Volume 2010, Article ID 853717, 15 pagesdoi:10.1155/2010/853717 Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains Hermann Douanla Department of
Trang 1Volume 2010, Article ID 853717, 15 pages
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,douanla@chalmers.se
Received 31 July 2010; Accepted 11 November 2010
Academic Editor: Gary Lieberman
Copyrightq 2010 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 Stekloff eigenvalue 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
∂
∂x i
a ij
x, x ε
∂u ε
∂x j
0, in Ω ε ,
N
i,j1
a ij
x, x ε
∂u ε
∂x j ν i λ ε u ε , on ∂T ε ,
u ε 0, on ∂Ω,
ε
S ε |u ε|2
dσ ε x 1,
1.1
where Ω is a bounded open set in N
x the numerical space of variables x x1, , x N,
with integer N ≥ 2 with Lipschitz boundary ∂Ω, a ij ∈ CΩ; L∞ N
y 1 ≤ i, j ≤ N, with
Trang 2the symmetry condition a ji a ij , the periodicity hypothesis: for each x ∈ Ω and for every
k ∈
N one has a ij x, yk a ij x, y almost everywhere in y ∈ N
y , and finally the ellipticity
condition: there exists α > 0 such that for any x ∈Ω
Re
N
i,j1
a ij
x, y
for all ξ ∈
N and for almost all y ∈ N
y , where|ξ|2 |ξ1|2 · · · |ξ N|2 The setΩε ε > 0 is a domain perforated as follows Let T ⊂ Y 0, 1 Nbe a compact subset in N
y with smooth boundary ∂T≡ S and nonempty interior For ε > 0, we define
t ε k ∈
N : εk T ⊂ Ω ,
T ε
k∈t ε
ε k T,
Ωε Ω \ T ε
1.3
In this setup, T is the reference hole, whereas εk T is a hole of size ε and T εis the collection
of the holes of the perforated domainΩε The family T εis made up with a finite number of holes sinceΩ is bounded Finally, ν ν i denotes the outer unit normal vector to ∂T ε ≡ S ε with respect toΩε
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 Taylor 4 , but the first homogenization results on this topic pertains to Vanninathan5 , where he considered eigenvalue problems for the laplace operator a ij δ ij 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 Svanstedt 7 , 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 Stekloff eigenvalue 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 resultTheorem3.7 by using the two-scale convergence method12–16 introduced by Nguetseng 15 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 Let Y 0, 1 N , and let F N be a given function space We denote by FperY the space of functions in Floc N that are Y-periodic and by F#Y the space of those functions
u ∈ FperY with Y uydy 0 Finally, the letter E denotes throughout a family of strictly
positive real numbers0 < ε ≤ 1 admitting 0 as accumulation point The numerical space N
and its open sets are provided with the Lebesgue measure denoted by dx dx1· · · dx N
Trang 3The 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 LetΩ be an open bounded set in N
x integer N ≥ 2 and Y 0, 1 N, the unit cube
Definition 2.1 A sequence u εε∈E ⊂ L2Ω is said to two-scale converge in L2Ω to
someu0 ∈ L2Ω × Y if, as E ε → 0,
Ωu ε xφ
x, x ε
dx −→
Ω×Y u0
x, y
φ
x, y
for all φ ∈ L2Ω; CperY.
Notation 1 We express this by writing u ε −−→ u 2s 0in L2Ω.
The following theorem is the backbone of the two-scale convergence method
Theorem 2.2 Let u εε∈E be a bounded sequence in L2Ω Then, a subsequence E can be extracted from E such that, as E ε → 0, the sequence u εε∈E two-scale converges in L2Ω to some u0 ∈
L2Ω × Y.
Here follows the cornerstone of two-scale convergence
Theorem 2.3 Let u εε∈E be a bounded sequence in H1Ω Then, a subsequence E can be extracted from E such that, as E ε → 0,
u ε −→ u0, in H1Ω-weak,
u ε −→ u0, in L2Ω,
∂u ε
∂x j
2s
−−→ ∂u0
∂x j ∂u1
∂y j , in L2Ω1≤ j ≤ N ,
2.2
where u0∈ H1Ω and u1∈ L2Ω; H1
#Y.
In the sequel, we denote by dσy y ∈ Y , dσ ε x x ∈ Ω, ε ∈ E, the surface measures
on S and S ε , respectively The surface measure of S is denoted by |S| The space of squared integrable functions, with respect to the previous measures on S and S ε are denoted by L2S and L2S ε , respectively Since the volume of S ε grows proportionally to 1/ε as ε → 0, we endow L2S ε with the scaled scalar product 17
u, v L2S ε ε
S ε u xvxdσ ε x u, v ∈ L2S ε. 2.3 Definition2.1then generalizes as
Trang 4Definition 2.4 A sequence u εε∈E ⊂ L2S ε is said to two-scale converge to some u0∈ L2Ω ×
S if as follows E ε → 0,
ε
S ε u ε xφ
x, x ε
dσ ε x −→
Ω×S u0
x, y
φ
x, y
dx dσ
y
2.4
for all φ ∈ CΩ; CperY.
The following result paves the way of the general version of Theorem2.2
Lemma 2.5 Let φ ∈ CΩ; C per Y Then, we have
ε
S ε
φx, x ε
2dσ ε x ≤ C φ 2
for some constant C independent of ε and, as E ε → 0,
ε
S ε
φx, x ε
2dσ ε x −→
Ω×S
φ
x, y 2
dx dσ
y
Proof The first part is left to the reader Let ϕ ∈ C Ω and ψ ∈ CperY We have
ε
S ε
ϕxψx ε2dσ ε x ε
k∈t ε
εkS
ϕxψx ε2dσ ε x. 2.7
Using the second mean value theorem, for any k ∈ t ε, we have
εkS
ϕxψx ε2dσ ε x ϕ x k2
εkS
ψx ε2dσ ε x 2.8
for some x k ∈ εk S ⊂ εk Y Hence,
ε
S ε
ϕxψx ε2dσ ε x ε
k∈t ε
εkS
ϕxψx ε2dσ ε x
ε
k∈t ε
ϕx k2
εkS
ψx ε2dσ ε x
ε
k∈t ε
ϕx k2
ε N−1
kS
ψ
y 2
dσ
y
S
ψ
y 2
dσ
y
k∈t ε
ε Nϕ
x k2
.
2.9
Trang 5But, as E ε → 0,
k∈t ε
ε Nϕ
x k2
−→
Ω
ϕ x2
and the proof is completed due to the density ofCΩ ⊗ CperY in CΩ; 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 by L2Ω; CperY.
Theorem2.2generalizes as follows
Theorem 2.7 Let u εε∈E be a sequence in L2S ε such that
ε
S ε
|u ε x|2
where C is a positive constant independent of ε There exists a subsequence E of E such that u εε∈E
two-scale converges to some u0∈ L2Ω; L2S in the sense of Definition 2.4
Proof Put F ε φ ε S ε u ε xφx, x/εdσ ε x for φ ∈ CΩ; CperY We have
F ε
φ ≤ Cε
S ε
φx, x ε
2dσ ε x
1/2
which allows us to view F εas a continuous linear form onCΩ; CperY Hence, there exists
a bounded sequence of measuresμ εε∈E such that F ε ε , φ Due to the separability of
CΩ; CperY there exists a subsequence E of E such that in the weak∗topology of the dual
ofCΩ; CperY we have μ ε → μ0 as E ε → 0 A limit passage E ε → 0 in 2.12 yields
μ0, φ ≤ C
Ω×S
φ
x, y 2
dx dσ
y 1/2
But μ0is a continuous form on L2Ω; L2S by density of CΩ; CperY in the later space, and there exists u0 ∈ L2Ω; L2S such that
μ0, φ
Ω×S u0
x, y
φ
x, y
dx dσ
y
2.14
for all φ ∈ CΩ; CperY, which completes the proof.
In the case whenu εε∈E is the sequence of traces on S ε of functions in H1Ω, 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
Trang 6Proposition 2.8 Let u εε∈E ⊂ H1Ω be such that
where C is a positive constant independent of ε and D denotes the usual gradient The sequence of traces of u εε∈E on S ε satisfies
ε
S ε |u ε x|2
and up to a subsequence E of E, it two-scale converges in the sense of Definition 2.4 to some
u0 ∈ L2Ω; L2S which is nothing but the trace on S of the usual two-scale limit, a function in
L2Ω; H1
#Y More precisely, as E ε → 0,
ε
S ε u ε xφ
x, x ε
dσ ε x −→
Ω×S u0
x, y
φ
x, y
dx dσ
y
,
Ωu ε xφ
x, x ε
dx dy −→
Ω×Y u0
x, y
φ
x, y
dx dy
2.17
for all φ ∈ CΩ; Cper Y.
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 function χ Gof
with
k∈ N
It follows from the closeness of T that Θ is closed in N y so that G is an open subset of N y
Next, let ε ∈ E be arbitrarily fixed, and define
V ε u ∈ H1Ωε : u 0 on ∂Ω . 3.3
We equip V ε with the H1Ωε-norm which makes it a Hilbert space We recall the following classical result18
Trang 7Proposition 3.1 For each ε ∈ E there exists an operator P ε of V ε into H1
0Ω with the following
properties:
i P ε sends continuously and linearly V ε into H1
0Ω;
ii P ε v|Ωε v for all v ∈ V ε ;
iii DP ε v L2 ΩN ≤ c Dv L2 ΩεN for all v ∈ V ε , where c is a constant independent of ε and
D denotes the usual gradient operator.
It is also a well-known fact that, under the hypotheses mentioned earlier in the introduction, the spectral problem1.1 has an increasing sequence of eigenvalues {λ k
ε}∞k1,
0 < λ1≤ λ2≤ λ3≤ · · · ≤ λ n
ε ,
It is to be noted that if the coefficients aε
ij are real valued then the first eigenvalue λ ε1 is
isolated Moreover, to each eigenvalue, λ k
ε is attached to an eigenvector u k
ε ∈ V εand{u k
ε}∞k1
is an orthonormal basis in L2S ε In the sequel, the couple λ k
ε , u k ε will be referred to as eigencouple without further ado
We finally recall the Courant-Fisher minimax principle which gives a usefulas will
be seen later characterization of the eigenvalues to problem 1.1 To this end, we introduce
the Rayleigh quotient defined, for each v ∈ V ε\ {0}, by
R ε v Ωε A ε Dv, Dv dx
where A ε is the N2-square matrixa ε
E k k ≥ 0 the collection of all subspaces of dimension k of V ε, the minimax principle is stated
as follows: for any k ≥ 1, the kth eigenvalue to1.1 is given by
λ k ε min
W∈E k
max
v∈W\{0} R ε v
max
W∈E k−1
min
v∈W⊥ \{0}R ε v
In particular, the first eigenvalue satisfies
λ1ε min
and every minimum in3.6 is an eigenvector associated with λ1
Now, let Q ε Ω \ εΘ This is an open set in N, andΩε \ Q εis the intersection of
Ω with 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 ∂Ω are of no effects
as regards the homogenization process since they are in arbitrary narrow stripe along the boundary
Lemma 3.2 see 19 Let K ⊂ Ω be a compact set independent of ε There is some ε0 > 0 such that
Ωε \ Q ε ⊂ Ω \ K for any 0 < ε ≤ ε0.
Trang 8Next, we introduce the space
1
0Ω × L2
Ω; H1
Endowed with the following norm
v
1 D x v0 D y v1
L2Ω×Y
v v0, v1 ∈
1 0
1
0 is a Hilbert space admitting F0∞ DΩ × DΩ ⊗ C∞
# Y as a dense subspace This being so, for
u, v ∈
1
1
0, let
aΩu, v N
i,j1
Ω×Y∗a ij
x, y ∂u0
∂x j ∂u1
∂y j
∂v0
∂x j ∂v1
∂y j
dx dy. 3.10
This defines a hermitian, continuous sesquilinear form on
1
1
0 We will need the following results.
Lemma 3.3 Fix Φ ψ0, ψ1 ∈ F∞
0 , and defineΦε:Ω → (ε > 0) by
Φε x ψ0x εψ1
x, x ε
If u εε∈E ⊂ H1
0Ω is such that
∂u ε
∂x i
2s
−−→ ∂u0
∂x i ∂u1
as E ε → 0, where u u0, u1 ∈
1
0, then
as E ε → 0, where
a ε u ε , Φ ε N
i,j1
Ωε a ε ij ∂u ε
∂x j
∂Φ ε
Proof For ε > 0, Φ ε∈ DΩ and all the functions Φε ε > 0 have their supports contained in a fixed compact set K ⊂ Ω Thanks to Lemma3.3, there is some ε0> 0 such that
Trang 9Using the decomposition Ωε Q ε∪ Ωε \ Q ε and the equality Q ε Ω ∩ εG, we get for
E ε ≤ ε0
a ε u ε , Φ ε N
i,j1
Ωε
a ij
x, x ε
∂u ε
∂x j
∂Φ ε
∂x i dx
N
i,j1
Q ε a ij
x, x ε
∂u ε
∂x j
∂Φ ε
∂x i dx
N
i,j1
Ω∩εG a ij
x, x ε
∂u ε
∂x j
∂Φ ε
∂x i dx
N
i,j1
Ωa ij
x, x ε
χ εG x ∂u ε
∂x j
∂Φ ε
∂x i dx
N
i,j1
Ωa ij
x, x ε
χ G
x ε
∂u ε
∂x j
∂Φ ε
∂x i dx.
3.16
Bear in mind that, as E ε → 0, we have see, e.g., 19, Lemma 2.4
N
i,j1
∂u ε
∂x j
∂Φ ε
∂x i
2s
−−→ N
i,j1
∂u0
∂x j ∂u1
∂y j
∂ψ0
∂x j ∂ψ1
∂y j
We also recall that a ij x, yχ G y ∈ CΩ; L2
perY1 ≤ i, j ≤ N and that Property 2.1 in Definition2.1still holds for f in CΩ; L2
perY instead of L2Ω; CperY whenever the
two-scale convergence therein is ensuredsee, e.g., 14, Theorem 15 Thus, as E ε → 0,
a ε u ε , Φ ε N
i,j1
Ωa ij
x, x ε
χ G
x ε
∂u ε
∂x j
∂Φ ε
∂x i dx
−→N
i,j1
Ω×Y a ij
x, y
χ G
y ∂u0
∂x j ∂u1
∂y j
∂ψ0
∂x j ∂ψ1
∂y j
dx dy
N
i,j1
Ω×Y∗a ij
x, y ∂u0
∂x j
∂u1
∂y j
∂ψ0
∂x j
∂ψ1
∂y j
dx dy
aΩu, Φ,
3.18
which completes the proof
Trang 10We now construct and point out the main properties of the so-called homogenized coefficients Let 1 ≤ j ≤ N, and fix x ∈ Ω Put
a x; u, v N
i,j1
Y∗
a ij
x, y ∂u
∂y j
∂v
∂y i dy,
l j x, v N
k1
Y∗a kj
x, y ∂v
∂y k dy
3.19
for u, v ∈ H#1Y Equipped with the seminorm
N u D y u L2Y∗ N
u ∈ H#1Y, 3.20
H#1Y is a pre-Hilbert space that is nonseparate and noncomplete Let H1
#Y∗ be its
separated completion with respect to the seminorm N· and i the canonical mapping of
H1
#Y into H1
#Y∗ We recall that
i H1
#Y∗ is a Hilbert space;
ii i is linear;
iii iH1
#Y is dense in H1
#Y∗;
iv iu H1Y∗ Nu for every u in H1
#Y;
v if F is a Banach space and l a continuous linear mapping of H1
#Y into F, then there exists a unique continuous linear mapping L : H1
#Y∗ → F such that l L ◦ i Proposition 3.4 Let j 1, , N, and fix x in Ω The noncoercive local variational problem
u ∈ H#1Y, ax; u, v l j x, v, ∀v ∈ H1
admits at least one solution Moreover, if χ j x and θ j x are two solutions,
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 H1
#Y∗ × H1
#Y∗ such that
Ax; iu, iv ax; u, v for all u, v ∈ H1
#Y Based on v above, we consider the antilinear
formlj x, · on H1
#Y∗ such that lj x, iu l j x, u for any u ∈ H1
#Y Then, χ j x ∈ H1
satisfies3.21 if and only if iχ j x satisfies
iχ j x∈ H1
#Y∗, Ax; i
χ j x, V
lj x, V , ∀V ∈ H1
#Y∗. 3.23
Butiχ j x is uniquely determined by 3.23 see, e.g., 20, page 216 We deduce that 3.21
admits at least one solution, and if χ j x and θ j x are two solutions, then iχ j x iθ j x,
...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... pertains to Vanninathan5 , where he considered eigenvalue problems for the laplace operator a ij δ ij Kronecker symbol in perforated domains, and combined... 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 Svanstedt 7 , Kaizu