Bloch wave method for homogenization
by M. Vanninathan
1
Homogenization Problem
Let us start with the operator
∂
∂
A≡−
akl (y)
∂yk
∂yl
!
y ∈ IRN .
Here the coefficients are assumed to be, as usual, Y -periodic (Y =]0, 2π[N ) and bounded.
The space of such functions is denoted by
o
n
∞
N
N
L∞
ZN .
# (Y ) = φ ∈ L (IR ); φ(y + 2πp) = φ(y), y ∈ IR a.e, p ∈ Z
We suppose that the operator is elliptic in the sense that there exists α > 0 such that
akl (y)ηk ηl ≥ α|η|2 ∀η ∈ IRN , y ∈ IRN a.e.
These are the usual assumptions on the operator A. In the present situation, we make one
additional assumption:
akl = alk
∀ k, l = 1, . . . N.
This will imply that the operator A defines an unbounded self-adjoint operator in L2 (IRN ).
Since the method to be presented below is spectral, there is no doubt that the above hypothesis is convenient to work with. (It is by no means necessary for homogenization).
For each ε > 0, let us introduce the operator
∂
Aε ≡ −
∂xk
∂
aεkl (x)
∂xl
!
with
aεkl (x)
= akl
x
, x ∈ IRN .
ε
Associated with Aε is the following boundary value problem posed on a bounded domain Ω
of IRN : find uε satisfying
Aε uε = f in Ω, uε = 0 on ∂Ω.
Here f is a given element in L2 (Ω) and ∂Ω denotes the boundary of Ω. Let us recall the
definition of the function space in which we seek the solution uε :
(
)
∂v
∈ L2 (Ω) ∀k = 1, . . . N .
H (Ω) = v ∈ L (Ω);
∂xk
1
2
1
Thanks to our hypotheses, above problem admits one and only one solution via Lax-Milgram
Lemma. Further, above problem can be given a weak formulation:
(
aε (uε , v) = (f, v) ∀v ∈ H01 (Ω),
uε ∈ H01 (Ω),
where we use the bilinear forms
R
aε (u, v) =
(f, v)
∂u ∂v
aεkl (x) ∂x
dx,
l ∂xk
Ω
R
=
f vdx .
Ω
Problem of homogenization is concerned with the study of asymptotic behaviour of uε as
ε → 0. It is easy to see that there is a uniform bound with respect to ε : kuε kH 1 (Ω) ≤ C and
this may be obtained by taking v = uε in the above weak formulation and using Poincaré
inequality. Hence, for a subsequence uε * u∗ in H01 (Ω)− weak. The main task is to
characterize the weak limit u∗ . (In particular, this will show the uniqueness of the weak
limit and hence the entire sequence converges). In this context, let us state the fundamental
result of homogenization:
Main Theorem. There are constant coefficients (qkl ) which depend on (akl (y)) such that
def
σkε (x) = akl xε
∀ k = 1, . . . N.
∂uε
(x)
∂xl
def
It follows that u∗ is the solution of
(
∗
* σk∗ (x) = qkl ∂u
(x) in L2 (Ω) − weak,
∂xl
∗
A∗ u∗ ≡ − ∂x∂ k qkl ∂u
= f (x), x ∈ Ω,
∂xl
∗
u
= 0 on ∂Ω.
The matrix (qkl ) called the homogenized matrix is defined in the following manner: Define
χk = χk (y) for k = 1, . . . , N, as the unique solution of the following cell problem:



We define


∂akl
∂yl
1
H# (Y
Aχk =
in IRN ,
χk ∈
), MY (χk ) ≡
qkl = MY
def
∂χl
akl + akm
∂ym
!
1
|Y |
R
Y
χk dy = 0.
∀ k, l = 1, . . . , N.
This is a nontrivial result because we know
aεkl * MY (akl ) in L∞ (Ω) − weak∗,
∂uε
∂xl
*
∂u∗
∂xl
in L2 (Ω) − weak,
2
and it is not generally true that the weak limit of products is equal to the product of their
weak limits because of the nature of their oscillations. We have seen several proofs of the
above result using
• asymptotic expansion method
• Tartar’s method of oscillating test functions
• Compensated Compactness method
• Gamma convergence method.
In this section, we will present a different approach. While the previous methods were all
based on physical space, the present one is based on Fourier techniques. It is then well-known
that it will be convenient to work in the entire space IRN . The principal steps of this method
are
(i) Spectral decomposition of the operator Aε in L2 (IRN ) using Bloch waves.
(ii) Transform the equation from the physical space to Bloch space.
(iii) Higher Bloch modes are negligible in the homogenization process.
(iv) Regularity of the lowest Bloch eigenvalue and the corresponding eigenvector.
(v) Their asymptotic properties and the passage to the limit.
We carry out the above programme step by step.
2
Fourier waves and spectral decomposition
To motivate things to come, let us consider the homogeneous medium described by akl (y) =
δkl ∀k, l = 1, . . . N. In this case, A = −∆ which defines an unbounded self-adjoint operator
in L2 (IRN ) with domain H 2 (IRN ). It is worth-while to recall how plane waves (also called
3
Fourier waves) eix·ξ are used in the spectral resolution of (−∆) in L2 (IRN ). The first step
is to define the Fourier transform (FT) for functions f = f (x) using plane waves:
fˆ(ξ) =
1
(2π)N/2
Z
f (x)e−ix·ξ dx, ξ ∈ IRN .
IRN
The main result concerning FT which is of interest to us is the following
Theorem. Let f ∈ L2 (IRN ) be arbitrary. Then
(i) fb is well defined as an element of L2 (IRN ).
(ii)
Parseval’s equality holds:
Z
|fˆ(ξ)|2 dξ =
IRN
(iii)
Z
|f (x)|2 dx .
IRN
Inverse formula also holds:
Z
1
f (x) =
fˆ(ξ)eix·ξ dξ, x ∈ IRN .
N/2
(2π)
N
IR
For a proof, the reader can refer to Rudin’s book [Ru]. Later we will deduce this result from
a more general one. We point out that the above result provides a spectral decomposition
of the operator (−∆). Indeed, though Fourier waves eix·ξ do not have finite energy, they can
be considered as generalized eigenfunctions of (−∆) because of the relation
(−∆)eix·ξ = |ξ|2 eix·ξ .
(2.1)
Since ξ ∈ IRN is arbitrary, this indicates that the spectrum of (−∆) is [0, ∞). The inverse
formula in the above Theorem can be interpreted as the resolution of the Identity operator
on L2 (IRN ) in terms of these generalized eigenfunctions. In particular, this shows that
o
eix·ξ ; ξ ∈ IRN is a generalized basis for L2 (IRN ). Further, one has the resolution of the
operator itself in terms of this basis:
n
∆f (x) =
Z
1
|ξ|2 fˆ(ξ)eix·ξ dξ,
(2π)N/2 N
x ∈ IRN ,
f ∈ H 2 (IRN ).
IR
This shows, in particular, that
d (ξ) = |ξ|2 fb(ξ),
−∆f
ξ ∈ IRN .
In other words, the differential operator −∆ goes over to the multiplication operator by the
polynomial |ξ|2 on the Fourier side.
4
3
Bloch waves
In the sequel, we are interested in generalizing the picture in §2 where (−∆) is replaced by
A. The counter part of plane waves are Bloch waves ψ which we introduce now.
Definition. Let η ∈ IRN . We say that a function ψ : IRN → C in (η, Y )−periodic if
ψ(y + 2πp) = e2πip·η ψ(y) ∀y ∈ IRN , p ∈ ZZ N .
The main reason for introducing the above definition is that the plane waves eiy·η enjoy this
property. Note that when η = 0, the above property is nothing but the usual periodicity.
Generalizing the relation (2.1), we introduce
Definition. Fix η ∈ IRN . Consider the eigenvalue problem for A with (η, Y ) peridicity
condition: Find λ = λ(η) ∈ C and ψ(η) = ψ(y, η) (non zero) such that
Aψ = λψ in IRN , ψ is (η, Y ) periodic.
The eigenvectors ψ are known as Bloch waves and the eigenvalues are called Bloch eigenvalues.
A couple of remarks follow. First of all, the (η, Y ) periodicity condition remains unchanged when we replace η by η+l, l ∈ ZZ N . Hence, without loss of generality, we can restrict
η to the cell Y 0 ≡ ] − 21 , 21 [N . Secondly, we seek ψ in the form ψ(y, η) = eiy·η φ(y, η). It can be
easily checked that the above spectral problem is transformed to another one which involves
the usual periodicity condition and which can be stated as follows: Find λ = λ(η) ∈ C and
φ = φ(y, η) (non-zero) satisfying
A(η)φ = λφ,
φ is Y − periodic,
where A(η) is a new operator associated with A.
∂
+ iηk
A(η) ≡ −
∂yk
!"
∂
akl (y)
+ iηl
∂yl
!#
,
and this is called the shifted operator in the literature.
Above eigenvalue problems parametrized by η ∈ IRN are resolved in the usual manner.
First step is to introduce function spaces in which we seek eigenvectors. We define
L2# (Y )
=
L2# (η, Y ) =
n
n
o
v ∈ L2loc (IRN ); v is Y − periodic ,
o
v ∈ L2loc (IRN ); v is (η, Y ) − periodic ,
5
n
o
1
1
H#
(Y )
= v ∈ Hloc
(IRN ); v is Y − periodic ,
n
o
1
1
H#
(η, Y ) = v ∈ Hloc
(IRN ); v is (η, Y ) − periodic .
The next step is to give a weak formulation of the problem in these function spaces. To this
end, let us introduce some bilinear forms.
a(u, v)
=
R
Y
a(η)(u, v) =
(u, v)
=
R
YR
∂u ∂v
dy,
akl (y) ∂y
l ∂yk
akl (y)
uvdy.
∂u
∂yl
+ iηl u
∂v
∂yk
+ iηk v dy,
Y
While the bilinear form a(., .) is classical, the bilinear form a(η) is not very familiar. We will
be interested in the following parametrized periodic boundary value problem:
1
Given F ∈ L2# (Y ), find u ∈ H#
(Y ) satisfying A(η)u = F in IRN , or equivalently,
1
a(η)(u, v) = (F, v) ∀ v ∈ H#
(Y ).
Resolution of the above problem via Lax-Milgram Lemma will naturally require a version of
Poincaré inequality. It is provided by our next result.
Lemma. There is a positive constant C such that
C{k∇vk0,Y + |η|kvk0,Y } ≤ k∇v + iηvk0,Y ≤ {k∇vk0,Y + |η| kvk0,Y }
1
for all v ∈ H#
(Y ) and η ∈ Y 0 .
Proof.
1
(i) The map v 7→ ve(y) = eiy·η v(y) defines an isomorphism between the spaces H#
(Y ) →
1
H#
(η, Y ). Further,
kvk0,Y
We work with ve instead of v.
k∇vek0,Y
= kvek0,Y ,
= k∇v + iηvk0,Y .
1
(ii) For smooth elements ve of H#
(η, Y ), we have the representation
(e
2πiηj
− 1)ve(y) = 2π
Z1
0
∂ ve
(y + 2πtej )dt ∀j = 1, . . . N.
∂yj
6
1
(iii) Since smooth elements are dense in H#
(η, Y ), it follows that, for some constant C > 0,
we have
1
C |η| kvek0,Y ≤ k∇vek0,Y ∀ ve ∈ H#
(η, Y ).
1
Remark. Above proof shows that for η 6= 0, k∇vek0,Y defines a norm on H#
(η, Y ) which is
equivalent to the original one, namely H 1 (Y ) norm. When η = 0, same assertion can’t be
made. However, we can assert that k∇vk0,Y is a norm equivalent to the quotient norm in
1
(Y )/C
H#
Above Lemma implies that the bilinear form a(η) is elliptic:
n
a(η)(v, v) ≥ C k∇vk20,Y + |η|2 kvk20,Y
Thus we can apply Lax-Milgram Lemma and get
o
1
∀ v ∈ H#
(Y ).
Theorem. (i) Let η 6= 0. Then given F ∈ L2# (Y ) arbitrary, there exists a unique solution u
satisfying
(
1
a(η)(u, v) = (F, v) ∀ v ∈ H#
(Y ),
1
u ∈ H# (Y ).
(ii) Let η = 0. Then given F ∈ L2# (Y ) such that
satisfying
(
R
F dy = 0, there exists a unique solution
Y
1
a(u, v) = (F, v) ∀v ∈ H#
(Y ),
1
u ∈ H# (Y )/C.
To solve the corresponding spectral problem, we consider the corresponding Green’s operator
1
G(η) : L2# (Y ) → H#
(Y ) ,→ L2# (Y ),
G(η)F = u.
1
Since the inclusion H#
(Y ) ,→ L2# (Y ) is compact, G(η) is a compact operator for η 6= 0.
(When η = 0, we replace the space L2# (Y ) by L2# (Y )/C)
Further, G(η) is self-adjoint because
1
a(η)(u, v) = a(η)(v, u) ∀ u, v ∈ H#
(Y ),
which is a consequence of our hypothesis that akl = alk . Thus applying the spectral theory
of compact self-adjoint operators, we arrive at
Theorem. Fix η ∈ Y 0 . Then there exist a sequence of eigenvalues {λm (η); m ∈ IN } and
corresponding eigenvectors {φm (y, η), m ∈ IN } such that
7
(i)
(ii)
A(η)φm (y, η) = λm (η)φm (y, η) ∀ m ∈ IN .
0 ≤ λ1 (η) ≤ λ2 (η) ≤ . . . → ∞. Each eigenvalue is of finite multiplicity. It is repeated
in the above sequence as many times as its multiplicity.
(iii) {φm (., η); m ∈ IN } is an orthonormal basis for L2# (Y ).
(iv) For φ in the domain of A(η), we have A(η)φ(y) =
∞
P
m=1
λm (η) (φ, φm (η))φm (y, η).
Above result gives the existence of Bloch eigenvalues and Bloch waves and describes
some of their properties. In the next section, we will see an application of these waves.
4
Bloch decomposition of L2(IRN )
In the last section, we obtained the spectral decomposition of A(η) considered with periodic
boundary condition. In the present section, we consider A as an unbounded self-adjoint
operator with dense domain contained in L2 (IRN ). We obtain its spectral decomposition in
a sense analogous to our analysis of §2, replacing plane waves by Bloch waves. Roughly
speaking, the results are as follows:
(i) {eiy·η φm (y, η); m ∈ IN , η ∈ Y 0 } forms a basis of L2 (IRN ) in a generalized sense.
(ii) L2 (IRN ) can be identified with L2 (Y 0 ; l2 (IN )).
(iii) A corresponds to a multiplication operator with λm (η) as multipliers.
Let us now state the precise result.
Theorem. (Bloch decomposition) Let g ∈ L2 (IRN ). Define mth Bloch coefficient of g as
follows.
Z
g(y)e−iy.η φm (y, η)dy,
(Bm g)(η) =
m ∈ IN , η ∈ Y 0 .
IRN
Then the following inverse formula holds:
g(y) =
Z X
∞
(Bm g)(η)eiy.η φm (y, η)dη.
Y 0 m=1
Further, we have Parseval’s equality:
Z
IRN
2
|g(y)| dy =
Z X
∞
Y 0 m=1
8
|Bm g(η)|2 dη.
Proof. First of all, we assume that g is smooth with compact support. The general case
can then be handled by density arguments. The proof is a three step process.
Step 1. We decompose g ∈ L2 (IRN ) in terms of (η, Y )- periodic functions. To this end, we
associate to g(y) a function of two variables g# (y, η) as follows:
X
g# (y, η) =
p∈Z
Z
or equivalently,
g(y + 2πp)e−i(y+2πp)·η
N
X
g# (y · η)eiy·η =
p∈Z
Z
g(y + 2πp)e−2πip·η .
N
First of all, we note the following properties w.r.t the variable y :
g# (y, η) is Y − periodic w.r.t y :
g# (y + 2πq, η) = g# (y, η) ∀ q ∈ ZZ N ,
or equivalently,
g# (y, η)eiy·η is (η, Y ) − periodic w.r.t. y.
Secondly, the right side of the above definition of g# eiy·η is a Fourier series w.r.t the variable
η ∈ Y 0 . From this observation, it follows then that
g# (y, η)eiy·η is Y 0 − periodic w.r.t η ∈ Y 0 ,
g(y) =
Z
Z
y ∈ IRN ,
g# (y, η)eiy·η dη,
Y0
|g# (y, η)|2 dη =
X
p∈Z
Z
Y0
|g(y + 2πp)|2 ,
y ∈ IRN .
N
Integrating the above equality w.r.t y ∈ Y, we obtain
Z Z
|g# (y, η)|2 dηdy =
Y Y0
Z
|g(y)|2dy.
IRN
Thus, we obtain a unitary transformation
L2 (IRN ) −→ L2 (Y 0 ; L2# (Y ))
g(y) 7−→ g# (y, η).
9
Step 2. In the previous step, we passed from the variable y ∈ IRN to a periodic variable
y ∈ Y. The canoical thing to do now is to decompose g# (y, η) in terms of the eigenbasis of
A(η) obtained in the previous section. We get
g# (y, η) =
∞
X
am (η)φm (y, η),
m=1
am (η) =
Z
g# (y, η)φm (y, η)dy,
Z
|g# (y, η)|2dy.
Y
∞
X
|am (η)|2 =
m=1
Y
To express the coefficient am (η) in terms of g, we use the definition of g# (y, η). Since φm (y, η)
is Y −periodic w.r.t y, we obtain
am (η) =
X Z
p∈Z
Z
N
g(y + 2πp)e−i(y+2πp)·η φm (y + 2πp, η)dy.
Y
The right side integral coincides with the definition of the mth Bloch coefficient and hence
am (η) = Bm g(η).
Step 3. To complete the proof it remains to combine the inverse relations and Parseval’s
equalities obtained in the previous two steps.
A bye product of the above decomposition is that A corresponds to an operator with
multipliers {λm (η); m ∈ IN , η ∈ Y 0 }. More precisely, we have
Corollary. For all g in the domain of A, we have
Ag(y) =
Z X
∞
λm (η)(Bm g)(η)eiy·η φm (y, η)dη.
Y 0 m=1
Proof. Simply combine the above inverse formula and the following relation:
A(eiy.η φm (y, η)) = eiy·η A(η)φm (y, η) = λm (η)eiy·η φm (y, η).
Example. Once again, let us consider the homogeneous medium defined by akl (y) =
δkl
k, l = 1 . . . N, so that A = −∆. In this case, the translated operator is given by
−∆ − 2iη · ∇ + |η|2 ,
10
η ∈ Y 0.
It is easily checked that its spectral elements can be indexed by m ∈ ZZ N :
Eigenvalues: λm (η) = |m + η|2 , m ∈ ZZ N , η ∈ Y 0 ,
Eigenvectors: φm (y, η) = cm eim·y , m ∈ ZZ N .
Note the following features of the homogeneity in the Bloch space:
(i)
(ii)
(iii)
(iv)
the indexing set ZZ N is a group under addition.
eigenvalues respect this addition: λm (η) = λ0 (m + η).
Eigenvalues are polynomials w.r.t η.
Eigenvectors do not depend on η.
Let us now consider the operator Aε in variable x introduced in §1 in the place of A.
These two operators are connected by the transformation y = x/ε. Let ξ denote the variable
dual to x. Just as we did for the operator A, we can introduce the eigenvalues λεm (η) and
eigenvectors φεm (x, ξ) of the operator Aε corresponding to εY −periodic condition:
Aε φεm (x, ξ) = λεm (ξ)φεm (x, ξ) m ∈ IN , x ∈ IRN , ξ ∈ IRN ,
φεm (x, ξ) is εY − periodic w.r.t x
0 ≤ λε1 (ξ) ≤ λε2 (ξ) ≤ . . . → ∞,
{φεm (x, ξ); m ∈ IN } is an orthogonal basis for L2# (εY ).
By homothecy, it is easy to check that the spectral elements of A and Aε are related:
x
λεm (ξ) = ε−2 λm (εξ), φεm (x, ξ) = φm ( , εξ).
ε
Further, ξ can be restricted to lie in ε−1 Y 0 just as η was confined to Y 0 . We set η = εξ.
Following is the Bloch decomposition theorem corresponding to Aε :
Theorem. Let g ∈ L2 (IRN ) be arbitrary. Define
ε
(Bm
g)(ξ) =
Z
ε
g(x)e−ix·ξ φm (x, ξ)dx,
m ∈ IN , ξ ∈ ε−1 Y 0 .
IRN
Then the following inverse formula holds:
g(x) =
Z
ε−1 Y 0
∞
X
ε
(Bm
g)(ξ)eix·ξ φεm (x, ξ)dξ.
m=1
Further, we have Parseval’s equality:
Z
IRN
2
|g(x)| dx =
Z
ε−1 Y 0
11
∞
X
m=1
ε
|Bm
g(ξ)|2 dξ.
Finally, for all g in the domain of Aε , we have
ε
A g(x) =
Z
ε−1 Y 0
5
∞
X
ε
λεm (ξ)(Bm
g)(ξ)eix·ξ φεm (x, ξ)dξ.
m=1
Regularity of the Ground State
In the previous section, we have obtained a simplified picture of the operator A, namely it
is a multiplication operator with eigenvalues {λm (η); m ∈ IN , η ∈ Y 0 } as multipliers. As one
may expect, the higher eigenvalues {λm (η); m ≥ 2, η ∈ Y 0 } do not play any role as they are
not excited in the homogenization process. Thus, we are reduced to consider the ground
state φ1 (y, η) and the corresponding energy λ1 (η). What matters in homogenization is
their regular behaviour near η = 0. Thus we have achieved an enormous reduction from the
original operator equation.
Regarding the regularity of spectral elements of A(η), we remark that our operator can
be considered to model some microscopic behaviour of crystals. From the physics of crystals,
we know that they exhibit intrinsic singularities. As a consequence, we do not expect too
much of global regularity of eigenvalues and eigenvectors. In the sequel, we establish two
types of results which are consistent with the regularity observed in the homogeneous case.
The first one is a global regularity result valid for all m ≥ 1 and η ∈ Y 0 . The second one is
local regularity of the ground state at η = 0.
Our first result is concerned with global regularity of all eigenvalues.
Theorem. For m ≥ 1, λm (η) is a Lipschitz function of η.
Proof. According to the Courant-Fischer min-max principle, we have
λm (η) = min
max
v∈F
dimF =m
a(η)(v, v)
,
(v, v)
1
where F ranges over all subspaces of H#
(Y ) of dimension= m. We notice that a(η)(v, v)
can be decomposed as
a(η)(v, v) = a(η 0 )(v, v) + R(v, η, η 0 ),
where
R=
Z
Y
Z
Z
∂v
∂v
0
0
dy + akl (ηl ηk − ηl0 ηk0 )|v|2 dy.
akl
(iηk − iηk )vdy + akl (iηl − iηl )v
∂yl
∂yk
Y
Y
12
By Cauchy-Schwarz inequality, R can be estimated. We have
Z
|R| ≤ c|η − η 0 | (|∇v|2 + |v|2 )dy.
Y
Using all these in the above min-max characterization, we get
λm (η) ≤ λm (η 0 ) + cm |η − η 0 |
for a suitable constant cm . Interchanging η and η 0 , we obtain
|λm (η) − λm (η 0 )| ≤ cm |η − η 0 |.
For homogenization purposes, above global regularity is not sufficient. We need a strong
local regularity of the ground state and the corresponding energy. This is stated in
def
Theorem. There is a small ball Bδ (0) = {η; |η| < δ} such that
(i)
(ii)
λ1 (η) is analytic for η ∈ Bδ (0).
There is a choice of the corresponding eigenvector φ1 (y, η) satisfying
1
η ∈ Bδ (0) → φ1 (., η) ∈ H#
(Y ) is analytic and
φ1 (y, 0) = |Y |−1/2 , a constant independent of y.
The complete proof is not easy and it needs new tools from Functional Analysis. That is
why we will not present it. (Interested readers can see the original article [C-V]. However, I
want to point out various phenomena and difficulties involved in the proof.
Above is a result in the so-called regular perturbation analysis of linear operators and
their spectra. This consists of studying the operator A(0) = A and see what of its properties
are sustained when we perturb A(0) to A(η) with |η| small. Sustained properties depend
also on the manner in which A(0) is perturbed. In our context, A(η) depends analytically
(infact, quadratically) on η. However this notion has to be properly formalized especially in
the case of unbounded operators acting in infinite dimensional spaces. (see [K] [R-S]). We
avoid such technicalities in the sequel.
Even though A(η) depends on η analytically, this, by no means, implies analytic dependence of eigenvalues of A(η) even in finite dimensional case. In fact, the dependence of
roots of a polynomial equation on its coefficients need not be analytic e-g: λ2 − η = 0. Roots
13
λ = ±η 1/2 have branch singularities. This is due to the multiplicity of the roots at η = 0.
To eliminate such situations, we must have simple roots.
The main property of A(0) = A on which the above Theorem rests is given in
Theorem. A is an unbounded self-adjoint operator in L2# (Y ) with zero as the lowest
eigenvalue (ground energy). It is simple/ non-degenerate in the sense that the corresponding
eigenvectors (ground states) form a one-dimensional space consisting of scalars.
This is just a fancy way of stating already known properties which can be stated in
terms of the associated bilinear form, namely
1
a(φ, ψ) = a(ψ, φ), a(φ, φ) ≥ 0, ∀ φ, ψ ∈ H#
(Y ),
a(φ, φ) = 0 iff φ ≡ constant.
We claim the non-degeneracy of the lowest eigenvalue is sustained under perturbations. Our
idea to prove this is to reduce the problem to the resolution of a nonlinear algebraic equation
and apply the classical Implicit Function Theorem (IFT). This programme is somewhat easy
to implement in the finite dimensional case and so we confine ourselves to it.
Assuming thus that A(η) is a family of self-adjoint matrices depending analytically on
η, we recall that λ(η) is an eigenvalue of A(η) iff
D(η, λ) ≡ det(A(η) − λI) = 0 when λ = λ(η).
This characterization of eigenvalues does not involve eigenvectors at all. Let us arrange the
eigenvalues of A(η) as usual:
λ1 (η) ≤ λ2 (η) ≤ . . .
These are nothing but the roots of the polynomial equation: D(η, λ) = 0. When η = 0, we
have seen above that λ1 (0) = 0 and it is simple. Another way of formulating simplicity for
self-adjoint matrices is that ∂D
(0, 0) 6= 0. We are in a perfect situation to apply IFT (stated
∂λ
below)
Implicit Function Theorem. [K-P] Let (η, λ) 7→ D(η, λ) ∈ IR be an analytic function
defined in a neighbourhood of (η, λ) = (0, 0) ∈ IRN × IR. Assume that D(0, 0) = 0 and
∂D
(0, 0) 6= 0. Then there exist δ > 0 and an analytic map η 7→ µ(η) defined for η ∈ Bδ (0)
∂λ
such that D(η, µ(η)) = 0 ∀η ∈ Bδ (0) and µ(0) = 0. Further µ(η) is the only solution of the
equation D(η, λ) = 0 lying near λ = 0.
14
We already know that the first eigenvalue λ1 (η) satisfies D(η, λ1 (η)) = 0 ∀η and
λ1 (0) = 0. Because of continuity, λ1 (η) is close to 0 when |η| is small. Thanks to the
uniqueness part of IFT, we conclude that λ1 (η) = µ(η) and in particular, λ1 (η) is analytic
and simple near η = 0.
Treatment of eigenvectors even in finite dimension is more difficult because unlike
eigenvalues, they are not uniquely determined. Even if we impose normalization condition
on them, there is still an arbitrary phase factor which has to be chosen properly so that we
have eigenvectors depending analytically on the parameter η.
Useful tool in this business is a classical notion from Complex Analysis, namely contour integration. Recall that this is used also in semi-group theory. Let us consider
1
P (η) =
2πi
Z
(A(η) − λ)−1 dλ,
Γr
where Γr is the circle {λ ∈ C; |λ| = r}. We choose r > 0 small enough and independent
of η ∈ Bδ (0) in such a way that |λ1 (η)| < r and |λm (η)| > r ∀ m ≥ 2. This is possible
because of the continuity of the eigenvalues and the fact that λ1 (0) is simple. In other words,
Γr encloses only the first eigenvalue λ1 (η) of A(η) for all η ∈ Bδ (0), all other eigenvalues
lie outside Γr . Since no eigenvalue lies on the circle Γr , (A(η) − λ)−1 exists and so the
above contour integral makes sense. Since the integrand is a matrix (or bounded operator
in the infinite dimensional case,) P (η) is a well defined matrix for η ∈ Bδ (0). What is its
significance? The integrand has a simple pole at λ = λ1 (η) and thus P (η) is nothing but the
residue at the pole.
Its main property that we require is stated next.
Theorem [R-S]. P (η) is a projection onto the eigenspace associated with λ1 (η).
It is easy to see that η ∈ Bδ (0) 7→ P (η) is analytic. Thus we have the following natural
choice of analytic eigenvectors:
φ1 (η) = P (η)φ1 (0) with φ1 (0) = |Y |−1/2 .
Since φ1 (η) → φ1 (0) as η → 0 we see φ1 (η) 6= 0 for |η| small. Thus, we may normalize it so
that kφ1 (η)k0,Y = 1.
This completes our discussion of the proof of the main result of this section.
15
6
Computation of derivatives of ground state
Having proved that the ground state is analytic near η = 0, it is our task to compute its
derivatives at η = 0. By this it is meant that a physical space description will be given
to derivatives taken with respect to Bloch space variable. Although all derivatives can be
computed in principle (cf. [C-O-V]), we require presently derivatives upto order two. We
will see a new expression and interpretation of the homogenized matrix a0 . In fact, we will
see that 2a0 coincides with the Hessien matrix of λ1 (η) at η = 0. Before proceeding further,
there is a need for proper normalization of ground state and this is what we do next.
Normalization Conditions. We have shown that there is a choice of eigenvectors φ1 (y, η)
that depend analytically on η in a small neighbourhood Bδ of η = 0. Since φ1 (., 0) is not
zero by our choice, we can meet the usual normalization condition
kφ1 (., η)k0,Y = 1,
η ∈ Bδ
(6.1)
at the cost of reducing the neighbourhood. This condition does not determine the eigenvector
uniquely; one still has the choice of the phase factor. Exploiting this choice, one can achieve
the following somewhat unusual condition also:
Im
Z
φ1 (y, η)dy = 0,
η ∈ Bδ .
(6.2)
Y
Indeed, there exist real-valued analytic functions α1 (η) and α2 (η) such that
Im
Z
(α1 (η) + iα2 (η))φ1 (y, η)dy = 0,
Y
where φ1 (., η) is the already chosen analytic branch. A natural choice for α1 (η) and α2 (η) is
α1 (η) = −Re
Z
φ1 (y, η)dy,
α2 (η) = Im
Y
Z
φ1 (y, η)dy.
Y
Such a choice might disturb (6.1) but it can be regained easily since (α1 (0), α2 (0)) 6= (0, 0).
We are now in a position to compute the derivatives of λ1 (η) and φ(., η) at η = 0. The
procedure consists of differentiating the above normalization condition and the eigenvalue
equation
(A(η) − λ1 (η))φ1 (., η) = 0.
16
(6.3)
We begin by recalling the expression of the shifted operator and noting that it can be
written in the form
A(η) ≡ A + iηk Ck + ηk ηl akl (y),
where
Ck φ ≡ −akj
∂
∂φ
−
(akj φ).
∂yj
∂yj
Step 1. (zeroth order derivatives). We simply recall that φ1 (y, 0) = |Y |−1/2 by our choice
and λ1 (0) = 0.
Step 2. (First order derivatives of λ1 ) Differentiate the equation once with respect to ηk to
obtain
Dk (A(η) − λ1 (η))φ1 (., η) + (A(η) − λ1 (η))Dk φ1 (., η) = 0.
(6.4)
Taking scalar product with φ1 (., η) in L2# (Y ), we get
hDk (A(η) − λ1 (η))φ1 (η), φ1 (η)i = 0.
Evaluate the above relation at η = 0 and use Dk A(0) = iCk . Since φ(0) = φ(y, 0) does not
depend as y, we get
∂
(akj φ(0))
∂yj
and hence its integral over Y vanishes because of periodicity. It follows therefore that
Ck φ(0) = −
Dk λ1 (0) = 0 ∀ k = 1, . . . , N.
Step 3. (First order derivatives of φ1 ) Evaluate (6.4) at η = 0 and we use λ1 (0) = 0, and
Dk λ1 (0) = 0. We get the following equation with respect to variable y for the derivative of
φ1 at η = 0 :
ADk φ1 (., 0) = −Dk A(0)φ1 (., 0)
which can be written as
ADk φ1 (., 0) = −iCk φ1 (., 0) = −iφ1 (., 0)Ck 1 = iφ1 (., 0)
∂
(akj ).
∂yj
(6.5)
This equation determines Dk φ1 (., 0) uniquely upto an additive constant which can be fixed
thanks to normalization conditions. Indeed, differentiating (6.1), (6.2) we get
RehDk φ1 (., η), φ1 (., η)i = 0,
R
Im Dk φ1 (y, η)dy = 0.
Y
17
(6.6)
A simple evaluation at η = 0 yields
Z
Dk φ1 (y, 0)dy = 0 ∀ k = 1, . . . , N.
(6.7)
Y
Comparing (6.5), (6.7) with the defining conditions of the test function χk (y), (cf. Section
1) we get
Dk φ1 (y, 0) = i|Y |−1/2 χk (y).
(6.8)
In particular, Dk φ1 (y, 0) is purely imaginary.
Step 4. (Second derivatives of λ1 ) This step is similar to Step 2. We differentiate (6.4) with
respect to ηl to obtain
2
[Dkl
(A(η) − λ1 (η))]φ1 (η) + [Dk (A(η) − λ1 (η))]Dl φ1 (η) + [Dl (A(η) − λ1 (η))]Dk φ1 (η) +
2
+(A(η) − λ1 (η))Dkl
φ1 (η) = 0.
Taking scalar product with φ1 (η) in L2# (Y ), we get
2
h[Dkl
(A(η) − λ1 (η))φ1 (η), φ1 (η)i + h[Dk (A(η) − λ1 (η))]Dl φ1 (η), φ1 (η)i
+h[Dl (A(η) − λ1 (η))]Dk φ1 (η), φ1 (η)i = 0.
2
Note that Dkl
A(η) = 2akl (y) ∀ k, l = 1, . . . N. Evaluating the above relation at η = 0 and
using the information obtained in the previous steps, we obtain
1
1 2
Dkl λ1 (0) =
2!
|Y |
Z
Y
1
akl (y)dy −
2|Y |
Z
(Ck χl (y) + Cl χk (y))dy = qkl ,
∀ k, l = 1, . . . , N,
Y
where qkl are precisely the homogenized coefficients defined in Section 1.
Above procedure can obviously be continued indefinitely to compute all derivatives of
λ1 (η) and φ1 (η) at η = 0. However, we won’t do it.
7
First Bloch transform tends to Fourier transform
ε
We recall that various Bloch transforms Bm
g(ξ), m ∈ IN are intrinsic quantities associated
with a periodic medium and Fourier transform ĝ(ξ) is similarly associated with a homoge-
neous medium. Heuristically speaking, the first Bloch transform B1ε g is defined in terms of
the first Bloch wave φ1 ( xε , εξ) eix·ξ which converges as ε → 0 to a multiple of the Fourier
18
wave eix·ξ since φ1 (y, 0) is a constant. In this section, we will formalize the above impression
and examine the sense in which B1ε g tends to ĝ. This is the first sign of the fact that periodic
media approximate homogeneous media in the homogenization limit. Since higher Bloch
ε
modes Bm
g, m ≥ 2 are not excited in the homogenization process, they do not figure in this
analysis.
Even though stronger result holds, we prove only a weak result which is sufficient for
our needs. As we shall see, its proof relies on some regularity properties of the first Bloch
mode φ1 .
Theorem 7.1. (i) If g ε * g in L2 (IRN ) weak then χε−1 Y 0 (ξ)B1ε g ε (ξ) * ĝ(ξ) in L2 (IRN )
weak provided there is fixed compact at K such that suppg ε ⊆ K ∀ ε.
(ii) if g ε → g in L2 (IRN ) strong then χε−1 Y 0 (ξ)B1ε g ε (ξ) → ĝ(ξ) in L2loc (IRN ) strong.
Proof. (i) We write for ξ in ε−1 Y 0 ,
B1ε g ε (ξ) =
Z
g ε (x)e−ix·ξ φ1
IRN
x
, 0 dx +
ε
Z
g ε (x)e−ix·ξ φ1
K
x
x
, εξ − φ1
,0
ε
ε
dx
Since φ1 (y, 0) = |Y |−1/2 = (2π)−N/2 , the first term is nothing but the Fourier transform of
g ε and so it converges to gb in L2 (IRN ) weak.
Applying Cauchy-Schwarz inequality, we can bound the second term from above by
ε
kg kL2 (IRN )
R
K
|φ1
x
, εξ
ε
− φ1
≤ Ckφ1 (y, εξ) − φ1 (y, 0)k0,Y
x
,0
ε
2
| dx
!1/2
where C depends on K.
Here is where some regularity of the first Bloch mode η 7→ φ1 (., η) ∈ L2# (Y ) when η
is near zero is required. Analyticity of this map is established in Section 6. What we need
here is simply the fact that the above map is Lipschitz near η = 0. We can then conclude
N
that the second term is bounded above by Cε|ξ| and thus converges to zero in L∞
loc (IRξ ).
This completes the proof of (i).
(ii) Let us first consider the case g ε = g where g ∈ L2 (IRN ) is with compact support. The
proof of (i) shows that B1ε g → g in L2loc (IRN ) strong.
To treat more general cases, let us consider the operator B1ε : L2 (IRN ) → L2 (IRN ).
19
Parseval’s equality in Bloch decomposition theorem (cf. Section 4) shows that kB1ε k ≤ 1. We
can now complete the proof (ii) by density arguements. Indeed, if g ∈ L2 (IRN ) is arbitrary,
we can approximate it by h ∈ L2 (IRN ) with compact support. The desired result follows via
triangle inequality applied to the relation
B1ε g − ĝ = B1ε (g − h) + (B1ε h − ĥ) + (ĥ − ĝ).
Finally, if g ε → g in L2 (IRN ) strong then the relation
B1ε g ε − ĝ = B1ε (g ε − g) + (B1ε g − ĝ)
shows that B1ε g ε → ĝ in L2loc (IRN ), thereby completing the proof of (ii)
Remark. The factor χε−1 Y 0 (ξ) in the above result was merely used to extend the relevant
functions by zero outside their domain of definition. It did not play any part in the proof
because we are interested in local convergence.
8
Proof of the Homogenization Theorem
The purpose of this section is to provide a proof of the main result of homogenization stated
in Section 1. All the preparations made in the previous section will come very handy in this
process.
Before talking up the rigorous proof, we give some heuristics. We consider the equation
in the whole space:
(
Aε uε = f in IRN ,
uε * u0 in H 1 (IRN ).
We pass to the limit in this equation in a heuristic manner to see how homogenized equation
is obtained in the present method. Since Bloch transform converts Aε into multiplication
operator (cf. Section 4), above equation is equivalent to
ε ε
ε
λεm (ξ)Bm
u (ξ) = Bm
f (ξ) m ∈ IN , ξ ∈ ε−1 Y 0
This is a set of equations in the Bloch space. Ignoring the equations corresponding to m ≥ 2
because they play no role in homogenization process, we concentrate on the case m = 1 :
λε1 (ξ)B1ε uε (ξ) = B1ε f (ξ),
20
ξ ∈ ε−1 Y 0 .
Recalling λε1 (ξ) = ε−2 λ1 (η), η = εξ and expanding λ1 (η) by Taylor’s formula around η = 0,
we get
"
#
1 ∂ 2 λ1
(0)ξk ξl + O(ε2 ξ 4 ) B1ε uε (ξ) = B1ε f (ξ).
2 ∂ηk ηk ηl
A simple passage to the limit yields
1 ∂ 2 λ1
(0)ξk ξl û0 (ξ) = fˆ(ξ), ξ ∈ IRN ,
2 ∂ηk ∂ηl
using the results of Section 7. Because the Hessien matrix of λ1 at η = 0 coincides with
twice the homogenized matrix, above equation is nothing but the homogenized equation in
the Fourier space, i.e., it is just the Fourier transform of the usual homogenized equation,
namely A0 u0 (x) = f (x).
It is to be noted that the above passage to the limit is more direct because no derivatives
are involved in the equation in the Bloch space. However, there is one flaw in our argument.
Strictly speaking, we cannot apply the results of Section 7 since uε does not have uniform
compact support. A natural way to overcome this difficulty is to use the cut-off function
technique to localize the equation. This is what we plan to carry out now. In doing so, we
can handle the case of bounded domains also.
Proof of the Main Theorem. Consider therefore a sequence uε such that uε * u0 in
H 1 (Ω) weak and Aε uε = f in Ω. Our aim is to show that u0 satisfies A0 u0 = f in Ω. There
are several steps in the proof.
Step 1. (localization). Let v ∈ D(Ω) be arbitrary. Then the localization vuε satisfies
Aε (vuε ) = vf + g ε + hε in IRN ,
(8.1)
where,
∂uε ∂v
∂2v ε
− aεkl
u,
∂xl ∂xk
∂xk ∂xl
∂aε ∂v ε
hε = − kl
u.
∂xk ∂xl
Note that g ε and hε correspond to terms containing zero and first order derivatives on aεkl
respectively.
g ε = −2aεkl
Step 2. (limit of LHS of (8.1)). As remarked earlier, we consider only the first Bloch
transform of LHS which is λε1 (ξ)B1ε (vuε ). Since v has compact support, we can pass to the
21
limit using the results of Section 7. Indeed, using the fact that vuε → vu0 in L2 (IRN ) strong,
we see that
1 2
d0 (ξ) in L2 (IRN ) strong.
χε−1 Y 0 (ξ)λε1 (ξ)B1ε (vuε )(ξ) → Dkl
λ1 (0)ξk ξl vu
loc
ξ
2
Step 3. (limit of B1ε g ε ) Note that g ε can be written as
g ε = −2σkε
∂2v ε
∂v
− aεkl
u,
∂xk
∂xk ∂xl
where we recall,
σkε
=
∂u
aεkl
ε
.
∂xl
Since σkε is bounded in L2 (Ω), there exists a convergent subsequence in L2 (Ω). Let σk0 be its
limit as well as its extension by zero outside Ω. Thus,
def
g ε * g 0 = −2σk0
∂2v 0
∂v
− MY (akl )
u in L2 (IRN ) weak.
∂xk
∂xk ∂xl
Consequently, by our results in Section 7, we have
χε−1 Y 0 B1ε g ε (ξ) * gb0 (ξ) in L2 (IRN ) weak.
By integration by parts, we get
0
ĝ (ξ) = |Y |
−1/2
Z
[−2σk0
IRN
∂v
∂v ∂u0
∂v −ix·ξ
+ MY (akl )
− iξk MY (akl )
]e
dx.
∂xk
∂xl ∂xk
∂xl
Step 4. (limit of B1ε hε ). Note that hε is uniformly supported in a fixed compact set and
bounded in H −1 (IRN ) but not in L2 (IRN ). Hence, we cannot apply directly Theorem 7.1.
However, following the idea of Theorem 7.1, we decompose
B1ε hε (ξ) =
Z
hε (x)eix·ξ φ1
IRN
x
, 0 dx +
ε
Z
hε (x)e−ix·ξ φ1
IRN
x
x
, εξ − φ1
,0
ε
ε
dx.
(8.2)
The main difference is that since hε is not bounded in L2 (IRN ), the contribution of the second
term cannot be neglected. (The proof of Theorem 7.1 shows that the second term tends to
zero if the sequence is bounded in L2 (IRN )). In fact, using the Taylor expansion of φ1 (y, η)
at η = 0 (which is valid), we see that the second term is equal to
−
Z
IRN
(
)
∂aεkl ∂v ε −ix·ξ ∂φ1 x
ue
ε
( , 0)ξj + O(ε2 ξ 2 ) dx,
∂xk ∂xl
∂ηj ε
22
which via integration by parts becomes,
ξj
Z
aεkl
IRN
∂v ε −ix·ξ ∂ 2 φ1
ue
∂xl
∂ηj ∂yk
x
, 0 dx + O(εξ).
ε
(8.3)
We claim that the above integral term converges in Lploc (IRN
ξ ) for all 1 ≤ p < ∞ to
MY
!
∂ 2 φ1
akl (y)
(y, 0) ξj
∂ηj ∂yk
Z
IRN
∂v 0 −ix·ξ
ue
dx.
∂xl
(8.4)
2
∂v ε ∂ φ1 x
Setting I ε (x) = aεkl (x) ∂x
u ∂ηj ∂yk ( ε , 0), we see that the above integral term is nothing but
l
IN
N
ε
1
ξj (2π)N/2 Iˆε (ξ). This is bounded in L∞
loc (IRξ ) since I is bounded in L (IRx ). Further, it is
easy to check Iˆε (ξ) → Iˆ0 (ξ) ∀ξ ∈ IRN where
0
I (x) = MY
!
∂v
∂ 2 φ1
(y, 0)
(x)u0 (x).
akl (y)
∂ηj ∂yk
∂xl
Thus ξj Iˆε (ξ) → ξj Iˆ0 (ξ) in Lploc (IRN
ξ ) for all 1 ≤ p < ∞.
The conclusion is that the second term of RHS of (8.2) converges in Lploc (IRN
ξ ) strongly
to (8.4).
Regarding the first term on the RHS of (8.2), we can write it, after doing integration
by parts, as
#
"
Z
∂v ε −ix·ξ
∂ 2 v ε ∂v ∂uε
−1/2
ε
|Y |
u +
− iξk
u e
dx.
akl
∂xk ∂xl
∂xl ∂xk
∂xl
N
IR
This can be handled exactly in the same way as the second term was handled just earlier.
The conclusion is that the first term on the RHS of (8.2) converges in Lploc (IRN ) for all
1 ≤ p < ∞ to
|Y |
−1/2
Z "
IRN
#
∂v
∂2v 0
∂v 0 −ix·ξ
MY (akl )
u + σl0
− iξk MY (akl )
u e
dx.
∂xk ∂xl
∂xl
∂xl
Doing integrating by parts in the first term, we see the above is equal to
|Y |
−1/2
Z
N
IR
σl0
!
∂v
∂v ∂u0 −ix·ξ
e
dx.
− MY (akl )
∂xl
∂xl ∂xk
23
(8.5)
Combining (8.4) and (8.5), we see that χε−1 Y 1 (ξ)B1ε hε (ξ) converges strongly in Lploc (IRN
ξ )
for all 1 ≤ p < ∞ to
−|Y |−1/2 MY
∂χj
akl
∂yk
!
(iξj )
Z
IRN
∂v 0 −ix·ξ
ue
dx+|Y |−1/2
∂xl
Z
IRN
!
∂v
∂v ∂u0 −ix·ξ
− MY (akl )
σl0
e
dx.
∂xl
∂xl ∂xk
(8.6)
Note that the minus sign in the first term is due the complex conjugation:
∂φ1
(y, 0) = −i|Y |−1/2 χj (y).
∂ηj
Step 5. (Limit of (8.1)). Taking the first Bloch transform of the equation (8.1), we can pass
to the limit using all previous steps. We obtain
c (ξ) − |Y |−1/2
A0d
(vu0 )(ξ) = vf
Z
σk0
IRN
∂v ix·ξ
e dx − (iξk )|Y |−1/2 qkl
∂xk
Z
IRN
u0
∂v −ix·ξ
e
dx.
∂xl
We call this localized homogenized equation in the Fourier space. The conclusion of the Main
Theorem are very easy consequences of this equation. We demonstrate it in the next step.
Step 6. Here, we finish the proof of the Main Theorem. In fact, taking the inverse Fourier
transform of the localized homogenized equation obtained in the previous step, we get
0
0
A (vu ) = vf −
σk0
∂v
∂
− qkl
∂xk
∂xk
∂v 0
u
∂xl
!
in IRN .
On the other hand, we can calculate A0 (vu0 ) directly using the definition of A0 :
!
∂2v 0
∂v ∂u0
∂
∂
(vu0 ) = −qkl
u − 2qkl
+ vA0 u0 in IRN .
A (vu ) ≡ −
qkl
∂xk
∂xl
∂xk ∂xl
∂xk ∂xl
0
0
A simple comparison between the two above relation yields
!
∂u0
∂v
v(A u − f ) = qkl
− σk0
in IRN .
∂xl
∂xk
0 0
Since the above relation is true for all v ∈ D(Ω), the desired conclusions follow. Indeed,
let us choose φ(x) = φ0 (x)einx·ω where ω is a unit vector in IRN and φ0 ∈ D(Ω) is fixed.
Letting n → ∞ in the resulting relation and varying the unit vector ω, we can easily deduce
0
successively that σk0 = qkl ∂u
in Ω ∀ k = 1, . . . , N and A0 u0 = f in Ω. This completes our
∂xl
discussion of the Main Theorem of Homogenization.
24
References
[C-V] C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM. J. Appl. Math., vol 57 (6), pp. 1639-1659, 1997.
[C-O-V] C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in Homogenization
and Applications, to appear.
[K] T. Kato, Perturbation Theory for linear operators, Springer-Verlag, Berlin, 1976.
[K-P] S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Birkhauser, Basel,
1992.
[R-S] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol IV, Analysis
of Operators, Academic Press, New York, 1978.
[R] W. Rudin, Functional Analysis, TMH Edition, New Delhi, 1974.
25