Geometric versus spectral convergence for the Neumann Laplacian under exterior perturbations of the domain

0.17 MB·English

by  Jose M. Arrieta

GEOMETRIC VERSUS SPECTRAL CONVERGENCE FOR
THE NEUMANN LAPLACIAN UNDER EXTERIOR
PERTURBATIONS OF THE DOMAIN
9
0 JOSE´ M.ARRIETA∗ ANDDAVIDKREJCˇIRˇ´IK†
0
2
Abstract. We analyze the behavior of the eigenvalues and eigenfunc-
n tionsoftheLaplaceoperatorwithhomogeneousNeumannboundarycon-
a ditions when the domain is perturbed. We show that if Ω0 ⊂ Ωǫ are
J bounded domains(although notnecessarilyuniformlybounded) andwe
know that the eigenvalues and eigenfunctions with Neumann boundary
9
conditioninΩǫconvergetotheonesinΩ0,thennecessarilywehavethat
2 |Ωǫ\Ω0|→0whileitisnotnecessarilytruethatdist(Ωǫ,Ω0)ǫ−→→00. As
a matter of fact we will construct an example of a perturbation where
]
P thespectrabehavecontinuouslybutdist(Ωǫ,Ω0)ǫ−→→0+∞.
S
.
h
t
a
m 1. Introduction
[ Thispaperisconcernedwiththebehavioroftheeigenvaluesandeigenfunc-

tionsoftheLaplaceoperatorinboundeddomainswhenthe domainundergoes
1
a perturbation. It is well known that if the boundary condition that we are
v
6 imposing is of Dirichlet type, the kind of perturbations that we may allow in
2 ordertoobtainthecontinuityofthespectraismuchbroaderthaninthecaseof
7 Neumann boundary condition. This is explicitly stated in the pioneer work of
4 Courantand Hilbert [5] and it has been subsequently clarifiedin many works,
. see [4, 2, 6] and reference therein among others. See also [8] for a generaltext
1
on different properties of eigenvalues and [9] for a study on the behavior of
0
9 eigenvalues and in general partial differential equations when the domain is
0 perturbed.
: In particular, with Dirichlet boundary condition we may consider the case
v where the fixed domainis a bounded “smooth”domain Ω ⊂RN, N ≥2, and
i 0
X the perturbed domain is Ω in such a way that Ω ⊂ Ω , that is we consider
ǫ 0 ǫ
exterior perturbation of the domain. We may have perturbations of this type
r
a where |Ω \Ω | ≥ η for some fixed η > 0 and still we have the convergence
ǫ 0
of the eigenvalues and eigenfunctions. Moreover, we may even have the case
|Ω \ Ω | → +∞ and still we have the convergence of the eigenvalues and
ǫ 0
eigenfunctions.
Date:29January2009.
∗Partiallysupported byGrants PHB2006-003-PC andMTM2006-08262 fromMEC and
by “Programa de Financiacio´n de Grupos de Investigacio´n UCM-Comunidad de Madrid
CCG07-UCM/ESP-2393. Grupo920894”andSIMUMAT-ComunidaddeMadrid,Spain.
†Partiallysupported by the Czech Ministryof Education, Youth and Sports within the
projectLC06002.
1
2 J.M.ARRIETA&D.KREJCˇIRˇ´IK
To obtain and example of this situation is not too difficult. If we consider
for instance Ω⊂R2, given by Ω =(0,1)×(−1,0) and
0
Ω (a)={(x,y):0<x<1,−1<y <a(1+sin(x/ǫ))}⊃Ω
ǫ 0
where a>0 is fixed, we can easily see that the eigenvalues and eigenfunctions
oftheLaplaceoperatorwithDirichletboundaryconditioninΩ convergetothe
ǫ
1
ones in Ω . Moreover|Ω |=|Ω |+ a(1+sin(x/ǫ))dx∼|Ω |+a for ǫ small
0 ǫ 0 0 0
enough. Moreover,itisnotdifficulttomodifythe exampleabovechoosingthe
R
constantadependentwithrespecttoǫinsuchawaythata(ǫ)→+∞andsuch
that we still get that the eigenvalues and eigenfunctions in Ω (a(ǫ)) converge
ǫ
to the ones in Ω and |Ω (a(ǫ))\Ω | → +∞. This example shows that the
0 ǫ 0
class of perturbations that we may allow to get the “spectral convergence” of
the Dirichlet Laplacian is very broad and that knowing that the eigenvalues
and eigenfunctions of the Dirichlet Laplacian converge does not have many
“geometrical” restrictions for the domains.
ThecaseofNeumannboundaryconditionismuchmoresubtle. Asamatter
offact,forthesituationdepictedaboveitisnottruethatthespectraconverge.
So we ask ourselves the following questions: if we have a domain Ω and
0
consider a perturbation of it given by Ω ⊂ Ω , where we assume that all the
0 ǫ
domainsaresmoothandboundedalthoughnotnecessarilyuniformlybounded
on the parameter ǫ, then if we have the convergence of the eigenvalues and
eigenfunctions,
ǫ→0
(Q1) should it be true that |Ω \Ω |−→0?
ǫ 0
ǫ→0
(Q2) should it be true that dist(Ω ,Ω )=sup dist(x,Ω )−→0?
ǫ 0 x∈Ωǫ 0
We will see that the answer to the first question is Yes and, surprisingly,
the answer to the second one is No.
Observethat,astheexampleaboveshows,theanswertobothquestionsfor
the case of Dirichlet boundary condition is No.
In Section 2 we recall a result from [1, 3] which provides a necessary and
sufficientconditionfortheconvergenceofeigenvaluesandeigenfunctionswhen
the domain is perturbed. In Section 3 we provide an answerto question (Q1)
and in Section 4 we provide an answer to question (Q2).
2. Characterization of spectral convergence of Neumann
Laplacian
In this section we give a necessary and sufficient condition for the con-
vergence of the eigenvalues and eigenfunctions of the Laplace operator with
Neumann boundary conditions. We refer to [1] and [3] for a general result in
this direction, even in a more general context than the one in this note. In
our particular case, we will consider the following situation: let Ω be a fixed
0
bounded smooth (Lipschitz is enough) open set in RN with N ≥2 and let Ω
ǫ
be a family of domains such that for each fixed 0<ǫ≤ǫ , Ω is bounded and
0 ǫ
smooth with Ω ⊂Ω .
0 ǫ
Let us define now what we mean by the spectral convergence. For 0 ≤
ǫ ≤ ǫ , we denote by {λǫ}∞ the sequence of eigenvalues of the Neumann
0 n n=1
Laplacian in Ω , always ordered and counting its multiplicity, and we denote
ǫ
by {φǫ}∞ a corresponding set of orthonormal eigenfunctions in Ω . Also,
n n=1 ǫ
GEOMETRIC VERSUS SPECTRAL CONVERGENCE 3
since weareconsideringdomains whichvarywith the parameterǫ andwewill
needto comparefunctions defined inΩ andinΩ , weintroduce the following
0 ǫ
space H1 = H1(Ω )⊕H1(Ω \Ω¯ ), that is χ ∈ H1 if χ ∈ H1(Ω ) and
ǫ 0 ǫ 0 ǫ |Ω0 0
χ|(Ωǫ\Ω¯0) ∈H1(Ωǫ\Ω¯0), with the norm
kχk2 =kχk2 +kχk2 .
Hǫ1 H1(Ω0) H1(Ωǫ\Ω¯0)
We have that H1(Ω ) ֒→ H1 and in a natural way we have that if χ ∈
ǫ ǫ
H1(Ω ) via the extension by zero outside Ω we have χ ∈ H1. Hence, with
0 0 ǫ
certain abuse of notation we may say that if χ ∈ H1, 0 ≤ ǫ ≤ ǫ , then
ǫ ǫ 0
χ −ǫ→→0 χ in H1 if kχ −χ k +kχ k −ǫ→→0 0.
ǫ 0 ǫ ǫ 0 H1(Ω0) ǫ H1(Ωǫ\Ω0)
Definition2.1. WewillsaythatthefamilyofdomainsΩ convergesspectrally
ǫ
toΩ asǫ→0iftheeigenvaluesandeigenprojectorsoftheNeumannLaplacian
0
behave continuously at ǫ = 0. That is, for any fixed n ∈ N we have that
λǫ → λ0 as ǫ → 0, and for each n ∈ N such that λ0 < λ0 the spectral
n n n n+1
projections Pǫ :L2(RN)→H1(Ω ), Pǫ(ψ)= n (φǫ,ψ) φǫ, satisfy
n ǫ n i=1 i L2(Ωǫ) i
sup{kPnǫ(ψ)−Pn0(ψ)kHǫ1, ψ ∈L2(RN)P,kψkL2(RN) =1}−ǫ→→0 0.
The convergence of the spectral projections is equivalent to the following:
for each sequence ǫ → 0 there exists a subsequence, that we denote again
k
by ǫ and a complete system of orthonormal eigenfunctions of the limiting
k
problem {φ0n}∞n=1 such that kφǫnk −φ0nkHǫ1k →0 as k →∞.
Inorderto write downthe characterization,weneedto considerthe follow-
ing quantity
|∇φ|2
τ = min ZΩǫ . (2.1)
ǫ
φ∈H1(Ωǫ) |φ|2
φ=0 in Ω0
ZΩǫ
Observe that τ is the first eigenvalue of the following problem with a combi-
ǫ
nation of Dirichlet and Neumann boundary conditions:
−∆u=τu, Ω \Ω¯ ,
ǫ 0
u=0, ∂Ω ,
 0
 ∂u =0, ∂Ω \∂Ω .
ǫ 0
∂n
We can prove the following,
Proposition 2.2. A necessary and sufficient condition for the spectral con-
vergence of Ω to Ω is
ǫ 0
ǫ→0
τ −→+∞. (2.2)
ǫ
We refer to [1] and [3] for a proof of this result.
Remark 2.3. The fact that Ω ⊂ Ω can be relaxed. It is enough asking
0 ǫ
that for each compact set K ⊂ Ω there exists ǫ(K) such that K ⊂ Ω for
0 ǫ
0<ǫ≤ǫ(K), see [3].
4 J.M.ARRIETA&D.KREJCˇIRˇ´IK
3. Measure convergence of the domains
In this section we provide an answer to the first question. Observe that in
ǫ→0
Proposition 2.2 we do not require that |Ω \Ω |−→0. However, we have the
ǫ 0
following
Corollary 3.1. In the situation above if Ω converges spectrally to Ω , then
ǫ 0
necessarily |Ω \Ω |−ǫ→→0 0.
ǫ 0
Proof. This result is proved in [3] but for the sake of completeness and since
it is a simple proof, we include it in here.
Ifthiswerenottruethenwewillhaveapositiveη >0andasequenceǫ →0
k
such that |Ω \Ω | ≥ η. Let ρ = ρ(η) be a small number such that |{x ∈
ǫk 0
RN \Ω , dist(x,Ω ) ≤ ρ}| ≤ η/2. This implies that |{x ∈ Ω , dist(x,Ω ) ≥
0 0 ǫk 0
ρ}| ≥ η/2. Let us construct a smooth function γ with γ = 0 in Ω , and
0
γ(x) = 1 for x ∈ RN \Ω with dist(x,Ω ) ≥ ρ. Then obviously γ ∈ H1(Ω )
0 0 ǫk
with k∇γkL2(Ωǫk) ≤ C and kγkL2(Ωǫk) ≥ (η/2)12. This implies that τǫk is
ǫ→0
bounded. Henceitisnottruethatτ −→+∞andtherefore,fromProposition
ǫ
2.2, we do not obtain the spectral convergence.
In particular, this result implies that the answer to question (Q1) is af-
firmative. That is, if we have the convergence of Neumann eigenvalues and
ǫ→0
eigenfunctions, necessarily we have that |Ω \Ω |−→0.
ǫ 0
4. Distance convergence of the domains
Inthis sectionwe willprovideananswerto question(Q2)and,asamatter
offact,wewillseethattheanswerisNo. Wewillprovethisbyconstructingan
exampleofafixeddomainΩ andasequenceofdomainsΩ withΩ ⊂Ω with
0 ǫ 0 ǫ
thepropertythatdist(Ω ,Ω )doesnotconvergesto0,buttheeigenvaluesand
ǫ 0
eigenfunctions of the Laplace operator with Neumann boundary conditions in
Ω converge to the ones in Ω , see Definition 2.1.
ǫ 0
Asamatteroffactin[3,Section5.2]averyparticularexampleofadumbbell
domain (two disconnected domains joined by a thin channel) is provided so
that the eigenvalues from the dumbbell converge to the eigenvalues of the
two disconnected domains and no spectral contribution from the channel is
observed. In this note we will obtain a family of channels for which the same
phenomena occurs, see Corollary 4.4, and will provide a proof, different from
the one given in [3].
Let us consider a fixed domain Ω ⊂ RN which satisfies that Ω ⊂ {x ∈
0 0
RN, x <0} and such that
1
Ω ∩{x=(x ,x′)∈R×RN−1,−1<x <1, |x′|≤ρ}
0 1 1
={x=(x ,x′)∈R×RN−1,−1<x <0, |x′|≤ρ}
1 1
for some fixed ρ>0.
We will construct Ω as Ω =int(Ω¯ ∪R¯ ), where R is given as follows
ǫ ǫ 0 ǫ ǫ
R ={(x ,x′)∈R×RN−1 : 0<x <L, |x′|<g (x )} (4.1)
ǫ 1 1 ǫ 1
GEOMETRIC VERSUS SPECTRAL CONVERGENCE 5
where the function g will be chosen so that g > 0, g ∈ C1([0,L]) and
ǫ ǫ ǫ
g → 0 uniformly on [0,L], see Figure 1. For the sake of notation we denote
ǫ
by Γǫ =∂R ∩{x =0} and Γǫ =∂R ∩{x =L}.
0 ǫ 1 L ǫ 1
x'
g (x)
ε 1
0 L x
1
Figure1. TheexteriorperturbationR . Thethicklinerefers
ǫ
tothesupplementaryDirichletconditionintheproblem(4.2),
while Neumann boundary conditions are imposed elsewhere.
Wereferto[12]forageneralreferenceonthebehaviorofsolutionsofpartial
differential equations on thin domains. See also the recent survey [7] for a
study on the spectrum of the Laplacian on thin tubes in various settings, and
for many related references.
Observe that if L is fixed then dist(Ω ,Ω ) = L for each 0 < ǫ ≤ ǫ .
ǫ 0 0
Moreover we will show that for certain choices of g we obtain the spectral
ǫ
convergenceofthe Laplaceoperator. To provethis results,we useProposition
2.2 and show that τ → +∞. Notice that τ , defined in (2.1) is the first
ǫ ǫ
eigenvalue of
−∆u=τu, R ,
ǫ
u=0, Γǫ,
 0 (4.2)
 ∂u =0, ∂R \Γǫ.
∂n ǫ 0
SincewehaveNeumannboundaryconditionsonthelateralboundaryofRǫ,
there clearly exist profiles of g for which τ remains uniformly bounded as
ǫ ǫ
ǫ → 0. In fact, a simple trial-function argument shows that τ ≤ π2/(2L)2
ǫ
6 J.M.ARRIETA&D.KREJCˇIRˇ´IK
whenever g (s)≥g (0) for every s∈[0,L]. The idea to get τ →+∞ consists
ǫ ǫ ǫ
in choosing a rapidly decreasing function s 7→ g (s), which enables one to
ǫ
get a large contribution to τ coming from the longitudinal energy due to the
ǫ
approaching Dirichlet and Neumann boundary conditions in the limit ǫ → 0.
Let us notice that a similar trick to employ the repulsive contribution of such
a combination of the boundary conditions have been used recently in [10] to
establish a Hardy-type inequality in a waveguide; see also [11] for eigenvalue
asymptotics in narrow curved strips with combined Dirichlet and Neumann
boundary conditions. In our case, we are able to show
Proposition 4.1. With the notations above, for any function γ ∈ C2([0,L])
satisfying
0<α ≤γ ≤α <1, γ˙(L)≤0, and γ¨ ≥α >0 (4.3)
0 1 2
for some positive numbers α , α and α , if we define g = γ1/ǫ we have that
0 1 2 ǫ
τ −ǫ→→0 0.
ǫ
In particular, applying Proposition 2.2 we obtain the convergence of the
eigenvalues and eigenfunctions of the Neumann Laplacian in Ω to the ones in
ǫ
Ω .
0
Remark 4.2. Observe that a function γ satisfying (4.3) necessarily satisfies
that γ˙(s)<0 for 0≤s<L. Hence, the function γ is decreasing.
Proof: Since τ is given by minimization of the Rayleigh quotient,
ǫ
|∇φ|2
τ = inf ZRǫ
ǫ
φ∈H1(Rǫ) |φ|2
φ=0 in Γǫ
0 ZRǫ
we analyze the integral |∇φ|2 for a smooth real-valued function φ with
Rǫ
φ=0 in a neighborhood of Γǫ. We have
R 0
L
|∇φ|2 = (|φx1|2+|∇x′φ|2)dx′dx1
ZRǫ Z0 Z|x′|<gǫ(x1)
Consideringthechangeofvariablesx =y ,x′ =g (y )y′ whichtransforms
1 1 ǫ 1
(x ,x′) ∈ R into (y ,y′) ∈ Q where Q is the cylinder Q = {(y ,y′) : 0 <
1 ǫ 1 1
y <L, |y′|<1}andperformingthischangeofvariablesintheintegralabove,
1
elementary calculations show that
2
N N
g˙ 1
|∇φ|2 = ϕ − ǫ y ϕ + |ϕ |2 gN−1dy
ZRǫ ZQ y1 gǫ i=2 i yi! gǫ2 i=2 yi  ǫ
X X
 
where ϕ(y)=φ y ,g (y )y′ .
1 ǫ 1
Writingthea(cid:0)boveexpress(cid:1)ionintermsofthenewfunctionψ(y)=gǫ(y1)N2−1ϕ(y)
so that
g(N−1)/2ϕ =ψ , i=2,...,N,
ǫ yi yi
N −1g˙
g(N−1)/2ϕ =− ǫ ψ+ψ ,
ǫ y1 2 g y1
ǫ
GEOMETRIC VERSUS SPECTRAL CONVERGENCE 7
we get,
|∇φ|2
ZRǫ
2
N N
N −1g˙ g˙ 1
= − ǫψ+ψ − ǫ y ψ + |ψ |2 dy
ZQ 2 gǫ y1 gǫ i=2 i yi! gǫ2 i=2 yi 
X X
 2 
N −1g˙ 2 g˙ N g˙
ǫ ǫ ǫ
= − ψ + ψ − y ψ −(N −1) ψψ
2 g y1 g i yi g y1
ZQ"(cid:18) ǫ (cid:19) ǫ i=2 ! ǫ
X
g˙2 N 1 N
+(N −1) ǫ y ψ ψ+ |ψ |2 dy
g2 i yi g2 yi
ǫ i=2 ǫ i=2 #
X X
N −1 2 g˙2 g˙
≥ ǫψ2−(N −1) ǫψψ
2 g2 g y1
ZQ"(cid:18) (cid:19) ǫ ǫ
g˙2 N 1 N
+(N −1) y ψ ψ+ ψ2 dy
g2 i yi g2 yi
ǫ i=2 ǫ i=2 #
X X
where we have used that (ψ − N y ψ g˙ǫ)2 ≥0. Via integration by parts
y1 i=2 i yigǫ
in the second and third term above, we get,
P
g˙ L g˙
−(N −1) ǫψψ dy = −(N −1) ǫ (ψ2) dy dy′
g y1 2g y1 1
ZQ ǫ Z|y′|<1Z0 ǫ
g˙ y1=L L g˙ .
= − (N −1) ǫ ψ2 + (N −1) ǫ ψ2dy dy′
1
2g 2g
Z|y′|<1 (cid:20) ǫ (cid:21)y1=0 Z0 (cid:18) ǫ(cid:19) !
g˙ (L) N −1 g¨ g˙2
=− (N −1) ǫ ψ2(L,y′)dy′+ ǫ − ǫ ψ2dy
2g (L) 2 g g2
Z|y′|<1 ǫ ZQ (cid:18) ǫ ǫ(cid:19)
and
g˙2 N L g˙2 N 1
(N −1) ǫ y ψ ψdy = (N −1) ǫ y (ψ2) dy′dy
g2 i yi g2 i2 yi 1
ZQ ǫ i=2 Z0 ǫ i=2Z|y′|<1
X X
L N −1g˙2
= ǫ ψ2−(N −1) ψ2dy′ dy .
2 g2 1
Z0 ǫ Z|y′|=1 Z|y′|<1 !
Hence if we require that g˙ (L)≤0, we have,
ǫ
N −1g¨ N −1 2 N −1 g˙2
|∇φ|2 ≥ ǫ − + ǫ ψ2dy
2 g 2 2 g2
ZRǫ ZQ" ǫ (cid:18) (cid:19) ! ǫ# (4.4)
L N −1g˙2 1 N
+ ǫ ψ2dy′ dy + ψ2 dy.
2 g2 1 g2 yi
Z0 ǫ Z|y′|=1 ! ZQ ǫ i=2
X
8 J.M.ARRIETA&D.KREJCˇIRˇ´IK
The last two terms in this expression can be written as
L 1 N −1
g2(y ) |∇y′ψ|2+ 2 g˙ǫ2(y1) ψ2 dy1
Z0 ǫ 1 Z|y′|≤1 Z|y′|=1 !
and we have that
N −1
|∇y′ψ|2+ 2 g˙ǫ2 ψ2 ≥ρ ψ2
Z|y′|≤1 Z|y′|=1 Z|y′|≤1
with ρ=ρ(y ) being the first eigenvalue of the problem
1
−∆y′ψ =ρψ, |y′|<1,
∂ψ N −1
 + g˙2(y )ψ =0, |y′|=1,
 ∂n 2 ǫ 1
wherendenotestheoutwardunitnormalvectorfieldtothe(N−2)dimensional

unit sphere S ={y′ ∈RN−1 : |y′|=1}.
1
We claim that if we denote by λ(η) the first eigenvalue of
−∆y′ψ =λψ, |y′|<1,
∂ψ
 +ηψ =0, |y′|=1,
 ∂n
we have that λ(ηη) → ||BS11|| as η →0, where B1 is the (N −1) dimensional unit
ball and S its surface, which satisfy |S | = (N −1)|B |. As a matter of fact
1 1 1
by standard continuity result we know that λ(η) → 0 and its eigenfunction
ψ , which is radially symmetric, converges to the constant function 1/ |B |,
η 1
which is the first eigenfunction of the Neumann eigenvalue problem. But
p
λ(η)= |∇y′ψη|2+η |ψη|2 ≥η |ψη|2
ZB1 ZS1 ZS1
which implies that
λ(η) |S |
≥ |ψ |2 → 1 .
η
η |B |
ZS1 1
Moreover, using ψ = 1/ |B | as a test function in the Rayleigh quotient
1
for λ(η), we immediately obtain λ(η) ≤ η|S1|. This proves our claim. In
p |B1|
particular, given δ > 0 small, we can choose η = η (δ) such that λ(η) >
0 0
(N −1−δ)η for 0<η ≤η .
0
Therefore, if we choose the function g such that g˙ (y ) → 0 uniformly in
ǫ ǫ 1
y ∈[0,L], we have that ρ(y )≥ (N−1)(N−1−δ)g˙2(y ) for ǫ small enough.
1 1 2 ǫ 1
Hence,
2
N −1g¨ N −1
|∇φ|2 ≥ ǫ −
2 g 2
ZRǫ ZQ(cid:26) ǫ (cid:20)(cid:18) (cid:19)
(N −1)(N −1−δ) N −1 g˙2
− + ǫ ψ2dy
2 2 g2
(cid:21) ǫ (cid:27)
N −1 g¨ N −1 g˙2
= ǫ − −(N −1−δ)+1 ǫ ψ2dy
2 g 2 g2
ZQ(cid:26) ǫ (cid:20) (cid:21) ǫ(cid:27)
GEOMETRIC VERSUS SPECTRAL CONVERGENCE 9
and observe that the number κ = N−1 −(N −1−δ)+1 is strictly less than
2
one for all values of N ≥2 choosing a fixed and small δ >0. If we denote by
g¨ g˙2
m = inf ǫ −κ ǫ
ǫ 0≤y1≤L(cid:18)gǫ gǫ2(cid:19)
then
N −1 N −1
|∇φ|2 ≥ m ψ2 = m φ2.
ǫ ǫ
2 2
ZRǫ ZQ ZRǫ
Consequently, τ ≥ N−1m .
ǫ 2 ǫ
Letusseethatwecanmakeachoiceofthefamilyoffunctionsg ,satisfying
ǫ
thetwopreviousconditionswehaveimposed,thatisg˙ (L)≤0andg˙ (y )→0
ǫ ǫ 1
uniformly in 0≤y ≤L such that m →+∞ as ǫ→0.
1 ǫ
Let us choose a function γ ∈ C2([0,L]) satisfying (4.3) and let g = γ1/ǫ.
ǫ
Then, we have
1 1 1 1
g˙ǫ = γ1ǫ−1γ˙ , g¨ǫ = ( −1)γ1ǫ−2γ˙2+ γ1ǫ−1γ¨,
ǫ ǫ ǫ ǫ
and simple calculations show that
g¨ g˙2 1 1 1 2 γ˙ 2 γ¨ α 1
ǫ −κ ǫ = ( −1)−κ + ≥ 2
g g2 ǫ ǫ ǫ γ ǫγ α ǫ
ǫ ǫ " (cid:18) (cid:19) #(cid:18) (cid:19) 0
for ǫ > 0 small enough so that 1(1 − 1)− κ 1 2 ≥ 0 . This shows that
ǫ ǫ ǫ
m →+∞ and it proves the proposition.
ǫ
(cid:0) (cid:1)
Remark 4.3. Now that we have been able to construct a thin domain R as
ǫ
in (4.1) such that τ −ǫ→→0 +∞, we can construct another thin domain R˜ such
ǫ ǫ
that its “length” goes to infinity, its width goes to zero and still τ˜ −ǫ→→0 +∞,
ǫ
where τ˜ is the first eigenvalue of (4.2) in R˜ instead of R .
ǫ ǫ ǫ
For this, let R be a thin domain constructed as in Proposition 4.1 and let
ǫ
ρ be a sequence with ρ → +∞ such that τǫ → +∞ and α1/ǫρ → 0. Define
ǫ ǫ ρ2 1 ǫ
ǫ
R˜ =ρ R , that is
ǫ ǫ ǫ
R˜ ={(x ,x′): 0<x <ρ L, |x′|<ρ g (x )},
ǫ 1 1 ǫ ǫ ǫ 1
then 0<ρ g (x )≤α1/ǫρ −ǫ→→0 0 and τ˜ = τǫ −ǫ→→0 +∞.
ǫ ǫ 1 1 ǫ ǫ ρ2
ǫ
Observe that if we require also a Dirichlet boundary condition in Γǫ, we
L
canrelaxtheconditionsonγ inProposition4.1andinparticularthecondition
γ˙(L)≤0 can be dropped. Hence, we can show,
Corollary 4.4. With the notations above, for any function γ ∈ C2([0,L])
satisfying
0<α ≤γ ≤α <1, and γ¨ ≥α >0
0 1 2
10 J.M.ARRIETA&D.KREJCˇIRˇ´IK
for some positive numbers α , α and α , if we define g = γ1/ǫ we have that
0 1 2 ǫ
τ˜ −ǫ→→0 0, where τ˜ǫ is the first eigenvalue of
ǫ
−∆u=τu, R ,
ǫ
u=0, Γǫ ∪Γǫ ,
 0 L
 ∂u =0, ∂R \(Γǫ ∪Γǫ).
∂n ǫ 0 L
Proof: This followseasily by a Neumann bracketing argument. More pre-
cisely, from the hypotheses, γ˙ is a strictly increasing function. Hence, either
γ is strictly monotone in (0,L), or there exists a unique L∗ ∈(0,L) such that
γ˙(L∗)=0.
Inthefirstcase,ifγ isdecreasing(respectivelyincreasing)wesubstitutethe
Dirichlet boundary condition at Γǫ (respectively at Γǫ) by a Neumann one.
L 0
Then the new eigenvalue problem gives rise to τ defined exactly in the same
ǫ
way as (4.2) (modulo possibly a mirroring of R ) and we have τ˜ ≥τ →+∞
ǫ ǫ ǫ
as ǫ→0.
In the second case, we cut the domain R in two domains R0 = R ∩{0 <
ǫ ǫ ǫ
x < L∗}, R1 = R ∩{L∗ < x < L}. We know that τ˜ ≥ inf{τ0,τ1}, where
1 ǫ ǫ 1 ǫ ǫ ǫ
τ0 and τ1 are the corresponding eigenvalues in R0 and R1 with a Neumann
ǫ ǫ ǫ ǫ
boundary condition imposed at the newly created boundary R ∩{x = L∗}
ǫ 1
onbothdomains. InbothdomainswecanapplyProposition4.1asinthefirst
case so that τ0,τ1 −ǫ→→0 +∞, which implies τ˜ →0.
ǫ ǫ ǫ
Remark 4.5. This corollary recovers and generalizes the results from Sec-
tion 5.2 in [3].
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