Emmanuel Frénod

Two-Scale Convergence

The concept of Two-Scale Convergence was introduced in two papers of Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Nguetseng | Asymptotic analysis for a stiff variational problem arising in mechanics[END_REF] in 1989. Then in 1992, Allaire [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF] produced a synthetic and very readable proof of the result.

How Homogenization brought the concept

The concept of Two-Scale Convergence emerged from questions of Periodic Homogenization.

Homogenization is a Mathematical Theory, or more precisely, an Asymptotic Analysis Theory that originates from Material Engineering, or more precisely, from understanding the way Constitutive Equation of composite material can be gotten from Constitutive Equation of each component of the concerned material and from their topological and geometrical distributions.

Microstructure

Material shape Figure 1.1: Composite material has a macroscopic shape and a microstructure. The ratio between the size of the microstructure and the size of the material is ε.

In order to make the purpose clear, we first consider the simplest -but rich enoughexample I know. (The explanation that follows does not aim to be mathematically rigorous. It clearly appeals to intuition and to non-rigorous vocabulary.) Imagine that we want to get the temperature field within a composite material which is in thermal equilibrium, knowing the temperature on its boundary. Symbolically, as represented in Figure 1.1 (in a bi-dimensional setting), the composite material has a macroscopic shape at a macroscopic size. Within it, heterogeneities are more or less periodically distributed with a periodicity -or a characteristic size -which is ε times smaller than its macroscopic size, where ε is a small parameter. This makes up what is usually called the microstructure of the composite material. Now, to achieve our goal, we contemplate the following Heat Equation ∇ • a ε (x, x ε )∇u ε = 0 within the material, u ε given on the boundary of the material, (1.1) that is supposed to describe how the temperature u ε is being splitted within the material from its distribution on the boundary. In this equation, a ε stands for the Thermal Diffusion Coefficient (it is the ratio Thermal Conduction over Calorific Capacity times material Density), ∇ and ∇• stand for the gradient and divergence operators. (If a unidimensional material is considered, x = x lives in R, if a bi-dimensional material is considered, x = (x, y) lives in R 2 and if a tridimensional material is considered, x = (x, y, z) lives in R 3 .)

The fact that a ε depends on x and x/ε needs to be understood in the following sense. Variable x is the dimensionless position, meaning that when used to describe the material at its macroscopic scale, the needed variations of variable x are of the order of 1. Beside this, the dependance of a ε in x/ε models the variation of the Thermal Diffusion at the microstructure scale. To illustrate this ability of x/ε-dependance to describe variations at the microscopic scale, I show the graph of some kinds of functions in one and two dimensions. In Figure (right) between -π and π.

1.2, is drawn the graph, between -π and π, of a ε (x, x/ε) = (1/2) sin(x) + 1 + ε cos(x/ε) for ε = 1/20, 1/40 and 1/80. Those functions have a variation at the macroscopic scale, which is described by the piece (1/2) sin(x)+1 and variations at much smaller scales, which are the microscopic variations. In this example, the microscopic variations can be qualified of being high frequency periodic oscillations with small amplitude. They are described by the term cos(x/ε) which needs to be multiplied by ε (explaining the presence of superscript ε in a ε ) to insure amplitude of size ε of those high frequency periodic oscillations. The next figure

(figure 1.3) shows the graph of a ε (x, x/ε) = a(x, x/ε) = (1/2) sin(x) + 1 + (1/2) cos(x/ε)
for the same values of ε as previously. Here, the macroscopic scale variation is always given by the piece (1/2) sin(x) + 1 and the variations at a smaller scale, making up the microscopic variations, are given by (1/2) cos(x/ε). (This term is not multiplied by ε, bringing the uselessness of superscript ε in a ε .) In this case the microscopic variations can be qualified of high frequency periodic oscillations with large amplitude or periodic Strong (right) between -π and π.

-4 -3 -2 -1 0 1 2 3 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 1.4: Graph of 1 2 (sin(x) + 1) cos( x ε
) for ε = 1/20 (left), 1/40 (center) and 1/80 (right) between -π and π.

Oscillations. Figure 1.4 shows the ability of a function depending on x and x/ε to describe situations with microscopic variations which are with modulated amplitude -with regions where they are Strong Oscillations and regions where they are not Strong. The drawn function is (1/2)(sin(x) + 1) cos(x/ε). In this expression, the microscopic variations, which are high frequency periodic oscillations, are described by factor cos(x/ε) and the modulated amplitude is (1/2)(sin(x) + 1). Figure 1.5 shows function

(1/2) cos(x) + 1 + (1/2)(sin(x) + -4 -3 -2 -1 0 1 2 3 4 -0.5 0 0.5 1 1.5 2 2.5 -4 -3 -2 -1 0 1 2 3 4 -0.5 0 0.5 1 1.5 2 2.5 -4 -3 -2 -1 0 1 2 3 4 -0.5 0 0.5 1 1.5 2 2.5 Figure 1.5: Graph of 1 2 cos(x) + 1 + 1 2 (sin(x) + 1) cos( x ε ) for ε = 1/20 (left), 1/40 (center)
and 1/80 (right) between -π and π.

1) cos(x/ε) for always the same values of ε. Those functions possesses both macroscopic scale variation and modulated amplitude high frequency oscillations as microscopic variations.

In every example above the microscopic scale variations are periodic. Yet, the x/ε-dependancy may also describe microscopic scale variations that are not periodic. This is illustrated in figure 1.6 where function (1/10) sin(x/3) + 1 + (1/4) cos(x/ε) sin((π/4)x/ε) is drawn.

Despite the fact that, for visibility reasons, the chosen value of ε is not very small (1/20), figures 1.7 and 1.8 show bi-dimensional functions with periodic Strong Oscillations (figure 1.7) and modulated amplitude in one direction (figure 1.8).

At the end of the day, functions writing a ε (x, x/ε) have the ability to describe a variety of coupling both macroscopic and microscopic variations which is wide enough.

It is practicable to introduce a variable (say ξ, which may be ξ, (ξ, υ) or (ξ, υ, ζ), depending on the dimension number) which describes the variations at the microscopic scale. This consists in considering that a ε in fact depends on two variables: a ε (x, ξ) and that the coefficient in equation (1.1) is

a ε (x, ξ = x ε ). (1.2)
Applying this practicable trick to the examples involved in figures above, the following

x 3 ) + 1 + 1 4 cos( x ε ) sin( π 4 x ε ) for ε = 1/20 (top), 1/40 (center)
and 1/80 (botom) between -3π and 3π.

x 2 + y 2 + sin(2x)(sin( y ε ) + 1) + (sin( x ε ) + 1) for ε = 1/20 on [0, 3] 2 .
formulas are gotten:

a ε (x, ξ) = 1 2 sin(x) + 1 + ε cos(ξ), in the case of figure 1.2, a ε (x, ξ) = a(x, ξ) = 1 2 sin(x) + 1 + 1 2 cos(ξ), in the case of figure 1.3, a ε (x, ξ) = a(x, ξ) = 1 2
(sin(x) + 1) cos(ξ), in the case of figure 1.4,

a ε (x, ξ) = a(x, ξ) = 1 2 cos(x) + 1 + 1 2
(sin(x) + 1) cos(ξ), in the case of figure 1.5,

a ε (x, ξ) = a(x, ξ) = 1 10 sin( x 3 ) + 1 + 1 4 cos(ξ) sin( π 4 ξ), a ε (x, y, ξ, υ) = a(x, y, ξ, υ) = x 2 + y 2 + 1 2
(sin(υ + 1) + (sin(ξ) + 1), and

a ε (x, y, ξ, υ) = a(x, y, ξ, υ) = x 2 + y 2 + 1 2 (sin(υ) + 1)(sin(ξ) + 1)
in the case of figure 1.7, and,

a ε (x, y, ξ, υ) = a(x, y, ξ, υ) = x 2 + y 2 + 1 2
(sin(υ) + 1)(sin(ξ) + 1), in the case of figure 1.8.

(1.3)

If ξ → a ε (x, ξ) is periodic, the microscopic scale variations are qualified of high frequency periodic oscillations.

Remark 1.1 Two-Scale Convergence is essentially designed to be used in the context of high frequency periodic oscillations.

Going back to the question we imagine we are interested in, we need to implement a numerical method, for instance a Finite Difference Method or a Finite Element Method, in order to compute an approximated solution of Partial Differential Equation (1.1). Since, x is a dimensionless variable, the domain on which (1.1) is set has a size which order of magnitude is 1. But, to get a reasonable result, we must choose a discretization step ∆x which is such that ∆x << ε. Otherwise, the effect of the microstruture is not taken into account, and the resulting computation has nothing to do with reality. Hence, if ε is very small, meaning that the microstructure is much smaller than the macroscopic size, the computation can be very expensive and even not feasible. For instance, if we consider a tridimensional material, with ε = 10 -3 then, with the constraint ∆x << ε, that we consider to be achieved with ∆x = 10 -1 ε, the order of magnitude of the number of degrees of freedom needed for the computation is (10 • 10 3 ) 3 = 10 12 . This is quite expensive. Leading such a computation may not be completely unreasonable if we are interested in knowing the intimate distribution of the temperature at the microstructure scale, but in most of the situations the tiny variations at this scale have no interest. In those cases, it is highly unreasonable to lead such a heavy computation to get the result.

Hence, we would like to have on hand an equation, which does not explicitly involve any microstructure, but which contains, or more precisely, which induces in its solution the average effect of the microstructure (which is described by the x/ε-dependance of a ε in 1.1). Denoting symbolically this equation by

Hu = 0, (1.4)
involving an operator H, with the constraint that u is close to u ε in some senses, we can expect to implement a numerical method to compute an approximated solution of (1.4)which is also an approximation of u ε -with a cost much smaller (because the constraint ∆x << ε is useless) than the one needed for the direct approximation of equation (1.1).

Homogenization Theory gathers a collection of methods that allows us to build operators H satisfying the required constraint.

The first Homogenization methods were set out by Engineers in the middle of the 1970s and then formalize by Mechanical Scientists. They are based on Asymptotic Expansion. Applying them in the case of equation (1.1) consists in writing

u ε (x) = U (x, x ε ) + εU 1 (x, x ε ) + ε 2 U 2 (x, x ε ) + . . . , (1.5)
with functions U (x, ξ), U 1 (x, ξ), U 2 (x, ξ), . . . periodic with respect to ξ, in injecting this expansion in equation (1.1), in identifying and gathering terms in factor of ε -2 , ε -1 , ε 0 , ε, ε 2 , . . . and then in deducing a set of equations:

H -2 U = 0, (1.6)
H -1 U 1 = I(U ), (1.7)
H 0 U 2 = I ′ (U, U 1 ), (1.8)
. . . .

Extracting information from those equations, we get well-posed equations for U , U 1 , U 2 , . . . . For a full understanding of the methods based on Asymptotic Expansion, I refer to the books by Sanchez-Palencia [START_REF] Sanchez-Palencia | Nonhomogeneous media and vibration theory[END_REF] and to the one by Bensoussan, Lions & Papanicolaou [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF].

Then, if we want to get a rigorous mathematical justification of the process just described, we need to prove results like

u ε (x) -U (x, x ε ) ? → 0, (1.9)
for a norm ? to be determined, or in a weaker sense,

u ε (x) -U (x, x ε ) ⇀ 0. (1.10)
If we want to get justification at higher orders, we need to prove convergence results like

  u ε (x) -U (x, x ε ) ε -U 1 (x, ξ)   → 0, (1.11)
in some senses,

1 ε 1 ε u ε (x) -U (x, x ε ) -U 1 (x, ξ) -U 2 (x, ξ) → 0, (1.12)
in some senses, and so on.

In the case of a Parabolic Partial Differential Equation like (1.1), with Dirichlet boundary conditions, convergence results of the kind of (1.9) can be brought out using the Maximum Principle and boundary estimates (see Bensoussan, Lions & Papanicolaou [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]). In other cases, the way is less straightforward, and appeals to Oscillating Test Functions used within a Weak Formulation of the Partial Differential Equation. Passing to the limit using Compensated Compactness like results (see Tartar [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF]) may give convergence results of the kind of (1.9) or (1.10). This method is called "Energy Method" or "Oscillating Test Function Method" and was designed by Tartar [START_REF] Tartar | Cours Peccot[END_REF] in collaboration with Murat [START_REF] Murat | H-convergence[END_REF] (see also [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]). For mathematical justification, works of Engquist have also to be consulted, in particular [START_REF] Engquist | Computation of oscillatory solutions to partial differential equations[END_REF].

The Weak Formulation with Oscillating Test Functions writes, in the case of equation

(1.1), Material ∇ • a ε (x, x ε )∇u ε (x) ϕ(x, x ε ) dx = 0, (1.13)
which yields, using the Stokes Formula,

Material a ε (x, x ε )∇u ε (x) • ∇ ϕ(x, x ε ) dx = Boundary Something, (1.14) or Material a ε (x, x ε )∇u ε (x) • ∇ x ϕ(x, x ε ) + 1 ε ∇ ξ ϕ(x, x ε ) dx = Boundary Something. (1.15)
In those integrals, ∇u ε converges generally in a weak sense only and this is the same for the other involved functions:

a ε (x, x ε ), ∇ x ϕ(x, x ε ) and ∇ ξ ϕ(x, x ε ).
It is well known that passing to the limit in a product of two weak converging sequences of functions is a nonstraightforward task. Hence, passing to the limit in (1.13), (1.14) or (1.15) is not so easy and consequently involves relatively sophisticated analytical methods (like Compensated Compactness results).

The situation just described is typical of the mathematical justification of homogenization results. Two-Scale Convergence offers an efficient framework to pass to the limit in such terms, in the case when oscillations are periodic. It is certainly possible to infer that Two-Scale Convergence emerges from those kind of questioning. Yet, as it will be illustrated by the example treated in section 3, Two-Scale Convergence is much more than a method to justify Asymptotic Expansion: it is a constructive Homogenization Method very well adapted to Singularly Perturbed Hyperbolic Equations.

Of course, we can try to provide an answer to question of the same type. For instance we can be interested in the time varying version of (1.1):

∂u ε ∂t -∇ • a ε (x, x ε
)∇u ε = 0 within the material, u ε (t, .) given on the boundary of the material, u ε (0, .) given at initial time within the material, (1.16) where t stands for a dimensionless time and where x has the same meaning as in equation (1.1).

We can also consider the following equations :

∂z ε ∂t - 1 ε ∇ • a(t, t ε , x)∇z ǫ = 1 ε ∇ • c(t, t ε , x), (1.17)
and

∂z ε ∂t - 1 ε 2 ∇ • a(t, t ε , x)∇z ǫ = 1 ε 2 ∇ • c(t, t ε , x), (1.18)
which are relevant models for the short-term and long-term dynamics of dunes on a seabed of a coastal ocean where tide is strong. In equation (1.17) and (1.18), z ε = z ε (t, x), where t stands is the dimensionless time and x is the bi-dimensional dimensionless position variable, is the dimensionless seabed altitude at time t and in position x. Those equations were widely studied in Faye, Frénod & Seck [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF][START_REF] Faye | Long term behaviour of singularly perturbed parabolic degenerated equation[END_REF].

We can also be interested in the following Vlasov equations :

∂f ε ∂t + v • ∇ x f ε + (E(x, t) + 1 ε v × B(x, t)) • ∇ v f ε = 0, (1.19)
and

∂f ε ∂t + v • ∇ x f ε + 1 ε v ⊥ • ∇ x f ε + (E(x, t) + 1 ε v × B(x, t)) • ∇ v f ε = 0, (1.20)
which are models involved in Tokamak Plasma physic. In those equations x ∈ R 3 stands for the dimensionless position, v ∈ R 3 for the dimensionless velocity and t for the dimensionless time. The solution

f ε = f ε (t, x, v)
which is also dimensionless is, at time t the density of ions in position x and with velocity v. Field E is the Electric field and field (1/ε)B(x, t) is the Strong Magnetic field. We denote by and ⊥ the directions parallel and perpendicular to this magnetic field.

Those two last equations involve neither oscillating coefficients nor any microstructure. Nevertheless, the strong magnetic field induces in the solution high frequency periodic oscillations. Equations (1.19) or (1.20) can be cast into the following framework of a singularly perturbed convection equation:

∂u ε ∂t + a • ∇u ε + 1 ε b • ∇u ε = 0, x ∈ R d , t > 0, (1.21)
by setting, in the case of (1.19),

a(x, v, t) = v E(t, x) and b(x, v) = 0 v × B(t, x) , (1.22)
and, in the case of (1.20),

a(x, v, t) = v E(t, x) and b(x, v) = v ⊥ v × B(t, x) . ( 1

A remark concerning periodicity

Two-Scale Convergence is well-adapted to the framework of high frequency periodic oscillations (or to cases that can be brought to this framework by an adequate transformation). But, it essentially does not work in non-periodic cases. Even in the case of oscillations with a period depending on the variable describing the macroscopic variation, it does not work. Many questions linked with non-periodic homogenization are essentially open.

A remark concerning weak-* convergence

Here, I give the proof of two important and representative results. They concern the characterization of the weak limit of functions with high frequency periodic oscillations. I will denote, for p = 1, . . . , ∞, by L p (R) the space of functions defined almost everywhere on R such that their p-th power is Lebesgue integrable, by L p # (R) the space of functions being in L p (R) and periodic of period 1, by C 0 # (R) the space of functions being continuous over R and periodic of period 1, by C ∞ (R) the space of functions being infinitely derivable over R, by D(R) the space of functions being in C ∞ (R) and compactly supported and by L p (R; C 0 # (R)) the space of functions mapping R to C 0 # (R) such that the p-th power of their norm is Lebesgue integrable.

The first result gives the asymptotic behavior, with respect to weak topology, of a periodic function applied in ξ = x/ε.

Lemma 1.1 Let ψ ∈ L ∞ # (R). Defining [ψ] ε by [ψ] ε (x) = ψ( x ε ), then [ψ] ε ⇀ 1 0 ψ(ξ) dξ in L ∞ (R) weak-*. (1.24) Remark 1.2 Convergence (1.24) means that for any function φ ∈ L 1 (R), R [ψ] ε (x) φ(x) dx → 1 0 ψ(ξ) dξ R φ(x) dx. (1.25)
Proof. The first step of the proof consists in noticing that, since the space

D(R) is dense in L 1 (R), it is enough to prove R [ψ] ε (x) ϕ(x) dx → 1 0 ψ(ξ) dξ R ϕ(x) dx. (1.26)
for any fixed ϕ ∈ D(R).

In a second step, fixing a D(R)-function ϕ, a real M such that [-M, M ] contains the support of ϕ is chosen. Then, the set of points {-M, -M +ε, -M +2ε, . . . , M +E(2M/ε)ε, M + (E(2M/ε)+1)ε}, where E stands for the integer part, is considered and the integral in (1.26) is shared in the following way:

R [ψ] ε (x) ϕ(x) dx = E(2M/ε)+1 i=1 -M +iε -M +(i-1)ε ψ( x ε ) ϕ(x) dx. (1.27)
In the third step, since ϕ is regular, it is possible to write that, for any i = 1, . . . ,

E(2M/ε)+ 1, for any x ∈ [-M + (i -1)ε, -M + iε], there exists a c i (x) ∈ [-M + (i -1)ε, x] such that ϕ(x) = ϕ(-M + (i -1)ε) + (x + M -(i -1)ε)ϕ ′ (c i (x)). Clearly, |(x + M -(i -1)ε)| ≤ ε and |ϕ ′ (c(x))| ≤ ϕ ′ ∞ . (1.28)
Using this in the sum of (1.27) yields

R [ψ] ε (x) ϕ(x) dx = E(2M/ε)+1 i=1 -M +iε -M +(i-1)ε ψ( x ε ) dx ϕ(-M (i -1)ε) + E(2M/ε)+1 i=1 -M +iε -M +(i-1)ε ψ( x ε ) (x + M -(i -1)ε)ϕ ′ (c i (x)) dx. (1.29)
In the last step, using the periodicity of ψ, the first term of (1.29) becomes

1 0 ψ(ξ) dξ ε E(2M/ε)+1 i=1 ϕ(-M (i -1)ε), (1.30)
which, because of Riemann like definitions of an integral, converges towards

1 0 ψ(ξ) dξ R ϕ(x) dx. (1.31)
as ε goes to 0. Beside this using (1.28), the second term of (1.29) is bounded in the following way:

E(2M/ε)+1 i=1 -M +iε -M +(i-1)ε ψ( x ε ) (x + M -(i -1)ε)ϕ ′ (c i (x)) dx ≤ 1 0 |ψ(ξ)| εdξ 2M + 1 ε ε ϕ ′ ∞ ≤ ε(2M + 1) 1 0 |ψ(ξ)| dξ ϕ ′ ∞ , (1.32)
and then converges to 0 as ε goes to 0. A careful watch to (1.27), (1.31) and (1.32) gives convergence 1.25. Since, this can be done for any ϕ ∈ D(R), the Lemma is proved.

In view of Lemma 1.1, and from the application point of view, it is cleaver to see the L ∞ (R) weak-* convergence as a way to generalize the concept of average value to functions which have non-periodic oscillations.

Hence finding H involved in (1.4) -or other similar question -may be translated into a mathematical framework as: "Find an equation satisfied by the weak-* limit of u ε ."

This explains why weak-* limit is a key-notion of Homogenization Theory

The second result characterizes the asymptotic behavior of a function depending on x and ξ, with a periodic dependence in ξ, and applied in ξ = x/ε. Notice that here more regularity with respect to ξ is needed than previously.

Lemma 1.2 Let ψ = ψ(x, ξ) ∈ L ∞ (R; C 0 # (R)) and define [ψ] ε by setting [ψ] ε (x) = ψ(x, x ε ). Then [ψ] ε ⇀ 1 0 ψ(x, ξ) dξ in L ∞ (R) weak-*.
(1.33)

Remark 1.3 Convergence (1.33) means that for any function φ ∈ L 1 (R), R [ψ] ε (x) φ(x) dx → R 1 0 ψ(x, ξ) dξ φ(x) dx. (1.34)
But, for the same reason as previously, if the following convergence

R [ψ] ε (x) ϕ(x) dx → R 1 0 ψ(x, ξ) dξ ϕ(x) dx, (1.35)
is proven for any ϕ ∈ D(R), it gives the Lemma.

Proof. (I chose to restrict the proof to the case when ψ ∈ C 0 (R; C 0 # (R)) to avoid technical arguments linked with Integration Theory.) The first stage of this proof consists in partitioning interval [0, 1] into m intervals of length 1/m, for any integer m. Then, fixing any value of m, the characteristic functions χ i , for i = 1 . . . , m, of all intervals are considered. They are extended by periodicity over R. The center ξ i of each interval is also considered. Using them, function ψm defined by

ψm (x, ξ) = m i=1 ψ(x, ξ i ) χ i (ξ), (1.36)
is built. For every x, ξ → ψm (x, ξ) is constant by intervals and, as m tends to infinity, ψm (x, ξ) → ψ(x, ξ) uniformly on every compact of R 2 .

(1.37)

Since, applying Lemma 1.1,

[χ i ] ε ⇀ 1 0 χ(ξ) dξ = 1 m in L ∞ (R) weak-*, (1.38)
it may be gotten:

[ ψm ] ε ⇀ m i=1 ψ(x, ξ i ) 1 m in L ∞ (R) weak-*, (1.39)
which is clearly

1 0 ψm (x, ξ) dξ.
Hence, as ε goes to 0,

[ ψm ] ε ⇀ 1 0 ψm (x, ξ) dξ in L ∞ (R) weak-*. (1.40)
The second stage of the proof consists in fixing ϕ ∈ D(R), and in showing that for any δ > 0, it is possible find an ε 0 , such that for any ε ≤ ε 0 ,

R [ψ] ε (x) ϕ(x) dx - R 1 0 ψ(x, ξ) dξ ϕ(x) dx ≤ δ.
(1.41)

The way to get this inequality consists in writing :

R [ψ] ε (x) ϕ(x) dx - R 1 0 ψ(x, ξ) dξ ϕ(x) dx = R [ψ] ε (x) -[ ψm ] ε (x) ϕ(x) dx + R [ ψm ] ε (x) - 1 0 ψm (x, ξ) dξ ϕ(x) dx + R 1 0 ψm (x, ξ) dξ - 1 0 ψ(x, ξ) dξ ϕ(x) dx ≤ R [ψ] ε (x) -[ ψm ] ε (x) |ϕ(x)| dx + R [ ψm ] ε (x) - 1 0 ψm (x, ξ) dξ ϕ(x) dx + R 1 0 ψm (x, ξ) -ψ(x, ξ) dξ |ϕ(x)| dx (1.42)
Because of uniform convergence (1.37), it is possible to fix an m such that

R [ψ] ε (x) -[ ψm ] ε (x) |ϕ(x)| dx ≤ δ 3 for any ε, (1.43) R 1 0 ψm (x, ξ) -ψ(x, ξ) dξ |ϕ(x)| dx ≤ δ 3 , (1.44)
and once this m is fixed, because of (1.40), it is possible to fix an ε 0 such that Since, this can be done for any ϕ ∈ D(R), the Lemma is proved.

R [ ψm ] ε (x) -
2 Two-Scale Convergence -Definition and Results

Definitions

There are several variants of the Two-Scale Convergence result, more or less well adapted to targeted applications and involving various functional spaces (see Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Nguetseng | Asymptotic analysis for a stiff variational problem arising in mechanics[END_REF], Allaire [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF], Amar I have chosen to present the Two-Scale Convergence in the framework set out in Frénod, Raviart & Sonnendrücker [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF] since this framework permits to select among the variables the ones that carry oscillations and the others. I begin by giving some notations. Definition 2.1 Let Ω be a regular domain in R n , L a separable Banach space and L ′ its topological dual space. Let q ∈ [1, +∞) and p ∈ (1, +∞] being such that 1/q + 1/p = 1. Let C 0 # (R n ; L) be the space of continuous functions R n → L and periodic of period 1 with respect to every variable and L p (Ω, L ′ ) be the space of (equivalence classes for the equivalence relation "= a.e." of measurable) functions f : Ω → L ′ such that the p-th power of the norm of

f : |f | p L ′ is Lebesgue integrable. Let L p # (R n ; L ′
) be the space of functions g : R n → L ′ such that the p-th power of the norm of f : |f | p L ′ is locally Lebesgue integrable and periodic of period 1.

(L p # (R n ; L ′ ) = (L p # (R n ; L)) ′ .) Spaces L q (Ω; L q # (R n , L)), L q (Ω; C 0 # (R n ; L)) and L p (Ω; L p # (R n , L ′ )) are also considered.
Now, I give the definition of Two-Scale Convergence.

Definition 2.2 A sequence (u ε ) = (u ε (x)) ⊂ L p (Ω; L ′ ) is said to Two-Scale converges to a profile U = U (x, ξ) ∈ L p (Ω; L p # (R n , L ′ )) if, for any function φ = φ(x, ξ) ∈ L q (Ω; C 0 # (R n ; L)
), the following convergence holds:

lim ε→0 Ω L ′ u ε (x), φ(x, x ε ) L dx = Ω [0,1] n L ′ U (x, ξ), φ(x, ξ) L dxdξ, (2.1)
where L ′ ., . L is the duality bracket between L ′ and L.

To end this definition subsection, I give two definitions of the Strong Two-Scale Convergence.

Definition 2.3 If p = q = 2, L is a Hilbert space and U ∈ L 2 (Ω; C 0 # (R n ; L ′ )), the sequence (u ε ) = (u ε (x)) ⊂ L 2 (Ω; L ′ ) is said to Strongly Two-Scale convergences to U = U (x, ξ) if lim ε→0 Ω u ε (x) -U (x, x ε ) 2 L ′ dx = 0. (2.2)
In (2.2) |.| 2 L ′ is the norm in L ′ -
which can be identified with L -associated with inner product L ′ ., . L ′ of L ′ -which is also the inner product L ., . L of L and the duality bracket L ′ ., . L between L ′ and L.

In Definition 2.3, the assumption that U is continuous with respect to ξ is to insure the ability to compute U (x, x/ε), which is not insured for a function which is defined only almost everywhere, since the measure of [START_REF] Nguetseng | Σ-Convergence of nonlinear parabolic operators[END_REF] gave a definition of Strong Two-Scale Convergence which involves less regularity:

{(x, x/ε), x ∈ R n } is 0 in R n × R n . Nguetseng & Woukeng
Definition 2.4 If p = q = 2, L is a Hilbert space and U ∈ L 2 (Ω; L 2 # (R n ; L ′ )), the sequence (u ε ) = (u ε (x)) ⊂ L 2 (Ω; L ′ ) is said to Strongly Two-Scale convergences to U = U (x, ξ) if ∀δ > 0, ∃ε 0 and Ũ ∈ L 2 (Ω; C 0 # (R n ; L ′ )) such that Ω [0,1] n U (x, ξ) -Ũ (x, ξ) 2 L ′ dxdξ ≤ δ 2 , and (2.3
)

Ω u ε (x) -Ũ (x, x ε ) 2 L ′ dx ≤ δ 2
, for every ε ≤ ε 0 .

Link with Weak Convergence

Two-Scale Convergence and weak-* convergence are strongly related. In fact Two-Scale Convergence may be seen are a generalization of weak-* convergence. This link expresses by the following Proposition.

Proposition 2.1 If a sequence (u ε ) ⊂ L p (Ω; L ′ ) Two-Scale converges to U ∈ L p (Ω; L p # (R n ; L ′ )), then u ε ⇀ [0,1] n U (., ξ) dξ weak-* in L p (Ω; L ′ ). (2.4)
Proof. Choosing test functions φ(x, ξ) = φ(x) -independent of the oscillating variable ξ -for any φ ∈ L q (Ω; L)) in the definition of Two-Scale Convergence is possible since L q (Ω; L)) ⊂ L q (Ω; C 0 # (R n ; L)). Doing this yields:

lim ε→0 Ω L ′ u ε (x), φ(x) L dx = Ω [0,1] n L ′ U (x, ξ), φ(x) L dxdξ = Ω L ′ [0,1] n U (x, ξ)dξ , φ(x) L dx. (2.5)
which is nothing but (2.4), proving the Proposition.

Remark 2.2

The last equality in (2.5) can be considered as trivial. Nevertheless, I give in this Remark the details bringing it.

For any fixed integer m, a partition of a subdomain ω m of Ω with mK(m) hypercubes of measure 1/m and a partition of [0, 1] n with m hypercubes of measure 1/m are considered.

If Ω is compact, ω m = Ω for every m and K(m) is constant and equal to the measure of Ω; if Ω is not compact, (ω m ) is a sequence of subdomains such that ω m ⊂ ω m+1 for every m, such that the measure of ω m is K(m) < +∞ and ∪ m∈N ω m = Ω. φ k , for k = 1, . . . , mK(m), stands for the value on the k-th hypercube of a piecewise constant function approximating φ and U k,l , for k = 1, . . . , mK(m) and l = 1, . . . , m, is the value on the tensor product of the k-th hypercube of ω m and the l-th hypercube of [0, 1] n of a piecewise constant function approximating U .

Ω [0,1] n L ′ U (x, ξ), φ(x) L dxdξ = lim m→+∞ mK(m) k=1 m =1 1 m 2 L ′ U k,l , φ k L = lim m→+∞ mK(m) k=1 1 m L ′ 1 m m =1 U k,l , φ k L = Ω L ′ [0,1] n U (x, ξ)dξ , φ(x) L dx. (2.6)

Injection Lemma

Now, I turn to the first important ingredient of Two-Scale Convergence which is the fact that taking functions of L q (Ω; C 0 # (R n , L)) in ξ = x/ε is a way to inject continuously this space in L q (Ω; L).

Lemma 2.1 If φ ∈ L q (Ω; C 0 # (R n ; L)), then for all ε > 0, function [φ] ε : Ω → L defined by [φ] ε (x) = φ(x, x ε ) (2.7)
is mesurable and satisfies

[φ] ε L q (Ω;L) ≤ φ L q (Ω;C 0 # (R n ;L)) . (2.8)
Proof. The first point is to see that φ ∈ L q (Ω; C 0 # (R n ; L)) if and only if there exists a set E of measure zero in Ω such that ∀x ∈ Ω \ E, ξ → φ(x, ξ) is continuous and periodic, (2.9)

∀ξ ∈ [0, 1] n , x → φ(x, ξ) is mesurable over Ω, (2.10)
x → sup

ξ∈[0,1] n |φ(x, ξ)| L is L q (Ω; R + ). (2.11)
Remark 2.3 This equivalence property is a consequence of integration theory but is not completely obvious. Nevertheless, I take it for granted. In this Remark, I just recall milestones that are needed to get it. Denoting L q (Ω; C 0 # (R n ; L)) means that measured space [Ω, completed of Borelian σ-algebra, Lebesgue measure] in considered. Since L is a separable Banach space, C 0 # (R n ; L) is also a separable Banach space. Hence, the following measurable space [C 0 # (R n ; L), Borelian σ-algebra] may be considered. Then, the use of the definition of strictly measurable function

Ω → C 0 # (R n ; L) is needed. (A function f is strictly measurable if there exists a set E ′ of measure zero in Ω such that f (Ω \ E ′ ) is included in a separable subset of C 0 # (R n ; L)
, and, if the inverse image by f of any set of the σ-algebra of C 0 # (R n , L) belongs to σ-algebra of Ω.) With this, the integration theory can be implemented, involving step functions Ω → C 0 # (R n , L) (which is quite long) and characterizations of integrable functions are gotten.

Among them, there is the following one: A function

f : Ω → C 0 # (R n ; L) is integrable if it is strictly measurable and if Ω f C 0 # (R n ;L) dx < +∞.
(2.12)

Finally, spaces L p (Ω; C 0 # (R n ; L)) and L q (Ω; C 0 # (R n ; L)) can be built. Characterization of L q (Ω; C 0 # (R n ; L)) by (2.9), (2.10) and (2.11) can also be gotten.

The second point consists in fixing ε. From (2.9), for all fixed ε,

[φ] ε (x) = φ(x, x ε ) is well defined on Ω \ E.
Now, the goal is to prove that [φ] ε is mesurable. For this purpose, for any fixed integer m, a partition of a subdomain ω m of Ω with mK(m) hypercubes of measure 1/m is considered. (If Ω is compact, ω m = Ω for every m and K(m) is constant and equal to the measure of Ω; if Ω is not compact, (ω m ) is a sequence of subdomains such that ω m ⊂ ω m+1 for every m, such that the measure of ω m is K(m) and ∪ m∈N ω m = Ω.) For every i = 1, . . . , mK(m), the center x i of the i-th hypercube and the characteristic function ι i of the i-th hypercube are considered. With this, function η ε m defined by

η ε m (x) = mK(m) i=1 x i ε ι i (x), (2.13)
is a step function approximating x ε .

Considering now function [φ]

ε m defined by

[φ] ε m (x) = φ(x, η ε m (x)), (2.14)
clearely,

∀x ∈ Ω \ E, [φ] ε m (x) → [φ] ε (x) as m → +∞, (2.15)
and, since

[φ] ε m (x) = φ(x, η ε m (x)) = mK(m) i=1 φ(x, x i ε ) ι i (x), (2.16)
and because of (2.10), [φ] ε m is a finite sum of measurable functions, hence is measurable. Hence, [φ] ε is almost everywhere the limit of a measurable function. Hence it is measurable.

The last point is the proof of (2.8). From (2.11) it is deduced that

φ q L q (Ω;C 0 # (R n ;L)) = Ω sup ξ∈[0,1] n |φ(x, ξ)| L q dx < +∞.
(2.17)

On the other hand

[φ] ε L q (Ω;L) = Ω φ(x, x ε ) q L dx ≤ Ω sup ξ∈[0,1] n |φ(x, ξ)| L q dx = φ q L q (Ω;C 0 # (R n ;L)) , (2.18)
which is (2.8), ending the proof of the Lemma.

This Injection Lemma is supplemented by a property giving information on the asymptotic behavior of [φ] ε as ε goes to zero. Proposition 2.2 Under the same assumption as in Lemma 2.1, function [φ] ε defined by (2.7) satisfies:

lim ε→0 [φ] ε L q (Ω;L) = lim ε→0 Ω φ(x, x ε ) q L dx = Ω |φ(x, ξ)| q L dxdξ = φ L q (Ω;L q (R n ;L)) . (2.19)
Proof. For any fixed integer m, a partition of [0, 1] n with m hypercubes of measure 1/m is considered. the center of the i-th hypercube is called ξ i and its characteristic function, extended by periodicity over R n is called χ i . With φ is associated the sequence of function φm defined by

φm (x, ξ) = m i=1 φ(x, ξ i ) χ i (ξ), (2.20)
The second step consists in considering [χ i ] ε defined by

[χ i ] ε (x) = χ i ( x ε
).

(2.21)

using Lemma 1.1, generalized to the multidimensional setting, the following is deduced:

[χ i ] ε ⇀ [0,1] n χ i (ξ) dξ = 1 m in L ∞ (Ω; R) weak-*, (2.22) ([χ i ] ε ) q ⇀ [0,1] n χ q i (ξ) dξ = 1 m in L ∞ (Ω; R) weak-*. (2.23) Hence lim ε→0 Ω φm (x, x ε ) q L dx = lim ε→0 Ω m i=1 |φ(x, ξ i )| q L ([χ i ] ε ) q dx = m i=1 1 m Ω |φ(x, ξ i )| q L dx = Ω [0,1] n φm (x, ξ) q L dxdξ. (2.24)
The goal of the third step is to get, as m → 0,

φm → φ in L q (Ω; C 0 # (R n ; L)). (2.25)
For this purpose, the function Γ m : Ω → R defined by

Γ m (x) = sup ξ∈[0,1] n φm (x, ξ) -φ(x, ξ) q L , (2.26) is used. Since ξ → φm (x, ξ) -φ(x, ξ) is piecewise continuous, sup ξ∈[0,1] n φm (x, ξ) -φ(x, ξ) q L = sup ξ∈[0,1] n ∩Q n φm (x, ξ) -φ(x, ξ) q L . (2.27)
Hence, Γ m is a supremum on an countable set of measurable functions and, as such, measurable. Moreover, Γ m (x) → 0 a.e. on Ω, (2.28)

0 ≤ Γ m (x) ≤ 2 sup ξ∈[0,1] n |φ(x, ξ)| q L , (2.29)
and sup ξ∈[0,1] n |φ(x, ξ)| q L is an integrable function. Hence, invoking the Lebesgue Dominated Convergence Theorem, it may be deduced that, as m → 0:

Γ m (x) → 0 in L 1 (Ω; R), (2.30)
and consequently (2.25). The last step consists in using (2.24) and (2.25) and writing

[φ] ε q L q (Ω;L) = Ω φ(x, x ε ) q L dx = Ω φ(x, x ε ) q L dx - Ω φm (x, x ε ) q L dx + Ω φm (x, x ε ) q L dx - Ω φm (x, ξ) q L dxdξ + Ω φm (x, ξ) q L dxdξ . (2.31)
The last term of the right hand side of this formula satisfies

Ω φm (x, ξ) q L dxdξ → Ω |φ(x, ξ)| q L dxdξ as m → +∞, (2.32)
the second term satisfies

Ω φm (x, x ε ) q L dx - Ω φm (x, ξ) q L dxdξ → 0 as ε → 0, (2.33)
and concerning the first term,

Ω φ(x, x ε ) q L dx - Ω φm (x, x ε ) q L dx ≤ Ω φ(x, x ε ) q L -φm (x, x ε ) q L dx ≤ Ω sup ξ∈[0,1] n φ(x, ξ) -φm (x, ξ) q L dx → 0 as m → +∞. (2.34)
Using these three convergence results in (2.31) give (2.19) and prove the proposition.

Two-Scale Convergence criterion

Once the Injection Lemma gotten, the following Theorem, which is important for Homogenization issues, may be proven relatively easily.

Theorem 2.1 If a sequence (u ε ) is bounded in L p (Ω; L ′ ), i.e. if u ε L p (Ω;L ′ ) = Ω |u ε (x)| p L ′ dx 1 p ≤ c, (2.35)
for a constant c independent of ε, then, there exists a profile U ∈ L p (Ω; L p # (R n ; L ′ )) such that, up to a subsequence, (u ε ) Two-Scale converges to U .

(2.36)

Proof. In a first stage of the proof, the Injection Lemma and assumption (2.35) is used to get that, for any function φ = φ(x, ξ) ∈ L q (Ω;

C 0 # (R n ; L)) ((1/p) + (1/q) = 1), Ω L ′ u ε (x), φ(x, x ε ) L dx ≤ c [φ] ε L q (Ω,L)
(2.37)

≤ c φ L q (Ω;C 0 # (R n ;L)) .
(2.38)

Hence the sequence (µ ε ) of applications, where

µ ε : L q (Ω; C 0 # (R n ; L)) → R φ → Ω L ′ u ε (x), φ(x, x ε ) L dx, (2.39) is bounded in the dual (L q (Ω; C 0 # (R n ; L))) ′ of L q (Ω; C 0 # (R n ; L)) which is a separable space.
Remark 2.4 The norm on (L q (Ω;

C 0 # (R n ; L))) ′ is µ = sup 0 =φ∈L q (Ω;C 0 # (R n ;L)) | µ, φ | φ L q (Ω;C 0 # (R n ;L))
.

(2.40)

Since (µ ε ) is the dual of a separable space, extracting a subsequence, there exists an application µ ∈ (L q (Ω;

C 0 # (R n ; L))) ′ µ ε ⇀ µ in (L q (Ω; C 0 # (R n ; L))) ′ weak-*. (2.41)
In particular, it implies

µ ε , φ → µ, φ , (2.42) for any φ ∈ L q (Ω; C 0 # (R n ; L)).
The beginning of the second stage of the proof consists in making ε → 0 in (2.37). The left hand side converges towards µ, φ and, according to Proposition 2.2, the right hand side converges towards c φ L q (Ω;L q (R n ;L)) . Hence, for every function in φ ∈ L q (Ω;

C 0 # (R n ; L)), µ, φ ≤ c φ L q (Ω;L q (R n ;L)) . (2.43) Knowing that L q (Ω; C 0 # (R n ; L)) is dense in L q (Ω; L q # (R n ; L ′ )) -whose dual is L p (Ω; L p # (R n ; L)
), applying the Riez Representation Theorem, it may be deduced that there exists

U ∈ L p (Ω; L p # (R n ; L ′ )) such that µ, φ = Ω [0,1] n L ′ U (x, ξ), φ(x, ξ) L dxdξ, (2.44)
and consequently, such that, up to a subsequence, as ε goes to zero,

Ω L ′ u ε (x), φ(x, x ε ) L dx → Ω [0,1] n L ′ U (x, ξ), φ(x, ξ) L dxdξ, (2.45)
which is exactly (2.1) of definition 2.2, proving (2.36) and so ending the proof.

Strong Two-Scale Convergence criterion

In this section p = q = 2 and L and L ′ are the same separable Hilbert space. In order to go on gradually, I begin by the following very simple result.

Lemma 2.2 If ψ = ψ(x, ξ) ∈ L 2 (Ω; C 0 # (R n ; L)) then the sequence of functions ([ψ] ε ) ⊂ L 2 (Ω; C 0 # (R n ; L)) defined by [ψ] ε (x) = ψ(x, x ε ), (2.46)
Strongly Two-Scale converges towards ψ.

Proof. Applying directly (2.19) of Proposition 2.2, it can be gotten:

Ω L ψ(x, x ε ), φ(x, x ε ) L dx → Ω [0,1] n L ψ(x, ξ), φ(x, ξ) L dxdξ, (2.47)
for all function φ ∈ L 2 (Ω; C 0 # (R n ; L)), meaning that ([ψ] ε ) Two-Scale converges towards ψ. Now, in view of (2.2) in Definition 2.3, since

Ω [ψ] ε (x) -ψ(x, x ε ) 2 L ′ dx → 0, (2.48)
is completely obvious, the Strong Two-Scale Convergence is insured.

As easyly, the following result may also be proven.

Proposition 2.3 If ψ is like in Lemma 2.2, [ψ] ε L 2 (Ω;L) = Ω L ψ(x, x ε ), ψ(x, x ε ) L dx 1 2 → Ω [0,1] n L ψ(x, ξ), ψ(x, ξ) L dxdξ 1 2 = ψ L 2 (Ω;L 2 # (R n ;L)) . (2.

49)

Now I will give a result, that was already evoked by Lemma 2.2, establishing the link between Strong Two-Scale Convergence and Two-Scale Convergence.

Theorem 2.2 If a sequence (u ε ) ⊂ L 2 (Ω; L)) Strongly Two-Scale Converges towards U and if U ∈ L 2 (Ω; C 0 # (R n ; L)), then it Two-Scale Converges towards U .
Proof. Considering the following quantity

I ε = Ω L u ε (x) -U (x, x ε ), φ(x, x ε ) L dx, (2.50)
for any function φ ∈ L 2 (Ω; C 0 # (R n ; L)), on the one hand this quantity satisfies

|I ε | ≤ Ω u ε (x) -U (x, x ε ) 2 L dx 1 2 Ω φ(x, x ε ) 2 L dx 1 2 → 0, (2.51)
as ε → 0, because of the Strong Two-Scale Convergence. On the other hand,

I ε = Ω L u ε (x), φ(x, x ε ) L dx - Ω L U (x, x ε ), φ(x, x ε ) L dx, (2.52)
and according to Lemma 2.2,

Ω L U (x, x ε ), φ(x, x ε ) L dx → Ω [0,1] n L U (x, ξ), φ(x, ξ) L dxdξ. (2.53)
Using (2.51) and (2.53) it is gotten that, as ε goes to 0,

Ω L u ε (x), φ(x, x ε ) L dx → Ω [0,1] n L U (x, ξ), φ(x, ξ) L dxdξ, (2.54)
i.e. (u ε ) Two-Scale Converges to U , ending the proof. Now, I give the important Theorem concerning Strong Two-Scale Convergence.

Theorem 2.3 If a sequence (u ε ) ⊂ L 2 (Ω; L) Two-Scale converges towards a profile U , if U ∈ L 2 (Ω; C 0 # (R n ; L)) and if lim ε→0 u ε L 2 (Ω;L) = U L 2 (Ω;L 2 ([0,1] n ;L) , (2.55)
then

(u ε ) Strongly Two-Scale converges to U , (2.56)
and, for any sequence

(v ε ) ⊂ L 2 (Ω; L) Two-Scale converging towards a profile V , L u ε , v ε L ⇀ [0,1] n L U (., ξ), V (., ξ) L dξ, in D ′ (Ω).
(2.57)

Proof. The proof of the first part of the Theorem consists just in computing:

Ω u ε (x) -U (x, x ε ) 2 L dx = Ω |u ε (x)| 2 L dx -2 Ω L u ε (x), U (x, x ε ) L dx + Ω U (x, x ε ) 2 L dx, (2.58)
and in passing to the limit, as ε goes to 0, using the assumptions of the Theorem,

lim ε→0 Ω u ε (x) -U (x, x ε ) 2 L dx = lim ε→0 Ω |u ε (x)| 2 L dx -2 Ω [0,1] n L U (x, ξ), U (x, ξ) L dxdξ + Ω [0,1] n |U (x, ξ)| 2 L dxdξ = 0. (2.59)
In order to prove the second part of the Theorem, for any test function ϕ ∈ D(Ω) the following quantity is computed:

D ′ L u ε , v ε L , ϕ D = Ω L u ε (x), v ε (x) L ϕ(x) dx = Ω L U (x, x ε ), v ε (x) L ϕ(x) dx - Ω L u ε (x) -U (x, x ε ), v ε (x) L ϕ(x) dx. (2.60) Since u ε (x) -U (x, x ε ) → 0, the second term of the right hand side is such that Ω L u ε (x) -U (x, x ε ), v ε (x) L ϕ(x) dx → 0, (2.61)
as ε goes to 0. A direct calculation gives the behavior of the first term as ε goes to 0:

Ω L U (x, x ε ), v ε (x) L ϕ(x) dx = Ω L v ε (x), U (x, x ε ) L ϕ(x) dx = Ω L v ε (x), ϕ(x)U (x, x ε ) L dx → Ω [0,1] n L V (x, ξ), ϕ(x)U (x, ξ) L dxdξ = Ω [0,1] n L V (x, ξ), U (x, ξ) L ϕ(x) dxdξ = Ω [0,1] n L U (x, ξ), V (x, ξ) L ϕ(x) dxdξ, (2.62)
coupling (2.60), (2.61) and (2.62) gives

D ′ L u ε , v ε L , ϕ D → Ω [0,1] n L U (x, ξ), V (x, ξ) L ϕ(x) dxdξ (2.63)
for any test function ϕ ∈ D(Ω), as ε goes to 0, i.e (2.57), ending the proof.

Application : Homogenization of linear Singularly Perturbed Hyperbolic Equations

Here, I show how to homogenize a linear Singularly Perturbed Hyperbolic Equation with a method based on Two-Scale Convergence. As said in the Introduction, this equation is related to Tokamak Plasma Physics. The setting is a very simplified one. A more general setting may be found in Frénod, Raviart & Sonnendrücker [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF] despite the presentation is different: In [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF], we used Two-Scale Convergence to justify Asymptotic Expansion while here the Two-Scale Convergence based method is used as a constructive Homogenization Method.

Equation of interest and setting

The considered equation is the following:

∂u ε ∂t + a • ∇u ε + 1 ε b • ∇u ε = 0, (3.1)
u ε |t=0 = u 0 . (3.2)
This equation is set for u ε = u ε (t, x) with x ∈ R d and t ∈ [0, T ), for a given T > 0.

Concerning a, it is assumed that a = a(x) does not depend on time t, is very regular and that its divergence ∇ • a is zero. (Those assumptions can be relaxed but it complicates calculations.) Concerning b the following assumptions (which can essentially not be relaxed) are done: b = b(x) = M x, where M is a matrix such that trM = 0, and such that τ → e τ M is periodic of period 1.

Remark 3.1 According to those assumptions, the divergence ∇ • b of b is zero, and since

X(τ ) = e τ M x is solution to ∂X ∂τ = M X(= b(X)), X(0) = x, (3.3)
the characteristics associated with operator (b • ∇) are periodic of period 1 and preserve the Lebesgue measure.

A priori estimate

Multiplying equation 3.1 by u ε and integrating over

R d gives d R d |u ε | 2 dx dt = 0, (3.4)
since

R d a • ∇u ε u ε dx = - R d a • ∇u ε u ε dx - R d ∇ • a u ε u ε dx = - R d a • ∇u ε u ε dx = 0. (3.5)
Integrating 3.4 from 0 to t yields

R d |u ε (t, .)| 2 dx = R d |u 0 | 2 dx, (3.6)
and consequently

u ε L 2 ([0,T );L 2 (R d )) = T 0 R d |u ε | 2 dxdt = T R d |u 0 | 2 dx. (3.7)
As a consequence, the following result can be claimed.

Lemma 3.1 If u 0 ∈ L 2 (R d ), then the sequence (u ε ) is bounded in L 2 ([0, T ); L 2 (R d )).
Hence, up to a subsequence

(u ε ) Two-Scale Converges to U = U (t, τ, x) ∈ L 2 ([0, T ); L 2 # ((R; L 2 (R d ))), (3.8)
u ε ⇀ u = 1 0 U (., τ, .) dτ in L 2 ([0, T ); L 2 (R d )) weak-*.
(3.9)

Weak Formulation with Oscillating Test Functions

From any function φ = φ(t, τ,

x) ∈ C 1 ([0, T ); C 1 # ((R; C 1 (R d ))) it is possible to define [φ] ε by [φ] ε (t, x) = φ(t, t ε , x). (3.10) Since ∂[φ] ε ∂t = ∂φ ∂t ε + 1 ε ∂φ ∂τ ε , (3.11)
multiplying (3.1) by [φ] ε and integrating the result by parts, the following Weak Formulation with Oscillating Test Functions is gotten:

T 0 R d u ε ∂φ ∂t ε + 1 ε ∂φ ∂τ ε + a • [∇φ] ε + 1 ε b • [∇φ] ε dxdt + R d
u 0 φ(0, 0, .) dx = 0.

(3.12)

Order 0 Homogenization -Constraint

Multiplying Weak Formulation with Oscillating Test Functions (3.12) by ε and passing to the limit using the Two-Scale Convergence, we obtain:

T 0 1 0 R d U ∂φ ∂τ + b • ∇φ dxdτ dt = 0, (3.13)
that is nothing but a weak formulation of

∂U ∂τ + b • ∇U = 0. (3.14)
This last equation says that U is constant along the characteristics of operator (b•∇). Hence the following Lemma is true.

Lemma 3.2 There exists a function V = V (t, y) ∈ L 2 ([0, T ); L 2 (R d )) such that U (t, τ, x) = V (t, e -τ M x).
Remark 3.2 The result of this Lemma may also be gotten by direct computations. For instance, Using such functions in Weak Formulation with Oscillating Test Functions (3.12) cancels the terms in factor of 1/ε:

∂(V (t, e -τ M x)) ∂τ + b • ∇(V (t, e -τ M x)) = ∇V (t, e -τ M x)) • ((-e -τ M )M x) + ((e -τ M )M x) • ∇V (t, e -τ M x)) = 0.
T 0 R d u ε ∂φ ∂t ε + a • [∇φ] ε dxdt + R d
u 0 φ(0, 0, .) dx = 0.

(3.17)

Passing to the limit yields

T 0 1 0 R d U (t, τ, x) ∂φ ∂t (t, τ, x) + a(x) • ∇φ(t, τ, x) dxdτ dt + R d u 0 φ(0, 0, .) dx = 0, (3.18)
and using expression of U in terms of V and of φ in terms of γ, since

∂φ ∂t (t, τ, x) = ∂γ ∂t (t, e -τ M x) and ∇φ(t, τ, x) = (e -τ M ) T ∇γ(t, e -τ M x), (3.19)
gives

T 0 1 0 R d V (t, e -τ M x) ∂γ ∂t (t, e -τ M x) + e -τ M a(x) • ∇γ(t, e -τ M x) dxdτ dt + R d u 0 (x) γ(0, x) dx = 0. (3.20)
In the first integral of the left hand side we make the change of variables (t, τ, x) → (t, τ, y = e -τ M x) which preserves the Lebesgue measure and which reverse transform is (t, τ, y) → (t, τ, x = e τ M y). It gives

T 0 1 0 R d V (t,
∂(u ε ) 2 ∂t + a • ∇(u ε ) 2 + 1 ε b • ∇(u ε ) 2 = 0, (3.32)
(u ε ) 2 |t=0 = u 2 0 . (3.33) Hence if u 2 0 is in L 2 (R d ), i.e. if u 0 ∈ L 4 (R d )
, it is possible to do the same for equation (3.32) as for (3.1) and find that (u ε ) 2 Two-Scale converges to a profile, called Z, and that Z is solution to

∂Z ∂t + 1 0 e -σM a(e σM x)dσ • ∇Z = 0, (3.34)
Z |t=0 = u 2 0 (e -τ M x), (3.35)
leading to the conclusion that Z = U 2 or

((u ε ) 2 ) Two-Scale Converges to U 2 . (3.36)
From (3.36), it is easy to get that

u ε L 2 ([0,T );L 2 (R d )) → U L 2 ([0,T );L 2 # ((R;L 2 (R d ))) , (3.37)
as ε → 0. Indeed, we only need to consider for any δ > 0 the regular function β δ = β δ (x) which is such that β δ (x) = 1 when |x| < 1/δ, β δ (x) = 0 when |x| > 1/δ + 1 and 0 ≤ β δ ≤ 1. Clearly from (3.36), for any δ,

T 0 R d (u ε ) 2 β δ dxdt → T 0 1 0 R d U 2 β δ dxdτ dt, (3.38)
and as δ → 0,

T 0 R d (u ε ) 2 β δ dxdt → u ε L 2 ([0,T );L 2 (R d )) , (3.39)
T 0 R d U 2 β δ dxdτ dt → U L 2 ([0,T );L 2 # ((R;L 2 (R d ))) . (3.40) Moreover, if u 0 is in C 0 (R d ) then, u ε ∈ C 0 ([0, T ); C 0 (R d )), U ∈ C 0 ([0, T ); C 0 # ((R; C 0 (R d ))) and V ∈ C 0 ([0, T ); C 0 (R d ))
. This can be directly deduced from the equations satisfied by those functions. Hence Theorem 2.3 can be invoked to deduce the next Lemma.

∂(u ε -[U ] ε ) ∂t + a • ∇(u ε -[U ] ε ) + 1 ε b • ∇(u ε -[U ] ε ) = -a - 1 0 e -σM a(e σM x)dσ • ∇[U ] ε , (3.42) (u ε -[U ] ε ) |t=0 = 0. (3.43)
Multiplying this equation by 1/ε we obtain

∂ u ε -[U ] ε ε ∂t + a • ∇ u ε -[U ] ε ε + 1 ε b • ∇ u ε -[U ] ε ε = - 1 ε a - 1 0 e -σM a(e σM x)dσ • ∇[U ] ε , (3.44) u ε -[U ] ε ε |t=0 = 0. (3.45)
The left hand side of this equation is the same as in (3.1) but the right hand side is in factor of 1/ε. Hence, in a second stage, we introduce a function

W 1 = W 1 (t, τ, y) such that W1 = W1 (t, τ, x) = W 1 (t, τ, e -τ M x), (3.46)
satisfies

∂ W1 ∂τ + b • ∇ W1 = -a - 1 0 e -σM a(e σM x)dσ • ∇U. (3.47) Because of (3.47), considering [ W1 ] ε = [ W1 ] ε (t, x) = W1 (t, t/ε, x), ∂[ W1 ] ε ∂t + a • ∇[ W1 ] ε + 1 ε b • ∇[ W1 ] ε = ∂ W1 ∂t ε + 1 ε ∂ W1 ∂τ ε + a • ∇[ W1 ] ε + 1 ε b • ∇[ W1 ] ε = ∂ W1 ∂t ε + a • ∇[ W1 ] ε - 1 ε a - 1 0 e -σM a(e σM x)dσ • ∇[U ] ε . (3.48) Subtracting (3.48) from (3.42) gives ∂ u ε -[U ] ε ε -[ W1 ] ε ∂t + a • ∇ u ε -[U ] ε ε -[ W1 ] ε + 1 ε b • ∇ u ε -[U ] ε ε -[ W1 ] ε = - ∂ W1 ∂t ε -a • ∇[ W1 ] ε , (3.49) u ε -[U ] ε ε -[ W1 ] ε |t=0 = -[ W1 ] ε |t=0 . (3.50)
The goal of the third stage is to give an expression of the function W 1 : Function W1 is solution of (3.47) if and only if W 1 is solution to

∂W 1 ∂τ = -a(e τ M y) - 1 0
e -σM a(e (σ+τ )M y)dσ • ∇U (t, τ, e τ M y).

(3.51)

Beside this, ∇U (t, τ, e τ M y) = (e -τ M ) T ∇ U (t, τ, e τ M y) = (e -τ M ) T ∇V (t, y), hence W 1 is solution to

∂W 1 ∂τ = -e -τ M a(e τ M y) - 1 0 e -(σ+τ )M a(e (σ+τ )M y) dσ • ∇V (t, y)
= -e -τ M a(e τ M y) -

- ∂ W1 ∂t ε -a • ∇[ W1 ] ε L ∞ ([0,T );L 2 (R d )) ≤ C 1 , (3.54)
for a constant C 1 not depending on ε.

3.9 Order 1 Homogenization -A priori estimate and convergence

Multiplying (3.49) by ((u ε -[U ] ε )/ε -[ W1 ] ε ), we get d R d u ε -[U ] ε ε -[ W1 ] ε 2 dx dt ≤ C 1 R d u ε -[U ] ε ε -[ W1 ] ε 2 dx 1 2 , (3.55)
from which an estimate can be gotten and expressed in the following Lemma.

Lemma 3.5 If u 0 ∈ (L 2 ∩ L 4 ∩ C 2 )(R d )
, then in addition to every already stated results, the sequences

u ε -[U ] ε ε -[ W1 ] ε , and consequently u ε -[U ] ε ε , (3.56) are bounded in L 2 ([0, T ); L 2 (R d ))
. Then, up to subsequences,

u ε -[U ] ε ε Two-Scale Converges to U 1 = U 1 (t, τ, x) ∈ L 2 ([0, T ); L 2 # ((R; L 2 (R d ))), (3.57)
u ε -[U ] ε ε -[ W1 ] ε Two-Scale Converges to U 1 -W1 , (3.58)
where W 1 is defined in (3.53) and W1 by (3.46).

Order 1 Homogenization -Constraint

For any Oscillating Test Function

φ = φ(t, τ, x) ∈ C 1 ([0, T ); C 1 # ((R; C 1 (R d ))
), it is possible to write the following Weak Formulation:

T 0 R d u ε -[U ] ε ε -[ W1 ] ε ∂φ ∂t ε + 1 ε ∂φ ∂τ ε + a • [∇φ] ε + 1 ε b • [∇φ] ε dxdt = T 0 R d - ∂ W1 ∂t ε -a • ∇[ W1 ] ε [φ] ε dxdt. (3.59)
Multiplying this equation by ε and passing to the limit yields the next constrain equation:

∂(U 1 -W1 ) ∂τ + b • ∇(U 1 -W1 ) = 0. (3.60)
Hence the following Lemma is true.

Lemma 3.6 There exists a function V 1 = V 1 (t, y) ∈ L 2 ([0, T ); L 2 (R d )) such that U 1 (t, τ, x)-W1 (t, τ, x) = V 1 (t, e -τ M x) or, in other words, such that

U 1 (t, τ, x) = V 1 (t, e -τ M x) + W 1 (t, τ, e -τ M x), (3.61)
where W 1 is defined in (3.53).

Order 1 Homogenization -Equation for V 1

Using now in (3.59) Oscillating Test Function φ(t, τ, x) = γ(t, e -τ M x) for any regular function γ = γ(t, y), the terms in factor of ε cancel and passing to the limit, it gives The integrand in the last term is the dot product of a symmetric matrix (not depending on τ ) applied to a vector with the τ -derivative of this same vector; so it is an exact τ -derivative. Consequently, the last term is zero. Beside this, integrating by parts the first piece of the first term of the right hand side of (3.69) gives:

T 0 1 0 R d V 1 (t, e -τ M

For the numerics

In the case when ε is small, computing a numerical approximation of (3.1) can be expensive in term of CPU time since it requires a time step which is small compared with ε.

If the result we just set out is reinterpreted, it can be claimed that u ε (t, x) ∼ V (t, e -τ M x) + ε V 1 (t, e -τ M x) + W 1 (t, τ, e -τ M x) |τ =t/ε , (3.73)

Figure 1 . 2 :

 12 Figure 1.2: Graph of 1 2 sin(x) + 1 + ε cos( x ε ) for ε = 1/20 (left), 1/40 (center) and 1/80

Figure 1 . 7 : 1 )Figure 1 . 8 :

 17118 Figure 1.7: Graph of x 2 + y 2 + 1 2 (sin( y ε ) + 1) + (sin( x ε ) + 1) (left) and of x 2 + y 2 + 1 2 (sin( y ε ) + 1)(sin( x ε ) + 1) (right) for ε = 1/20 on [0, 3] 2 .

1 0

 1 ψm (x, ξ) dξ ϕ(x) dx ≤ δ 3 for any ε ≤ ε 0 . (1.45) Using (1.43), (1.43) and (1.45) in (1.42) gives the sought formula (1.41).

[ 4 ]

 4 , Casado-Díaz & Gayte [6], Frénod, Raviart & Sonnendrücker [14], Nguetseng & Woukeng [25] and Nguetseng & Svanstedt [24]). They are in fact very close to each other in what concerns what they claim and their proofs all follow the same routine based on these two phases : • A continuous injection Lemma, • A compactness Theorem, Remark 2.1 In 2005, Pak [26] made a important improvement in the Two-Scale Convergence Theory adapting it to manifolds and differential forms.

(3. 15 ) 3 . 5

 1535 Order 0 Homogenization -Equation for V From any regular function γ = γ(t, y) ∈ C 1 ([0, T ); C 1 (R d )), φ defined by φ(t, τ, x) = γ(t, e -τ M x) is regular and satisfies ∂φ ∂τ + b • ∇φ = 0. (3.16)

eTheorem 3 . 1 1 0e. 24 ) 3 . 3 1 0ee 1 0e 1 0e 1 0 1 0e 1 0U 3 . 3 + 1 0ee 1 0u 0

 311243311111133110 y) ∂γ ∂t (t, y) + e -τ M a(e τ M y) • ∇γ(t, y) dydτ dt + R d u 0 (y) γ(0, y) dy = 0, -τ M a(e τ M y) dτ • ∇γ(t, y) dydt + R d u 0 (y) γ(0, y) dy = 0, (3.22)Which says: Under assumption of Lemma 3.1, function V (t, y) linked by Lemma 3.2 with the Two-Scale limit U (t, τ, x) of (u ε ) is solution to∂V ∂t + -σM a(e σM y) dσ • ∇V = 0,(3.23)V |t=0 = u 0 . (3Remark Clearly, the solution of (3.23) and (3.24) is unique. As a consequence, the whole sequence (u ε ) converges (Two-Scale towards U , and weak-* towards u)3.6 Order 1 Homogenization -Preparations: equation for U and uBecause of the linearity of the problem, it is possible to deduce from (3.23) an equation for U also. Indeed, since ∇U (t, τ, x) = (e -τ M ) T ∇V (t, e -τ M x) or ∇V (t, e -τ M x) = (e τ M ) T ∇U (t, τ, x), writing (3.23) in y = e -τ M x, we obtain that0 = ∂ V (t, e -τ M x) ∂t + -σM a(e σM e -τ M x)dσ • ∇V (t, e -τ M x) -σM a(e (σ-τ )M x)dσ • ∇U = ∂U ∂t + (τ -σ)M a(e (σ-τ )M x)dσ • ∇U = ∂U ∂t + -σM a(e σM x)dσ • ∇U,(3.25)the last equality being gotten from periodicity of σ → e σM . Now, since ( e -σM a(e (σ)M x)dσ) does not depend on τ and because of (3.9), integrating(3.25) gives∂u ∂t + -σM a(e σM x)dσ • ∇u = 0.(3.26)Finally, since u(0, x) = (0, τ, x) dτ and U (0, τ, x) = V (0, e -τ M x) = u 0 (e -τ M x), the following Lemma is true Lemma Under assumption of Lemma 3.1, the Two-Scale limit U (t, τ, x) of (u ε ) and its weak-* limit u are solutions to ∂U ∂t -σM a(e σM x)dσ• ∇U = 0,(3.27)U |t=0 = u 0 (e -τ M x), -σM a(e σM x)dσ • ∇u = 0,(3.29)u |t=0 = (e -τ M x) dσ. (3.30) 3.7 Order 1 Homogenization -Strong Two-Scale convergence of U Because ∂(u ε ) 2 ∂t = 2u ε ∂u ε ∂t and ∇(u ε ) 2 = 2u ε ∇u ε , (3.31) multiplying (3.1) by 2u ε , we obtain that (u ε ) 2 is solution to:

Lemma 3 . 4

 34 If u 0 ∈ (L 2 ∩ L 4 ∩ C 0 )(R d ), then in addition to every already stated results, (u ε ) Strongly Two-Scale Converges to U.(3.41)Having this result, we know that (u ε -[U ] ε ) → 0, we now can show more: (u ε -[U ] ε )/ε Two-Scale Converges.

3. 8

 8 Order 1 Homogenization -Function W 1 In a first stage, from equation (3.1), (3.14) and (3.27) we deduce

1 0e 0 e 1 0e

 101 -σM a(e σM y) dσ • ∇V (t, y), (3.52) (using once again periodicity of τ → e τ M ) which is:W 1 (t, τ, y) = -τ -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • ∇V (t, y). (3.53) This allows us to compute [ W1 ] ε . In particular in (3.50), [ W1 ] ε |t=0 = 0 and if u 0 is regular (for instance in C 2 (R 2 )) in addition to assumptions of Lemma 3.4, because of equation (3.27) V satisfies, it is easily gotten that

V 1 1 0ee 1 0ee 1 0e 1 0ee 1 0ee -σM a(e σM y) dσ τ 0 e 1 0e 1 0-[∇∇V ] τ 0 e 1 0e 1 0e 1 0- 1 0e 0 e 1 0e(∇V ) dτ + 1 0e 0 e 1 0ee 1 0ee 1 0e 0 -e 1 0e 1 0e 0 e 1 0e(e 1 0e 1 0e

 1111110110111101101110110111 x) ∂γ ∂t (t, e -τ M x) + e -τ M a(x) • ∇γ(t, e -τ M x) dxdτ dt = ∂t a(x) • ∇ W1 γ(t, e -τ M x)dxdτ dt. (3.62) Making in (3.62) the change of variables (t, τ, x) → (t, τ, y = e -τ M x) gives (t, y) ∂γ ∂t (t, y) + e -τ M a(e τ M y) • ∇γ(t, y) dydτ dt = τ M a(e τ M y) • ∇W 1 γ(t, y)dydτ dt, (3.63) which is the weak formulation of ∂V 1 ∂t + -σM a(e σM y)dσ • ∇V 1 = τ M a(e τ M y) • ∇W 1 dτ, (3.64) V 1|t=0 = 0. (3.65) Now, it remains to express the right hand side of (3.64) using expression (3.53) of W 1 . For this we need to compute the time derivative of ∇V and the Jacobian matrices of -( τ 0 e -σM a(e σM y) dσ -τ 1 0 e -σM a(e σM y) dσ) and of ∇V (i.e. the Hessian matrix of V ). First, using the equation (3.23) satisfied by V , -σM a(e σM y) dσ T (∇V ) -[∇∇V ] -σM a(e σM y) dσ . -σM a(e σM y) dσ -τ -σM a(e σM y) dσ ∇ -σM a(e σM y) dσ T -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • (∇V ) dτ + -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • -σM a(e σM y) dσ dτ. (3.67) On another hand, e -τ M a(e τ M y) • ∇W 1 dτ = -τ M a(e τ M y) • ∇ τ -σM a(e σM y) dσ -τ -σM a(e σM y) dσ T -τ M a(e τ M y) • [∇∇V ]τ -σM a(e σM y) dσ -τ -σM a(e σM y) dσ = -σM a(e σM y) dσ -τ -σM a(e σM y) dσ e -τ M a(e τ M y) • (∇V ) dτ + -σM a(e σM y) dσ -τ -σM a(e σM y) dσ• e -τ M a(e τ M y) dτ.(3.68)As a consequence, the right hand side of (3.64) expresses as1 ∂W 1 ∂t -e -τ M a(e τ M y) • ∇W 1 dτ = -σMa(e σM y) dσ -τ -σM a(e σM y) dσ e -τ M a(e τ M y)-∇ -σM a(e σM y) dσ τ -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • e -τ M a(e τ M y) --σM a(e σM y) dσ dτ.(3.69)

e 1 0e 1 0e 0 e 1 0e 1 0∇ 1 0e 0 e 1 0e 0 e 1 0e 1 0e 1 0∇ 1 0e 1 0eTheorem 3 . 2 1 0e 1 0∇e 1 0eRemark 3 . 4

 110111010111113211134 -σM a(e σM y) dσ -τ -σM a(e σM y) dσ e -τ M a(e τ M y) -∇ -σM a(e σM y) dσ τ -σM a(e σM y) dσ -τ -σM a(e σM y) dσ • (∇V ) dτ = -e -τ M a(e τ M y) --σM a(e σM y) dσ τ -σM a(e σM y) -∇ -σM a(e σM y) dσ τ -σM a(e σM y) dσ + ∇ -σM a(e σM y) dσ τ -σM a(e σM y) dσ • (∇V ) dτ = ∇ e -τ M a(e τ M y) τ 0 e -σM a(e σM y) + -σM a(e σM y) dσ τ -σM a(e σM y) dσ • (∇V ) dτ (3.70) Using this and injecting in (3.64), allows us to claim the following Theorem. Under assumption of Lemma 3.5, function V 1 (t, y) linked by Lemma 3.6 with the Two-Scale limit U 1 (t, τ, x) of (u ε -[U ] ε )/ε is solution to ∂V 1 ∂t + -σM a(e σM y)dσ • ∇V 1 = (3.71) e -τ M a(e τ M y) τ 0 e -σM a(e σM y) dτ -σM a(e σM y) dσ -σM a(e σM y) dσ • (∇V ) Uniqueness of the solution of (3.71) and (3.72) leads that the whole sequence (u ε -[U ] ε )/ε converges.

  Order 0 Homogenization -Constraint . . . . . . . . . . . . . . . . . . . . . . . 3.5 Order 0 Homogenization -Equation for V . . . . . . . . . . . . . . . . . . . . 3.6 Order 1 Homogenization -Preparations: equation for U and u . . . . . . . . . 3.7 Order 1 Homogenization -Strong Two-Scale convergence of U . . . . . . . . . 3.8 Order 1 Homogenization -Function W 1 . . . . . . . . . . . . . . . . . . . . . 3.9 Order 1 Homogenization -A priori estimate and convergence . . . . . . . . . 3.10 Order 1 Homogenization -Constraint . . . . . . . . . . . . . . . . . . . . . . . 3.11 Order 1 Homogenization -Equation for V 1 . . . . . . . . . . . . . . . . . . . . 3.12 For the numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	1 Introduction
	1.1 On Two-Scale Convergence first statements

1 Introduction 1.1 On Two-Scale Convergence first statements . . . . . . . . . . . . . . . . . . . 1.2 How Homogenization brought the concept . . . . . . . . . . . . . . . . . . . . 1.3 A remark concerning periodicity . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A remark concerning weak-* convergence . . . . . . . . . . . . . . . . . . . . . 2 Two-Scale Convergence -Definition and Results 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Link with Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Injection Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Two-Scale Convergence criterion . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Strong Two-Scale Convergence criterion . . . . . . . . . . . . . . . . . . . . . 3 Application : Homogenization of linear Singularly Perturbed Hyperbolic Equations 3.1 Equation of interest and setting . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A priori estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Weak Formulation with Oscillating Test Functions . . . . . . . . . . . . . . . 3.4

where V is given as the solution of equation (3.23) and (3.24) which neither contain or generate small oscillations in its solution, W 1 is explicitly given in terms of V by (3.53) and where V 1 is also solution of a problem without oscillation: (3.71) and (3.72).

This approach can be used to build numerical methods called Two-Scale Numerical Methods. Such an approach was used in Ailliot, Frénod & Monbet [1], Frénod, Mouton & Sonnendrücker [13], Frénod, Salvarani & Sonnendrücker [15] and Mouton [20].


Vivekananda Umesh

Mrcp

MBBS Zita-Rose Manjalay

MBBS Jeban Ganesalingham

MBBS Jacqueline Simms

MD Christopher E Shaw

PhD Nigel Leigh

PhD Martin R Turner

PhD Ammar Al-Chalabi
email: ammar.al-chalabi@kcl.ac.uk

Frcp Dipstat

Low index-to-ring finger length ratio in sporadic ALS supports pre-natally defined motor neuronal vulnerability

Background:

The aetiology of apparently sporadic ALS is unknown but prenatal factors are known to influence disease development. In both men and women motor neurons require testosterone for survival and axonal regeneration after injury, and androgen insensitivity leads to a form of motor neuron degeneration in men. Reduction in the ratio of index to ring finger length (2D:4D ratio) is considered a surrogate marker for high prenatal testosterone levels in both men and women. We therefore tested the hypothesis that prenatal testosterone irrespective of gender is an independent risk factor for the development of ALS later in life, and that would be reflected in a lower 2D:4D ratio in both men and women with ALS.

Methods: Patients and unrelated control individuals attending a specialist tertiary referral centre for ALS were studied. A digital camera was used to photograph hands. Finger lengths were measured by four independent scorers blind to case-control status, and the mean 2D:4D ratio derived. Analysis was by linear regression and receiver-operator curve (ROC) analysis.

Results: Controlling for differences in sex ratio between groups, the 2D:4D ratio was lower for people with ALS (n=47) than controls (n=63) (r = -0.25, 2-tailed p = 0.009).

Conclusions: ALS patients have a lower 2D:4D ratio, consistent with higher prenatal circulating levels of testosterone, and possibly a prenatal influence of testosterone on motor neuron vulnerability in later life.

.

INTRODUCTION

ALS is a neurodegenerative disease of upper and lower motor neurons resulting in relentlessly progressive weakness with death from respiratory failure. Median survival is two years from symptom onset. The aetiology of apparently sporadic ALS is unknown. Outside families with known mutations, there is currently no knowledge of the 'at risk' population for ALS, although genetic factors are known to contribute to sporadic ALS disease susceptibility [START_REF] Al-Chalabi | An estimate of amyotrophic lateral sclerosis heritability using twin data[END_REF] and athleticism has been proposed to be an environmental risk factor [START_REF] Harwood | Physical activity as an exogenous risk factor in motor neuron disease (MND): a review of the evidence[END_REF] .

Testosterone plays an important role in the normal function of motor neurons, which express androgen receptors. A degenerative disease of motor neurons, X-linked spinobulbar muscular atrophy (Kennedy's disease), results from a trinucleotide repeat expansion in the androgen receptor gene [START_REF] Spada | Androgen receptor gene mutations in X-linked spinal and bulbar muscular atrophy[END_REF] . There is relative sparing of the oculomotor nuclei and Onuf's nucleus [START_REF] Sobue | X-linked recessive bulbospinal neuronopathy. A clinicopathological study[END_REF] , which is characteristic in ALS too. Kennedy's is thought to result from a combination of toxic gain-of-function as well as a loss of normal function of the androgen receptor. Neonatal androgen exposure in rats increases the ability of cortical neurons to respond to injury in adulthood [START_REF] Forgie | Manipulation of gonadal hormones in neonatal rats alters the morphological response of cortical neurons to brain injury in adulthood[END_REF] , and motor neuron numbers in the spinal nucleus of the bulbocavernosus in gerbils are increased by intrauterine testosterone exposure [START_REF] Forger | Intrauterine position affects motoneuron number and muscle size in a sexually dimorphic neuromuscular system[END_REF] . Studies of androgen insensitivity in transgenic mouse models of Kennedy's disease crossed with testicular feminization (androgen receptor null) mice, demonstrate that androgen receptor transactivation is critical for adult neuronal health and survival and occurs in direct proportion to the sensitivity of the androgen receptor [START_REF] Thomas | Loss of endogenous androgen receptor protein accelerates motor neuron degeneration and accentuates androgen insensitivity in a mouse model of Xlinked spinal and bulbar muscular atrophy[END_REF] .

Although there is no evidence for an association between alleles coding for larger androgen receptor trinucleotide repeats and ALS [START_REF] Garofalo | Androgen receptor gene polymorphisms in amyotrophic lateral sclerosis[END_REF] , testosterone is known to mediate motor neuron survival, dendritic length [START_REF] Costa | Diagnosis, pathogenesis and therapeutic targets in amyotrophic lateral sclerosis[END_REF] and axonal regeneration after injury [START_REF] Perez | Trophic effects of androgen: receptor expression and the survival of laryngeal motor neurons after axotomy[END_REF] , as well as being antiapoptotic [START_REF] Brooks | A cell culture model for androgen effects in motor neurons[END_REF] , which are all contemporary themes in the current understanding of ALS pathophysiology [START_REF] Rothstein | Current hypotheses for the underlying biology of amyotrophic lateral sclerosis[END_REF] , and it is reasonable to assume the androgen receptor is required for this protection to be mediated . Male sex is associated with higher prenatal levels of testosterone [START_REF] Manning | Second to fourth digit ratio and male ability in sport: implications for sexual selection in humans[END_REF] and also with an increased risk of ALS [START_REF] Johnston | Amyotrophic lateral sclerosis in an urban setting: a population based study of inner city London[END_REF] . We hypothesized that a high prenatal testosterone level rather than male sex itself is an independent risk factor for the development of ALS in later life. There is evidence that the ratio of the index and ring finger lengths (2D:4D ratio) is sexually dimorphic because it is dependent on prenatal testosterone levels, with males having on average a lower ratio (relatively longer ring fingers) [START_REF] Manning | The ratio of 2nd to 4th digit length: a predictor of sperm numbers and concentrations of testosterone, luteinizing hormone and oestrogen[END_REF] . We therefore tested the hypothesis that prenatal testosterone, as an independent risk factor for the development of ALS later in life, would be reflected in a lower 2D:4D ratio for both men and women with ALS.

METHODS

Clinical methods

Consecutive patients and unrelated individuals attending a specialist tertiary referral centre for ALS were invited to participate. The diagnosis of ALS was made by two consultant neurologists after exclusion of other conditions. The study was ethically approved by the Joint South London and Maudsley and Institute of Psychiatry Research Ethics Committee and all participants provided written informed consent prior to inclusion.

Finger length measurements

A digital camera was used to photograph hands. Photographs were taken with the hand supinated and the fingers flattened to full extension by a sheet of Perspex with the camera placed over the center of the Perspex sheet. Digit length was measured from the basal crease of the digit to the tip using the measurement tool in Adobe Photoshop. This measurement is known to show a high degree of repeatability [START_REF] Manning | Digit ratio. A pointer to fertility, behaviour and health[END_REF] . The right hand was chosen for measurement where possible. People with faint creases and those with contractures would not be reliably measurable in this study. Rather than selecting patients for these features before entry, which might be biased and arbitrary, we invited consecutive patients to participate and then used an objective rule to exclude hands with poor measurability. Images were measured by four independent scorers (UV, JG, ZM, JS) who were blinded to case-control status. Photographs with contractures resulting in finger flexion or supination were identified by flattening and whitening of skin at the finger tips and thenar eminence by the Perspex sheet, and excluded from analysis. Those in which the measurements made by the two scorers differed by more than one standard deviation were excluded from analysis. Remaining measurements were used and the average ratio taken as the result for that subject.

Statistical methods

Digit ratio was transformed into parametric data using a ladder of powers model and analysed with a paired t-test. Analyses were repeated without outliers, i.e. ratios that lay outside one standard deviation of the normal distribution, in case of bias introduced by extreme measurements. To confirm the sex ratio between patient and control groups did not affect the results we also performed linear regression with gender included as a covariate.

SPSS was used for calculations.

RESULTS

There were 141 subjects photographed (73 patients, 68 controls). Twenty one with contractures were excluded. The remaining 120 photographs were scored. Of these, ten could not be measured by one of the four scorers. The remaining 110 ratios were used for analysis. There were 47 cases (33 males, 70 %) and 63 controls (20 males, 32 %).There was no overall difference in 2D:4D ratio between males and females (1-tailed p = 0.41).

However, the 2D:4D ratio was lower for the ALS group versus controls (p=0.007), and in linear regression with sex as a covariate this difference was independent of the sex ratio difference between the groups. The ROC curve (Figure 1) showed that the 2D:4D ratio was a discriminator of case-control status (area under the curve = 0.70, p = 0.0001), with a sensitivity of 72% and specificity of 65%, a positive predictive value of 61% and negative predictive value of 71%.

DISCUSSION

This study has demonstrated that a group of ALS patients have a lower 2D:4D ratio in comparison with a control group, and crucially that this is independent of sex and so not explained by the higher proportion of males in the patient group. This finger length ratio is thought to be a marker of high prenatal testosterone levels and our results are therefore consistent with the hypothesis that testosterone levels during development modify the subsequent risk of ALS. The digit ratio has no use as a screening tool, but the ROC curve is another way of understanding the difference between the case and control 2D:4D ratios.

One limitation of this study was the exclusion of those with contractures. We attempted to limit bias by inviting every patient to participate and then analyzing photographs for measurability using an objective measure of repeatability. Nevertheless the palms with contractures are more likely to be from cases, so the analysis could not be performed in a truly blinded fashion. There is however no reason to believe that those with contractures are more likely to have higher digit ratios.

The ratio of the index and ring finger lengths (2D:4D ratio) is sexually dimorphic [START_REF] Baker | Anthropological notes on the human hand[END_REF] , with males having on average a lower ratio (relatively longer ring fingers) [START_REF] Phelps | Relative index finger length as a sex-influenced trait in man[END_REF] . There was no overall effect of sex on 2D:4D ratio in this study, which might be regarded as surprising given the role of androgens in determining sexual differentiation. However, the effect of sex on 2D:4D ratio is not strong because of the large variance of the sexual dimorphism of finger lengths, and larger sample sizes than we have used are required to detect it [START_REF] Baker | Anthropological notes on the human hand[END_REF] .

Several studies hypothesize that it is prenatal testosterone levels and sensitivity to the effects of testosterone, rather than sex, that are linked most strongly with 2D:4D ratio 15;19 , and the use of 2D:4D in investigating the effects of prenatal androgens has been reviewed in comparison to other established methods [START_REF] Mcintyre | The use of digit ratios as markers for perinatal androgen action[END_REF] . While direct evidence for prenatal testosterone effects on 2D:4D digit ratio is not strong, a study of 33 children who had amniocentesis and were subsequently followed, supported this interpretation showing a non-significant trend towards a correlation with intrauterine testosterone and a significant association with the ratio of testosterone to estradiol [START_REF] Lutchmaya | 2nd to 4th digit ratios, fetal testosterone and estradiol[END_REF] . Although another study did not find a relationship between second and third trimester testosterone and 2D:4D ratio in females [START_REF] Hickey | Maternal and umbilical cord androgen concentrations do not predict digit ratio (2D:4D) in girls: a prospective cohort study[END_REF] , strong indirect evidence is that women and men with congenital adrenal hyperplasia, which leads to greatly elevated androgen levels in utero, have a low 2D:4D ratio 23;24 . Thus 2D:4D ratio is likely to be a surrogate marker for intrauterine testosterone exposure. Because high prenatal testosterone is associated with sporting ability [START_REF] Manning | Endurance running and digit ratio (2D:4D): implications for fetal testosterone effects on running speed and vascular health[END_REF] , and to 2D:4D ratio in women [START_REF] Manning | Digit ratio. A pointer to fertility, behaviour and health[END_REF] , it is conceivable that this is a mechanism by which the association between ALS and sporting prowess has been observed. 26 27 The 2D:4D ratio does not predict adult circulating testosterone levels [START_REF] Honekopp | Second to fourth digit length ratio (2D:4D) and adult sex hormone levels: new data and a meta-analytic review[END_REF] and is therefore compatible with the observation that people with ALS have low levels of free testosterone [START_REF] Militello | The serum level of free testosterone is reduced in amyotrophic lateral sclerosis[END_REF] .

The concept of intrauterine factors influencing later disease in adult life is now firmly established [START_REF] Barker | The fetal and infant origins of adult disease[END_REF] , and there is evidence that intrauterine testosterone affects neuronal function in adulthood.

High intra-uterine testosterone levels might reflect pre-existing androgen insensitivity, but our reported association between low 2D:4D ratio and ALS nonetheless supports the possibility that very early factors in embryonic development might influence adult-onset motor neuronal degeneration.

FIGURE 1. The ROC curve for 2D:4D ratio as a predictor of ALS-control status.

The optimal discriminatory position in ROC space is the top left corner, which has 100% sensitivity and specificity. The diagonal line shows the expected plot for a random, nondiscriminatory test.

ACKNOWLEDGEMENTS

We thank the UK Medical Research Council and the Motor Neurone Disease Association of Great Britain and Ireland for support. AAC also received support from the Myrtle Sketchley Fund.