . , N, as the solutions of the following problems:

−∆vεj = 0 in Ωε,

∂vεj

∂n= vεnj on ∂T ε,

vεj = 0 on ∂Ω,

(4.18)

where nj denotes the jth component of the normal unit vector. The variationalformulation of this problem is:

Find vεj ∈ V ε such that∫

Ωε

∇vεj · ∇Ψ dx =

∫

∂T ε

vεnjΨ ds ∀Ψ ∈ V ε.(4.19)

Using these auxiliary functions, the integral I4 can be rewritten as follows:

I4 = r(ε)−(N−1)εN

∫

Ωε

∇vεj (x)·∇

(∂ϕ

∂xj(x)e−iξ·x

[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

])dx,

then

|I4|≤

∣∣∣∣r(ε)−(N−1)εN

∫

Ωε

∇vεj (x)·

(∇

∂ϕ

∂xj(x)

)e−iξ·x

[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

]dx

∣∣∣∣

+

∣∣∣∣r(ε)−(N−1)εN

∫

Ωε

∇vεj (x) · ξ

∂ϕ

∂xj(x)e−iξ·x

[φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

]dx

∣∣∣∣

+

∣∣∣∣r(ε)−(N−1)εN

∫

Ωε

∇vεj (x) ·

∂ϕ

∂xj(x)e−iξ·x∇

([φε

1 (r(ε);x; ξ) − φε1 (r(ε);x; 0)

])dx

∣∣∣∣ .

Applying Lemma 3.9, we obtain that

|I4|≤C(1 + |ξ|)r(ε)−(N−1)εN‖vεj‖H1(Ωε)

∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)−φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

,


where the constant C is independent of ε. Then, due to the variational formulation(4.19) and Lemma 3.11, it follows that

|I4| ≤ C(1 + |ξ|)‖vε‖H1(Ωε)

∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)− φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

.

Using that the application η 7→ φ1

(r(ε)

ε ; ·; η)∈ H1

#(Y ⋆ε ) is Lipschitz and that

r(ε)−(N−1)εNvε is bounded in H1(Ωε), we obtain

|I4| ≤ C(1 + |ξ|)|ξ|

(r(ε)

ε

)N−1

.

Thus, we conclude the convergence to zero of the integral I4 in L∞loc(R

N ).Hence, from all the previous convergence of the integrals I1, . . . , I4 it follows

that

limε→0

r(ε)−(N−1)εN

∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds =γ|∂T |

(2π)Nϕ(ξ). (4.20)

Due to this previous limit, the identity (4.8) becomes

δklξkξl(ϕv)(ξ) −γ|∂T |

(2π)Nϕ(ξ) = −2∇ϕ · ∇v(ξ) − (∆ϕ)v(ξ),

which can be written

(−ϕ∆v)(ξ) −(

γ|∂T |

(2π)Nϕ

)(ξ) = 0. (4.21)

With this last equation, called in the literature the localized homogenized equationin the Fourier space, the conclusion of our result is an easy consequence. Moreprecisely, taking the inverse Fourier transform of (4.21), we obtain

ϕ

(−∆v −

γ|∂T |

(2π)N

)= 0 in RN .

Since the above relation is valid for all ϕ in D(Ω), we conclude that v is the solutionof the equation (2.7).

4.2. Second step of the proof

Let us assume that limε→0

r(ε)N−2ε−NG( r(ε)ε ) = 0. Due to Lemma 3.11, it is easy

to prove that the sequence

(r(ε)

ε

)−N/2

G( r(ε)

ε

)1/2P εvε

remains bounded in

H10 (Ω). Then, we can extract a subsequence, that we shall still denote by(

r(ε)ε

)−N/2

G( r(ε)

ε

)1/2P εvε

, such that

(r(ε)

ε

)−N/2

G( r(ε)

ε

)1/2P εvε v weakly in H1

0 (Ω), as ε → 0. (4.22)


Our goal in what follows is to use the Bloch waves method in order to prove thatv = 0. To do this, we begin by using the arguments outlined above leading to (4.3)to deduce that for all ξ ∈ ε−1Y ′ :

λε1(r(ε); ξ)

(Bε

1

(ϕ|Ωε

( r(ε)ε

)−N/2G

( r(ε)ε

)1/2P εvε

))(ξ)

−( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds

= −2Bε1

(˜(∇ϕ)|Ωε

( r(ε)ε

)−N/2G

( r(ε)ε

)1/2∇(P εvε)

)(ξ)

− Bε1

(˜(∆ϕ)|Ωε

( r(ε)ε

)−N/2G

( r(ε)ε

)1/2P εvε

)(ξ).

Passing to the limit and using (4.4), we get

δklξkξl(ϕv)(ξ) − limε→0

( r(ε)ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

∂(ϕvε)


1 (r(ε); s; ξ) ds

= −2∇ϕ · ∇v(ξ) − (∆ϕ)v(ξ). (4.23)

In the sequel, we are going to study the previous boundary integral that wedecompose into the sum of the following four integrals:

I1=( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

γϕ(s)e−iξ·sφε1 (r(ε); s; 0) ds,

I2=( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

∂ϕ

∂n(s)vε(s)e−iξ·sφε

1 (r(ε); s; 0) ds,

I3=( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

γϕ(s)e−iξ·s[φε

1 (r(ε); s; ξ) − φε1 (r(ε); s; 0)

]ds,

I4=( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2∫

∂T ε

∂ϕ

∂n(s)vε(s)e−iξ·s

[φε

1 (r(ε); s; ξ) − φε1 (r(ε); s; 0)

]ds.

The first integral I1, can be directly bounded as follows:

|I1| ≤ C( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2|Y ⋆

ε |−1/2|∂T ε|.

Using (4.14), we obtain that

|I1| ≤ C(r(ε)N−2ε−NG

( r(ε)ε

))1/2

.

Therefore, we conclude that I1 converges to zero, as ε tends to zero.In order to pass to limit in the integral I2, we use similar arguments as in the

first step of the proof, then we get

|I2| ≤ C(1 + |ξ|)|Y ⋆ε |−1/2

( r(ε)ε

)−N/2G

( r(ε)ε

)1/2‖P εvε‖H1(Ω)

(r(ε)

ε

)N/2

.


In this case, we have that the bounded sequence in H1(Ω) is ( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2

P εvε, then we deduce that if |ξ| ≤ M , the integral I2 is bounded by C(1 +

M)(

r(ε)ε

)N/2

. Hence, we conclude that I2 converges to zero in L∞loc(R

N ), as ε → 0.

For the third integral I3, we use the arguments outlined in the first step of theproof and we obtain

|I3| ≤ C(1 + |ξ|)( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2‖vε‖H1(Ωε)∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)− φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

.

Here, we use the fact that the sequence( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2vε is bounded in

H1(Ωε) and the Lipschitz continuity property in H1-norm of the first Bloch eigen-vector in order to obtain that

|I3| ≤ C(1 + |ξ|)|ξ| ε.

If |ξ| ≤ M, we see that the integral is bounded by C(1+M)Mε, and in particular,converges to zero in L∞

loc(RN ), as ε → 0.

Finally, we estimate the integral I4 following the same methodology from thefirst step of the proof, with the correspondent changes. We get that

|I4| ≤ C(1 + |ξ|)( r(ε)

ε

)−N/2G

( r(ε)ε

)1/2‖vε

j‖H1(Ωε)∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)− φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

,

where the constant C is independent of ε, and the functions vεj are defined in

(4.18). Now, due to the variational formulation (4.19) and Lemma 3.11, it followsthat

|I4| ≤ C(1 + |ξ|)r(ε)G( r(ε)

ε

)−1∥∥∥∥φ1

(r(ε)

ε; ·; εξ

)− φ1

(r(ε)

ε; ·; 0

)∥∥∥∥H1(Y ⋆

ε )

,

then we conclude the convergence to zero of I4 in L∞loc(R

N ), as ε → 0.Therefore, the identity (4.23) becomes

(−∆(ϕv))(ξ) = −2∇ϕ · ∇v(ξ) − (∆ϕ)v(ξ).

Taking the inverse Fourier transform, we deduce that −ϕ∆v = 0 in RN . Then,we conclude the second step of the proof because the function ϕ is arbitrary inD(Ω).

Acknowledgments

The first author was partially supported by grants Fondecyt 1030943 1070148 andEcos C04E07. The second author was partially supported by grants Fondecyt


1050332 and Fondap on Applied Mathematics. The third one was supported bygrant Fondecyt 3070029 and by fellowship of Center for Mathematical Modelling,University of Chile.

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Jaime OrtegaDepartamento de Ciencias Basicas, Facultad de CienciasUniversidad del Bıo-BıoCasilla 447Campus Fernando MayChillanandCentro de Modelamiento MatematicoUMR 2071, CNRS–UCHILEFacultad de Ciencias Fısicas y MatematicasUniversidad de ChileSantiagoChilee-mail: [email protected]

Jorge San MartınDepartamento de Ingenierıa Matematica andCentro de Modelamiento MatematicoUMR 2071, CNRS–UCHILEFacultad de Ciencias Fısicas y MatematicasUniversidad de ChileCasilla 170/3-Correo 3SantiagoChilee-mail: [email protected]

Loredana SmarandaCentro de Modelamiento MatematicoUMR 2071, CNRS–UCHILEFacultad de Ciencias Fısicas y MatematicasUniversidad de ChileCasilla 170/3-Correo 3SantiagoChileandDepartment of Mathematics

Faculty of Mathematics and Computer Science

University of Pitesti

Romania

e-mail: [email protected]

(Received: December 20, 2006)

Published Online First: August 6, 2007

bloch wave homogenization of a non-homogeneous neumann problem

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