Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap

0.19 MB·English

by  Andrii Khrabustovskyi

Gaps in the spectrum of the Neumann Lapla-
cian generated by a system of periodically dis-
tributed trap
Andrii Khrabustovskyi1 2 3, Evgeni Khruslov2
3 Abstract. The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic
1 unbounded domain Ωε depending on a small parameter ε > 0. The domain has the form Ωε = Rn Sε,
0 whereSεisanεZn-periodicfamilyoftrap-likescreens. Weprovethatforanarbitrarilylarge Lthespec\trum
2
has just one gapin [0,L] when εsmall enough, moreover when ε 0this gap converges to some interval
n →
whoseedgescanbecontrolledbyasuitablechoiceofgeometryofthescreens. Anapplication tothetheory
a
J of2D-photonic crystalsisdiscussed.

4
1 Keywords: periodicdomain,NeumannLaplacian,spectrum, gaps

]
P
S
.
h Introduction
t
a
m Itiswell-known(see,e.g.,[1–3])thatthespectrumofself-adjointperiodicdifferentialoperators
has a band structure, i.e. it is a union of compact intervals called bands. The neighbouring bands
[
mayoverlap,otherwisewehaveagapinthespectrum(i.e. anopenintervalthatdoesnotbelongto
1
thespectrumbutitsends belongtoit). In general theexistenceofspectral gapsisnotguaranteed.
v
6 For applications it is interesting to construct the operators with non-void spectral gaps since
2
their presence is important for the description of wave processes which are governed by differen-
9
2 tial operators under consideration. Namely, if the wave frequency belongs to a gap, then the cor-
1. responding wave cannot propagate in the medium without attenuation. This feature is a dominant
0 requirement for so-called photonic crystals which are materials with periodic dielectric structure
3
attractingmuchattentioninrecent years (see, e.g., [4–6]).
1
: In the present work we derive the effect of opening of spectral gaps for the Laplace operatorin
v
i Rn (n 2)perforated by afamilyofperiodicallydistributedtraps on whichweposetheNeumann
X ≥
boundaryconditions. Thetrapsaremadefrominfinitelythin screens(seeFig. 1). Inthecasen = 2
r
a thisoperatordescribesthepropagationofthe H-polarizedelectro-magneticwavesinthedielectric
medium containing a system of perfectly conducting trap-like screens (see the remark in Section
3).
We describe the problem and main result more precisely. Let ε > 0 be a small parameter. Let
Sε = Sε be a union of periodically distributed screens Sε in Rn (n 2). Each screen Sε is an
i Zn i i ≥ i
(n 1)∈-dimensional surface obtained by removing of a small spherical hole from the boundary of
S
−
an-dimensionalcube. Itissupposedthatthedistancebetweenthescreens isequal to ε,thelength
1DepartmentofMathematics,KarlsruheInstituteofTechnology,Germany
2MathematicalDivision,B.VerkinInstituteforLowTemperaturePhysicsandEngineeringoftheNationalAcad-
emyofSciencesofUkraine
3Correspondence to: Department of Mathematics, Karlsruhe Institute of Technology, Kaiserstrasse 89-93, Karl-
sruhe76133,Germany
E-mail:[email protected],[email protected]
1
2
of their edges is equal to bε, while the radius of the holes is equal to dε n if n > 2 and e 1/dε2 if
n−2 −
n = 2. Here d (0, ),b (0,1)are constants.
∈ ∞ ∈
Sε
i
ε
Fig. 1. Thesystemofscreens Sε
i
By ε we denote the Neumann Laplacian in the domain Rn Sε. Our goal is to describe the
A \
behaviourofitsspectrum σ( ε)as ε 0.
A →
The main result of this work is as follows (see Theorem 1.1). For an arbitrarily large L the
spectrumσ( ε)hasjustonegapin[0,L]whenεissmallenough. Whenε 0thisgapconverges
A →
to some interval (σ,µ) depending in simple manner on the coefficients d and b. Moreover (see
Corollary 1.1) with a suitable choice of d and b this interval can be made equal to an arbitrary
preassignedintervalin(0, ).
∞
The possibility of opening of spectral gaps by means of a periodic perforation was also inves-
tigated in [7]. Here the authors studied the spectrum of the Neumann Laplacian in R2 perforated
by Z2-periodic family of circular holes. It was proved that the gaps open up when the diameter
d of holes is close enough to the distance between their centres (which is equal to 1). However,
the structure of the spectrum in [7] differs essentially from that one in the present paper. Namely,
when d 1thespectrumconverges (uniformlyon compactintervals)toasequenceofpoints.
→
Various examples of scalar periodic elliptic operators in the entire space with periodic coeffi-
cients were presented in [8–18]. In these works spectral gaps are the result of high contrast in
(someof)thecoefficients oftheoperator.
The outline of the paper is as follows. In Section 1 we describe precisely the operator ε and
A
formulatethemainresultofthepaper(Theorem 1.1)describingthebehaviourof σ( ε)asε 0.
A →
Theorem1.1isprovedinSection2. Finally,inSection3onaformallevelofrigourwediscussthe
applicationsto thetheory of2D photoniccrystals.
1. Setting of theproblemand themain result
Letn N 1 and letε > 0. Weintroducethefollowingsets:
∈ \{ }
B = x = (x ,...,x ) Rn : b/2 < x < b/2, i ,where b (0,1)isaconstant.
1 n i
• { ∈ − ∀ } ∈
Dε = x ∂B : x x0 < dε , where x0 = (0,0,...,0,b/2) and dε is defined by the fol-
• ∈ | − |
lowingformula:
n o
2
dεn 2, n > 2,
−
dε = 1 (1)
ε 1exp , n = 2.
 − −dε2!
Here d > 0 isaconstant. It issupposedthatε issmallenoughso thatdε < b/2
3
Sε = ∂B Dε
• \
Fori Zn weset:
∈
Sε = ε(Sε +i) (2)
i
and (seeFig. 1)
Ωε = Rn Sε (3)
\ i
i Zn 
Now we define precisely the Neumann Laplacian[∈in Ωε. We denote by ηε[u,v] the sesquilinear
form in L (Ωε)whichisdefined bytheformula
2
ηε[u,v] = ( u, v¯)dx (4)
∇ ∇
Z
Ωε
n
∂u ∂v
and the definitionaldomain dom(ηε) = H1(Ωε). Here ( u, v) = . The form ηε[u,v] is
∇ ∇ ∂x ∂x
k k
k=1
X
denselydefinedclosedandpositive. Then(see,e.g.,[19,Chapter6,Theorem2.1])thereexiststhe
uniqueself-adjointand positiveoperator ε associatedwiththeform ηε,i.e.
A
( εu,v) = ηε[u,v], u dom( ε), v dom(ηε) (5)
A L2(Ωε) ∀ ∈ A ∀ ∈
It follows from (5) that εu = ∆u in the generalized sense. Using a standard regularity theory
A −
(see,e.g,[20,Chapter5])itiseasytoshowthateachu dom( )belongstoH2 (Ωε),furthermore
∈ A loc
∂u
= 0forany smoothΓ ∂Ωε.
∂n ⊂
Γ
W(cid:12) edenotebyσ( ε)thespectrumof ε. Todescribethebehaviourof σ( ε)asε 0weneed
(cid:12)
(cid:12) A A A →
som(cid:12)eadditionalnotations.
(cid:12)
In thecase n > 2 wedenoteby cap(T) thecapacity ofthedisc
T = x = (x ,...,x ) Rn : x < 1, x = 0
1 n n
{ ∈ | | }
Recall (see, e.g,[21])that itisdefined by
cap(T) = inf w2dx
w |∇ |
RZn
wheretheinfimumistakenoversmoothandcompactlysupportedinRn functionsequalto1onT.
Weset
cap(T)dn 2
−
, n > 2, σ
σ = 4bn µ = (6)
 πd 1 bn
2b2, n = 2, −
It isclearthatσ < µ. 
Thebehaviourof σ( ε)as ε 0is describedby thefollowingtheorem.
A →
Theorem 1.1. Let L be an arbitrary number satisfying L > µ. Then the spectrum of the operator
ε hasthefollowingstructurein[0,L] when ε is smallenough:
A
σ( ε) [0,L] = [0,L] (σε,µε) (7)
A ∩ \
4
where theinterval(σε,µε)satisfies
limσε = σ, limµε = µ (8)
ε 0 ε 0
→ →
Corollary 1.1. For an arbitrary interval (σ,µ) (0, ) there is a family Ωε of periodic un-
ε
bounded domains in Rn such that for an arbitrar⊂y num∞ber L satisfying L > {µ th}e spectrum of the
corresponding Neumann Laplacian ε has just one gap in [0,L] when ε is small enough and this
A
gapconverges to theinterval(σ,µ)asε 0.
→
(6)
Proof. It is easy to see that the map (d,b) (σ,µ) is one-to-one and maps (0, ) (0,1) onto
7→ ∞ ×
(σ,µ) R2 : 0 < σ < µ . Theinversemap isgivenby
∈
n no24σ(1 σµ 1)(cap(T)) 1, n > 2,
d = − − − − b = n1 σµ 1 (9)
2σπ 1(1 σµ 1), n = 2, − −
ThenthedomainΩε conspid−ered−inTh−eorem1.1withd andbbeingdefipnedbyformula(9)satisfies
therequirementsofthecorollary. (cid:3)
Remark1.1. ItwillbeeasilyseenfromtheproofofTheorem1.1thatthemainresultremainsvalid
for an arbitrary open domain B which is compactly supported in the unit cube ( 1/2,1/2)n and
−
whose boundary contains an open flat subset on which we choose the point x0. In this case the
coefficients σ and µ are defined again by formula (6) but with B instead of bn (here by we
| | | · |
denotethevolumeofthedomain).
Remark 1.2. One can guess that in order to open up m > 1 gaps we have to place m screens Sε,
1
Sε,...,Sε in the cube ( 1/2,1/2)n. However the proof of this conjecture is more complicated
2 m −
comparingwith thecase m = 1. We proveitin ournextwork. Below weonly announcetheresult
forthecase m = 2.
Let B and B be arbitrary open cuboids which are compactly supported in ( 1/2,1/2)n. On
1 2
−
∂B and ∂B wechoosethepoints x1 and x2 correspondingly. Wesupposethat thesepointsdo not
1 2
belongtotheedgesofcuboids. LetDε andDε beopenballswiththeradiidε anddε andthecentres
1 2 1 2
at x1 and x2 correspondingly. Here dε, j = 1,2 are defined by formula (1) but with d instead of d
j j
(d > 0 are constants). For i Zn, j = 1,2weset Sε = ε(Sε +i)and finally
j ∈ ij j
Ωε = Rn Sε
\ ij
i Zn,j=1,2 
By ε wedenotetheNeumannLaplacian inΩε.∈ [ 
A
We introduce the numbers σ , j = 1,2 by formula (6) with d and B instead of d and bn
j j j
| |
correspondingly. We supposethat d are such thattheinequality σ < σ holds. Finallywe define
j 1 2
thenumbers µ by theformula
j
1
µ = ρ +ρ +σ +σ +( 1)j (ρ +ρ +σ +σ )2 4(ρ σ +ρ σ +σ σ )
j 1 2 1 2 1 2 1 2 1 2 1 2 1 2
2 − −
where ρ = σ (cid:16)B (1 B ) 1. It isnot hardptocheck that σ < µ < σ < µ . (cid:17)
j j j j − 1 1 2 2
| | −| |
Now, let L be an arbitrary number satisfying L > µ . Then σ( ε) has just two gaps in [0,L]
2
A
when ε is small enough, moreover the edges of these gaps converge to the intervals (σ ,µ ) as
j j
ε 0. Also it is easily to showthat thepointsσ ,µ can be controlled by a suitablechoiceof the
j j
→
numbers d and thecuboids B .
j j
5
2. ProofofTheorem 1.1
We present the proof of Theorem 1.1 for the case n 3 only. For the case n = 2 the proof is
≥
repeated word-by-word withsomesmallmodifications.
In whatfollowsbyC,C ... wedenotegenericconstantsthat donot dependon ε.
1
By u wedenotethemean valueofthefunctionv(x) overthedomain B,i.e.
B
h i
1
u = u(x)dx
B
h i B
| | Z
B
Recall that by B we denotethevolumeofthedomain B.
| |
2.1. Preliminaries. Weintroducethefollowingsets(seeFig. 2):
Y = x Rn : 1/2 < x < 1/2, i .
i
{ ∈ − ∀ }
Yε = Y Sε.
\
F = Y B.
\
Let Aε betheNeumann Laplacianin ε 1Ωε. It isclearthat
−
σ( ε) = ε 2σ(Aε) (10)
−
A
Aε is an Zn-periodic operator, i.e. Aε commutes with the translations u(x) u(x+i), i Zn. For
7→ ∈
us it is more convenient to deal with the operator Aε since the external boundary of its period cell
isfixed (itcoincides with∂Y).
Dε
F
Sε
B
Fig. 2. Theperiod cell Yε
In view of the periodicity of Aε the analysis of the spectrum σ(Aε) is reduced to the analysis
of the spectrum of the Laplace operator on Yε with the Neumann boundary conditions on Sε and
so-called θ-periodicboundaryconditionson ∂Y. Namely,let
Tn = θ = (θ ,...,θ ) Cn : k θ = 1
1 n k
{ ∈ ∀ | | }
For θ Tn we introduce the functional space H1(Yε) consisting of functions from H1(Yε) that
∈ θ
satisfythefollowingconditionon∂Y:
k = 1,n : u(x+e ) = θ u(x) for x = (x ,x ,..., 1/2,...,x ) (11)
k k 1 2 n
∀ −
k-th↑place
where e = (0,0,...,1,...,0).
k
6
By ηθ,ε we denote the sesquilenear form defined by formula (4) (with Yε instead of Ω) and
the definitional domain H1(Yε). We define the operator Aθ,ε as the operator acting in L (Yε) and
θ 2
associatedwiththeform ηθ,ε, i.e.
(Aε,θu,v) = ηε,θ[u,v], u dom(Aε,θ), v dom(ηε,θ)
L2(Yε) ∀ ∈ ∀ ∈
The functions from dom(Aθ,ε) satisfy the Neumann boundary conditions on Sε, condition (11) on
∂Y and thecondition
∂u ∂u
k = 1,n : (x+e ) = θ (x) for x = (x ,x ,..., 1/2,...,x ) (12)
k k 1 2 n
∀ ∂x ∂x −
k k
Theoperator Aθ,ε haspurelydiscretespectrum. Wedenoteby λθ,ε thesequenceofeigenval-
k k N
ues of Aθ,ε writtenintheincreasingorderand repeated according to th∈eirmultiplicity.
n o
The Floquet-Bloch theory (see, e.g., [1–3]) establishes the following relationship between the
spectraoftheoperators Aε and Aθ,ε:
∞
σ(Aε) = L , where L = λθ,ε (13)
k k k
k=1 θ Tn
[ [∈ n o
Thesets L arecompact intervals.
k
Also we need the Laplace operators on Yε with the Neumann boundary conditions on Sε and
either the Neumann or Dirichlet boundary conditions on ∂Y = ∂Yε Sε. Namely, we denote by
\
ηN,ε (resp. ηD,ε)thesesquilinearformin L (Yε)defined byformula(4)(with Yε insteadofΩε)and
2
the definitional domain H1(Yε) (resp. H1(Yε) = u H1(Yε) : u = 0 on∂Yε Sε ). Then by AN,ε
0 ∈ \
(resp. AD,ε)wedenotetheoperatorassociatedwiththeform ηN,ε (resp. ηD,ε), i.e.
n o
b
(Aε, u,v) = ηε, [u,v], u dom(Aε, ), v dom(ηε, )
∗ L2(Yε) ∗ ∀ ∈ ∗ ∀ ∈ ∗
where is N (resp. D).
∗
The spectra of the operators AN,ε and AD,ε are purely discrete. We denote by λN,ε (resp.
k k N
λD,ε ) the sequence of eigenvalues of AN,ε (resp. AD,ε) written in the increasningoo∈rder and
k k N
repeate∈d according totheirmultiplicity.
n o
From the min-maxprinciple(see, e.g., [1, ChapterXIII]) and the enclosure H1(Yε) H1(Yε)
⊃ θ ⊃
H1(Yε)onecan easilyobtaintheinequality
0
k N, θ Tn : λN,ε λθ,ε λD,ε (14)
b ∀ ∈ ∀ ∈ k ≤ k ≤ k
In this end of this subsection we introduce the operators which will be used in the description
of the behaviour of λN, λD and λθ as ε 0. By ∆N (resp. ∆D, ∆θ) we denote the operator which
k k k → F F F
acts in L (F) and is defined by the operation ∆, the Neumann boundary conditions on ∂B and the
2
Neumann (resp. Dirichlet, θ-periodic) boundary conditions on ∂Y. By ∆ we denote the operator
B
which acts in L (B) and is defined by the operation ∆ and the Neumann boundary conditions on
2
∂B. Finally, we introducethe operators AN, AD, Aθ which act in L (F) L (B) and are defined by
2 2
⊕
thefollowingformulae:
∆N 0 ∆D 0 ∆θ 0
AN = F , AD = F , Aθ = F
− 0 ∆B − 0 ∆B − 0 ∆B
! ! !
7
We denote by λN (resp. λD , λθ ) the sequence of eigenvalues of AN (resp. AD, Aθ)
k k N k k N k k N
writtenin theincreas∈ing orderandr∈epeated∈according totheirmultiplicity. It isclearthat
n o n o n o
λN = λN = 0, λN > 0 (15)
1 2 3
λD = 0, λD > 0 (16)
1 2
λθ = λθ = 0, λθ > 0 ifθ = (1,1,...,1) (17)
1 2 3
λθ = 0, λθ > 0 ifθ , (1,1,...,1) (18)
1 2
2.2. Asymptotic behaviour of Dirichlet eigenvalues. We start from the description of the as-
ymptoticbehaviouroftheeigenvaluesoftheoperator AD,ε as ε 0.
→
Lemma 2.1. Foreach k N onehas
∈
limλD,ε = λD (19)
k k
ε 0
→
Furthermore
λD,ε σε2 asε 0 (20)
1 ∼ →
where σis definedbyformula(6).
Proof. We startfrom theproofof (19). Itis based onthefollowingabstracttheorem.
Theorem. (Iosifyanetal.[22])Let ε, 0 beseparableHilbertspaces,let ε : ε ε, 0 :
H H L H → H L
0 0 be linear continuous operators, im 0 0, where is a subspace in 0.
H → H L ⊂ V ⊂ H V H
SupposethatthefollowingconditionsC C hold:
1 4
−
C . The linear bounded operators Rε : 0 ε exist such that Rεf 2 ̺ f 2 for any
f 1 . Here̺ > 0 isa constant. H → H k kHε ε→→0 k kH0
∈ V
C . Operators ε, 0 are positive, compact and self-adjoint. The norms ε are bounded
2 ( ε)
L L kL kLH
uniformlyin ε.
C . Forany f : εRεf Rε 0f 0.
3 ε
∈ V kL − L kH ε→0
C . For any sequence fε ε such that s→up fε < the subsequence ε ε and w
4 ε ′
∈ H ε k kH ∞ ⊂ ∈ V
exist suchthat εfε Rεw 0.
ε
kL − kH ε=−ε→0
Then foranyk N ′→
∈
µε µ
k ε→0 k
→
where µε and µ aretheeigenvaluesoftheoperators ε and 0,whicharerenumberedin
{ k}∞k=1 { k}∞k=1 L L
theincreasingorder andwith accountof theirmultiplicity.
Let us apply this theorem. We set ε = L (Yε), 0 = L (F) L (B), ε = (AD,ε + I) 1,
2 2 2 −
H H ⊕ L
0 = (AD +I) 1 (here I is the identity operator), = 0, the operator Rε : 0 ε is defined
−
L V H H → H
by theformula
f (y), y F,
[Rεf](y) = F ∈ f = (f , f ) 0 (21)
F B
f (y), y B, ∈ H
 B
ObviouslyconditionsC (with̺ =1)andC ∈hold(namely, ε 1).
Let us verify condition C1 . Let f = (f , f )2 0. We set fkεL=kLR(εHfε,) v≤ε = εfε. By vε and vε
3 F B ∈ H L F B
wedenotetherestrictionsof vε onto F and Bcorrespondingly. It isclearthat
vε 2 + vε 2 = vε 2 2 fε 2 = 2 f (22)
k FkH1(F) k BkH1(B) k kH1(Yε) ≤ k kL2(Yε) k kH0
8
Wedenote
H1(F) = w H1(F) : w = 0
0 ∈ |∂Y
Since H1(F) H1(B) is compactly embednded into 0 then dueoto estimate (22) there is a subse-
0 ⊕ b H
quenceε ε andv H1(F), v H1(B)suchthat
′ ⊂ F ∈ 0 B ∈
b
vε v weakly in H1(F)and stronglyin L (F)
vbFε ε=−ε→′→0 vF weakly in H1(B)and stronglyin L2(B) (23)
B ε=−ε→0 B 2
′→
Onehastheintegralequality
vε, wε +vεwε fεwε dy = 0, wε H1(Yε) (24)
∇ ∇ − ∀ ∈ 0
YZε (cid:20) (cid:21)
(cid:0) (cid:1)
b
Weintroducetheset
W = (w ,w ) C (F) C (B) : w = 0, supp(w ) supp(w ) x0 = ∅
F B ∞ ∞ F ∂Y F B
∈ ⊕ | ∪ ∩{ }
(cid:26) (cid:27)
(here as usual by supp(f) we denote the closure of the(cid:0)set x : f(x) , 0 ).(cid:1) Let w = (w ,w )
F B
{ }
be an arbitrary function from W. We set wε = Rεw. It follows from the definition of W that
supp(wε) Dε = ∅when ε issmallenoughand therefore wε C (Yε) H1(Yε).
∩ ∈ ∞ ∩ 0
Wesubstitutewε into(24)andtakingintoaccount(23)wepasstothelimitin(24)asε = ε 0.
′
→
As aresultweobtain b
v , w +v w f w dy+ v , w +v w f w dy = 0 (25)
F F F F F F B B B B B B
∇ ∇ − ∇ ∇ −
ZF (cid:20) (cid:21) ZB (cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
Since W is a dense subspace of H1(F) H1(B) then (25) is valid for an arbitrary (w ,w )
0 ⊕ F B ∈
H1(F) H1(B). Itfollowsfrom(25)that ∆Dv +v = f and ∆ v +v = f andconsequently
0 ⊕ − F F F F − B B B B
b
v = 0f, where v = (v ,v ) (26)
F B
b L
We remark that v do not depend on the subsequence ε and therefore the whole sequence (vε,vε)
′ F B
converges to (v ,v ) as ε 0. ConditionC followsdirectly from (21), (23)and (26). Obviously
F B 3
→
conditionC was provedduringtheproofof C .
4 3
Thustheeigenvaluesµε oftheoperator ε convergestotheeigenvaluesµ oftheoperator 0 as
k L k L
ε 0. ButλD,ε = (µε) 1 1,λ = (µ ) 1 1 thatimplies(19).
→ k k − − k k − −
Nowwefocusontheproofof (20). Let vD,ε betheeigenfunctionof AD,ε thatcorrespondstothe
1
eigenvalueλD,ε and satisfies
1
vD,ε = 1 (27)
k 1 kL2(Yε)
vD,ε 0 (28)
h 1 iB ≥
Onehasthefollowinginequalities
vD,ε 2 C vD,ε 2 (29)
k 1 kL2(F) ≤ k∇ 1 kL2(F)
vD,ε vD,ε 2 C vD,ε 2 (30)
k 1 −h 1 iBkL2(B) ≤ k∇ 1 kL2(B)
B vD,ε 2 vD,ε 2 1 (31)
| |·|h 1 iB| ≤ k 1 kL2(B) ≤
9
Here the first one is the Friedrichs inequality, the second one is the Poincare´ inequality and the
thirdoneistheCauchy inequality. Furthermoreonehas
vD,ε 2 = λD,ε vD,ε 2 + vD,ε vD,ε 2 + B vD,ε 2 (32)
k∇ 1 kL2(Yε) 1 k 1 kL2(F) k 1 −h 1 iBkL2(B) | |·|h 1 iB|
Below wewillprove(see inequalit(cid:16)y(56))that (cid:17)
λD,ε Cε2 (33)
1 ≤
Then itfollowsfrom (27), (29)-(33)that
vD,ε 2 Cε2 (34)
k∇ 1 kL2(Yε) ≤
vD,ε 2 Cε2 (35)
k 1 kL2(F) ≤
vD,ε vD,ε 2 Cε2 (36)
k 1 −h 1 iBkL2(B) ≤
Thelastinequalitycan bespecified. Namely,onehas
1 = vD,ε 2 + vD,ε vD,ε 2 + B vD,ε 2 (37)
k 1 kL2(F) k 1 −h 1 iBkL2(B) | |·|h 1 iB|
Then inviewof(35)-(37)
vD,ε 2 B 1 Cε2
|h 1 iB| −| |− ≤
Finally,takingintoaccount(28)weget:
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
lim vD,ε B 1/2 2 = 0 (38)
ε 0k 1 −| |− kL2(B)
→
Now we construct a convenient approximation vD,ε for the eigenfunction vD,ε. We consider the
1 1
followingproblem
∆ψ = 0 inRn T (39)
\
ψ = 1 in ∂T (40)
ψ(x) = o(1)as x (41)
| | → ∞
Recall that T = x Rn : x < 1, x = 0 , obviously ∂T = T. It is well-known that this problem
n
{ ∈ | | }
hastheuniquesolutionψ(x)satisfying ψ2dx < . Moreoverithasthefollowingproperties:
|∇ | ∞
Rn T
R\
ψ C (Rn T) (42)
∞
∈ \
ψ(x ,x ,...,x ,x ) = ψ(x ,x ,...,x , x ) (43)
1 2 n 1 n 1 2 n 1 n
− − −
cap(T) = ψ2dx (44)
|∇ |
RZn T
\
Thefirst twopropertiesimply:
∂ψ
= 0 in x Rn : x = 0 T (45)
n
∂x { ∈ }\
n
∂ψ ∂ψ
+ = 0 inT (46)
∂x ∂x
n(cid:12)(cid:12)xn=+0 n(cid:12)(cid:12)xn=−0
Furthermorethefunction ψ(x) satisfies(cid:12) theestimat(cid:12)e(see, e.g, [23,Lemma2.4]):
(cid:12) (cid:12)
(cid:12) (cid:12)
Dαψ(x) C x2 n α for x > 2, α = 0,1,2 (47)
− −
| | ≤ | | | | | |
10
WedefinethefunctionvD,ε bytheformula
1
1 x x0 x x0
ψ − ϕ | − | , x F
v1D,ε(x) = 2√√1B|B|− 2 √1dBε ψ! x −dεx0l ϕ !|x−l x0| , x ∈∈ B∪Dε (48)
where ϕ : R → Ris atwice-con|ti|nuously| d|iff erentiab!lef unction!such that
ϕ(ρ) = 1 as ρ 1/2and ϕ(ρ) = 0as ρ 1, (49)
≤ ≥
l isan arbitrary constantsatisfying
1
0 < l < min 1 b,b (50)
4 { − }
Here we also supposethat ε is smallenough so that dε < l/2. It is easy to see that the constructed
functionvD,ε(x) belongstodom(AD,ε)in view(40),(42), (45), (46), (49), (50).
1
Takingintoaccount(44), (47)weobtain:
vD,ε 2 4 1cap(T)dn 2 B 1ε2 = σε2 (ε 0) (51)
k∇ 1 kL2(Yε) ∼ − − | |− →
∆vD,ε 2 Cε4 (52)
k 1 kL2(Yε) ≤
Since vD,ε = 0 on ∂Y and vD,ε B 1/2 = 0 on ∂B x : x x0 l (here [...] means the
1 1 − | |− int \ | − | ≤ int
value of the function when we approach ∂B from inside of B) we have the following Friedrichs
(cid:2) (cid:3) n o
inequalities
vD,ε 2 C vD,ε 2 (53)
k 1 kL2(F) ≤ k∇ 1 kL2(F)
vD,ε B 1/2 2 C vD,ε 2 (54)
k 1 −| |− kL2(B) ≤ k∇ 1 kL2(B)
It followsfrom (51), (53), (54)that
vD,ε 2 1 (ε 0) (55)
k 1 kL2(Yε) ∼ →
Usingthemin-maxprinciple(see, e.g., [1, ChapterXIII]) and takinginto account (51), (55) we
get
vD,ε 2
λD,ε = vD,ε 2 k∇ 1 kL2(Yε) cap(T)dn 2 B 1ε2 = σε2 (ε 0) (56)
1 k∇ 1 kL2(Yε) ≤ vD,ε 2 ∼ − | |− →
k 1 kL2(Yε)
Nowletus estimatethedifference
wε = vD,ε vD,ε (57)
1 − 1
Onehas
wε 2 2 vD,ε 2 + vD,ε 2 +2 vD,ε B 1/2 2 + B 1/2 vD,ε 2
k kL2(Yε) ≤ k 1 kL2(F) k 1 kL2(F) k 1 −| |− kL2(Bε) k| |− − 1 kL2(Bε)
and thusinviewof((cid:16)35), (38), (51), (53), (5(cid:17)4)we(cid:16)get (cid:17)
wε 2 Cε2 (58)
k kL2(Yε) ≤
SubstitutingtheequalityvD,ε = vD,ε +wε into(56)and integratingbyparts weobtain
1 1
vD,ε 2
wε 2 2(∆vD,ε,wε) + k∇ 1 kL2(Yε) vD,ε 2
k∇ kL2(Yε) ≤ 1 L2(Yε) vD,ε 2 −k∇ 1 kL2(Yε)
 k 1 kL2(Yε) 