On Burnett coefficients in periodic media in low contrast regime

23
On Burnett coefficients in periodic media in low contrast regime C. Conca, 1,2,a J. San Martín, 1,2,b L. Smaranda, 2,3,c and M. Vanninathan 4,d 1 Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile 2 Centro de Modelamiento Matemático, UMR 2071 CNRS-UChile, Casilla 170/3, Correo 3, Santiago 8370459, Chile 3 Department of Mathematics, University of Piteşti, Romania 4 TIFR-CAM, Yelahanka, Bangalore 560065, India Received 26 October 2007; accepted 11 April 2008; published online 12 May 2008 In this work, we consider low contrast periodic media and we study the dependence of the effective or homogenized tensor and the dispersion tensor in terms of the microstructure. We treat both one-dimensional structures and some laminated struc- tures in higher dimension. Interesting properties of the sign of these coefficients are found. Surprisingly, these depend on the microstructure only through the local proportion parameter, and in some cases, they do not depend on the microstructure at all. © 2008 American Institute of Physics. DOI: 10.1063/1.2919066 I. INTRODUCTION In this paper, as in Conca et al., 5 we consider periodic media with a small period , and once again, we are interested in Burnett coefficients. This is an important aspect in the study of the acoustic wave propagation in such media since various physical constants associated with wave propagation such as reflection, refraction, transmission, dispersion coefficients are included in Burnett coefficients. Inside the class of periodic media, the ones with low or weakly contrast are extremely important see Milton, 8 and Torquato 12 and we deal with them here. Roughly speaking, low contrast media consist of mixture of two homogeneous materials whose propagating proper- ties are close to each other. One of them is considered a fixed background medium and the other one is a small perturbation from the first one. Let denote the volume proportion of the pertur- bation in the background medium. It is well known that the low energy acoustic wave propagation in the effective or homog- enized medium provides a good approximation to the propagation of sufficiently long waves in the original periodic medium. If the waves are short, then the above approximation is poor and we need a refined approximation of the periodic medium. As shown in earlier works of Conca et al., 4,5 this can be obtained from the asymptotic expansion of the first Bloch eigenvalue associated with the periodic medium of unit period near the wave number =0, 1 = 1 2! 1 2 0 2 + 1 4! 1 4 0 4 + ¯ . Let us give some details. First of all, we have the following expansion for the first Bloch eigen- value 1 of the -periodic medium in terms of the corresponding wave number = -1 , a Electronic mail: [email protected]. b Electronic mail: [email protected]. c Author to whom correspondence should be addressed. Electronic mail: [email protected]. d Electronic mail: [email protected]. JOURNAL OF MATHEMATICAL PHYSICS 49, 053514 2008 49, 053514-1 0022-2488/2008/495/053514/23/$23.00 © 2008 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

Transcript of On Burnett coefficients in periodic media in low contrast regime

On Burnett coefficients in periodic media in low contrastregime

C. Conca,1,2,a� J. San Martín,1,2,b� L. Smaranda,2,3,c� and M. Vanninathan4,d�

1Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas yMatemáticas, Universidad de Chile2Centro de Modelamiento Matemático, UMR 2071 CNRS-UChile, Casilla 170/3, Correo 3,Santiago 8370459, Chile3Department of Mathematics, University of Piteşti, Romania4TIFR-CAM, Yelahanka, Bangalore 560065, India

�Received 26 October 2007; accepted 11 April 2008; published online 12 May 2008�

In this work, we consider low contrast periodic media and we study the dependenceof the effective or homogenized tensor and the dispersion tensor in terms of themicrostructure. We treat both one-dimensional structures and some laminated struc-tures in higher dimension. Interesting properties of the sign of these coefficients arefound. Surprisingly, these depend on the microstructure only through the localproportion parameter, and in some cases, they do not depend on the microstructureat all. © 2008 American Institute of Physics. �DOI: 10.1063/1.2919066�

I. INTRODUCTION

In this paper, as in Conca et al.,5 we consider periodic media with a small period �, and onceagain, we are interested in Burnett coefficients. This is an important aspect in the study of theacoustic wave propagation in such media since various physical constants associated with wavepropagation �such as reflection, refraction, transmission, dispersion coefficients� are included inBurnett coefficients. Inside the class of periodic media, the ones with low �or weakly� contrast areextremely important �see Milton,8 and Torquato12� and we deal with them here. Roughly speaking,low contrast media consist of mixture of two homogeneous materials whose propagating proper-ties are close to each other. One of them is considered a fixed background medium and the otherone is a small perturbation from the first one. Let � denote the volume proportion of the pertur-bation in the background medium.

It is well known that the �low energy� acoustic wave propagation in the effective �or homog-enized� medium provides a good approximation to the propagation of sufficiently long waves inthe original periodic medium. If the waves are short, then the above approximation is poor and weneed a refined approximation of the periodic medium. As shown in earlier works of Conca et al.,4,5

this can be obtained from the asymptotic expansion of the first Bloch eigenvalue associated withthe periodic medium of unit period near the wave number �=0,

�1��� =1

2!�1

�2��0��2 +1

4!�1

�4��0��4 + ¯ .

Let us give some details. First of all, we have the following expansion for the first Bloch eigen-value �1

���� of the �-periodic medium in terms of the corresponding wave number �=�−1�,

a�Electronic mail: [email protected]�Electronic mail: [email protected]�Author to whom correspondence should be addressed. Electronic mail: [email protected]�Electronic mail: [email protected].

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�1���� = �−2�1���� =

1

2!�1

�2��0��2 +1

4!�2�1

�4��0��4 + ¯ . �1.1�

The first term is alone important for waves with wave number � such that �2���4=o�1� as �→0.We need to include the second term too for waves, such that �2���4=O�1�, as �→0, but �4���6=o�1�, as �→0. The first term in the series expansion �1.1� involves the homogenized matrix12�1

�2��0� which is well known to be positive definite. The second term involves the tensor�1

�4��0�, which was referred to as dispersion tensor associated with the periodic medium and whichwas surprisingly shown to be negative semidefinite �see Ref. 5�. Some other examples where thehigher-order terms play a significant role are presented, for instance, in Bakhvalov andPanasenko,1 and Cherednichenko and Smyshlyaev.3

Let us now assume that the periodic medium consists of two-phase material with low contrast.More precisely, let �0�0 be the constant coefficient representing the background isotropic homo-geneous medium and �1�0 be the corresponding coefficient for the perturbed medium. Weassume the relation

�1 = �1 + ���0,

with ��R, ���1 denoting the contrast parameter.Our interest in this work is to study the dependence of the homogenized matrix and the

negative dispersion tensor as a function of �. Indeed, we will expand them as a power series in �and give explicit expressions for the coefficients of various terms of the expansion which in turnyield necessary and sufficient conditions for their signs.

Even though the coefficients themselves depend on the microstructure as expected, their signs,somewhat surprisingly, depend only on the volume proportion parameter �. The significance of thecoefficients is pretty obvious: they represent the contributions of the microstructure at variousorders of �. Though, in principle, we can deal with all the coefficients appearing in the expansion,we treat only the first four of them since they are of interest to engineers �see Ref. 12, p. 526�.

By proposing the following expansions for the first two terms of �1.1� with respect to �,

1

2!�1

�2��0��2 = �k=0

�kq�k����2

and

1

4!�1

�4��0��4 = �k=0

�kd�k����4,

we establish in this paper the following inequalities:

q�0� � 0, q�1� � 0, q�2� � 0, q�3� � 0, q�4� � 0, �1.2�

d�0� = 0, d�1� = 0, d�2� � 0, �1.3�

d�3� � 0 if and only if � �23 , d�4� � 0 if and only if � �

12 . �1.4�

The result d�0�=0 can be easily understood because for �=0, the medium is homogeneous and,hence, the dispersion coefficient d is zero. Similarly, we can understand that q�0��0 because thehomogenized matrix q is positive definite whatever be the microstructure.

One can understand that d�1�=0 because d has a structure of “perfect square” �see the repre-sentation �3.4� below�. By using analogous reasons, one can easily explain that q�1� is positive �seethe representation �3.1��. The inequality d�2��0 can be understood because it is the first nonzerocoefficient of the dispersion, which is negative irrespective of the microstructure. The sign of q�2�

can also be understood if one admits the connection between q�2� and H-measure in the periodic

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medium �see Ref. 8, p. 294�. Indeed, the inequality q�2��0 is connected with the fact thatH-measure is positive. At this point, it may be worthwhile to recall that the sign of the H-measureis crucially used in the H-measure approach for obtaining bounds on the homogenized coefficients�see Tartar,11 p. 225�. It is a surprise to note that the coefficients in �1.2� �starting from �2� changetheir signs in an alternating pattern. We are not able to provide an explanation to this phenomenonfrom a physical point of view.

Our arguments below will show that the inequalities �1.2� and �1.3� hold irrespective ofdimensions and without any hypothesis on �. On the other hand, we firstly prove the inequalities�1.4� in one dimension, and secondly in higher dimensions, but with coefficients varying only inone direction �under the so-called laminated microstructure hypothesis�. Since these coefficientsdepend on the microstructure, so do their signs. Our finding in this paper is that this dependenceis only through the local proportion parameter �. It is worth remarking that this parameter plays acrucial role in various optimal design problems involving microstructures �see Ref. 8 and Muratand Tartar9�. When �=0, it is easy to see that d and, hence, d�3� and d�4� vanish. It is a surprise toobserve that as soon as � is positive and small, the coefficients d�3� and d�4� pick up opposite signs.

Results analogous to �1.4� in higher dimensions are open. However, we study laminatedstructures in Sec. VI. More precisely, two examples referred to as longitudinal and orthogonalcases are treated with the techniques of this paper. �General case seems to be out of the scope ofthis work�. We find that the longitudinal case is very similar to the one dimensional case �see Sec.VI B�. However, the orthogonal case �Sec. VI C� very differently behaves. At the level of thehomogenized coefficient, we find q�j�=0 for all j�2. At the level of the dispersion coefficient, ourcomputations show some universal characteristics of the sign of the coefficients d�0�, d�1�, d�2�, d�3�,and d�4� which are independent of theta and the microstructure. This is a big surprise compared to�1.3� and �1.4�.

II. PRELIMINARIES

Let us now introduce some notations adapted in this work. We consider the operator

A =def

−d

dy���y�

d

dy�, y � R , �2.1�

where the coefficient satisfies

� � L#�Y� with Y = �0,2 �,

i.e., � = ��y� is a Y-periodic bounded measurable function defined on R

and given by ��y� = �0�TC�y� + �1�T�y�, y � Y ,

�2.2�

where we recall that �1= �1+���0. Here, T is a measurable subset of Y and �T�y� denotes thecharacteristic function of T. Further, we introduce �= �T� / �Y� and the bilinear form associated withthe previous operator,

a�u,v� = Y

��y�du

dy�y�

dvdy

�y�dy .

For each ��0, we also consider the �-periodic operator A� defined by

A� =def

−d

dx����x�

d

dx� with ���x� =

def

�� x

��, x � R . �2.3�

In homogenization theory, it is usual to refer to x and y the slow and the fast variables, respec-tively. They are related by y=x /�.

Our results are based on Bloch waves � associated with the operator A which we define now.Let us consider the following spectral problem parametrized by ��R: find �=�����R and �=��y ;�� �not identically zero�, such that

053514-3 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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�A��· ;�� = ������· ;�� in R, ��· ;�� is ��;Y�-periodic,i.e.,

��y + 2 m;�� = e2 im���y ;�� ∀ m � Z, y � R .�2.4�

Next, by the Floquet theory, we define ��y ;��=e−iy���y ;��, and �2.4� can be rewritten in terms of� as follows:

A���� = �� in R, � is Y-periodic. �2.5�

Here, the operator A��� is called the translated operator and is defined by

A��� = e−iy�Aeiy�.

It is well known �see Bensoussan et al.2 and Conca et al.6� that for each ��Y�=�− 12 , 1

2 �, the abovespectral problem �2.5� admits a discrete sequence of eigenvalues �m���. Their associated eigen-functions �m�y ;�� �referred to as Bloch waves� enable us to describe the spectral resolution of A�an unbounded self-adjoint operator in L2�R�� in the orthogonal basis �eiy��m�y ;�� �m�1,��Y� .

Let us introduce Bloch waves at the �-scale,

�m� ��� = �−2�m���, �m

� �x;�� = �m�y ;��, �m� �x;�� = �m�y ;�� ,

where the variables �x ,�� and �y ,�� are related by y=x /� and �=��. Observe that �m� �x ;�� is

�Y-periodic �in x� and �−1Y� periodic with respect to �. In the same manner, �m� �· ;�� is ��� ;�Y�

periodic because of the relation �m� �x ;��=eix��m

� �x ;��. Note that the dual cell at �-scale is �−1Y�,and hence we take � to vary in �−1Y� in the sequel.

We consider a sequence u� bounded in H1�R� and f �L2�R� satisfying

A�u� = f in R . �2.6�

We assume that u�⇀u weakly in H1�R�. The homogenization problem consists of passing to the

limit, as �→0, in the previous equation and obtain the equation satisfied by u, namely,

Qu � − qd2u

dx2 = f in R , �2.7�

where q is a constant defined by 12�1

�2��0� �see Ref. 4�.

III. HOMOGENIZED COEFFICIENT AND NEGATIVE DISPERSION COEFFICIENT

The purpose of this section is to present expressions for the homogenized matrix and thenegative dispersion coefficient which will in turn be used later.

These representations will be in terms of test functions to be introduced below. This givesphysical space interpretation of these coefficients which are defined in terms of the derivatives ofthe first Bloch eigenvalue at �=0. These derivatives of the first eigenvalue and eigenfunction at�=0 exist thanks to the regularity property established in Conca and Vanninathan.7 In fact, weknow that there exists �0�0 such that the first eigenvalue �1��� is an analytic function on B�

= �� � �� � ��0 , and there is a choice of the first eigenvector �1�y ;�� satisfying

� � �1�· ;�� � H#1�Y� is analytic on B�0

, �1�y ;0� = �Y�−1/2 =1

�2 �N/2 .

Apart from the above result of regularity on the Bloch spectrum, the following propositionwas also proven in Ref. 4.

Proposition 3.1: We have the relations

�1�0� = 0, �1�1��0� = 0,

1

2!�1

�2��0� = q,1

4!�1

�4��0� = d ,

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where q is the homogenized coefficient defined by

q =def 1

�Y�a�X�1� + y,X�1� + y� , �3.1�

with test function X�1� defined by the following cell problem:

AX�1� =d�

dyin R ,

X�1� � H#1�Y�, MY�X�1�� =

def 1

�Y�Y

X�1��y�dy = 0,�3.2�

or equivalently

a�X�1� + y,v� = 0 ∀ v � H#1�Y� . �3.3�

Next, d is the negative dispersion coefficient which admits the following representation:

d =def

−1

�Y�a�X�2� −

1

2�X�1��2, X�2� −

1

2�X�1��2� , �3.4�

where X�2� is the solution of the problem

AX�2� = �� − q� − CX�1�. �3.5�

Here, the operator C is defined by

�Cv��y� =def

− ��y�dvdy

�y� −d

dy���y�v�y�� . �3.6�

The weak formulation of the problem �3.5� is as follows:

a�X�2�,v� = Y

�� − q�v − c�X�1�,v� ∀ v � H#1�Y� , �3.7�

where the bilinear form c�· , · � associated with the operator C is given

c�u,v� = − Y

�du

dyv +

Y

�udvdy

.

IV. EXPANSIONS IN TERMS OF THE CONTRAST PARAMETER

In this section, we give the expansion of a�· , · �, c�· , · �, X�1�, X�2�, q, and d in terms of powerseries in the contrast parameter �.

A. Expansion of a„· , · …

By taking into account the dependence of ��y� on �, we see that the bilinear form a�· , · � canbe written as

a�u,v� = a�0��u,v� + �a�1��u,v� ,

where

053514-5 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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a�0��u,v� = �0Y

du

dy

dvdy

and

a�1��u,v� = �0T

du

dy

dvdy

.

B. Expansion of c„· , · …

In the same manner, we have that the bilinear form c�· , · � can be written as

c�u,v� = c�0��u,v� + �c�1��u,v� ,

where

c�0��u,v� = − �0Y

du

dyv + �0

Y

udvdy

,

c�1��u,v� = − �0T

du

dyv + �0

T

udvdy

.

C. Expansion of X„1…

By proposing the ansatz

X�1� = X�1��0� + �X�1�

�1� + �2X�1��2� + �3X�1�

�3� + �4X�1��4� + ¯

and recalling that X�1� satisfies Eq. �3.3� and finally identifying the various power of �, we arriveat the following equations:

a�0��X�1��0�,v� + a�0��y,v� = 0 ∀ v � H#

1�Y� , �4.1�

a�0��X�1��1�,v� + a�1��X�1�

�0�,v� + a�1��y,v� = 0 ∀ v � H#1�Y� , �4.2�

a�0��X�1��2�,v� + a�1��X�1�

�1�,v� = 0 ∀ v � H#1�Y� , �4.3�

a�0��X�1��3�,v� + a�1��X�1�

�2�,v� = 0 ∀ v � H#1�Y� , �4.4�

a�0��X�1��4�,v� + a�1��X�1�

�3�,v� = 0 ∀ v � H#1�Y� . �4.5�

It is easy to see that each of the above equations has a unique solution X�1��j� in H#

1�Y� with the

property MY�X�1��j� �=0 ∀j� �0,1 ,2 ,3 ,4 .

D. Expansion of q

By using the expansion of X�1� obtained above, it is straightforward to get an expansion for q.More precisely, writing

q = q�0� + �q�1� + �2q�2� + �3q�3� + �4q�4� + ¯

and recalling the representation �3.1� for q, we get that

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q�0� =1

�Y�a�0��X�1�

�0� + y,X�1��0� + y� , �4.6�

q�1� =1

�Y��2a�0��X�1�

�1�,X�1��0� + y� + a�1��X�1�

�0� + y,X�1��0� + y�� , �4.7�

q�2� =1

�Y��2a�0��X�1�

�2�,X�1��0� + y� + 2a�1��X�1�

�1�,X�1��0� + y� + a�0��X�1�

�1�,X�1��1��� , �4.8�

q�3� =1

�Y��2a�0��X�1�

�3�,X�1��0� + y� + 2a�0��X�1�

�2�,X�1��1�� + 2a�1��X�1�

�2�,X�1��0� + y� + a�1��X�1�

�1�,X�1��1��� ,

�4.9�

q�4� =1

�Y��2a�0��X�1�

�4�,X�1��0� + y� + 2a�0��X�1�

�3�,X�1��1�� + 2a�0��X�1�

�2�,X�1��2�� + 2a�1��X�1�

�3�,X�1��0� + y�

+ 2a�1��X�1��2�,X�1�

�1��� . �4.10�

E. Expansion of X„2…

By following the same procedure, we propose the ansatz

X�2� = X�2��0� + �X�2�

�1� + �2X�2��2� + �3X�2�

�3� + �4X�2��4� + ¯ ,

and we find the following equations at various powers of � : ∀v�H#1�Y�:

a�0��X�2��0�,v� =

Y

��0 − q�0��v − c�0��X�1��0�,v� , �4.11�

a�0��X�2��1�,v� + a�1��X�2�

�0�,v� = �0T

v − q�1�Y

v − c�0��X�1��1�,v� − c�1��X�1�

�0�,v� , �4.12�

a�0��X�2��2�,v� + a�1��X�2�

�1�,v� = − q�2�Y

v − c�0��X�1��2�,v� − c�1��X�1�

�1�,v� , �4.13�

a�0��X�2��3�,v� + a�1��X�2�

�2�,v� = − q�3�Y

v − c�0��X�1��3�,v� − c�1��X�1�

�2�,v� , �4.14�

a�0��X�2��4�,v� + a�1��X�2�

�3�,v� = − q�4�Y

v − c�0��X�1��4�,v� − c�1��X�1�

�3�,v� . �4.15�

It is easy to see that each of the above equations has a unique solution X�2��j� in H#

1�Y� with the

property MY�X�2��j� �=0 ∀j� �0,1 ,2 ,3 ,4 .

F. Expansion of d

By calling Z=X�2�−12 �X�1��2, we rewrite the representation formula �3.4� as follows:

053514-7 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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d = −1

�Y�a�Z,Z� .

Now, we expand Z=Z�0�+�Z�1�+�2Z�2�+�3Z�3�+¯, then since

�X�1��2 = �X�1��0��2 + 2�X�1�

�0�X�1��1� + �2�2X�1�

�0�X�1��2� + �X�1�

�1��2� + ¯ , �4.16�

we get

Z�0� = X�2��0� − 1

2 �X�1��0��2, Z�1� = X�2�

�1� − X�1��0�X�1�

�1�,

Z�2� = X�2��2� − 1

2 �2X�1��0�X�1�

�2� + �X�1��1��2�, Z�3� = X�2�

�3� − �X�1��0�X�1�

�3� + X�1��1�X�1�

�2�� .

We expand

d = d�0� + �d�1� + �2d�2� + �3d�3� + �4d�4� + ¯ .

With these notations, we get the following expressions for d�j�, j� �0,1 ,2 ,3 ,4 :

d�0� = −1

�Y�a�0��Z�0�,Z�0�� , �4.17�

d�1� = −1

�Y��2a�0��Z�0�,Z�1�� + a�1��Z�0�,Z�0��� , �4.18�

d�2� = −1

�Y��2a�0��Z�0�,Z�2�� + a�0��Z�1�,Z�1�� + 2a�1��Z�0�,Z�1��� , �4.19�

d�3� = −1

�Y��2a�0��Z�0�,Z�3�� + 2a�0��Z�1�,Z�2�� + 2a�1��Z�0�,Z�2�� + a�1��Z�1�,Z�1��� , �4.20�

d�4� = −1

�Y��2a�0��Z�0�,Z�4�� + 2a�0��Z�1�,Z�3�� + a�0��Z�2�,Z�2�� + 2a�1��Z�0�,Z�3�� + 2a�1��Z�1�,Z�2��� .

�4.21�

V. THE MAIN RESULT

The aim of this section is to prove the inequalities �1.2�–�1.4�.

A. Proof of the inequalities „1.2…

Since a�0��y ,v�=0 for all v�H#1�Y�, we can easily deduce from Eq. �4.1� that X�1�

�0� is identi-

cally zero. We use this kind of argument in the various representation formulas of Sec. IV D. Wehave

q�0� =1

�Y�a�0��y,y� = �0 � 0,

and since X�1��1� is Y-periodic,

053514-8 Conca et al. J. Math. Phys. 49, 053514 �2008�

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q�1� =1

�Y��2a�0��X�1�

�1�,y� + a�1��y,y�� = �0� � 0.

Moreover, by using Eq. �4.2� with the function X�1��1�, we get that

q�2� = −1

�Y�a�0��X�1�

�1�,X�1��1�� � 0.

Next, in the formula �4.9�, we use Eq. �4.3� with test function X�1��1� to deduce that

q�3� =1

�Y�a�1��X�1�

�1�,X�1��1�� � 0.

Next, in the formula �4.10�, we use Eq. �4.3� with test function X�1��2� and Eq. �4.2� with test function

X�1��3�, then we deduce

q�4� = −1

�Y�a�0��X�1�

�2�,X�1��2�� � 0.

The above procedure is very general and shows that the signs of the coefficients q�j� with j�2 alternate in any dimension.

B. Some preliminary lemmas

In this subsection, we establish some crucial relations which use the one-dimensional natureof the problem.

Lemma 5.1: We have the relations

��y�dX�1�

dy�y� = q − ��y� , �5.1�

dX�2�

dy= − X�1�. �5.2�

Proof: By integrating Eq. �3.2�, we get the relation

��y�dX�1�

dy�y� = − ��y� + � ,

for some constant ��R. This constant can be obtained by taking the average of the previousrelation over the periodic cell Y, and the result is that � is equal to the homogenized coefficient q.Then, we get the relation �5.1�.

By using the relation �5.1� in Eq. �3.5� and integrating, we obtain

�dX�2�

dy= − �X�1� + � ,

for some constant �. By dividing by � and taking average over the periodic cell Y, we get �=0and hence the required relation is obtained. �

By identifying the coefficients of � j in the relation �5.2� stated in Lemma 5.1, we obtain thefollowing result.

053514-9 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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Corollary 5.2:

dX�2��j�

dy= − X�1�

�j� ∀ j = 0,1, . . . .

Corollary 5.3: We have the following identities:

T

X�2��1� = −

Y

�X�1��1��2

and

TC

X�2��1� =

Y

�X�1��1��2.

Proof: In Eq. �4.12�, we choose the test function X�2��1� and we get

a�0��X�2��1�,X�2�

�1�� = �0T

X�2��1� − c�0��X�1�

�1�,X�2��1�� .

By using the identity �5.2�, we get the first required result. The second result is a direct conse-quence of the fact that the average of X�2�

�1� is zero over Y. �

Corollary 5.4: The following identity holds:

dX�1��1�

dy= �� in TC

− �1 − �� in T .�

Proof: By identifying the coefficient of � in the relation �5.1� stated in Lemma 5.1, we deducethat

�0

dX�1��1�

dy+ �0�T

dX�1��0�

dy= q�1� − �0�T.

By using the results that X�1��0�=0 and q�1�=�0�, we obtain the desired identity. �

Lemma 5.5: The following recurrence formula holds:

X�1��j+1� = − �1 − ��X�1�

�j� ∀ j = 1,2, . . . . �5.3�

Proof: In order to prove the relation for j=1, first we consider Eq. �4.2�, which can berewritten as follows:

Y

dX�1��1�

dy

dvdy

+ Y

�Tdvdy

= 0 ∀ v � H#1�Y� .

Now, we consider Eq. �4.3� which can be rewritten as follows:

Y

dX�1��2�

dy

dvdy

+ Y

�T

dX�1��1�

dy

dvdy

= 0 ∀ v � H#1�Y� ,

then, by using Corollary 5.4, we get

Y

dX�1��2�

dy

dvdy

− �1 − ��Y

�Tdvdy

= 0 ∀ v � H#1�Y� .

Therefore, we obtain the recurrence relation for j=1, that is,

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X�1��2� = − �1 − ��X�1�

�1�.

Let us now assume that the relation �5.3� is true for some j. Let us prove that this relationremains true for j+1. To this end, we write the equations for X�1�

�j+1� and X�1��j+2� as follows:

Y

dX�1��j+1�

dy

dvdy

+ Y

�T

dX�1��j�

dy

dvdy

= 0 ∀ v � H#1�Y� �5.4�

and

Y

dX�1��j+2�

dy

dvdy

+ Y

�T

dX�1��j+1�

dy

dvdy

= 0 ∀ v � H#1�Y� .

By using the induction hypothesis, we have that

dX�1��j+1�

dy= − �1 − ��

dX�1��j�

dy,

then

Y

dX�1��j+2�

dy

dvdy

− �1 − ��Y

�T

dX�1��j�

dy

dvdy

= 0 ∀ v � H#1�Y� .

This relation and �5.4� imply that

X�1��j+2� = − �1 − ��X�1�

�j+1�.

As direct consequences of Corollary 5.4 and Lemma 5.5, we get the following result.Corollary 5.6: For any j=1,2 , . . ., we have

X�1��j� = �− 1� j−1�1 − �� j−1X�1�

�1�

and

dX�1��j�

dy= ��− 1� j−1��1 − �� j−1 in TC

�− 1� j�1 − �� j in T .�

Remark 5.7: By using Corollary 5.6, one can give the following expressions for the coefficientsq�2�, q�3�, and q�4�:

q�2� = − �0��1 − ��, q�3� = �0��1 − ��2, q�4� = − �0��1 − ��3. �5.5�

Lemma 5.8: We have that

T

�X�1��1��2 = �

Y

�X�1��1��2

and

TC

�X�1��1��2 = �1 − ��

Y

�X�1��1��2.

Proof: Equation �4.2� can be rewritten as follows:

053514-11 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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Y

dX�1��1�

dy

dvdy

+ Y

�Tdvdy

= 0 ∀ v � H#1�Y� .

If we consider the test function v defined by

dvdy

= �X�1��1��2 −

1

�Y�Y

�X�1��1��2 in R ,

v � H#1�Y�,

1

�Y�Yv = 0,

then the above equation becomes

Y

d

dy� 1

3 �X�1��1��3� −

1

�Y�Y

�X�1��1��2 ·

Y

dX�1��1�

dy+

Y

�T�X�1��1��2 −

�T��Y�Y

�X�1��1��2 = 0.

Since X�1��1� is Y-periodic, we obtain the first required result.

The second identity of Lemma is a direct consequence of the first one. �

C. Proof of the inequalities „1.3…

By using the facts that X�1��0�=0 and q�0�=�0 in Eq. �4.11�, we easily deduce that X�2�

�0�=0. Then

follows that

Z�0� = 0, �5.6�

and consequently d�0�=d�1�=0. Moreover, the expression �4.19� simplifies to

d�2� = −1

�Y�a�0��Z�1�,Z�1�� � 0.

Further, by using Corollary 5.2, we get the following expression of the coefficient d�2� in term ofX�1�

�1�:

d�2� = −�0

�Y�Y

�X�1��1��2. �5.7�

D. Proof of the inequalities „1.4…

1. Expression for d„3…

First of all, if we use the fact that Z�0�=0, then the expression �4.20� of d�3� is reduced to thefollowing one:

d�3� = −1

�Y��2a�0��Z�1�,Z�2�� + a�1��Z�1�,Z�1��� .

The first term of the previous identity contains −�1 / �Y��a�0��Z�1� , �X�1��1��2� which we call nonlinear

term. We will now show that this term vanishes. To this end, we note that Z�1�=X�2��1� and we use

Corollary 5.2 to get that

−1

�Y�a�0��Z�1�,�X�1�

�1��2� =�0

�Y�Y

X�1��1� d

dy�X�1�

�1��2 = 0.

Consequently, d�3� can be written as

053514-12 Conca et al. J. Math. Phys. 49, 053514 �2008�

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d�3� = −1

�Y��2a�0��X�2�

�1�,X�2��2�� + a�1��X�2�

�1�,X�2��1��� .

By using Eq. �4.13� with test function X�2��1�, the above expression is reduced to

d�3� =1

�Y�a�1��X�2�

�1�,X�2��1�� +

2

�Y��c�0��X�2�

�1�,X�1��2�� + c�1��X�1�

�1�,X�2��1��� . �5.8�

By using Corollaries 5.2, 5.3, and 5.6, we get

c�0��X�2��1�,X�1�

�2�� = − 2�0�1 − ��2T

X�2��1� + 2�0��1 − ��

TCX�2�

�1� = 2�0�1 − ��Y

�X�1��1��2

and

c�1��X�1��1�,X�2�

�1�� = �0�1 − ��T

X�2��1� − �0

T

�X�1��1��2 = − �0�1 − ��

Y

�X�1��1��2 − �0

T

�X�1��1��2.

By assembling all these relations in �5.8� and using again Corollary 5.2, we obtain

d�3� =�0

�Y��1 − 2��

T

�X�1��1��2 +

2�0

�Y��1 − ��

TC�X�1�

�1��2.

By using Lemma 5.8, the above identity gives the following expression:

d�3� =�0

�Y��2 − 3��

Y

�X�1��1��2. �5.9�

Then, it easily follows that d�3��0 if and only if ��23 .

2. Expression for d„4…

By using the fact that X�1��0�=X�2�

�0�=0 in �4.21�, we get that

d�4� = −1

�Y��2a�0��X�2��3� − X�1�

�1�X�1��2�,X�2�

�1�� + a�0��X�2��2� −

1

2�X�1�

�1��2,X�2��2� −

1

2�X�1�

�1��2�+ 2a�1��X�2�

�2� −1

2�X�1�

�1��2,X�2��1��� .

First nonlinear term in d�4�.

2a�0��X�1��1�X�1�

�2�,X�2��1�� = 2�0

Y�dX�1�

�1�

dyX�1�

�2� + X�1��1�

dX�1��2�

dy�dX�2�

�1�

dy.

By invoking Corollary 5.2 and integrating by parts, we get

2a�0��X�1��1�X�1�

�2�,X�2��1�� = − �0

Y

�X�1��1��2

dX�1��2�

dy.

We use again Corollary 5.6 and Lemma 5.8 to deduce that this first nonlinear term is zero.Second nonlinear term in d�4�. An application of Corollary 5.2 gives that

053514-13 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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a�0��X�2��2�,�X�1�

�1��2� = − �0Y

X�1��2� d

dy��X�1�

�1��2� .

Analogous arguments as in the first nonlinear term will show that this second nonlinear term isalso zero.

Third nonlinear term in d�4�. We have

− 14a�0���X�1�

�1��2,�X�1��1��2� = − �0

Y

�X�1��1��2�dX�1�

�1�

dy�2

.

We use Corollary 5.6 and Lemma 5.8 and we obtain

− 14a�0���X�1�

�1��2,�X�1��1��2� = − �0��1 − ��

Y

�X�1��1��2.

Fourth nonlinear term in d�4�. Straightforward computations as before, we get

a�1���X�1��1��2,X�2�

�1�� = 2�0��1 − ��Y

�X�1��1��2.

By using the above information, we deduce that

d�4� =1

�Y��− 2a�0��X�2��3�,X�2�

�1�� − a�0��X�2��2�,X�2�

�2�� − 2a�1��X�2��2�,X�2�

�1�� + �0��1 − ��Y

�X�1��1��2� .

We use �4.14� with test function X�2��1�, then we deduce that

d�4� =1

�Y��2�c�0��X�1��3�,X�2�

�1�� + c�1��X�1��2�,X�2�

�1��� − a�0��X�2��2�,X�2�

�2�� + �0��1 − ��Y

�X�1��1��2� .

�5.10�

Now, we are going to concentrate on the first term of the previous expression. By using thedefinitions of c�0��· , · � and c�1��· , · �, and Corollaries 5.2 and 5.6, we have

c�0��X�1��3�,X�2�

�1�� = − 2�0Y

dX�1��3�

dyX�2�

�1� = − 2�0�− �1 − ��3T

X�2��1� + ��1 − ��2

TCX�2�

�1�� ,

c�1��X�1��2�,X�2�

�1�� = − �0�1 − ��2T

X�2��1� − �0

T

X�1��2�X�1�

�1�.

By using Corollary 5.3, we get that

c�0��X�1��3�,X�2�

�1�� = − 2�0�1 − ��2Y

�X�1��1��2, �5.11�

c�1��X�1��2�,X�2�

�1�� = �0�1 − ��2Y

�X�1��1��2 − �0

T

X�1��2�X�1�

�1�.

Finally, due to Corollary 5.6, we obtain

c�1��X�1��2�,X�2�

�1�� = �0�1 − ��Y

�X�1��1��2. �5.12�

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By using the identities �5.11�, �5.12�, and �5.10�, we get

d�4� =1

�Y��− a�0��X�2��2�,X�2�

�2�� − �0�1 − ���2 − 5��Y

�X�1��1��2� .

In order to compute a�0��X�2��2� ,X�2�

�2��, we use Corollaries 5.2 and 5.6, then

a�0��X�2��2�,X�2�

�2�� = �0�1 − ��2Y

�X�1��1��2.

With this, we obtain the following expression:

d�4� = −3�0

�Y��1 − ���1 − 2��

Y

�X�1��1��2. �5.13�

Therefore, we conclude that d�4��0 if and only if ��12 .

Remark 5.9: The expressions of the coefficients d�i�, i� �2,3 ,4 , given in (5.7), (5.9), and(5.13) depend on the microstructure through the integral �Y�X�1�

�1��2. One can give explicit formulas

for these coefficients in some particular cases. For instance, for a given n�N*, if we consider amultilayered mixture of the two phases, that is, T=�k=0

n−1k /n�Y� , ��k+�� /n�Y��, then

Y

�X�1��1��2 =

�Y�3

12n2�2�1 − ��2,

and hence,

d�2� = −�0

12n2 �Y�2�2�1 − ��2, d�3� =�0

12n2 �Y�2�2�1 − ��2�2 − 3�� ,

d�4� = −�0

4n2 �Y�2�2�1 − ��3�1 − 2�� .

The above computations show that the coefficients decrease and tend to zero as the number ofpieces becomes large. This is a well known phenomenon in the literature known under the name“size effect” on the homogenized coefficient. The dispersion coefficient too enjoys similar prop-erty.

VI. LAMINATED STRUCTURES

One generalization of the one-dimensional setting corresponds to the so-called laminatedcomposite materials. These kinds of materials have different behaviors depending on the directionof wave number �. The purpose of this section is to show that if � is parallel to the direction oflamination, the treatment of the sign of the dispersion coefficient is completely analogous to theone-dimensional case. In addition, we study the case where � is orthogonal to the direction oflamination in isotropic laminated materials. The treatment of the sign of the dispersion coefficientin this particular laminated material is slightly easy, but different from the one-dimensional treat-ment. Laminated materials in the general case are not covered here as their treatment requiresdifferent techniques.

A. Setting of the problem

Let us denote by A the divergence-type operator in the N-dimensional case, which is analo-gous to the operator A defined in �2.1� for the one-dimensional case, that is,

053514-15 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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A =def

−�

�yk��kl�y�

�yl�, y � RN, �6.1�

where the coefficient satisfies

�kl � L#�Y� with Y = �0,2 �N, �kl = �lk ∀ k,l = 1, . . . ,N

and

∃C � 0 such that �kl�y��k�l � C���2 ∀ � � RN, y � Y a.e.

The corresponding bilinear form is denoted by a�· , · � and is given by

a�u,v� = Y

�kl�y��u

�yk�y�

�v�yl

�y�dy .

Let us now recall the homogenized and dispersion coefficients in the N-dimensional case,which depend on the parameter ��RN �see Ref. 5�,

qkl�k�l =1

�Y�a���1� + y · �,��1� + y · �� , �6.2�

dklmn�k�l�m�n = −1

�Y�a���2� −

1

2���1��2,��2� −

1

2���1��2� , �6.3�

where the test functions ��1� and ��2� are defined by the following cell problems:

A��1� = �k��kl

�ylin RN,

��1� � H#1�Y�,

1

�Y�

Y

��1��y�dy = 0,�6.4�

A��2� = ��kl − qkl��k�l − C��1� in RN,

��2� � H#1�Y�,

1

�Y�

Y

��2��y�dy = 0.�6.5�

Here, the operator C is defined by

�Cv��y� =def

− �k�kl�y��v�yl

�y� − �k�

�yl��kl�y�v�y�� .

If we now consider laminated composite materials, the geometry of the problem varies only ina single direction. In other words, the matrix �kl depends only on a single space variable, say y1,as

�kl�y� = �kl�y1� ∀ k,l = 1, . . . ,N .

By using this hypothesis, we obtain that the test functions ��1� and ��2�, defined in �6.4� and �6.5�,are dependent only on variable y1, that is,

��1��y� = ��1��y1� ,

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��2��y� = ��2��y1� .

Therefore, the cell problem �6.4� becomes

−�

�y1��11�y1�

���1�

�y1�y1�� = �k

��k1

�y1�y1� in R ,

��1� � H#1�Y�,

1

�Y�Y

��1��y1�dy1 = 0,

�6.6�

where Y is the reference cell defined in �2.2�. It is clear that the solution of this equation is

��1� = �kX�1�,k, �6.7�

where X�1�,k is solution of the following problem:

−�

�Y1��11�Y1�

�X�1�,K

�Y1�Y1�� =

��K1

�Y1�Y1� IN R ,

X�1�,k � H#1�Y�,

1

�Y�Y

X�1�,k�y1�dy1 = 0.

�6.8�

Remark 6.1: Let us observe that if k=1, then X�1�,1=X�1�, where X�1� is solution of problem(3.2).

Since �6.7� holds, then the homogenized coefficients, given in �6.2�, can be written as follows:

qkl =1

�Y�a�X�1�,k + yk,X�1�,l + yl� . �6.9�

Now, by integrating equation �6.8�, we get the relation

− �11�y1��X�1�,k

�y1�y1� = �k1�y1� + K ,

for some constant K�R. This constant can be obtained by taking the average in the previousrelation over the reference cell Y, and the result is

K = −

MY��k1

�11�

MY� 1

�11� .

Then, we get

�11�y1��X�1�,k

�y1�y1� =

MY��k1

�11�

MY� 1

�11� − �k1�y1� , �6.10�

which is analogous to Eq. �5.1�.Since X�1�,k is solution of Eq. �6.10�, the identity �6.9� becomes

053514-17 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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qkl = MY��kl −�k1�l1

�11� +

MY��k1

�11�MY� �l1

�11�

MY� 1

�11� ∀ k,l = 1, . . . ,N , �6.11�

which is the classical formula of homogenized coefficients in laminated structures �see Tartar10�.Let us now give the necessary formulas to compute the dispersion coefficient of laminated

materials. First, we have that the operator C applied to functions v depending only on y1 reducesto

�Cv��y1� = − �k�k1�y1��v�y1

�y1� − �k�

�y1��k1�y1�v�y1�� =

def

�k�Ckv��y1� .

Then, the cell problem �6.5� becomes

−�

�y1��11�y1�

���2�

�y1�y1�� = ��kl − qkl��k�l − CkX�1�,l�k�l in R ,

��2� � H#1�Y�,

1

�Y�Y

��2��y1�dy1 = 0.

�6.12�

It is easy to see that the solution of this cell problem is

��2� = �k�lX�2�,kl, �6.13�

where X�2�,kl solves

−�

�y1��11�y1�

�X�2�,kl

�y1�y1�� = ��kl − qkl� − CkX�1�,l in R ,

X�2�,kl � H#1�Y�,

1

�Y�Y

X�2�,kl�y1�dy1 = 0.

�6.14�

Remark 6.2: Let us observe that if k= l=1, then X�2�,11=X�2�, where X�2� is the solution ofproblem (3.5).

Since �6.13� holds, then the dispersion coefficient, given in �6.3�, can be written as follows:

dklmn = −1

�Y�a�X�2�,kl −

1

2X�1�,kX�1�,l,X�2�,mn −

1

2X�1�,mX�1�,n� . �6.15�

Let us now integrate Eq. �6.14�. To this end, we use the definition of the operator Ck, Eq.�6.10�, and the identity �6.11� in order to rewrite �6.14� as follows:

−�

�y1��11�y1�

�X�2�,kl

�y1�y1�� =

�y1��k1X�1�,l� + ��kl −

�k1�l1

�11� − MY��kl −

�k1�l1

�11�

+ ��k1

�11− MY��k1

�11��ql1. �6.16�

Let us now define Mkl�y1�, for all k , l=1, . . .N, as solution of the following problem:

053514-18 Conca et al. J. Math. Phys. 49, 053514 �2008�

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�Mkl

�y1= ��kl −

�k1�l1

�11� − MY��kl −

�k1�l1

�11� + ��k1

�11− MY��k1

�11��ql1 in R ,

Mkl � H#1�Y�,

1

�Y�Y

Mkl�y1�dy1 = 0.�6.17�

By using this definition, we integrate Eq. �6.16� and we obtain

− �11�y1��X�2�,kl

�y1�y1� = �k1�y1�X�1�,l�y1� + Mkl�y1� + K ,

for some constant K�R. This constant is obtained by taking the average of the previous relationover Y and the result is

K = −

Y

�k1

�11X�1�,l +

Y

1

�11Mkl

Y

1

�11

.

Therefore, we have

− �11�y1��X�2�,kl

�y1�y1� = �k1�y1�X�1�,l�y1� + Mkl�y1� −

Y

�k1

�11X�1�,l +

Y

1

�11Mkl

Y

1

�11

. �6.18�

One can use Eqs. �6.10� and �6.18� in relation �6.15� in order to give a more explicit formulafor the dispersion coefficient in laminated materials. We will do it in the next two subsections fortwo particular cases.

B. Longitudinal case

Let us now consider waves such that the wave number � is parallel to the direction oflamination e1= �1,0 , . . . ,0�, that is,

� = �1e1.

By using this hypothesis, the homogenized and dispersion coefficients are reduced to thefollowing formulas:

qkl�k�l = q11�12, dklmn�k�l�m�n = d1111�1

4.

Since �6.9� holds, we have

q11 =1

�Y�a�X�1�,1 + y1,X�1�,1 + y1� , �6.19�

then, by using Remark 6.1, we obtain qkl�k�l=q�12, where q is defined in �3.1�.

On the other hand, since �6.15� holds, we get

d1111 = −1

�Y�a�X�2�,11 −

1

2�X�1�,1�2,X�2�,11 −

1

2�X�1�,1�2� , �6.20�

then, by using Remarks 6.1 and 6.2, we obtain dklmn�k�l�m�n=d�14, where d is defined in �3.4�.

053514-19 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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These two formulas �6.19� and �6.20� imply that the homogenized and dispersion coefficientsof laminated materials are equal to the corresponding coefficients in the one-dimensional case,provided the wave number is parallel to the direction of lamination. Thus, the conclusions on theirsigns remain the same as in one-dimensional case.

C. Orthogonal case

In this subsection, we study the signs of the principal terms of homogenized and Burnettcoefficients which appear in the expansions with respect to the parameter � for isotropic laminatedmaterials as the wave number � is orthogonal to the direction of lamination. In this case, theconductivity matrix and the wave number are given by

�kl�y1� = ��y1��kl ∀ k,l = 1, . . . ,N �6.21�

and

� = �k=2

N

�kek, �6.22�

where ��kl� denotes the identity matrix and � is given in �2.2�.By using hypothesis �6.21�, let us compute the test functions X�1�,k and X�2�,kl. First, Eq. �6.10�

yields

X�1�,k = �X�1� if k = 1

0 if k � 2,� �6.23�

where X�1� is solution of Eq. �3.2�. Next, Eq. �6.17� becomes

�Mkl

�y= 0 if k � l

0 if k = l = 1

� − MY��� if k = l � 2.� �6.24�

Finally, Eq. �6.18� of X�2�,kl can be rewritten as follows:

− ��y1��X�2�,kl

�y1�y1� =

0 if k � l

��y1�X�1��y1� if k = l = 1

M22�y1� −

Y

1

�M22

Y

1

if k = l � 2.� �6.25�

Let us now compute the homogenized coefficient. We use hypothesis �6.22�, Eq. �6.23� in�6.9�, and we obtain

qkl�k�l = MY������2.

Now, by taking into account the dependence of ��y1� on �, we see that the previous expressionbecomes

qkl�k�l = �0���2 + ��0����2.

This last formula gives us the same inequalities as �1.2�, with q�j�=0∀ j�2.In the sequel, we compute the dispersion coefficient. To do this, we use the hypothesis �6.22�,

the identities �6.23� and �6.25�, in Eq. �6.15� and obtain

053514-20 Conca et al. J. Math. Phys. 49, 053514 �2008�

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dklmn�k�l�m�n = −1

�Y�a�X�2�,22,X�2�,22����4.

Since �6.25� holds and X�2�,22 is a Y-periodic function, the previous expression of the dispersioncoefficient becomes

dklmn�k�l�m�n = −�MY� �M22�2

�� −

�MY�M22

���2

MY� 1

�� ����4. �6.26�

Now, since ��y1�=�0�1+��T�y1�� and MY���=�0�1+���, the solution M22 of Eq. �6.17� canbe written as

M22�y1� = − �0�M22�y1� ,

where M22 is solution of the following problem:

�M22

�y1= � − �T

in R ,

M22 � H#1�Y� , MY�M22� = 0.

� �6.27�

By using this definition, let us compute the terms appearing in the right hand side of the identity�6.26�. We have

MY� �M22�2

�� = �2�0MY� �M22�2

1 + ��T� = MY��M22�2��0�2 − MY��M22�2�T��0�3�1 − � + ¯ �

and

MY�M22

�� = − �MY� M22

1 + ��T� = − MY�M22�� + MY�M22�T��2�1 − � + �2 − ¯ �

= MY�M22�T��2

1 + �.

By using the previous formula and the fact that 1 /MY�1 /��=�0�1+�� / �1+ �1−����, we deduce

�MY�M22

���2

MY� 1

�� = �MY�M22�T��2 �0�4

�1 + ���1 + �1 − ����

= �MY�M22�T��2�0�4�1 − �2 − ��� − �1 − ���2 + ¯ � .

Therefore, formula �6.26� becomes

dklmn�k�l�m�n = ���4�− MY��M22�2��0�2 + MY��M22�2�T��0�3 − �MY��M22�2�T�

− �MY�M22�T��2��0�4 + ¯ � .

This last formula gives us the following inequalities, which are similar to �1.3� and �1.4�, with

d�0� = 0, d�1� = 0, d�2� = − �0MY��M22�2� � 0, �6.28�

053514-21 On Burnett coefficients in low contrast regime J. Math. Phys. 49, 053514 �2008�

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d�3� = �0MY��M22�2�T� � 0, d�4� = �0��MY�M22�T��2 − MY��M22�2�T�� � 0. �6.29�

In fact, in order to establish the sign of the coefficient d�4�, we use Cauchy–Schwarz’s inequalityas follows:

d�4� =�0

�Y�2��T

M22dy1�2

− �Y�T

�M22�2dy1� ��0

�Y�2��T�T

�M22�2dy1 − �Y�T

�M22�2dy1�= − �1 − ���0MY��M22�2�T� � 0.

General feature of �6.28� and �6.29� is that the signs alternate. Let us observe that, similar to�1.3�, inequalities �6.28� are independent of �. We recall that in the one-dimensional setting,inequalities �1.4� depend on the local proportion parameter, however, inequalities �6.29� are inde-pendent of �. This remark shows one important difference between the one-dimensional settingand higher dimensions.

Remark 6.3: By comparing the definition of M22 given in (6.27) with the property of theone-dimensional function X�1�

�1� from Corollary 5.4, we deduce that these two functions are the

same. Therefore, we use Lemma 5.8 in order to reduce the expressions of d�3� and d�4� given in(6.29) as follows:

d�3� =�0

�Y��

Y

�M22�2, d�4� =�0

�Y�2�T

M22�2

−�0

�Y��

Y

�M22�2.

In addition, if we consider the particular case of a multilayered mixture of the two phases, asin Remark 5.9, we have

Y

�M22�2 =�Y�3

12n2�2�1 − ��2, T

M22 = 0,

and hence,

d�2� = −�0

12n2 �Y�2�2�1 − ��2, d�3� =�0

12n2 �Y�2�3�1 − ��2, d�4� = −�0

12n2 �Y�2�3�1 − ��2.

ACKNOWLEDGMENTS

The first author �C. C.� thanks the MICDB for partial support through Grant No. ICM P05-001-F. The second one �J. S. M.� was partially supported by Grants Fondecyt 1050332 and Fondapon Applied Mathematics. The third author �L. S.� was supported by Grant Fondecyt 3070029 andfellowship of Center for Mathematical Modeling, University of Chile. The authors are grateful tothe anonymous referees for significant improvements in the revised version of the manuscript.

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