asymptotic behaviour of spectrum of laplace–beltrami operator on riemannian manifolds with complex...

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Asymptotic behaviour of spectrumof Laplace–Beltrami operator onRiemannian manifolds with complexmicrostructureAndrii Khrabustovskyi aa Mathematical Division , B. Verkin Institute for Low TemperaturePhysics and Engineering, National Academy of Sciences ofUkraine , Kharkiv, UkrainePublished online: 21 Sep 2010.

To cite this article: Andrii Khrabustovskyi (2008) Asymptotic behaviour of spectrum ofLaplace–Beltrami operator on Riemannian manifolds with complex microstructure, ApplicableAnalysis: An International Journal, 87:12, 1357-1372, DOI: 10.1080/00036810802213249

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Applicable AnalysisVol. 87, No. 12, December 2008, 1357–1372

Asymptotic behaviour of spectrum of Laplace–Beltrami operator

on Riemannian manifolds with complex microstructure

Andrii Khrabustovskyi*

Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering,National Academy of Sciences of Ukraine, Kharkiv, Ukraine

Communicated by A. Piatnitski

(Received 28 December 2007; final version received 18 May 2008)

The article deals with a convergence of the spectrum of the Laplace–Beltramioperator �" on a Riemannian manifold depending on a small parameter "4 0.This manifold consists of a domain ��R

n with a large number of small ‘holes’whose boundaries are glued to the boundaries of the n-dimensional spheres withsmall truncated segment. The number of the ‘holes’ increases, as "! 0, while theirradii tend to zero. We prove that the spectrum converges to the spectrum of thehomogenized operator having (in contrast to �") a non-empty essential spectrum.

Keywords: homogenization; Laplace–Beltrami operator; spectrum; Riemannianmanifold

AMS Subject Classifications: 35B27; 35P20; 58G25; 58G30

1. Introduction

The aim of this article is to study the asymptotic behaviour of the spectral problem

��"u" ¼ �"u", in M",

u" ¼ 0, on @M",ð1:1Þ

as "! 0. Here �" is the Laplace–Beltrami operator, M" is a n-dimensional Riemannianmanifold with complex microstructure depending on ". It is constructed in the followingway. Let � be a bounded domain in R

n, and fD"i : i ¼ 1, . . . ,Nð"Þg be a system of balls

(‘holes’) in � depending on ", �" ¼ �n [Nð"Þi¼1 D"

i : Suppose that the boundary of each ‘hole’D"

i is glued to the boundary of B"i (‘bubble’) – the n-dimensional sphere with smalltruncated segment. Thus, we obtain a manifold M" (Figure 1):

M" ¼ �"[

[Nð"Þi¼1

B"i

!:

More precise description of M" will be specified later in Section 2.

*Email: [email protected]

ISSN 0003–6811 print/ISSN 1563–504X online

� 2008 Taylor & Francis

DOI: 10.1080/00036810802213249

http://www.informaworld.com

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On M" we introduce a Riemannian metric that coincides with the flat Euclidean metricon �" and coincides with the spherical metric on the ‘bubbles’.

It is supposed that the centres of the ‘holes’ form a periodic lattice with theperiod ", their radii are much smaller than the distance between them and are of orderexp(� 1/"2) (n¼ 2) or "n=ðn�2Þ (n4 2). The radii of ‘bubbles’ are equivalent to ".Clearly, the total volume of the ‘bubbles’ is bounded from above and from belowuniformly in ".

The following result is obtained. The spectrum Sp(��") of the problem (1.1) convergesto the spectrum of an operator A in the Hausdorff sense. We give an explicit description ofthis operator later on. Now we only note that A acts on a two-vector-functions fromL2(�)�L2(�), A is self-adjoint, but in contrast to the operator ��", the spectrum of theoperator A is not purely discrete. Namely, if the radii of the ‘bubbles’ are identical, theessential spectrum contains one point. We also consider the manifold with different‘bubbles’ – in this case the essential spectrum contains a segment.

We also generalize our result to a manifold consisting of two copies of the domain �"

and a system of the n-dimensional spheres with two small truncated segments (Figure 2).The topological type of M" increases as "! 0.

Figure 1. The manifold M".

Figure 2. The manifold M".

1358 A. Khrabustovskyi

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Spectral questions in homogenization have been studied by many authors, see,

e.g. [1–7]. In particular, in [3,5] the spectrum of the homogenized operator is not purely

discrete, but its structure differs from the spectrum of the operator A in the present article.Homogenization problems on a manifolds with complex microstructure have been

studied in [2,8–11]. In particular, homogenization of semi-linear parabolic equations on

a manifold described above (Figure 1) has been studied in [9] in the case of the identical

radii of ‘bubbles’. The behaviour of the spectrum of the Laplace–Beltrami operator has

been studied in [2]. The structure of the manifolds constructed in [2] differs from the

structure of the manifold M" and in contrast to the present article, the homogenized

operator have purely discrete spectrum.We remark that various physical problems can be reduced to a study of

homogenization problems on Riemannian manifolds depending on a small parameter.

Namely, in paper [8], the authors apply their results for studying the asymptotic behaviour

of coloured particles, which wander in the domain with many small obstacles reflecting the

particles and changing their colour. The results in [10,11] are interpreted in terms of

general relativity.In the present article, we study proper oscillation of the membrane M" with complex

‘bubbles-like’ structure.The article is organized as follows. In Section 2 we describe the structure of the

manifold M" and formulate the main result of this article (Theorem 2.2). It is proved in

Section 3. In Section 4 we study the case of the ‘bubbles’ with different radii (Theorem 4.1).

In Section 5, we generalize our results for to a manifold with increasing topological

type (Theorem 5.1).

2. The problem setting and the main result

Let � be a bounded domain in Rn (n� 2) and fD"

i : i ¼ 1::Nð"Þg be the system of disjoint

balls in � depending on a small parameter ". Let x"i be the centre of the ball D"i and let its

radius be equal to d ". Suppose that points x"i form a periodic lattice with period ",i.e. x"i ¼ "

Pnj¼1 ejz

ij, where {ei, i¼ 1, . . . , n} is an orthonormal basis in R

n, zij 2 Z.We consider the following domain with ‘holes’

�" ¼ �n[Nð"Þi¼1

D"i :

Suppose that each ‘hole’ D"i is glued to the truncated n-dimensional sphere (‘bubble’) with

the radius b"

B"i ¼ ~x ¼ ð�1, �2, . . . , �nÞ : �1 2 ½0, 2��, �i 2 ½0,��, i ¼ 2, . . . , n� 1, �n 2 ½arcsind "

b",��

� �,

where �1, . . . , �n are the spherical coordinates.Namely, we identify @D"

i and the boundary of B"i :

@B"i ¼ x 2 B"i :�n ¼ arcsind"

b"

� �:

Applicable Analysis 1359

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As a result we obtain a manifold M" (the fragment is represented on the Figure 1)

M" ¼ �"[

[Nð"Þi¼1

B"i

!:

The boundary of M" coincides with @�. We denote by ~x points of this manifold. If the

point ~x belongs to �", sometimes we write x instead of ~x having in mind a corresponding

point in �.Clearly, M" can be covered by a system of charts and compatible local coordinates

fx1, . . . , xng ! ~x 2M" can be introduced.We suppose that M" is equipped by the metric g" that coincides with the flat Euclidean

metric on �" and coincides with the spherical metric on the ‘bubbles’.We give more precise description of this metric at the points, where the boundary of the

‘holes’ is glued to the boundary of the ‘bubbles’. We introduce local coordinates (x1, . . . , xn)

in a small neighbourhood of @D"i as follows. Let (�1, . . . , �n�1, r) be the spherical coordinates

in�with the origin at x"i . Here �1, . . . , �n�1 are the angular coordinates, r is the distance to x"i

(in particular, r¼ d" for the points of @D"i ).We set xj¼ �j ( j¼ 1, . . . , n� 1), xn¼ r� d" (xn� 0)

for ~x 2 �" and xn¼�b"(�n� �

") (xn5 0), where �" ¼ arcsinðd "=b"Þ, for ~x 2 B"i . Then the

components of the corresponding metric tensor g"��ðx1, . . . , xnÞ have the following form:

g"�� ¼

g1"�� ¼ ��ðxn þ d"Þ2Qn�1

j¼�þ1

sin2 xj, xn � 0,

g2"�� ¼ ��ðb"Þ2 sin2

xnb"� �"

� � Qn�1j¼�þ1

sin2 xj, xn 5 0,

8>>><>>>: � ¼ 1, n� 1, g"n� ¼ �n�

(for �¼ n� 1 we setQn�1

j¼�þ1 sin2 xj :¼ 1). Here, �� is the Kronecker’s delta.

Remark 1 It is clear that the components of the metric tensor g"�� introduced above are

continuous but not differentiable functions. In particular, they do not allow us to calculate

the curvature at the points of @D"i . Nevertheless, this tensor can be approximated by

a smooth tensor g"�� that differs from g"�� only in a small neighbourhood of @D"i . Namely,

we define g"�� by the formula

g"��ðx1, . . . , xnÞ ¼ g1"��ðx1, . . . , xnÞ � ’�ðxnÞ þ g2"��ðx1, . . . , xnÞ � ð1� ’�ðxnÞÞ,

where ’�(xn) is a smooth positive function equal to 1 for xn� � and equal to 0 for xn��,�¼ �(")4 0. The obtained tensor is smooth, so the curvature can be calculated everywhere.

Clearly, the curvature tends to infinity at the points of @D"i as �! 0.

In order to simplify our calculations, we will further consider the piecewise

smooth tensor g"��. However, all results (in particular, the formula (2.5) below) are

still valid for the smooth tensor g"�� if �(") converges to 0 sufficiently fast as "! 0

(see, also, [8, subsection 3.2], [11, section 2], for similar assertions).Our main assumption is about the radii d " of the ‘holes’:

d " ¼exp �

1

a"2

� �, n ¼ 2,

a"n=n�2, n4 2,

8<: ð2:1Þ

where a is a positive constant.


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We suppose that the ‘bubbles’ radii b" are the following

b" ¼ b � ", b is a positive constant. ð2:2Þ

It is easy to see that the total volume of the ‘bubbles’ is bounded from above andfrom below, namely lim"!0

PNð"Þi¼1 jB

"i j ¼ bn � !n � j�j, where !n is the volume of the

n-dimensional unit sphere.We consider the following eigenvalue problem

��"u" ¼ �u", ~x 2M", u" ¼ 0, ~x 2 @M", ð2:3Þ

where �" is the Laplace–Beltrami operator on M", which have the following form on thelocal coordinates

�" ¼1ffiffiffiffiGp "

Xn�,�¼1

@

@x�

ffiffiffiffiGp "

g��"@

@x�

� �:

Here, G" ¼ det g"��, g��" are the components of the tensor inverse to g"��.

Let 05 �"1 � �"2 � �

"3 � � � � � �

"k � � � � ! 1 be the eigenvalues of the problem (2.3),

written with account of their multiplicity and u"1, u"2, . . . , u"k . . . be the corresponding

eigenfunctions such that ðu"k, u"l Þ0" ¼ �kl. We denote by Sp(��") the spectrum of the

problem (2.3) : Spð��"Þ ¼ f�"i , i ¼ 1, . . . ,1g:The goal of this article is a description of asymptotic behaviour of Sp(��") as "! 0.

Notice that the Dirichlet boundary condition on @M" is irrelevant; it can be replaced byany other one.

We use the following notations:

R"i ¼ ~x 2 �" :d" � jx� x"i j5"

2

n o,

G"i ¼ R"i [ B"i ,

S"i ¼ ~x 2 �" : jx� x"i j ¼"

2

n o @G"i :

We denote by �" the first eigenvalue of the Dirichlet problem

��"v" ¼ �"v", ~x 2 G"i , v" ¼ 0, ~x 2 S"i : ð2:4Þ

and let � ¼ lim"!0 �": It is possible to show (see, e.g. [8,9] for close assertions) that � is

finite and equal to

� ¼

a

4b2, n ¼ 2,

n� 2

2�an�2!n�1

bn!n, n4 2:

8>><>>: ð2:5Þ

Note, that in spite of the fact that the volume of G"i converges to zero as "! 0, �" does notblow up because of a weak connection between B"i and R"i .

We introduce the following functional spaces:

L2(M") is the Hilbert space of real-valued functions on M" with the norm

ku"k0" ¼

ZM"

ðu"Þ2d ~x

� �1=2

,


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where d ~x ¼ffiffiffiffiffiffiG"p

dx1, . . . , dxn is the volume form on M";H is the Hilbert space of real-valued two-vector-functions from L2(�)�L2(�) with

the norm

kUk0 ¼

Z�

hðu1ðxÞÞ

2þ ðu2ðxÞÞ

2idx

� �1=2

, U ¼u1

u2

� �,

where the weight ¼ bn!n.

Definition 2.1 Let A"�R be a set depending on a positive parameter ". We say that A"

converges as "! 0 in the Hausdorff sense to the set A if the following conditions hold:

(A) if �"2A" and lim"!0 �" ¼ �, then �2A,

(B) for any �2A there is �"2A" such that lim"!0 �" ¼ �.

Now, we formulate the main theorem.

THEOREM 2.2 The spectrum Sp(��") converges to the spectrum SpA of the self-adjoint

operator A :H!H in the Hausdorff sense. A is defined by the operation

AU ¼��u1 þ �ðu1 � u2Þ

�ðu2 � u1Þ

� �, U ¼

u1

u2

� �ð2:6Þ

and the boundary condition

u1 ¼ 0, x 2 @�: ð2:7Þ

Before investigating the limit lim"!0 Spð��"Þ at first, we describe the spectrum of the

operator A. Namely, the following simple proposition is valid:

PROPOSITION 2.3 Let 05"1 � "2 �

"3 � � � � �

"k � � � � ! 1 be the eigenvalues of

the Laplace operator in L2(�) with Dirichlet boundary condition, f1, f2, . . . , fk . . . be the

corresponding eigenfunctions. Then the spectrum of the operator A (2.6) and (2.7) has

the following structure:

SpðAÞ ¼ f�0g [ f��k , k ¼ 1, 2, 3. . .g [ f�þk , k ¼ 1, 2, 3. . .g, ð2:8Þ

where

�0 ¼ �, �k ¼1

2k þ �þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk þ �þ �Þ

2� 4�k

q� �:

The points �k , k ¼ 1, 2, 3, . . . , belong to the discrete spectrum, �0 is a point of the essential

spectrum:

05 ��1 � ��2 � � � � � �

�k ! �0 ¼ � 5 �þ �5 �þ1 � �

þ2 � � � � � �

þk !1:

The corresponding eigenfunctions have the form

Uk ¼ fk1

�ð� � �k Þ�1

� �:


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3. Proof of the Theorem 1.2

3.1. Verification of the condition (A)

Let �"2 Sp(��") converge to some �2R. We prove that � belongs to the spectrum of the

operator A (2.6) and (2.7).Let u" be the eigenfunction corresponding to �" such that ku"k20" ¼ 1 (and so

kr"u"k20" ¼ �"). In order to describe the behaviour of u" in �" as "! 0, we

introduce an extension operator �"1 : H1ðM"Þ ! H1ð�Þ with the following properties:

8u"2H1(M")

1: �"1u"ðxÞ ¼ u"ð ~xÞ, 8x 2 �",

2: k�"1u"kH1ð�Þ � �1ku

"kH1ð�"Þ, where �1 does not depend on ":ð3:1Þ

Such operator exists, see [8].Also, we introduce the operator �"

2 : L2ðM"Þ ! L2ð�Þ

�"2uðxÞ ¼

u"i , x 2 �"i,

0, x 2 �nS

i�"i,

(

where �"i is the cube with the centre in x"i and the side-length ", u"i ¼ ð1=jB"i jÞRB"iu" d ~x

is the average value of the function u" over the domain B"i . �"2u" describes the behaviour

of u" inS

i B"i as "! 0. It is easy to see that the following inequality holds:

k�"2u"kL2ð�Þ � �2ku

"kL2ðM"Þ, where �2 does not depend on ": ð3:2Þ

Using (3.1), (3.2) and the imbedding theorem, we obtain the subsequence (still denoted

by ") such that

�"1u"! u1 2 H1

0ð�Þ strongly in L2ð�Þ,

�"2u" * u2 2 L2ð�Þ weakly in L2ð�Þ:

We have two cases of the limit function u1.

Case I u1 6¼ 0. One has the following integral equality for any w" from the definitional

domain of the operator �" with Dirichlet boundary condition:

�

ZM"

�w"u" d ~x ¼ �"ZM"w"u" d ~x: ð3:3Þ

We choose the following test-function w":

w"ð ~xÞ ¼

w1ðxÞ, ~x 2 �"nS

i R"i ,

w1ðxÞ þw1ðx

"i Þ � w1ðxÞ

�jx� x"i j

8d"

� �þw2ðx

"i Þ � w1ðx

"i Þv"i ð ~xÞ�

jx� x"i j

"

� �, ~x 2 R"i ,

w2ðx"i Þ þ

w1ðx

"i Þ � w2ðx

"i Þ1� v"i ð ~xÞ

, ~x 2 B"i :

8>>>>>>>><>>>>>>>>:ð3:4Þ


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Here w1ðxÞ 2 C10 ð�Þ,w2ðxÞ 2 C1ð�Þ are arbitrary functions, �(r) is a smooth function

equal to 1 as 0 � r � 1=4 and equal to 0 as r � 1=2, v"i is the eigenfunction of the

problem (2.4) corresponding to �" and normalized by the conditionZB"i

v"i ð ~xÞd ~x ¼ jB"i j:

At first we investigate the integrals in (3.3) over the domain �". One has:

XNð"Þi¼1

ZR"

i

�w1ðx

"i Þ � w1ðxÞ

�jx� x"i j

8d"

� �� u"ðxÞdx

��

�XNð"Þi¼1

ZR"

i[D"

i

r"w1ðx

"i Þ � w1ðxÞ

�jx� x"i j

8d"

� �� r"�"

1u"ðxÞ

�� dxþ C

XNð"Þi¼1

jD"i j � C1

XNð"Þi¼1

jD"i [ suppr

"�j ! 0, "! 0: ð3:5Þ

We have for an arbitrary smooth function f (x)

XNð"Þi¼1

ZR"

i

�w2ðx

"i Þ � w1ðx

"i Þv"i ð ~xÞ�

jx� x"i j

"

� �� 2dx5C, ð3:6Þ

XNð"Þi¼1

ZR"

i

�w2ðx

"i Þ � w1ðx

"i Þv"i ð ~xÞ�

jx� x"i j

"

� �� f ðxÞdx

¼XNð"Þi¼1

f ðx"i Þw2ðx

"i Þ � w1ðx

"i Þ Z

@D"i

@v"i@~n

dsx þ �oð1Þ

¼XNð"Þi¼1

�""nf ðx"i Þðw2ðx"i Þ � w1ðx

"i ÞÞ þ �oð1Þ

! �

Z�

f ðxÞðw2ðxÞ � w1ðxÞÞdx, "! 0: ð3:7Þ

(dsx is an area form on @�.) We have used the following estimates proved in [9]

jD�v"i ðxÞj �

C"2ln jx� x"i j

ð�Þ, n ¼ 2,

C"n

jx� x"i jn�2þjaj

, n4 2,

8><>: ~x 2 R"i and jx� x"i j �"

4, ð3:8Þ

ZR"

i

jv"i ðxÞj2dx � C"nþ2: ð3:9Þ

It follows from (3.6) and (3.7) that �PNð"Þ

i¼1 �fðw2ðx�i Þ � w1ðx

�i ÞÞv

�i ðxÞ�ððjx� x"i jÞ="Þg dx

weakly converges to �(w1�w2). Using this fact, (3.1) and (3.5), we obtain

lim"!0

Z�"

��w"u" dx ¼

Z�

��w1u1 þ �ðw1 � w2Þu1

dx: ð3:10Þ


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In the same way, we obtain

lim"!0

Z�"

�"w"u" dx ¼

Z�

�w1u1dx: ð3:11Þ

Now, we investigate the behaviour of the integrals in (3.3) over the union of the

‘bubbles’. We have:

XNð"Þi¼1

ZB"i

�"hw1ðx

"i Þ � w2ðx


iu"ð ~xÞd ~x

¼ �XNð"Þi¼1

u"iw1ðx

"i Þ � w2ðx

"i Þ Z

@B"i

@v"

@~ndsx þ �oð1Þ

¼ �"XNð"Þi¼1

u"i ðw1ðx"i Þ � w2ðx

"i ÞÞ"

n þ �oð1Þ

¼ �"

Z�

Q"�w1 � w2

ðxÞ ��2u

"ðxÞdxþ �oð1Þ, "! 0,

where the operator Q" :L2(�)!L2(�) is defined by the formula

Q"wðxÞ ¼wðx"i Þ, x 2 �"i,

0, x 2 �nS

i�"i:

(Here we use the following estimate proved in [9]:Z

B"i

jv"i ðxÞj2dx ¼ jB"i j þOð"nþ1Þ: ð3:12Þ

It is clear that Q"[w1�w2](x) strongly converges to w1(x)�w2(x). Therefore,

lim"!0

XNð"Þi¼1

ZB"i

��nw"1ð ~xÞ � w2ðx


ou" d ~x ¼ �

Z�

u2ðxÞðw2ðxÞ � w1ðxÞÞdx:

ð3:13Þ

In a similar manner, we obtain

lim"!0

XNð"Þi¼1

ZB"i

�"w"u" d ~x ¼

Z�

�w2u2dx: ð3:14Þ

Thus, from (3.10), (3.11), (3.13) and (3.14) we conclude that u1 and u2 satisfy to the

equality:Z�

h��w1u1 þ �u1ðw1 � w2Þ þ �u2ðw2 � w1Þ

idx ¼ �

Z�

hu1w1 þ u2w2

idx ð3:15Þ

for any functions w1 2 C10 ð�Þ, w22C1(�).

It follows easily from (3.15) that � is an eigenvalue of A.

Case II u1¼ 0. We prove that lim"!0 �" ¼ �.


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In order to prove this, we construct a suitable approximation v" for the eigenvalue u".

We represent the eigenvalue u" in the form u"¼ v"� g"þw". Here

v" ¼

0, ~x 2 �"nS

i R"i ,

u"i v"i ðxÞ�

jx� x"i j

"

� �, ~x 2 R"i ,

u"i v"i ð ~xÞ, ~x 2 B"i :

8>>><>>>: ð3:16Þ

Recall, that u"i is the average value of the function u" over the domain B"i ; v"i and � are

introduced in the Case I. g" is the projection of v" onto the linear span of the eigenfunctions

u1, . . . , un"�1, where n" is a number of the eigenvalue �":

g" ¼Xn"�1k¼1

u"kðv", u"kÞ0":

From Poincare inequality for the ‘bubble’ B"i , one has

1 ¼ lim"!0ku"k20" ¼ lim

"!0

XNð"Þi¼1

ðu"i Þ2jB"i j: ð3:17Þ

Using (3.17) and the properties of v"i (3.8), (3.9) and (3.12), we have

kr"v"k20" ¼ �XNð"Þi¼1

ðu"i Þ2jB"i j þ �oð1Þ !

"!0�, ð3:18Þ

kv"k20" ¼XNð"Þi¼1

ðu"i Þ2jB"i j þ �oð1Þ !

"!01, ð3:19Þ

k�"v"k20" � C, ð3:20Þ

1

2ku" � v"k20" � ku

"k20,�" þ kv"k20,�" þ

XNð"Þi¼1

nku" � u"i k

20,B"

iþ kv" � u"i k

20,B"

i

o!"!0

0: ð3:21Þ

Using the Bessel’s inequality and (3.21), one has

kg"k20" ¼Xn"�1k¼1

jðv", u"kÞ0"j2 ¼

Xn"�1k¼1

jðv" � u", u"kÞ0"j2 � ku" � v"k20" !

"!00, ð3:22Þ

kr"g"k20" ¼Xn"�1k¼1

�"kjðv" � u", u"kÞ0"j

2 � �"ku" � v"k20" !"!0

0: ð3:23Þ

Now, we estimate the remainder w". It follows from (3.21) and (3.22) that

kw"k0" ! 0, "! 0:

It is clear that v" ¼ v" � g" is orthogonal to the eigenfunctions u"1, . . . , u"n"�1. Therefore,

we have

�" kr"u"k20" �krv"k20"kv"k20"


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and so

kr"w"k2 � 2jðr"v",r"w"Þ0"j þkr"v"k20"kv"k20"

� kr"v"k20"

��

¼ jð�"v",w"Þ0"j þ jðr"g",r"w"Þ0"j þ

kr"v"k20"kv"k20"

� kr"v"k20"

��:

Using (3.18)–(3.23), we obtain that

lim"!0kr"w"k0" ¼ 0: ð3:24Þ

Therefore, from (3.18), (3.23) and (3.24) we have

�" kr"u"k20" ¼ kr"v"k20" þ �oð1Þ ! �, "! 0:

The fulfilment of the condition (A) is proved.

3.2. Verification of the condition (B)

Let � belong to the spectrum of the operator A. We prove that there exists �"2 Sp(��")

such that lim"!0 �" ¼ �.

Proving this indirectly we assume the opposite. Then the subsequence (still denoted

by ") exists and a positive number � exists such that

distð�, Spð��"ÞÞ4 �: ð3:25Þ

Since �2 Sp(A) there exists the function f ðxÞ ¼ ð f1ðxÞf2ðxÞÞ 2 H, such that f =2 Im(A� �I),

where I is the identical operator.We consider the following problem

��"u" � �u" ¼ f ", ~x 2M", u" ¼ 0, ~x 2 @M", ð3:26Þ

where f "ð ~xÞ 2 L2ðM"Þ:

f "ð ~xÞ ¼

f1ðxÞ, ~x 2 �",

1

j�"i j

Z�"i

f2ðxÞdx, ~x 2 B"i :

8<:In consequence of (3.25), the problem (3.26) has the unique solution u"ð ~xÞ 2 H1

0ðM"Þ

such that

ku"k20" �k f "k20"�� C1,

where C1 does not depend on ". Moreover, it is easy to see that

kru"k20" � 2�k f "k0" � ku

"k0" þ �ku"k20"

�� C2,

where C2 does not depend on ".


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Then there exists a subsequence (still denoted by ") such that

�"1u"! u1 2 H1

0ð�Þ strongly in L2ð�Þ,

�"2u" * u2 2 L2ð�Þ weakly in L2ð�Þ:

Using the same calculations as in Case I of the previous section, we obtain

AU� �U ¼ f, U ¼u1

u2

� �:

Therefore, f2 Im(A� �I) and we obtain a contradiction. The fulfilment of

condition (B) and so Theorem 2.2 are proved.

4. Case of the ‘bubbles’ with different radii

We consider the manifold M" constructed in the same way as in Section 2. We again

suppose that the radii of the ‘holes’ D"i satisfy (2.1), while the radii of the ‘bubbles’ are

the following

b"i ¼ bðx"i Þ",

where b(x) be a positive function from C1ð�Þ.We denote

ðxÞ ¼bðxÞ

n� !n, �ðxÞ ¼

a

4bðxÞ

2 , n ¼ 2,

n� 2

2�an�2!n�1bðxÞ

n!n

, n4 2:

8>>><>>>:Let H be the Hilbert space of real-valued two-vector-function from L2(�)�L2(�) with

the scalar product

ðu, vÞ0 ¼

Z�

�u1ðxÞv1ðxÞ þ ðxÞu2ðxÞv2ðxÞ

dx:

We have the analogous to the Theorem 2.2 result.

THEOREM 4.1 The spectrum Sp(��") of the problem (2.3) converges in the Hausdorff sense

to the spectrum SpA of the self-adjoint operator A : H!H defined by the operation

U ¼��u1 þ �ðxÞðxÞðu1 � u2Þ

�ðxÞðu2 � u1Þ

� �, U ¼

u1

u2

� �, ð4:1Þ

and the boundary condition (2.7).

The proof is based on the following

LEMMA 4.2 The essential spectrum �ess(A) of the operator A contains the segment

½minx2��ðxÞ, maxx2��ðxÞ�.

Proof Let �0 is a point of the segment ½minx2��ðxÞ, maxx2��ðxÞ�. Then there exists

the point x0 2 � such that �(x0)¼ �0. We denote 0¼ (x0). First, we suppose that

x02 int�.


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In order to prove that �02 �ess(A), we construct the sequence Un2H with the following

properties: (1) Un is bounded, (2) Un is noncompact, (3)ðA� �0IÞUn !n!1

0 strongly in H.

We denote by A0 the operation

A0U ¼��u1 þ �00ðu1 � u2Þ

�0ðu2 � u1Þ

� �, U ¼

u1

u2

� �ð4:2Þ

Let Bn be the ball with the centre in the point x0 and the radius 1=n. Let nk be the kth

eigenvalue of the Laplace operator on the domain Bn with Dirichlet boundary conditions.

Let enk be the corresponding eigenfunction, such that ðenk, enl ÞL2ðBnÞ ¼ �kl: We redefine enk by

zero in �\Bn.We denote

�nk ¼1

2nk þ �00 þ �0 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn

k þ �00 þ �0Þ2� 4�0n

k

q� �:

Analogous to Proposition 2.3, we have that �nk is the eigenvalue of the operator defined by

the operation A0 and Dirichlet boundary condition on @Bn. The corresponding

eigenfunction has the form:

Unk ¼ ckne

nk

1

�0ð�0 � �nkÞ�1

� �, ckn is a constant:

We choose the constant factor ckn in such a way as to have kUnkk0 ¼ 1: Then one has

ðcknÞ2¼ kenkk

2L2ð�Þþ kenkk

2L2ð�Þ

�0�0 � �nk

� �2" #�1

:

It is easy to obtain the inequalities

minx2�

ðxÞ � kenkk2L2� max

x2�ðxÞ, ð4:3Þ

j�nk � �0j �M1

nk

, M1 does not depend on n and k, ð4:4Þ

ckn �M2

nk

, M2 does not depend on n and k: ð4:5Þ

Now we redefine Unk by zero in �\Bn and set

Un ¼ Unk�nðjx� x0jÞ, k ¼ kðnÞ,

where �n(r)¼�(nr), �(r) is a smooth function equal to 1 as r � ð1=2Þ and equal to 0 as

r� 1. The number k¼ k(n) we choose later. It is clear that Un is from the definitional

domain of the operator A.It is easy to see that kUnk0 � kU

nk0 ¼ 1 and exists constant C4 0 such that

C � kUnk0. On the other hand, Un converges nearly everywhere to zero (since the support

of Un tightens to the point x0). Therefore, Un is non-compact.

We have

kðA� �0IÞUnk0 � kðA0 � �0IÞU

nk0 þ kðA� A0ÞUnk0: ð4:6Þ


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It is obvious that the second term in (4.6) converges to zero as n!1

kðA� A0ÞUnk0 �M max

x2Bn

�j�ðxÞðxÞ � �00j þ j�ðxÞ � �0j

��

M

n:

Now we estimate the first term. Using the inequalities (4.3)–(4.5), we have

kðA0 � �0IÞUnk0 � k

ðA0 � �0IÞU

nk

��nk0 þ k2c

knr

"enk � r"�n þ ckne

nk ��

"�nkL2ð�Þ

� j�nk � �0j þMnffiffiffiffiffiffink

p þn2

nk

!:

We choose k¼ k(n) such that nk � n3. Then the first term also converges to zero.

We obtain that (A� �0I) Un! 0, n!1 and so �0 is the point of �ess(A).

We have proved that any �0 such that �0¼ �(x0) with x02 int� belongs to the essential

spectrum. The set of such �0 is dense in the segment ½minx2��ðxÞ, maxx2��ðxÞ�:Since the essential spectrum is a closed set we conclude that Lemma is valid for any

point of this segment.

Proof of Theorem 4.1 The fulfilment of the condition (B) is proved in the similar way as it

proved in Theorem 2.2. As for the property (A), similar to the proof of the Theorem 2.2

computations show that if �"2 Sp(��") converges to some �, then � is an eigenvalue of

the operator A (Case 1) or � 2 ½minx2��ðxÞ, maxx2��ðxÞ� (Case 2) and therefore, by

Lemma 4.2 � belongs to the essential spectrum of the operator A.

5. One generalization

We consider the manifold M" whose structure differs from the manifold in previous

sections. Namely, let �"1 and �"

2 be the two copies (‘sheets’) of the domain with ‘holes’ �"

constructed in Section 2. We denote by @D"ki, k¼ 1, 2 the boundary of the ith ‘hole’ on the

kth ‘sheet’. Let B"i be the n-dimensional sphere of radius b" with two truncated small

segments (‘bubble’):

B"i ¼

(x ¼ ð�1, �2, . . . , �nÞ : �12 ½0, 2��, �i 2 ½0,��, i ¼ 2, . . . , n� 1,

�n 2 arcsind "

b", �� arcsin

d"

b"

� �):

Recall, that d" is the radius of the ‘holes’ D"ki. Suppose that the boundaries of the ‘holes’

@D"1i and @D

"2i are connected by means of the ‘bubble’ B"i . Namely, we identify @D"

1i and the

first component of the boundary of B"i

½@B"i �1 ¼ x 2 B"i : �n ¼ arcsind "

b"

� �:

and identify @D"2i and the second component of the boundary of B"i

½@B"i �2 ¼ x 2 B"i : �n ¼ �� arcsind "

b"

� �:


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As a result, we obtain a manifold M" (the fragment is represented on Figure 2)

M" ¼ �"

1 [[Nð"Þi¼1

B"i

![�"

2:

We denote by ~x points of this manifold. If the point ~x belongs to �"k, we assign the pair

(x, k) to ~x where x is the corresponding point in �.We introduce on M" a metric tensor g"��ð ~xÞ such that the metric is the flat Euclidean

metric on �"1 [�"

2 and the spherical metric on B"i outside some small neighbourhood of the

boundary of B"i .We suppose that the radii d " of the ‘holes’ satisfy (2.1) and the ‘bubbles’ radii have the

form (2.2).We consider the eigenvalue problem (2.3) on this manifold. An asymptotic behaviour

of the spectrum Sp(��") of this problem as "! 0, is described by the following theorem.

THEOREM 5.1 The set Sp(��") converges in the Hausdorff sense to the spectrum SpA of

the operator A defined by the operation

U ¼

��u1 þ �ðu1 � vÞ

�ð2v� u1 � u2Þ,

��u2 þ �ðu2 � vÞ

0B@1CA, U ¼

u1

v

u2

0B@1CA

and the boundary conditions u1¼ u2¼ 0, x2 @�, where � is defined by the formula (2.5),

¼ bn!n.The spectrum of the operator Sp (A) has the structure (2.8), where �0 ¼ 2�,

�k ¼ 1=2ðk þ �þ 2� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk þ �þ 2�Þ2 � 8�k

qÞ:

Proof The proof is similar to that of Theorem 2.2. We only underline the main

difference – a choice of a test-function w" in the integral equality (3.3). Namely,

w"ð ~xÞ ¼

wkðxÞ, ~x 2 �"knS

i R"ki, k ¼ 1, 2,

wkðxÞ þwkðx

"i Þ � wkðxÞ

�jx� x"i j

8d"

� �þv"kið ~xÞ � wkðx

"i Þ�jx� x"i j

"

� �, ~x 2 R"ki, k ¼ 1, 2,

Wðx"i Þ þv1ið ~xÞ �Wðx"i Þ

�ð�nÞ

þv2ið ~xÞ �Wðx"i Þ

�ð�� nÞ, ~x ¼ ð�1, . . . , �nÞ 2 B"i :

8>>>>>>>>>><>>>>>>>>>>:Here R"ki ¼ f ~x ¼ ðx, kÞ 2 �"

k : d" � jx� x"i j5 "=2g ðk ¼ 1, 2Þ, w1,w2ðxÞ 2 C10 ð�Þ,WðxÞ 2

C1ð�Þ, �(r) is a smooth function equal to 1 as 0 � r � ð1=4Þ and equal to 0 as

r � ð1=2Þ, �(�) is a smooth function equal to 1 as 0 � � � ð�=6Þ and equal to 0 as

� � ð�=3Þ, v"1i and v"2i are the following functions:

�"v"1i ¼ 0, ~x 2 R"1i [bB1i,

v"1i ¼ w1ðx"i Þ, ~x 2 S"1i,

v"1i ¼Wðx"i Þ, ~x 2 �"i [bB2i;

8><>:�"v"2i ¼ 0, ~x 2 R"2i [

bB2i,

v"2i ¼ w2ðx"i Þ, ~x 2 S"2i,

v"2i ¼Wðx"i Þ, ~x 2 �"i [bB1i,

8><>:


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S"ki ¼n

~x ¼ ðx, kÞ 2 �"k: jx� x"i j ¼

"

2

o, k ¼ 1, 2,

bB"1i ¼ ~x ¼ ð�1, . . . , �nÞ 2 B"i : arcsind"

b"� �n 5

�

2

� �,

bB"2i ¼ ~x ¼ ð�1, . . . , �nÞ 2 B"i :�

25 �n � �� arcsin

d"

b"

� �,

�"i ¼ ~x ¼ ð�1, . . . , �nÞ 2 B"i : �n ¼�

2

n o:

Acknowledgements

The author is grateful to Prof. E.Ya. Khruslov for the statement of this problem and the attentionhe paid to this work. The work is partially supported by the Grant of NASU for the youngscientists No. 20207.

References

[1] M. Briane, Convergence of the spectrum of a weakly connected domain, Ann. Mat. Pur. Appl.CLXXVII(IV) (1999), pp. 1–35.

[2] G. Dal Maso, R. Gulliver, and U. Mosco, Asymptotic spectrum of manifolds of increasing

topological type. Prep. SISSA 78/2001/M.[3] F. Fleury and E. Sancez-Palencia, Asymptotic and spectral properties of the acoustic vibrations

of a body perfored by narrow chanels, Bull. Sci. Math. 110(2) (1986), pp. 149–176.

[4] O.A. Oleinik, G.A. Yosifian, and A.S. Shamaev, Mathematical Problems in Elasticity andHomogenization, North-Holland, Amsterdam, 1992.

[5] V. Rybalko, Vibrations of elastic systems with a large number of tiny heavy inclusions,

Asymptotic Anal. 32(11) (2002), pp. 27–62.[6] A.S. Shamaev, Spectral problems in the homogenization theory and G-convergence, Dokl. Akad.

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Acad. Sci. 90 (1981), pp. 239–271.[8] L. Boutet de Monvel and E.Ya. Khruslov, Averaging of the diffusion equation on Riemannian

manifolds of complex microstructure, Trudy Moscow Mat. Obshch. 58 (1997), pp. 137–161.

[9] L. Boutet de Monvel, I.D. Chueshov, and E.Ya. Khruslov, Homogenization of attractors forsemilinear parabolic equation manifolds with complicated microstructure, Ann. Mat. Pur. Appl.CLXXVII(IV) (1997), pp. 297–322.

[10] L. Boutet de Monvel and E.Ya. Khruslov, Homogenization of harmonic vector fields onRiemmanian manifold with complicated microstructure, Math. Phys. Analys. Geom. 1(1) (1998),pp. 1–22.

[11] A. Khrabustovskyi, Klein-Gordon equation as a result of wave equation averaging on the

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asymptotic behaviour of spectrum of laplace–beltrami operator on riemannian manifolds with complex...

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