Planar waveguide with "twisted" boundary conditions: small width

0.28 MB·English

by  D. Borisov

Planar waveguide with “twisted” boundary
conditions: small width
Denis Borisova, Giuseppe Cardoneb
a) Institute of Mathematics of Ufa Scientific Center of RAS, Chernyshevskogo
st. 112, 450008, Ufa, Russia Federation
Bashkir State Pedagogical University, October St. 3a, 450000 Ufa, Russian
Federation; [email protected]
2 b) University of Sannio, Department of Engineering, Corso Garibaldi, 107,
1
82100 Benevento, Italy; [email protected]
0
2

n
Abstract
a
J
Weconsideraplanarwaveguidewith“twisted”boundaryconditions. Bytwist-

9 ingwemeanaspecialcombinationofDirichletandNeumannboundaryconditions.
Assuming that the width of the waveguide goes to zero, we identify the effective
]
(limiting) operator as the width of the waveguide tends to zero, establish the
P
uniform resolvent convergence in various possible operator norms, and give the
A
estimates for the rates of convergence. We show that studying the resolvent con-
.
h vergence can be treated as a certain threshold effect and we present an elegant
t techniquewhich justifies such point of view.
a
m
[ 1 Introduction

2
v Inthispaperwestudyamodelofaplanarwaveguidewithtwistedboundaryconditions.
7 The waveguide is modeled by a strip of a small width. In this domain we consider the
8 LaplacianwithaspecialcombinationoftheDirichletandNeumanncondition,seefig.1.
7
The parameterL introducedonfig.1 is assumedto be either fixedordefined asL=εℓ
1
forafixedℓ,whereεisthewidthofthestrip. Ourmainaimistostudy theasymptotic
.
2 behavior of the resolvent of such operator as the width of the waveguide tends to zero.
1
There is a vast literature devoted to the study of various elliptic operator in thin
1
bounded domains. Not aiming to cite all existing papers and books, we just mention
1
: the books of S.A. Nazarov and G.P. Panasenko [38], [41], see also the references in
v
these books and other papers of these authors. In these works the most attention was
i
X paid to the case of Neumann problems and the behavior of the spectrum was studied.
r Similar studies but for Dirichlet problems were made in [9], [10], [11], [13], [23], [24],
a
[27], [32], [33], [34], [37], [39], [40]. The uniform resolvent convergence for Dirichlet
Laplacian in a thin bounded two-dimensional domain was established in [27], while
the multidimensional case was treated in [10]. We also mention the paper [20], where
the Laplace-Beltramioperatorwas studied on a bounded manifold shrinking to a finite
graph.
The case of unbounded thin domains was considered, too. Here we can refer to
the papers [2], [18], [21], [22], [25], [26], [33], [35], [36]. In all cases the geometry of
the thin domain was nontrivial in the sense that it could not be treated just by the
separation of variables. The first and on of the most popular examples is a curved
infinite thin strip or tube. Such model was studied in [18] with Dirichlet condition in
two- and three-dimensional case. Two-dimensional case with Dirichlet and Neumann
conditions on the opposite sides of the strip was considered in [36]. The main results
of the mentioned papers are the estimates for the number of bound states and their
asymptotic expansions. The uniform resolvent convergence to the effective operator
1
wasalsoestablished. Amoregeneralcase,namely,acurvedinfinite tube withatorsion
was studied in [35]. Here the complete asymptotic expansions for the eigenvalues were
constructed. The curved thin tube whose cross-section has a hole was treated in [2].
The quasi-classical approximation was considered and the operator was multiplied by
the square of a small parameter characterizing the width of the tube. The asymptotic
expansionsfortheeigenvaluesandtheeigenfunctionswereconstructed. Onemoreresult
was the asymptotic expansion to the solution of the initial evolution problem.
One more example of an infinite thin domain is a thin domain with variable width.
It was studied in [25]. The uniform resolvent convergence to an effective operator was
established as well as the estimates for the rate of convergence. One more result of
[25] is two-terms asymptotics for the first eigenvalues. Similar results but in the case
of periodically curved thin strip were established in [26]. Thin domains obtained as
appropriate approximations of various graphs were treated in [21], [22]. In [22] the
resolvent convergence and the effective operator were studied, while in [21] the study
was devoted to the asymptotic behavior of the resonances. Various physical aspects of
the elliptic operators in thin domains were discussed in [3], [4]. We also mention the
review [33], where one can find further information of the state-of-art in the studies of
thin domains.
The models similar to our were studied in [1], [14], [15], [16], [29]. The first four
papers are devoted to the model of a thin bent waveguide. The curvature describing
the bending was scaled together with the width in the same fashion as we rescale our
waveguide in the case L = εℓ. In [29] one more similar model was considered. Here
the waveguide was three-dimensional and the nontrivial geometry came from localized
twisting which was scaled together with the width as the bending in [1], [14], [15], [16].
The operator in [1], [14], [16], [29] was the Dirichlet Laplacian, while in [15] it was the
RobinLaplacian. Themainresultof[1],[14],[15],[16],[29]istheconvergencetheorems.
Namely, the effective operator was found and the uniform resolvent convergence was
proven. In [16] the estimates for the rate of convergence were established under some
additional restrictions for the resolvent’s domain. It was also shown in [1], [14], [15],
[16] that the effective operator can involve nontrivial boundary condition instead of
bending, ifcertainone-dimensionaloperatorspossesseseigenvalueorresonanceatzero.
The main difference of our model in comparison with [1], [14], [15], [16] is that in-
steadofbendingweconsideraspecial“twisted”combinationofDirichletandNeumann
boundary conditions. It must be said that we have borrowedthe idea of such combina-
tionfrom[17]. InthecaseL=εℓourmodelcanbealsoconsideredasatwo-dimensional
analogue of that in [29].
Let us describe our main results. In the case L = εℓ we can rescale the strip
to that with a fixed width and the boundary conditions imposed on fixed parts of
the boundary. Under such rescaling the original resolvent ( (ε) Eε 2 λ) 1, E =
Hεℓ − − − −
const, λ = const, becomes ε2( (1) E ε2λ) 1. Here (ε) denotes the operator we
Hℓ − − − HL
consider. As E we choose the threshold of the essential spectrum and we consider the
originalquestion onthe resolventconvergenceas a certainthreshold effect. Our results
show that the asymptotic behavior of the original resolvent highly depends on the
spectralpropertiesofthe thresholdE ofthe rescaledoperator. The ideaoftreatingthe
resolvent convergence for perturbed elliptic operators as a certain threshold effect has
beenrecentlydevelopedintheseriesofpapersbyM.Sh.BirmanandT.A.Suslinaforthe
homogenization problems, see, for instance, [5], [6], [42]. Although we study a problem
of a completely different nature and we employ an essentially different technique, we
showthatinourcasethe effective operatoralsoappearsas aresultofcertainthreshold
effect. Namely, the form of the effective operator depend on whether the considered
threshold is a virtual level or not. If it is not, the effective boundary condition for
the effective operator is the Dirichlet one. If the virtual level is present, the effective
boundary condition becomes more complicated. The last fact is in a good accordance
with the results of [1], [14], [15], [16] since our virtual levels play the same role as the
aforementionedzeroeigenvaluesorresonancesin[1], [14],[15], [16]. Thesamesituation
2
Figure 1: Waveguide with combined boundary conditions
occursin[29]. Namely,ifwerescalethetubeconsideredin[29]toafixedone,weobtain
the fixed tube with a fixed twisting. It is known that such model has no virtual levels
at the threshold of the essential spectrum and the Hardy inequality is valid, see [19].
This is why the effective operator in [29] involves the Dirichlet condition.
IfLisfixed,weagainmaketheaforementionedrescalingofthestrip,anditleadsus
(1)
to the operator , i.e., for this operator the length of the overlap of the Neumann
HLε−1
conditions increases unboundedly as ε +0. And as in the first case, we reduce the
→
questionondeterminingtheeffectiveoperatortostudyingthespectralpropertiesofthe
thresholds of certain fixed operator.
In addition to identifying the effective operator, we prove the uniform resolvent
convergence of the perturbed operator to the effective one. Moreover, we establish
the estimates for the rates of convergence. These results are obtained for two possible
operator norms in which we can consider the resolvent convergence. Namely, these are
the norms of the operators acting in L and from L into W1. We also observe that in
2 2 2
[1], [14], [16], [29] the uniform resolvent convergence was established only in the sense
of L norm.
2
Theapproachweuseisquiteelegant. Thecoreisthetechniquepresentedin[12]for
studying the same model but in the case of the fixed width. It comes originally from
the papers [7], [28], where it was used to study the behavior of the discrete eigenvalues
emerging from the essential spectrum. In [12] it was adapted also for the considered
model of the waveguide of a fixed width with twisted boundary conditions. In this
paper we apply the adapted technique to study the resolvent convergence and this
is for the first time that this approach is used for such study. Its main content is as
follows. InthecaseL=εℓitallowsustomakeananalyticcontinuationoftheresolvent
( (1) E ε2λ) 1 inavicinityofthethresholdoftheessentialspectrum. Andthenwe
Hℓ − − −
give the description of the possible singularities of this continuation at the threshold.
Exactly the last description determines the effective boundary condition and the rates
oftheresolventconvergence. IfLisfixed,weagainemploythesameapproachbutwith
the combination of some ideas of [30], [31]. We also mention that one of the effective
approachesof studying the asymptotic behavior of the solutions to the problems in the
thin domains is the method of matching of asymptotic expansions, see, for instance,
[38]. Nevertheless, this method does not work in our case since it requires a quite high
smoothnessofthe solutionto the limiting problemthatisnotthe caseforourproblem.
In conclusion we note that our approach is quite universal and can be employed in
studying various similar problems.
2 Formulation of the problem and the main result
Let x = (x ,x ) be the Cartesian coordinates in R2, ε be a small positive parameter,
1 2
and Π(ε) := x : 0 < x < ε be an infinite strip of the width ε, where ε is a small
2
{ }
positiveparameter. GivenanumberL>0,wepartitiontheboundaryofΠ(ε) asfollows,
γ(ε) := x:x >L,x =0 x:x < L,x =ε , Γ(ε) :=∂Π(ε) γ(ε).
L { 1 2 }∪{ 1 − 2 } L \ L
3
In this paper we consider the Laplacian in Π(ε) subject to the Dirichlet boundary
condition on γ(ε) and to the Neumann one on Γ(ε), cf. fig. 1. We define this operator
L L
asassociatedwiththesymmetriclower-semiboundedsesquilinearform( u, v)
∇ ∇ L2(Π(ε))
on W1 (Π(ε),γ(ε)), where the symbol W1 (Ω,S) is the Sobolev space of the functions
2,0 L 2,0
in W1(Ω) vanishing on S. We denote the introduced operator as (ε).
2 HL
Ourmaingoalistostudytheresolventconvergenceof (ε) asε +0. Weconsider
HL →
two cases. In the first case we let L = εℓ, where ℓ > 0 is a fixed number independent
of ε. In the second case L > 0 is fixed and independent of ε. The structure of the
effective (limiting) operator depends strongly on L and to formulate the main results
we introduce additional notations.
Considerthe operator (1). Itwasshownin[12,Th. 2.2]thatthereexistsinfinitely
Hℓ
many critical values 0 < ℓ < ℓ < ... < ℓ < ... such that for ℓ (ℓ ,ℓ ] the
1 2 n n n+1
∈
operator (1) has precisely n isolated eigenvalues. Given a function f L (Π(ε)), we
Hℓ ∈ 2
introduce two projections,
ε ε
( f)(x ,ε):= f(x)χ(ε)(x)dx , ( f)(x ,ε):= f(x)χ(ε)(x)dx ,
Pε 1 1 2 Pε,L 1 1,L 2
Z Z
0 0
2 πx
2
sin , x >L,
2 πx ε 2ε 1
sin 2, x >0, r
χ(1ε)(x):=r2ε cosπ2xε2, x1 <0, χ(1ε,L)(x):= √1ε, |x1|<L,
ε 2ε 1 2 πx
r cos 2, x < L.
By and  we denote the norm of an operatorarctiεng res2pεectivel1y in−L (Π(ε))
0 1 2
andkfr·okmL (Πk(ε·)k)inW1(Π(ε)). ThesymbolDwillbeemployedtoindicatethedomain
2 2
of an operator.
Assume first that L = εℓ and let us introduce the effective operator. It is the
Schr¨odinger operator on the axis
d2
eff := (2.1)
H −dx2
1
subject to certain boundary condition. The type of this condition depends on ℓ. If ℓ
is noncritical, the boundary condition is the Dirichlet one at zero, i.e., in this case the
domain of the effective operator is given by the identity
D( eff)= u W1(R):u(0)=0 W2(R ) W2(R ).
H { ∈ 2 }∩ 2 + ∩ 2 −
If ℓ = ℓ is critical, we have two subcases. Namely, for odd n there is no boundary
n
conditions at all and eff is the usual Schr¨odinger operator (2.1) having W2(R) as the
H 2
domain. For even n the boundary conditions are the most complicated and interesting
ones. Namely,inthis casethedomainconsistsofthe functionsinW2(R ) W2(R )
W1(R) satisfying the boundary conditions 2 + ∩ 2 − ∩
2
u(+0)= u( 0), u(+0)= u( 0). (2.2)
′ ′
− − − −
Now we are in the position to formulate our first main result.
Theorem 2.1. Assume L = εℓ and λ C R. Then for sufficiently small ε the
∈ \
1
resolvent (ε) π2 λ − is well-defined and
Hεℓ − 4ε2 −
(cid:16) (cid:17)
(cid:13) Hε(εℓ)− 4πε22 −λ −1−χ(1ε) Heff −λ −1Pε(cid:13) 6Cε1/2
(cid:13)(cid:18) (cid:19) (cid:13)0
(cid:13) (cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
4
for critical ℓ, and
(cid:13) Hε(εℓ)− 4πε22 −λ −1−χ(1ε) Heff −λ −1Pε(cid:13) 6Cε3/2
(cid:13)(cid:18) (cid:19) (cid:13)0
(cid:13)(cid:13)(cid:13)(cid:13) Hε(εℓ)− 4πε22 −λ −1−χ(1ε)(cid:0)Heff −λ(cid:1)−1Pε(cid:13)(cid:13)(cid:13)(cid:13) 6Cε1/2
(cid:13)(cid:18) (cid:19) (cid:13)1
(cid:13) (cid:0) (cid:1) (cid:13)
for noncritical ℓ(cid:13). Here the constants C are independent of ε(cid:13)but depend on λ and ℓ.
(cid:13) (cid:13)
Suppose now that L is fixed and independent of ε. Here we study the convergence
oftheresolventatthe pointEε 2+λ, λ C R,whereE =0orE =π2/4. Wechoose
−
∈ \
E in this way since 0 and π2/(4ε2) are exactly the eigenvalues associated with the first
transversalmodes on the cross-sections of the waveguides for x <L and x >L.
1 1
| | | |
It turns out that in the considered case the approximating operator is of different
nature than in Theorem2.1 an it also depends substantially on E. Namely, by eff we
H
denote the operator (2.1) subject to the Dirichlet condition at x = L, i.e., it has the
1
±
domain
D( eff)= u W1(R):u( L)=0, u(L)=0
H { ∈ 2 − }
W2( , L) W2( L,L) W2(L,+ ).
∩ 2 −∞ − ∩ 2 − ∩ 2 ∞
By we denote the characteristic function of the segment [ L,L], while is the
0 π2/4
charEacteristic functions of R [ L,L]. − E
\ −
Theorem 2.2. Assume L is fixed, λ C R, and E = 0 or E = π2/4. Then for
∈ \
sufficiently small ε the resolvent HL(ε) −Eε−2−λ −1 is well-defined and satisfies the
inequalities
(cid:0) (cid:1)
1
(cid:13) HL(ε)− εE2 −λ − −χ(1ε) Heff −λ −1EEPε(cid:13) 6Cε3/2,
(cid:13)(cid:18) (cid:19) (cid:13)0
(cid:13) (cid:0) (cid:1) (cid:13)
(cid:13)(cid:13)(cid:13) HL(ε)− εE2 −λ −1−χ(1ε) Heff −λ −1EEPε(cid:13)(cid:13)(cid:13) 6Cε1/2.
(cid:13)(cid:18) (cid:19) (cid:13)1
(cid:13) (cid:0) (cid:1) (cid:13)
Here the consta(cid:13)nts C are independent of ε but depend on λ a(cid:13)nd ℓ.
(cid:13) (cid:13)
Let us discuss the main results. The fact that the action of the effective operator is
determinedby the identity (2.1)is quite expectable andthe mainnontrivialityis inthe
boundarycondition. InthefirstcasetheassumptionL=εℓmeansthatthedomainΠ(ε)
with partition of the boundary into the subsets γ(ε), Γ(ε) can be rescaled to the fixed
εℓ εℓ
domainΠ(1) withthefixedpartitionγ(1),Γ(1). Andsinceafterrescalingweinfactstudy
ℓ ℓ
the convergence of the operator (1) π2 ε2λ −1, the effective boundary condition
H − 4 −
atzerois determined by the operator (1) inthe rescaleddomainΠ(1). More precisely,
(cid:0) Hℓ (cid:1)
theeffectiveboundaryconditiondependsonthespectralstructureofthethresholdπ2/4
of the essential spectrum of (1). If ℓ is critical, it was shown in [12] that there exists
Hℓ
a virtuallevelat the threshold of the essentialspectrum, see problem (3.1) below. And
the structure of this solution determines the effective boundary condition in this case.
If ℓ is non-critical, virtual levels are absent, and exactly this fact implies the effective
Dirichlet boundary condition at zero.
Similar situation occurs, if L is fixed. In this case we again rescale the domain
Π(ε) to Π(1). The sets γ(ε) and Γ(ε) become γ(1) and Γ(1) , while the resolvent
L L Lε−1 Lε−1
becomes ( (1) E ε2λ) 1. As ε goes to zero, we employ the ideas of [30], [31] this
HLε−1− − −
resolvent behaves approximately as a direct sum of two Laplacians in Π(1) subject to
the Neumann condition on x : x = 1 x : x < 0,x = 0 and to the Dirichlet
2 1 2
{ }∪{ }
condition on x : x > 0,x = 0 . This operator again has no virtual levels associated
1 2
{ }
5
with both the spectral points E = 0 and E = π2/4, see Lemmas 3.2, 3.3, 3.4. And
again this fact finally yields the Dirichlet condition at x = L.
1
±
We also observe that in the case L being fixed, there is one more factor in the
E
E
approximation for the original resolvent. And due to the presence of this function for
each f L (Π(ε)) we have
2
∈
( eff λ) 1 f (x )=0, x >L,
− 0 ε 1 1
H − E P | |
(cid:0)(Heff −λ)−1Eπ2Pε(cid:1)f (x1)=0, |x1|<L.
4
(cid:0) (cid:1)
It means that the action of the effective resolvent is nontrivial only as x < L for
1
| |
E =0 and as x >L for E =π2/4.
1
| |
As we see, in both cases L = εℓ and L = const the effective Dirichlet condition
appearsoncethereisnovirtuallevelsfortherescaledoperatoratthe pointE,whereE
is involvedinthe perturbed resolvent. The presenceofthe virtuallevelgivesriseeither
to nontrivial boundary conditions (2.2) or to the absence of the boundary conditions.
We believethatsuchinfluence ofvirtuallevelsonthe effectiveboundaryconditionsisa
generalfactoccurringnotonlyinourmodel. Inamorecomplicatedmodelthethreshold
can be also an (embedded) eigenvalue. We conjecture that in this case the projection
to the associated eigenfunction will be the leading term in the asymptotic expansion
for the resolvent. And we also conjecture that the leading term in the asymptotic
expansion of the perturbed resolvent will be a pole and the mentioned projection will
be the associated residue.
3 Preliminaries
In this section we collect a series of auxiliary results which will be employed in the
proofs of Theorems 2.1, 2.2.
Let Π(ε) := Π(ε) x : x < a , Π := Π(1) x : x > a . In what follows
a ∩{ | 1| } ±a ∩{ ± 1 ± }
as ε=1 we omit the superscript (1) in the notations related to the identity ε=1. For
instance, Π=Π(1), Π =Π(1).
a a
In [12, Th. 2.3] a criterium for ℓ to be critical for the operator was proven.
ℓ
H
Namely, the number ℓ=ℓ is critical, if and only if the boundary value problem
n
π2 ∂φ
n
∆φ = φ in Π, φ =0 on γ , =0 on Γ (3.1)
− n 4 n n ℓn ∂x ℓn
2
has a bounded solution belonging to W1(Π ) for each a>0 and satisfying the asymp-
2 a
totics
πx
φn(x)=sin 2 + e−√2πx1 , x1 + . (3.2)
2 O → ∞
This solution is unique. For even n it is od(cid:0)d w.r.t.(cid:1)the symmetry transformation
(x ,x ) ( x ,1 x ), (3.3)
1 2 1 2
7→ − −
and is even for odd n. The first auxiliary lemma describes certain properties of the
functions φ .
n
Lemma 3.1. The identities
1 1
π π
φ (0,x )sin x dx =( 1)n 1 φ (0,x )cos x dx ,
n 2 2 2 − n 2 2 2
2 − 2
Z Z
0 0
1 1
∂φ π ∂φ π
n(0,x )sin x dx =( 1)n n(0,x )cos x dx ,
2 2 2 2 2 2
∂x 2 − ∂x 2
Z 1 Z 1
0 0
6
1 ℓn
∂φ π π
n
(0,x )sin x dx φ (x ,0)dx =0,
2 2 2 n 1 1
∂x 2 − 2
Z 1 Z
0 0
1 0
∂φ π π
n
(0,x )cos x dx + φ (x ,1)dx =0,
2 2 2 n 1 1
∂x 2 2
Z 1 Z
0 −ℓn
1 ℓn
π π 1
φ (0,x )sin x dx + x φ (x ,0)dx = ,
n 2 2 2 1 n 1 1
2 2 2
Z Z
0 0
1 0
π π ( 1)n 1
−
φ (0,x )cos x dx x φ (x ,1)dx = −
n 2 2 2 1 n 1 1
2 − 2 2
Z Z
0 −ℓn
hold true.
Proof. The first two identities follow easily from the parity of φ under the symmetry
n
transformation(3.3). Theotherscanbeobtainedbyintegratingbypartsintheintegrals
π π2 π π2
sin x ∆+ φ dx=0, cos x ∆+ φ dx=0,
2 n 2 n
2 4 2 4
ΠZ+a (cid:18) (cid:19) ΠZ−a (cid:18) (cid:19)
π π2 π π2
x sin x ∆+ φ dx=0, x cos x ∆+ φ dx=0.
1 2 n 1 2 n
2 4 2 4
ΠZ+a (cid:18) (cid:19) ΠZ−a (cid:18) (cid:19)
One shouldalso take into consideration(3.2) and the parity of φ under (3.3) and pass
n
then to the limit as a + .
→ ∞
The restof this sectionis devotedto the study of two auxiliaryproblems which will
be employedin the proofofTheorems 2.1,2.2. Leth=h(x) be a compactly supported
function belonging to L (Π), µ be a small complex parameter. Denote Γ := x:x =
2 + 2
{
1 , Γ := x : x < 0, x = 0 , γ := x : x > 0, x = 0 . Consider the boundary
1 2 1 2
} − { } ∗ { }
value problem
∂v
( ∆ E+µ2)v =h in Π, v =0 on γ , =0 on Γ Γ , (3.4)
+
− − ∗ ∂x2 ∪ −
with E =0 or E =π2/4. We assume that the solution behaves at infinity as
v(x)=c+(µ)e−qπ42+µ2x1sinπx2 + e−q9π42+µ2x1 , x1 + ,
2 O → ∞
(cid:18) (cid:19) (3.5)
v(x)=c (µ)eµx1 + e√π2+µ2x1 , x ,
1
− O →−∞
(cid:16) (cid:17)
for E =0, and
v(x)=c+(µ)e−µx1sinπx2 + e−√2π2+µ2x1 , x1 + ,
2 O → ∞
(cid:16) (cid:17) (3.6)
v(x)=c (µ)eiπ2x1 + eq3π42+µ2x1 , x1 ,
− O →−∞
(cid:18) (cid:19)
for E =π2/4. In both case c (µ) are some constants.
±
In the case E = 0 the solvability of the similar problem but with the Dirichlet
condition on Γ was studied in [8, Sec. 5] (if one assumes in [8] that d = π and the
+
right hand side in [8, eq. (5.1)] is even w.r.t. x ). The technique used in this paper is
2
thesameasthatin[12,Sec. 4]. ThetypeoftheboundaryconditiononΓ iscompletely
+
inessential for this technique. This is why all the results established in [8, Sec. 5] are
valid in our case up to minor changes related to the other boundary condition on Γ .
+
We formulate the needed results below without adducing the proofs.
Let a>0 be such that the support of h lies in Π .
a
7
Lemma 3.2. Let E =0, µ be complex and sufficiently small, b>0 be a fixed number.
Then the problem (3.4), (3.5) is uniquely solvable. The operator mapping the function
h into the solution of the problem (3.4), (3.5) is bounded as that from L (Π ) into
2 a
W1(Π )andisholomorphic inµ. As x >athesolution(3.4),(3.5)canberepresented
2 b | 1|
as
v(x)= ∞ c+m(µ)e−qπ2(m−12)2+µ2x1sinπ m− 12 x2, x1 >a,
m=1 (cid:18) (cid:19)
X (3.7)
v(x)=c−0(µ)eµx1 + ∞ c−m(µ)e√π2m2+µ2x1sinπ m− 12 (1−x2), x1 6−a,
m=1 (cid:18) (cid:19)
X
where the coefficients c are holomorphic in µ and satisfy the uniform in µ estimate
±m
∞
|c−0(µ)|2+ m |c+m(µ)|2+|c−m(µ)|2 6Ckhk2L2(Πa). (3.8)
m=1
X (cid:0) (cid:1)
Inthe caseE =π2/4onecanagainemploythe same technique from[8, Sec. 5]. All
the calculations remain true the same up to some minor changes. The only substantial
change is the proof of an analogue of Lemma 5.3 from [8]. Although the idea of the
proofisthesame,thechangesarenotsominor. Thisiswhyweprovebelowthe needed
statement.
Lemma 3.3. The problem (3.4), (3.6) with E = π2, µ = 0 and h = 0 has the trivial
4
solution only.
Proof. Suppose such solution exists. In the same way as in [8, Lm. 4.2] one can check
that in a vicinity of zero this solution behaves as
θ
v(x)=αr1/2sin + (r), r 0,
2 O →
where (r,θ) are the polar coordinates associated with x. Bearing this fact and (3.4),
(3.6) in mind, we take any fixed a>0 and integrate by parts as follows,
π2 ∂v
0= x v ∆ dx
1
− 4 ∂x
Z (cid:18) (cid:19) 1
Πa
(3.9)
1 ∂2v ∂v ∂ x1=a ∂v 2
= x v (x v) dx +2 dx.
Z0 (cid:18) 1 ∂x21 − ∂x1∂x1 1 (cid:19)(cid:12)(cid:12)(cid:12)x1=−a 2 ΠZa (cid:12)(cid:12)(cid:12)∂x1(cid:12)(cid:12)(cid:12)
By the separation of variables for x <0 we ca(cid:12)(cid:12)n represent v as (cid:12) (cid:12)
1
v(x)=c (0)eiπ2x1 +v⊥(x),
−
where
1
v⊥(x)dx2 =0 for each x1 ( ,0).
∈ −∞
Z
0
We substitute this representationinto (3.9) and pass to the limit as a + . It yields
→ ∞
iπ c (0)2 ∂v 2 ∂v
⊥
0= | − | +2 dx+2 dx.
− 2 ∂x ∂x
Z (cid:12) 1(cid:12) Z (cid:12) 1(cid:12)
Π∩{x:x1<0} (cid:12)(cid:12) (cid:12)(cid:12) Π∩{x:x1<0} (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Hence,
∂v ∂v
⊥
c (0)=0, =0 as x <0, =0 as x >0.
1 1
− ∂x1 ∂x1
It implies that the function v is independent of x and thus v 0.
1
≡
8
All other arguments of [8, Sec. 5] can be easily adapted to the problem (3.4), (3.5)
with E = π2. The result is formulated in the next
4
Lemma 3.4. Let E = π2/4, µ be complex and sufficiently small, b > 0 be a fixed
number. Then the problem (3.4), (3.6) is uniquely solvable. The operator mapping the
function h into the solution of the problem (3.4), (3.6) is bounded as that from L (Π )
2 a
into W1(Π ) and is holomorphic in µ. As x > a the solution (3.4), (3.6) can be
2 b | 1|
represented as
πx
v(x)=c+0(µ)e−µx1sin 22
+ ∞ c+m(µ)e−qπ2(m−21)2−π42+µ2x1sinπ m− 12 x2, x1 >a,
m=2 (cid:18) (cid:19)
X (3.10)
v(x)=c−0(µ)eiqπ42−µ2x1
+ ∞ c−m(µ)eqπ2m2−π42+µ2x1sinπ m− 12 (1−x2), x1 6−a,
m=2 (cid:18) (cid:19)
X
where the coefficients c are holomorphic in µ and satisfy the uniform in µ estimate
±m
(3.8).
4 Proof of Theorem 2.1
In this section we prove Theorem 2.1. Given f L (Π(ε)), we denote
2
∈
u := (ε) π2 λ −1f.
ε Hεℓ − 4ε2 −
(cid:18) (cid:19)
We rescale the variables x xε 1 keeping the notation “x” for the new rescaled
−
7→
variables. It leads us to another representation for u ,
ε
π2 −1
u =ε2u (ε), u := ε2λ f(ε). (4.1)
ε ε ε ℓ
· H − 4 − ·
(cid:18) (cid:19)
In what follows we employ exactly this representation.
e e
Atournextstepwe introducethe functionu asthe solutionto the boundaryvalue
1
problem
π2 e
∆ ε2λ u =f in Π,
1
− − 4 −
(cid:18) (cid:19) (4.2)
∂u
u =0 on γ x:x =0, 0<xe <1 , 1 =0 on Γ .
1 0 1 2 0
∪{ } ∂x
2
The function u can be constructed explicitly by the separateion of variables,
e 1
∞
u (x,ε)= U (x,ε)χ (x), (4.3)
1 m m
m=1
X
e +∞e−km|x1−t1| e−km(x1+t1)
− f (εt )dt , x >0,
 2k m 1 1 1
Um(x1,ε)= Z00 e−km|x1−t1| mekm(ε√−λ)(x1+t1) (4.4)
− f (εt )dt , x <0,
m 1 1 1
2k
−Z∞ m√2sinπ m 1 x , x >0,
1 2 − 2 2 1
E :=π2 m , χ (x):= (cid:18) (cid:19)
m m
(cid:18) − 2(cid:19) √2sinπ m 1 (1 x2), x1 <0,
− 2 −
(cid:18) (cid:19)

9
1
k :=ε√ λ, k := E E ε2λ, f (εx ):= χ (x)f(εx)dx .
0 m m 1 m 1 m 2
− − −
Z
p 0
where the branch of the square root in the definition of k is fixed by the requirement
m
√1=1.
The series (4.3) converges in W2(Π x : x = 0 ) W1(Π) that can be shown by
2 \{ 1 } ∩ 2
analogy with [8, Lm 3.1]. It also clear that the Parsevalidentity
∞
f 2 =ε2 f (ε ) 2 (4.5)
k kL2(Π(ε)) k m · kL2(R)
m=1
X
holds true. Hereinafter by C we indicate various inessential constants independent of ε
and f.
Lemma 4.1. Let a>0 be a fixed number. The estimates
u U χ 6Cε 1 f , (4.6)
k 1− 1 1kW21(Π) − k kL2(Π(ε))
kUe1χ1kW21(Πa) 6Cε−3/2kfkL2(Πε) (4.7)
hold true.
Proof. The functions U solve the problems
m
−Um′′ +(Em−E1−ε2λ)Um =fm(ε·), x∈R\{0}, Um(0)=0,
and therefore satisfy the equations
kUm′ k2L2(R±)+(Em−E1−ε2λ)kUmk2L2(R±) =(fm(±)(ε·),Um)L2(R±).
In its turn, they imply
kUmkL2(R±) 6 Ekfm(εE·)kL2ε(RR±e)λ, kUm′ kL2(R±) 6 √Ekfm(εE·)kL2(εR±R)eλ. (4.8)
m− 1− m− 1−
These estimates and (4.5) yield (4.6).
To prove the second estimate, we represent U as follows,
1
x1sinhµt sinhµx +∞
U1(x,ε)=e−µy1 1f1(εt1)dt1+ 1 e−µt1f1(εt1)dt1, x1 >0
µ µ
Z Z
0 x1 (4.9)
x1 0
sinhµy sinhµt
U (y,ε)= 1 eµt1f (εt )dt +eµx1 1f (εt )dt , x <0,
1 1 1 1 1 1 1 1
− µ µ
Z Z
−∞ x1
where µ = ε√ λ. Employing this representation, the Parseval identity (4.5), and the
−
Schwarz inequality, one can prove easily the estimate (4.7).
We construct u as
ε
u =u ξ +u , (4.10)
ε 1 1 2
whereξ =ξ (x )eisaninfinitelydifferentiablecut-offfunctionbeingoneas x >ℓ+2
1 1 1 1
and vanishing as x <ℓ+1. In vieew ofe(4.2) thee function u is given by th|e f|ormulas
1 2
| |
u2 =(Hℓ−E1−ε2λ)−1g, g :=f −(−∆−E1−eε2λ)u1ξ ∈L2(Π).
It follows from Lemma 4.1 that
e e
u 6Cε 3/2 f . (4.11)
k 1kW21(Πℓ+2) − k kL2(Π(ε))
e 10