R Tahraoui

Anisotropic asymptotic behaviour for second order differential operators in non-divergential form

Keywords: 35J25, 35K20, 35B50

For any matrix

where

satisfies a differential condition.

I -Introduction.

In this work we study the asymptotic behaviour with respect to x of the solution u(•) of the following equation

1 -T r A(x)D 2 u = f (x), x ∈ R N , N 3 lim ||x||→+∞ u(x) = 0, u(x) > 0, ∀x ∈ R N
where f (•) is a bounded positive function with a compact support. The matrix A(x) = a ij (x) ij is bounded symmetric and uniformly elliptic that is to say : for any x ∈ R N α||y|| 2 A(x)y, y) β||y|| 2 ∀y ∈ R N

where α and β are real constant such that 0 < α < β, D 2 u stands for Hessian matrix of u(•), T r(B) stands for the trace of the matrix B and x → ||x|| stands for the euclidian norme of R N . The result depends to the behaviour of the function [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] ∧(x) := ∧ A (x) := T r A(x)

2 ||x|| 2 A(x).x, x
as ||x|| goes to infinity. Without hypotheses concerning the matrix A(•) we prove the following result : for any ε > 0 there exist three real positive constants c 1 := c 1 (ε), c 2 := c 2 (ε) and R := R(ε) such that for any x ∈ R N , ||x|| R, we have where

c 1 ||x|| 2(θ-1-ε) u(x) c 2 ||x|| 2(θ-1-ε)
∧(A, B)(x) = T r √ B -1 .A( √ B.x). √ B -1 2 ||x|| 2 √ B -1 .A( √ B.x). √ B -1 x, x .
Let us remark that θ > 1. This function ∧(•) plays a central role in our work. It is called the function of the spectral dispersion of the matrix A(•), in short the spectral dispersion of A. This name is justified by the following inequalities :

1 < N 2 . α β N 2 λ 1 (A) λ N (A) ∧(x) N 2 λ N (A) λ 1 (A) N 2 . β α .
These inequalities give an idea of the distribution of the spectrum of the matrix A(•) :

oscillation (∧) N 2 sup x λ N A(x) λ 1 A(x) -inf x λ 1 A(x) λ N A(x) β 2 -α 2 α.β . N 2
where the ordered real numbers 0 < λ 1 (A) • • • λ N (A) stand for the eigenvalues of A(•).

In some sens the previous result is optimal. Indeed in the case A is the identity matrix (or equivalently a constant matrix) we have θ = θ = N 2 > 1 and we obtain the classical optimal result : u(x) ∼ c ||x|| N -2 as ||x|| goes to infinity. Let us specify that there is no evident link between the asymptotic behaviour of ∧(•) and the asymptotic behaviour of A(•), as ||x|| goes to infinity. Indeed, for instance, there exist matrices A(•) such that, for any x, A(x) belongs to Q(α, β) = {M symmetric matrix / αI M βI} and has oscillating coefficients and for which the associated function ∧(•) goes to some constant as ||x|| goes to infinity. And there exist matrices A(•) having a function ∧(•) of spectral dispersion which is oscillating as ||x|| goes to infinity i.e. which satisfies :

lim R→+∞ sup ||x|| R ∧ A(x) -inf ||x|| R ∧ A(x) > 0,
with the notation ∧ A(x) := ∧(x). These two examples are explained by the fact that there are many matrices belonging to Q(α, β) and having the same function ∧(•) satisfying :

N 2 . α β ∧(x) N 2 . β α ∀x.
In section II we will give theses results with some details. To give a classification of this problem, we introduce the function which follows :

B ∈ Q(α, β) -→ J ∞ (B) = lim R→+∞ J R (B)
where

J R (B) = sup ||x|| R λ N B -1 .A(x) -inf ||x|| R λ 1 B -1 .

A(x) .

There exists

B ∞ belonging to Q(α, β) such that 0 J ∞ (B ∞ ) J ∞ (B) ∀B ∈ Q(α, β). If J ∞ (B ∞
) = 0, the matrix A(•) goes to a constant matrix as ||x|| goes to infinity and after a change of variables by a suitable transformation in [START_REF] Crandall | Lions User's guide to viscosity solutions of second order partial differential equation Bull[END_REF] we find that ∧(x) goes to the constant N 2 as ||x|| goes to infinty. And roughly speeking the asymptotic behaviour of u(•) is given by [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] : u(x) ∼ c ||x|| N -2 This result is also given by [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] in the exterior problem with f = 0 i.e. u(x) = 1 ||x|| N -2 . If J ∞ (B ∞ ) = 0, we have two cases to study. i) after a change of variable in (1) by a suitable transformation, if necessary, the behaviour of ∧(•) is radially oscillating as ||x|| goes to infinity. Let us suppose that this behaviour is given by, for instance, γ(||x|| 2 ). Then, under some hypothesis, the work in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] give the result :

u(x) ∼ c.v(||x|| 2 ) ∀x, ||x|| R,
for R > 0 large enough and where

v(||x|| 2 ) = +∞ ||x|| 2 exp - s R 2 γ(σ) σ dσ ds
-( cf remark 4.1 section III ) -.

ii) the function ∧(•) is oscillating anisotropically in a neighborhood of the infinity. This case is the aim of our study. Under a suitable hypothesis called (H) and satisfied by ∧(•) in a neighborhood of the infinity, we prove the following result :

u(x) ∼ c(x) ||x|| 2 ∧(x)-1 , 0 < c 0 c(x) c 1 , ∀x, ||x|| R ,
where R > 0 is large enough.

Let us specify that our hypothesis (H) is satisfied by the constant functions. A particulary interesting case is the following : Let us suppose that ∧(•) verifies ( 2)

∧(λx) = ∧(x) ∀x, ||x|| R, R large enough ∀λ 1 .
The relation ( 2) is satisfied, for instance, in the following case : There exists R > 0 large enough such that for any x 0 , ||x 0 || R we have A(λx 0 ) = A(x 0 ) ∀λ 1. Under some conditions, the result of this example can be explained as following : ∀x, ||x|| R we have

∀ λ 1 u(λx) = c(λx) |λ| 2 ∧(x)-1
where the funtion c(•) is such that 0 < c 0 c(x) c 1 . Our assumption (H) is a differential inequality satisfied by the function ∧(•). Roughly speaking

∧(•) satisfies c(x).T r A(x)D 2 ∧ + b(x). A(x)D∧, D ∧ + a(x). A(x)D∧, x ∈ - c(c, b, a) 1 + ||x|| 2(∧(x)+δ(x)) , c(c, b, a) 1 + ||x|| 2(∧(x)+δ(x))
for any x, ||x|| R > 0, where c(•), b(•) and a(•) stands for some given functions and δ(x) δ 0 > 0 such that δ 0 > θ -θ. Finally, note that after the publication of our work in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF], we discovered that when the matrix A(•) is constant at infinity -(i.e. there exists a constant matrix A 0 such that [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] for the exterior domain with f = 0 and ∧(•) was used. But this result [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] does not apply in the case ∧(x)

lim ||x||→+∞ A(x) = A 0 ) -,the behaviour u(x) = 1 ||x|| N -2 was established in
:= ∧(A)(x) = N 2
, ∀x, and A is an oscillating matrix -(cf proposition 3 section II)

-, where u(•) behaves like 1 ||x|| N -2 as ||x|| goes to infinity. Finally we note that the most part of the paper [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] is devoted to the question of the well_posedness of problems like [START_REF] Crandall | Lions User's guide to viscosity solutions of second order partial differential equation Bull[END_REF]. The paper is organised as follow. A wide part of this paper is devoted to justify our assumption (H). That is the goal of the preparatory part in section II. In the third section, we give a classification of asymptotic behaviour of the matrix x -→ A(x) when ||x|| tends to infinity and we shall need later. And we give a new presentation of previous results [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF]. Section IV will be devoted to establish the asymptotic behavior of the solution u(•) of (1) when the function ∧(•) oscillates anisotropically as ||x|| goes to infinity. The main result is obtained under the key hypothesis (H). The guiding idea of our method is to consider that A(•) and ∧(•) are two independant datas and to construct two functions u i (•) -(i = 1, 2)respectively solution of

(P i ) -T r(A i D 2 u i ) = f in R N u i (x) > 0, lim ||x||→+∞ u i (x) = 0, i = 1, 2 .
and having the same asymptotic behaviour as the solution u(•) of (1). The matrices A i (•) -(i = 1, 2) -are two adequates matrices satisfying : A 1 (x) = A 2 (x) = A(x) for any x, ||x|| R and for R > 0 large enough. Next we construct a super-solution H 1 (•) of (P 1 ) and a subsolution H 2 (•) of (P 2 ) . Next we establish that H 1 (•) and H 2 (•) have the same asymptotic behavior. And the final result follows from the comparison principle [START_REF] Crandall | Lions User's guide to viscosity solutions of second order partial differential equation Bull[END_REF], [START_REF] Protter | Principles in Differential Equations[END_REF].

II -Preliminary results preparing to justify the main assumption (H).

Let us give two real positive numbers α and β, 0

< α < β such that N 2 > β α . Let us set Q(α, β) = M ∈ S N ×N + / α||x|| 2 (M x, x) β||x|| 2 , ∀x ∈ R N
where S N ×N + stands for the set of symmetric definite positive matrices. For any x and y belonging to R N , (x, y) stands for the inner product of R N and ||x|| the associated norm of x. Let us give a function A : x -→ A(x) ∈ Q(α, β). In order to simplify the text we note

A(x) ∈ Q(α, β) ∀x ∈ R N or A(•) ∈ Q(α, β).
We have the following result.

Theorem 1. Let E be a subset of R N . Let ∧(•) be a function satisfying N.α 2β ∧(x) N.β 2α , ∀x ∈ E ⊆ R N \ {0}.
Then for any x ∈ E there exists an uncountable sub-set M(x) of Q(α, β) such that for any

A(x) ∈ M(x), T r A(x) 2 ||x|| 2 . A(x).x, x = ∧(x) ∀x ∈ E.

This means that ∧(•) is the spectral dispersion of A(•).

Proof . To proove this result we need two steps.

First step. For any x ∈ E let us give a partition

I 1 (x), I 2 (x), I 3 (x) of the set [1, 2, • • • , N ]. Let us set γ j (x) := i∈I j (x)
x 2 i , n j = card I j (x) , ∀j = 1, 2, 3 .

First, we look for a diagonal matrix A(•) belonging to Q(α, β) such that for any

x ∈ E a ii (x) = d j , ∀i ∈ I j (x) , ∀j ∈ [1, 2, 3] .
The unknown real numbers d j satisfies α d j β. Let us give an arbitrary function θ(•) such that αN θ(x) βN , ∀x ∈ E. The real numbers d j satisfy the following equations :

(3)

T rA := θ = 3 i=1 n i .d i , 3 i=1 λ i .d i = θ 2∧
where

∧(x) = T r A(x) 2 (A(x).x, x) ||x|| 2 and λ i (x) = γ i (x) ||x|| 2 , 3 i=1 λ i = 1
To solve the equations (3) we consider the following problem for any fixed x ∈ E :

(P) : inf J(d) = 3 i=1 d 2 i / d = (d 1 , d 2 , d 3 ) , α d i β , 3 i=1 n i d i = θ , i λ i d i = θ 2∧
This problem has a unique solution. So we found a diagonal matrix A(•) that the spectral dispersion is ∧(•).

Second step. To find an unreduced form of A(•), we introduce an arbitrary orthogonal transformation whose the matrix is B(•). Let us set θ(x) := θ B -1 (x).x and ∧(x) = ∧ B -1 (x).x , where θ(•) is the arbitrary function introduced in the first step. We denote by D(x) the diagonal matrix founded in the previous step and corresponding to the two functions θ(x) and ∧(x). After some elementary computations the matrix A(x) = B -1 (x).D B(x).x .B(x) satisfies :

∧(x) = ∧ A (x) := T r A(x) 2 (A(x).x, x) ||x|| 2
∀x ∈ E and thus A(x) solve our problem. In addition

x -→ A(x) is regular in R N \ {0}.
For any A(•) ∈ Q(α, β) we denote by ∧(A)(•) the associated spectral dispersion. There is no evident link between the asymptotic behavior of A(•) and the asymptotic behavior of

∧(A)(•) = ∧(•).
The following results show this fact.

Corollary 2. For any k ∈ N α β , N β α there exists a non constant matrix A(•) belonging to Q(α, β) such that ∧(A)(x) = k, ∀ x = 0. In addition A(•) does not have a limit when ||x|| goes to infinity. In fact we can prove a more precise result. Proposition 3. There exists a matrix A(•) having oscillating coefficients and such that ∧(A)(x) = constant for any x.

Proof . First, we are going to construct a such matrix in R 3 , and next we give its extension in R N . For this let us set y := (t, x) = (t, x 1 , x 2 ) ∈ R 3 . Let us give two functions y -→ a(y) such that 0 < 1 a(y) 2, ∀ y, for instance, and x -→ δ(x) = sin x 1 . sin x 2 . Let us set

γ(x) :=    δ(x) 2 x 1 .x 2 ||x|| 2 ∀ x = (x 1 , x 2 ) ∈ R 2 , x = 0; 0 if x = 0
The searched matrix is

A 0 (y) =   a(y) + γ(x) λ(x) 0 λ(x) a(y) -γ(x) 0 0 0 a(y)   where λ(x) = - δ(x) 2 . x 2 1 -x 2 2 ||x|| 2 , ∀ x ∈ R 2 , x = 0 and λ(0) = 0. Indeed after some computations we obtain that ∧(A 0 )(y) := ∧(y) = 3 2 , ∀ y ∈ R 3 .
In the case N > 3 we chose a(y) = a 0 , ∀ y ∈ R 3 and, for instance, the searched matrix is

A =        A 0 0 • • • 0 0 0 A 0 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • A 0 0 0 0 • • • 0 I       
where I is some identity matrix (n, n).

Proposition 4 . There exists a definite positive matrix A such that x -→ ∧(A)(x) is an oscillating function as ||x|| goes to infinity.

Proof . We can chose the following example

A(y) = A(x, t) =   k + λ 2 (x) λµ 0 λµ k + µ 2 (x) 0 0 0 k   where y = (x, t) = (x 1 , x 2 , t) ∈ R 3 , k = constant > 0, λ(x) = x 2 ||x|| cos(||x||) + 2 , µ(x) = x 1 ||x|| cos(||x||) + 2 .
We have

∧(A)(y) = ∧(y) = 3k + (cos(||x||) + 2) 2 2k
. This case of ∧(•) is interesting for the section III-B.

Approximation lemma. Let us give two regular matrices A 1 (•) and A 2 (•) belonging to Q(α, β). Then for any R 1 and R 2 , 0 < R 1 < R 2 there exists a regular matrix A(•) belonging to Q(α, β) such that :

A(x) = A 1 (x) ∀x, ||x|| R 1 , A 2 (x) ∀x, ||x|| R 2 .
In addition if d

∧ A 1 (x) c ∀x, ||x|| R, d ∧ A 2 (x) c ∀x, ||x|| R, we have d ∧ A (x) c ∀x, with R 1 < R < R 2 .
Proof. Let us give a regular increasing function Ψ : R + -→ [0, 1] such that Ψ(t) = 0, ∀t R 1 ; Ψ(t) = 1, ∀t R 2 . Let us set ϕ(x) = Ψ(||x||). Then we are going to prove that the matrix A(x) = 1 -ϕ(x) A 1 (x) + ϕ(x)A 2 (x) is the matrix that is our solution. Indead A(•) is regular and satisfies α||x|| 2 A(x).x, x β||x|| 2 and thus A(•) ∈ Q(α, β). The last result follows from the following remark : for any real numbers (a

1 , b 1 , a 2 , b 2 , c) ∈ (R +, * ) 5 such that a 1 b 1 c, a 2 b 2 c we have ta 1 +sa 2 tb 1 +sb 2 c, for any (t, s) ∈ (R + ) 2 such that tb 1 + sb 2 > 0.
The second inequality can be established in the same way.

Proposition 5. Let M be a symmetric positive definite matrix. Let us consider the halfstraight line of matrices

D(O, M ) = B/B = t.M , t > 0 . Then we have D(O, M ) ∩ Q(α, β) = ∅ if and only if β α λ N (M ) λ 1 (M )
. In addition we have

D(O, M ) ∩ Q(α, β) = B = t.M / t ∈ α λ 1 (M ) , β λ N (M )
.

If β α = λ N (M ) λ 1 (M ) we have D(O, M ) ∩ Q(α, β) = B = α λ 1 (M ) .M = B = β λ N (M ) .M .
And in this case D(O, M ) is called an extremal ray passing through the zero matrix O.

Proof . It is sufficient to remark that tM belongs to Q(α, β) if and only if

α t λ 1 (M ) • • • λ N (M ) β t .
Let us consider the set P of supporting hyperplanes of Q(α, β) passing through the zero matrix O. We set :

Q c (α, β) := P ∈P P ∩ Q(α, β) = T ∈ Q(α, β) / α = λ 1 (T ) λ i λ N (T ) = β i = 2, • • • , N -1
Theorem 6. Let V be a (N, N )-symmetric matrix. Then there exists a matrix A belonging to Q(α, β) such that T r(A.V ) = 0 if and only if (4)

   βλ N (V ) -αλ 1 (V ) βλ 1 (V ) -αλ N (V ) 0 (β -α)|||V ||| 2 βλ N (V ) -αλ 1 (V ) T rV if T rV 0 (β -α)|||V ||| 2 βλ 1 (V ) -αλ N (V ) T rV if T rV 0
Proof . Let us consider the matrices plan Vect{I, V } generated by the two matrices I and V . We have Vect{I,

V } ∩ Q(α, β) ∩ Q c (α, β) = {W 1 , W 2 } ⊆ Q c (α, β) i.e. λ 1 (W i ) = α, λ N (W i ) = β i = 1, 2 and α λ j (W i ) β, ∀ i = 1, 2 ; ∀ j = 2, • • • , N -1.
We have aI + bV = W 1 with a and b belonging to R. This entails that V and W have the sames eigenspaces. We equip then R N with the basis of the eigenvectors of V . We obtain

a = αλ N (V ) -βλ 1 (V ) λ N (V ) -λ 1 (V ) , b = β -α λ N (V ) -λ 1 (V ) > 0 , α λ i (W 1 ) = a + bλ i (V ) β, i = 2, • • • , N -1. And thus T r(W 1 .V ) = aT rV + b|||V ||| 2 .
In the same way we have cW 1 + dV = W 2 . The matrices W 1 , V and W 2 have the same eigenspaces. And as previously we obtain :

d = β 2 -α 2 βλ 1 (V ) -αλ N (V ) , c = βλ N (V ) -αλ 1 (V ) αλ N (V ) -βλ 1 (V ) , α λ i (W 2 ) = cλ i (W 1 ) + dλ i (V ) β , i = 2, • • • , N -1. Since VectV Q(α, β) -Q(α, β) = ∅, we have c 0. And thus T (V.W 2 ) = acT rV + (d + bc)|||V ||| 2 i.e. after some calculations T r(V.W 2 ) = βλ N (V ) -αλ 1 (V ) λ N (V ) -λ 1 (V ) .T rV - (β -α).|||V ||| 2 λ N (V ) -λ 1 (V ) .
It is easy to see that a necessary and sufficient condition that there exists

A ∈ Q(α, β) such that T r(A.V ) = 0 is that we have (5) T r(W 1 .V ).T r(W 2 .V ) 0, and c 0.
And (5) is equivalent to (4).

Remarks 1. 4) is satisfied, where S N ×N stands for the set of symmetric matrices.

1) Let us set

X := M ∈ S N ×N / T r(M.V ) = 0 . Then any M belonging to X ∩ Q(α, β) satisfies T r(V.M ) = 0 that is to say there exist many M belonging to Q(α, β) such that T r(M.V ) = 0 if (
2) Let V and R be two (N, N )-symmetric matrices. We can prove that there exists

A ∈ Q(α, β) such that T r(AV ) = T r(AR) = 0 if we have : (6)            |||V ||| 2 .|||R||| 2 -(R : V ) 2 = 0 and 1 - 1 N |||(T rV ).R -(T rR).V ||| 2 |||V ||| 2 .|||R||| 2 -(R : V ) 2 1 N - (α + β) 2 + (N -2)(α 2 + β 2 ) 2 (α + β) 2 (α 2 + β 2 ) + (N -2)(α 2 + β 2 ) 2
The proof follows from a straightforward calculation and classical idea.

Application. Let B(O, ε) be the open ball of R N , of radius ε, centred at O. Let us give a C 2 bounded function ϕ : R N \ B(O, ε) -→ R +, * such that ϕ(x) ϕ 0 > 0 ∀ x,
where ϕ 0 is a real positive number. Let us define the following matrix

x -→ M (ϕ)(x) := D 2 ϕ -γ(x)∇ϕ(x) ⊗ ∇ϕ(x) + kH x, ∇ϕ(x)
where

H x, ∇ϕ(x) ij = ϕ(x) 2 x i ||x|| 2 . ∂ϕ ∂x j + x j ||x|| 2 . ∂ϕ ∂x i ij , k is a real number and γ(•) is a given function defined on R N \ B(O, ε).
In section IV we use the following corollary to justify our essential hypothesis (H).

Corollary 7. Suppose that the matrix M (ϕ) verify the differential inequalities (4) of the theorem 6. Then there exists

x -→ A(x) ∈ Q(α, β) such that -T r(AD 2 ϕ) -γ(x) A∇ϕ, ∇ϕ + k ||x|| 2 A∇ϕ, x = 0 ∀ x ∈ R N \ B(O, ε)
Proof . For any B 1 and B 2 belonging to Q(α, β), we have :

(B 1 V, V ) = T r(B 1 .C 1 ) ∀ V = (v 1 , • • • , v N ) ∈ R N with C 1 = V ⊗ V = (v i .v j ); and T r B 2 .H(x, ∇ϕ) = 1 ||x|| 2 B 2 ∇ϕ, x
From this we apply the theorem 6 to obtain the result.

Remark 2. If we add the additional condition that ϕ(•) is the spectral dispersion of the matrix A that we are looking for, we can use (6) instead of corollary 7, with

V = M (ϕ) and R = I -2 ϕ(x) ||x|| 2 x ⊗ x, to conclude that there is A belonging to Q(α, β) such that T r(A.V ) = T r(A.R) = 0 if V and R satisfy (6). Remark 3. Given a matrix x -→ A(x) ∈ Q(α, β),
sup ||x|| R ϕ(x) -lim R→+∞ inf ||x|| R ϕ(x) > 0 ) -and such that we have (7) -T r(AD 2 ϕ) -γ(x) A∇ϕ, ∇ϕ + k.ϕ ||x|| 2 A∇ϕ, x = 0 for any x, ||x|| > R > 0.
Since any constant function is a solution of (7), it seems difficult to answer this question in general. However there are some encouraging examples which show that (7) has nontrivial solutions in general. A particular case of (7) is the following equation that we will encounter later -(section IV)-:

(8) -u + Log ||x|| 2 .|∇u| 2 + u x.∇u ||x|| 2 = 0, ||x|| > R.
Proposition 8. The equation ( 8) has many nontrivial solutions, for R > 0 large enough.

Proof . The result is technical, but it is straightforward. So we will give the essential steps of the proof. We construct a family of solutions u(•) in the form u(x) = ϕ Log(||x|| 2 ) .

We establish an o.d.e. satisfied by the function t -→ ϕ(t), where t plays the role of Log(||x|| 2 ) > t 0 > 1. We solve this o.d.e. by setting

ϕ(t) = f v(t) where (9) f (σ) = - 2 c Arctg σ c + d ∀ σ σ 0 > 0 with c > 0 and d > π 2 + N -1 2 . The function v(•) is solution of the following Cauchy-Lipschitz problem : (10) v (t) = G t, v(t) , ∀ t t 0 > 1 v(t 0 ) = v 0 , v 0 > σ 0 > 0 where G(t, σ) = t t 0 g(θ) dθ - N -2 2 σ + F (σ) ∀ σ t 0 and ∂G ∂σ (t, σ) N -2 2 + π 2 + d ; g(•)
is an arbitrary continuous function with compact support and F (•) is a primitive of f (•) :

F (σ) = - 2σ c Arctg σ c + d.σ + 1 c Log 1 + σ c 2 .
Thus we obtain :

(11) u(x) = - 2 c Arctg v Log(||x|| 2 ) + d is a solution of equation (8) in R N \ B(0, R)
where c,d and g are somewhat arbitrary datas.

III -Isotropic asymptotic behavior. This section gives a new presentation of the results already established in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] and allows us to understand the classification of the behavior at infinity of matrices and associated solutions. As we have already mentioned, the asymptotic behavior of the solution u(•) of ( 12)

   -T r A(x)D 2 u = f (x) in R N , lim ||x||→+∞ u(x) = 0 , u(x) > 0 ∀ x , with f (x) 0 , supportf = compact,
depends on the behavior of the function

∧(A)(x) = T r A(x) 2 ||x|| 2 (A(x).x, x)
. From the results of section II there is no evident links between the behavior of A(•) and that of ∧(A)(•). However there are some precise situations to be distinguished.

First case. The matrix A(•) is constant. By a change of variables x = B -1 y with B = √ A -1 , the equation (12) becomes (13) -ω(y) = g(y) = f (B -1 y) u(x) = ω(Bx) , lim ||y||→+∞ ω(y) = 0 , ω(y) > 0 .
For (13) the asymptotic behavior of ω(•) is known -(cf [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] for instance) -:

ω(y) c ||y|| N -2 . And thus u(x) c ||x|| N -2 , since ∧(I)(x) = N 2 ∀ x.
Second step. The matrix A(•) is not constant. In this case we see that it suffices to examine the behavior at infinity of A(•) and ∧(A)(•) [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF]. In the following our aim is to examine the oscillations of A(•) and

∧(A)(•). The matrix A(•) belongs to Q(α, β). Let B be a constant matrix belonging to Q(α, β). For any x, B -1 A(x) has n eigenvalues λ i (x) = λ i (B -1 A) such that α β λ 1 (x) • • • λ N (x) β α
where for any x ∈ R N , any i,

1 i N , λ i (x) = λ i B -1 A(x)
stands for the i th eigenvalue of the following problem :

A(x).r = λ i (x)B.r
where r ∈ R N is an eigenvector associated to λ i (x). For any R > 0, we introduce the following function

J R : Q(α, β) -→ R + defined by J R (B) = sup ||x|| R λ N B -1 A(x) -inf ||x|| R λ 1 B -1 A(x) . Proposition 9. The set of functions J R (•) R is 2β 2 α 3 -lipschitz. Proof . It is sufficient to prove that the function B -→ sup ||x|| R λ N B -1 A(x) and B -→ inf ||x|| R λ 1 B -1 A(x)
are uniformly lipschitz. For any y = 0 we have

B 1 y, y A(x)y, y - B 2 y, y A(x)y, y |||B 1 -B 2 ||| α or again - |||B 1 -B 2 ||| α + inf y =0 B 1 y, y A(x)y, y inf y =0 B 2 y, y A(x)y, y inf y =0 B 1 y, y A(x)y, y + |||B 1 -B 2 ||| α i.e. ∀ x ∈ R N , - |||B 1 -B 2 ||| α + 1 λ N B -1 1 A(x) 1 λ N B -1 2 A(x) 1 λ N B -1 1 A(x) + |||B 1 -B 2 ||| α .
This implies

sup ||x|| R λ N B -1 1 A(x) -sup ||x|| R λ N B -1 2 A(x) β α 2 |||B 1 -B 2 ||| α
From the same way we show that

inf ||x|| R λ 1 B -1 1 A(x) -inf ||x|| R λ 1 B -1 2 A(x) β α 2 |||B 1 -B 2 ||| α
and thus the result follows : for any R > 0

J R (B 1 ) -J R (B 2 ) 2β 2 α 3 |||B 1 -B 2 ||| Proposition 10. J R (•) converges uniformly to J ∞ (•) in Q(α, β) as R goes to infinity. In addition J ∞ (•) is 2β 2 α 3 -lipschitz and there exists B ∞ belonging to Q(α, β) such that, for any R > 0, we have 0 J ∞ (B ∞ ) J ∞ (B) J R (B) , ∀ B ∈ Q(α, β).
Proof . It is easy. We use Ascoli-Arzelà's theorem, the compactness of Q(α, β) and the fact that R -→ J R (•) is decreasing. Thus it follows that there exists

B ∞ ∈ Q(α, β) such that 0 J ∞ (B ∞ ) = inf B∈Q(α,β) J ∞ (B) J R (B) , ∀ B ∈ Q(α, β). Definition 1. The real number J ∞ (B ∞ ) = inf B∈Q(α,β) J ∞ (B)
is called the amplitude of the oscillations of the spectrum of A(•) as ||x|| goes to infinity.

Study of a classification :

We have two cases to examine.

A -First case : the amplitude J ∞ (B ∞ ) is null. We have :

0 = J ∞ (B ∞ ) J R (B ∞ ) ∀ R > 0 , (14) lim R→+∞ sup ||x|| R λ N B -1 ∞ A(x) = γ 1 ∞ lim R→+∞ inf ||x|| R λ 1 B -1 ∞ A(x) = γ 2 ∞ , since the sequences R -→ sup ||x|| R λ N B -1 ∞ A(x) and R -→ inf ||x|| R λ 1 B -1 ∞ A(x)
are monotonic and bounded. Thus we have :

lim R→+∞ J R (B ∞ ) = γ 1 ∞ -γ 2 ∞ = J ∞ (B ∞ ) = 0 , that is to say (15) γ ∞ := γ 1 ∞ = γ 2 ∞ Proposition 11. Let us assume that J ∞ (B ∞ ) = 0. Then we have 1) lim R→+∞ sup ||x|| R T r B -1 ∞ A(x) = N γ ∞ , 2) lim R→+∞ sup ||x|| R |||B -1 ∞ A(x) -γ ∞ .I||| = 0, lim R→+∞ sup ||x|| R |||A(x) -γ ∞ B ∞ ||| = 0 .
Proof . Le us prove the first result. We have :

N. inf ||x|| R λ 1 B -1 ∞ A(x) inf ||x|| R T r B -1 ∞ A(x) sup ||x|| R T r B -1 ∞ A(x) N. sup ||x|| R λ N B -1 ∞ A(x)
By definition of J R (•), we obtain :

0 sup ||x|| R T r B -1 ∞ A(x) -inf ||x|| R T r B -1 ∞ A(x) N.J R (B ∞ ) .
And from ( 14), (15) and the proposition 10, the first resultat is obtained. Let us prove the second result. By the very definition of λ i B -1 ∞ A(x) , i = 1, . . . , N , we have :

∀ y = 0 , inf ||x|| R λ 1 B -1 ∞ A(x) A(x).y, y B ∞ y, y sup ||x|| R λ N B -1 ∞ A(x) ,
and for any

i 0 ∈ [1, . . . , N ] (16) ∀ x inf ||t|| R λ 1 B -1 ∞ A(t) λ i 0 B -1 ∞ A(x) sup ||t|| R λ N B -1 ∞ A(t) .
And thus, for any y = 0 and any x, we have :

A(x).y, y -λ i 0 (x) B ∞ y, y B ∞ y, y J R (B ∞ ) , 1 β . A(x) -λ i 0 (x)B ∞ y, y ||y|| 2 J R (B ∞ ) ,
that is to say, for any x, we obtain

sup y =0 A(x) -λ i 0 (x)B ∞ y, y ||y|| 2 βJ R (B ∞ ) .
And since the matrix A(x) -λ i0 (x)B ∞ is symmetric we obtain

sup ||x|| R |||A(x) -λ i 0 (x)B ∞ ||| β.J R (B ∞ ) .
This entails

sup ||x|| R |||A(x) -γ ∞ B ∞ ||| β.J R (B ∞ ) + sup ||x|| R λ i0 (x) -γ ∞ .|||B ∞ ||| .
And from ( 16), ( 14) and (15) we obtain

(17) lim R→+∞ sup ||x|| R |||A(x) -γ ∞ B ∞ ||| = 0 .
And the second result 2) follows from

|||B -1 ∞ A(x) -γ ∞ I||| ≡ |||B -1 ∞ A(x) -γ ∞ B ∞ ||| |||B -1 ∞ |||.|||A(x) -γ ∞ B ∞ ||| .
Corollary 12. We have

lim R sup ||x|| R T r B -1 ∞ A(x) 2 ||x|| 2 . B -1 ∞ A(x).x, x = N 2 = lim R sup ||x|| R T r Â(x) 2 ||x|| 2 . Â(x).x, x where Â(x) = BA(B -1 x)B , B = B -1 ∞ .
Proof . For any x we have :

B -1 ∞ A(x).x, x = B -1 ∞ A(x) -γ ∞ I .x, x + γ ∞ I.x, x .
This entails

B -1 ∞ A(x).x, x ||x|| 2 γ ∞ -sup ||x|| R |||B -1 ∞ A(x) -γ ∞ I||| and B -1 ∞ A(x).x, x ||x|| 2 γ ∞ + sup ||x|| R |||B -1 ∞ A(x) -γ ∞ I||| .
Then it is easy to obtain

1 2 sup ||x|| R T r B -1 ∞ A(x) . γ ∞ + sup ||x|| R |||B -1 ∞ A(x) -γ ∞ I||| -1 sup ||x|| R T r B -1 ∞ A(x) 2 ||x|| 2 B -1 ∞ A(x)x,x 1 2 sup ||x|| R T r B -1 ∞ A(x) . γ ∞ -sup ||x|| R |||B -1 ∞ A(x) -γ ∞ I||| -1
.

Thanks to proposition 11 and passing to the limit as R goes to infinity the first result follows. Now let us prove the second equality. We have

BAB -B -1 ∞ A = BAB -γ ∞ I + γ ∞ I -B -1 ∞ A = B A -γ ∞ B -2 B + γ ∞ I -B -1 ∞ A = B A -γ ∞ B ∞ B + γ ∞ I -B -1 ∞ A ,
and thus

|||BAB -B -1 ∞ A||| |||B||| 2 |||A -γ ∞ B ∞ ||| + |||γ ∞ I -B -1 ∞ A||| .
From this last inequality and using the proposition 11, we prove the second equality in the same way as in the first one.

The following definition is justified by the previous result.

u(x) ∼ c ||x|| N -2
for ||x|| large enough.

Proof . In (12) we use the following change of variables :

x = B -1 y i.e. y = Bx with B = B -1 ∞ .
We obtain the following equation

(18) -T r Â(y)D 2 ω(y) = f (B -1 y) , u(x) = ω(B.x) , lim ||y||→+∞ ω(y) = 0 , ω(y) > 0 ,
where

Â(y) = B.A B -1 y .B .
From proposition 11 and corollary 12 the matrix Â(y) satisfies

T r Â(y) 2 ||y|| 2 . Â(y).y, y = N 2 , ∀x , ||x|| R .
We conclude by the result of [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF], [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] and the fact that u(•) and ω(•) have the same asymptotic behaviour.

B -Second case : the amplitude J ∞ (B ∞ ) is positive. The matrix A(•) is not constant at infinity. We call it oscillating matrix. The amplitude of its oscillations is measured by the real number J ∞ (B ∞ ). We have two possibilities to examine.

i) the behavior of the spectral dispersion is radial that is to say there exists a radial function

x -→ γ ||x|| 2 such that R -→ ϕ(R) = sup ||x|| R ∧ (x) -γ ||x|| 2 verifies : (18.1) lim R ϕ(R) = 0
with x -→ γ ||x|| 2 is oscillating at infinity. In this case, under some hypothesis about ϕ(•), the behavior of the solution u(•) of ( 12) is known [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] : there exists a real number R > 0 large enough such that for any x, ||x||

R we have u(x) ∼ c.v(||x|| 2 ). The function v(•) is given explicitly by (18.2) v(||x|| 2 ) = +∞ ||x|| 2 exp - s R 2 γ(σ) σ dσ ds
For more details we can see Theorem 3.1 and remark 4.1 hereafter.

ii) the function of spectral dispersion x -→ ∧(x) is anisotropically oscillating at infinity. In the sequel our goal is to study this case. In the sequel we consider the constructed matrix  and the associated equation (18). To simplify the notations we denote Â(•) by A(•) and we identify (18) to (12

) with f (B -1 •) identified to f (•).
(18.3) -T rA B (y)D 2 ω(y) = f √ B.y ω(y) > 0 , lim ||y||→+∞ ω(y) = 0 , A B (y) = √ B -1 A √ B.y . √ B -1
have the same asymptotic behaviour since u(x) = ω √ B -1 .x .

Theorem 13.1. Without any hypothesis about ∧(•), for any ε > 0, there exist R 0 := R 0 (ε) > 0,

c 1 := c 1 (ε) > 0, c 2 := c 2 (ε) > 0 such that the solution u(•) of (12) satisfies : for any x, ||x|| R 0 c 1 ||x|| 2(∧ ∞ -1+ε) u(x) c 2 ||x|| 2(∧ ∞ -1-ε)
where

∧ ∞ = inf B∈Q(α,β) lim R→+∞ sup ||x|| R ∧(A, B)(x) ∧ ∞ = sup B∈Q(α,β) lim R→+∞ inf ||x|| R ∧(A, B)(x) with ∧(A, B)(x) = T r A B (x) 2 ||x|| 2 . A B (x).x,x
where

A B (x) = √ B -1 .A √ B.x . √ B -1 .
Before to give the proof of this result, let us remark that from the very definition of ∧(A, B)

we have ∧ ∞ > 1 since N 2 > β α .
Proof . Let us prove the right hand side inequality. From the very definition of ∧ ∞ we can see that for any ε > 0 there exist B 0 = B 0 (ε) ∈ Q(α, β) and r 0 = r 0 (ε) such that : for any

x, ||x|| r 0 we have ∧ ∞ -ε ∧(A, B 0 )(x).
After the change of variables y = B -1 0 .x, we first prove our inequality for the function ω(•) defined by

u(x) = ω B -1 0 .x or ω(y) = u B 0 .y .
This function satisfies

(18.3.1) -T r A B0 (x)D 2 ω(x) = f √ B 0 .x ω(x) = ω ε (x) > 0 , ω(x) = u √ B 0 .x , lim ||x||→+∞ ω(x) = 0 ,
since problems (12) and (18.3) have the same asymptotic behaviour. For this let us set, for any x ∈ R N ,

f (x) = sup f ( √ C.x) 4(A C x, x) / C ∈ Q(α, β) .
There exists a radial positive function h(•), with compact support [0, ρ 2 0 ] such that h(||x|| 2 ) f (x) ∀x ∈ R N . Increasing r 0 if necessary, we give the following positive, continuous and increasing function :

(18.3.2) ∧(r) =    ∧ ∞ -ε , ∀ r r 2 0 , l(r) , ∧ ∞ -ε l(r) αN 2β > 1 , ∀ r < r 2 0 with l(||x|| 2 ) ∧(A, B 0 )(x) ∀ x , ||x|| r 0 ,
and let us consider ω rad (•) solution of the following differential equation

(18.3.3) -1 σ(r) σ(r)ω rad (r) = h(r) , ω rad (0) = 0 , ω rad (r) > 0 , lim r→+∞ ω rad (r) = 0 , with σ(r) = exp r 1 ∧(θ) θ dθ .
From the very definition of ∧(•) we have ∧(||x|| 2 ) ∧(A, B 0 )(x) ∀ x, and using the same methode as in section IV -B1 -first step or as in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF], we can prove that ω r (x) := ω rad (||x|| 2 ) is a super-solution of (18.3) that is to say :

(18.3.4) -div A B0 D 2 ω r (x) f ( √ B 0 .x) , ω r (x) > 0 , lim ||x||→+∞ ω r (x) = 0 .
And from the comparison principle [START_REF] Crandall | Lions User's guide to viscosity solutions of second order partial differential equation Bull[END_REF], [START_REF] Protter | Principles in Differential Equations[END_REF], we obtain :

(18.3.5) ∀ x ∈ R N , 0 ω(x) ω r (x) = +∞ ||x|| 2 1 σ(θ) θ 0 σ(t)h(t) dt dθ .
But we have :

∀ θ , θ r 0 (ε) > ρ 0 , θ 0 σ(t)h(t) dt = ρ 0 0 σ(t)h(t) dt ||σ|| ∞ .||h||| ∞ .ρ 0 c 0 (h) where c 0 (h) is a constant independant of ε; ∀ θ , θ r 0 (ε) > ρ 0 , σ(θ) = exp θ 1 ∧(r) r dr = exp r 0 1 l(r) r dr . θ r 0 ∧ ∞ -ε .
And thus for any x, ||x|| r 0 we obtain :

0 ω(x) c 0 (h).r ∧ ∞ -ε 0 exp r0 1 l(r) r dr . 1 ||x|| 2(∧ ∞ -1-ε) , 0 ω(x) c ε .c 0 (h) ||x|| 2(∧ ∞ -1-ε) , ∀ x , ||x|| r 0 (ε) .
By the reverse change of variables x = B -1 0 t we have u(t) = ω B -1 0 .t ; and we obtain

u(t) c ε .c 0 (h) || B -1 0 t|| 2(∧ ∞ -1-ε)
.

And since

√ α β 2 I B -1 0 √ β α 2 I, there exists c(α, β) > 0 such that (18.3.6) u(t) c ε .c(α, β).c 0 (h) ||t|| 2(∧ ∞ -1-ε) for any t, ||t|| 2 r 2 0 . α 2 √ β .
In the same way we prove the left hand side inequality.

Let us set ∧

B (r) = inf ||x|| r ∧(A, B)(x), ∀ r 0.
Corollary 13.2 Let us suppose that the matrix A(•) is such that the decreasing function

r -→ ϕ(r) = sup B∈Q(α,β) [∧ ∞ -∧ B (r)] satisfies the following : exp +∞ 1 ϕ(r)
r dr < +∞. Then we obtain the following result : there exist some constants c 1 (α, β, h) > 0, r 0 > 0, such that

u(x) c 1 (α, β, h) ||x|| 2(∧ ∞ -1) ∀ x , ||x|| r 0 .
Proof : From our hypothesis we have, for any B ∈ Q(α, β)

∧ ∞ -ϕ(r) ∧ B (r) ∧(A, B)(x) , ∀ x , ||x|| r .
In (18.3.1) we use B instead of B 0 and in (18.3.2) we set ∧(r) := ∧ ∞ -ϕ(r) ∀ r 0. Then (18.3.5) becomes :

0 ω B (x) := ω(x) c 0 (h). +∞ ||x|| 2 dθ σ(θ) , ∀ ||x|| ρ 0 , with σ(θ) = θ ∧ ∞ .exp - θ 1 ϕ(r) r dr ,
that is to say 1) .

0 ω(x) exp +∞ 1 ϕ(r) r dr . c 0 (h) ||x|| 2(∧ ∞ -
That is to say

0 ω(x) c 1 (h) ||x|| 2(∧ ∞ -1) ,
where c 1 (h) is a constant independant on B ∈ Q(α, β). And as previously, (18.3.6) becomes

u(x) c(h).c(α, β) ||x|| 2(∧ ∞ -1) ∀ ||x|| > r 0 ,
for some r 0 > ρ 0 .

We have a similar result to estimate u(•) from below : u(x) c 2 (α, β, h) / ||x|| 2(∧∞-1) .

Remark 4.1 Contrary to what is stated in theorems 21 and 22 of [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF], the result obtained in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] is the following : for any ε > 0 there exist r dr < +∞. And in this case the result of [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] follows from the same idea of corollary 13.2 i. e.

R 0 := R 0 (ε) > 0, c 1 (ε) > 0 and c 2 (ε) > 0 such that c 1 (ε)v 1 (||x|| 2 ) u(x) c 2 (ε)v 2 (||x|| 2 ) ∀ x , ||x|| R 0 with v 1 (||x|| 2 ) = +∞ ||x|| 2 exp - s R 2 γ(σ) + ε σ dσ ds v 2 (||x|| 2 ) = +∞ ||x|| 2 exp - s R 2 γ(σ) -ε σ dσ ds .

u(x)

∼ +∞ ||x|| 2 exp - s R 2 γ(σ) σ dσ ds
for ||x|| large enough.

IV -Anisotropic asymptotic behavior.

In the sequel we denote again by A(x) the matrix

Â(x) = B -1 ∞ .A √ B ∞ .x . B -1 ∞
, where B ∞ is given by proposition 10. From the very definition of A(•), the amplitude of its spectrum is positive. Let us set :

(18.4)        ∧ = lim R sup ||x|| R T r A(x) 2 ||x|| 2 A(x).x,x ∧ = lim R inf ||x|| R T r A(x) 2 ||x|| 2 A(x).x,x
Let δ 0 be a positive real number close enough to ∧ -∧ such that δ 0 > ∧ -∧. Let 2η 0 be a positive real number very small with respect to δ 0 -(∧ -∧) > 0. And let us give η > 0, 0 < η < η 0 -( arbitrary small) -. From (18.4) there exists R η = R ∞ such that for any x, ||x|| R ∞ we have (18.5)

∧(η) = ∧ -η ∧(x) ∧ + η = ∧(η)
where

∧(x) = T r A(x) 2 ||x|| 2 A(x).x, x .
Let us consider the operator

A(•) = ij a ij (x) ∂ 2 (•) ∂x i ∂x j . From operator A(•) we introduce a family of radial operators x ∈ R N -→ A r (x)(•) = d 2 (•) dr 2 + ∧(x) r d dr (•) = 1 r ∧(x) d dr r ∧(x) d dr (•) ,
where r > 0 and x -→ ∧(x) is the function of the spectral dispersion associated to the matrix A(x). The link between A(•) and A r (x)(•) is the following : for any regular and radial function Ψ(x) = ϕ(||x|| 2 ) we have

AΨ 4 ij a ij x i x j = ϕ (||x|| 2 ) + ∧(x) ||x|| 2 ϕ (||x|| 2 ) = A ρ (x)ϕ(ρ) /ρ=||x|| 2 .
Remark 5. Let us point out that for some technical reasons the variable r plays the role of ||x|| 2 , instead of ||x|| as usually.

A -Method of sub and supersolution.

Let us give a positive radial function r -→ g(r) which will be choosen later in a suitable way. For any -(but fixed)-x belonging to R N , with ||x|| large enough, we consider a family of radial problems :

(P x ) : (19) -A r (x)h(x, r) = -h (x, r) -∧(x) r h (x, r) = g(r) , r > 0 h (x, 0) = 0 , lim r→+∞ h(x, r) = 0 , h(x, r) > 0 where g satisfies +∞ 1
σ ∧ g(σ) dσ < +∞ with ∧ given by (18.4) section IV. The unique solution r -→ h(x, r) of ( 19) is given by

(20) h(x, r) := h g (x, r) = +∞ r 1 s ∧(x) s 0 σ ∧(x) g(σ) dσ ds . If ∧(•) = constant ∀ x , ||x|| R 0 , the function h g (x, ||x|| 2 ) taks the form h g (||x|| 2 ) = h(||x|| 2 ).
And let us point out that in this case x -→ h(||x|| 2 ) gives the behavior at infinity of the solution of (12) [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF]. So roughly speaking, our idea is to look for a subsolution σ, -( respectivement a supersolution σ)-of (12) in the form

(21) H(x) = H g (x) = h g (x, ||x|| 2 ) = h(x, ||x|| 2 ) .
How to construct these functions? For this it is sufficient to choose two functions g 1 and g 2 in a suitable way such that σ(•) = H g1 (•) and σ(•) = H g2 (•) have the same behavior as ||x|| goes to infinity. Next we establish the result thanks the comparison principle.

B-Construction of sub and supersolution of (12). B0 -Preliminary results.

Let us introduce the following functions that we need later. For any x ∈ R N , any r > 0 and any suitable positive function g(•) : R + -→ R + , we set

K 1 = K 1 (x, r, g) = +∞ r 1 s ∧(x) s 0 Log σ s .σ ∧(x) g(σ) dσ ds K 2 = K 2 (x, r, g) = +∞ r 1 s ∧(x) s 0 Log σ s 2 .σ ∧(x) g(σ) dσ ds K 3 = K 3 (x, r, g) = - 1 r ∧(x) s 0 Log σ r .σ ∧(x) g(σ) dσ .
To simplify the presentation we use the short notation K i , i = 1, 2, 3. Since g(•) 0 we have K 1 0, K 2 0 and K 3 0. Proposition 14. The following estimates are true for r > 0 large enough :

1) 0 -K 1 = k. r ∧+1 1 -r ∧+1 0 (∧ + 1)(∧ -1) . Log r r ∧-1 + 1 r ∧-1 2) 0 K 2 = k. r ∧+1 1 -r ∧+1 0 (∧ + 1)(∧ -1) . (Log r) 2 r ∧-1 + 1 r ∧-1 3) 0 K 3 = k. r ∧+1 1 -r ∧+1 0 ∧ + 1 . Log r r ∧-1 + 1 r ∧-1 where ∧ = ∧(x) and g(r) = k.χ [r0,r1] (r), k > 0 and χ [r0,r1] (r) = 1 if r 0 r r 1 and χ [r0,r1] (r) = 0 if r / ∈ [r 0 , r 1 ]. The proof is straightforward.
Proposition 15. Let us assume the spectral dispersion satisfies :

||∇ ∧ || L ∞ R N \B(0,R0) + ||D 2 ∧ || L ∞ R N \B(0,R0) c
for R 0 large enough. Then we have the following : for any ε > 0 there exist R ε R 0 and c ε > 0 such that (23)

K 2 . A(x)∇∧, ∇ ∧ + K 1 .T r A(x)D 2 ∧ + 4K 3 . A(x)∇∧, x c ε . ||g|| ∞ ρ ∧+1 1 -ρ ∧+1 0 ||x|| 2(∧(x)-1-ε)
for any x, ||x|| R ε , r = ||x|| 2 and for any positive function g(•) such that support g ⊆ x / ρ 0 ||x|| ρ 1 and where ∧ and ∧ are given by (18.4) section IV.

Proof . It follows easily from the proposition 14. Remark 6. Without another assumption on ∧(•) the left-hand side of (23) tends to zero as ||x|| goes to infinity. But we will see that the behavior of (23) at infinity is not sufficient to prove our result. So we will assume that the left-hand side of (23) goes to zero faster than 1 ||x|| 2(∧(x)-1-ε) as ||x|| goes to infinity. The results of section II allow us to show that the supposed assumptions are suitable for a large classe of matrices.

From [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] and theorem 13.1 and for some fixed ε > 0 small enought, we have

a ||x|| 2(∧-1+ε) u(x) b ||x|| 2(∧-1-ε)
for ||x|| sufficiently large and with 0 < a < b. These inequalities prove that u(•) has a behavior of the form u(x) = c(x) ||x|| δ(x) where the function δ(•) and c(•) satisfy :

0 < c 0 c(x) c 1 , 2(∧ -1) δ(x) 2(∧ -1) , Log c 0 -Log u(x) Log (||x||) δ(x) Log c 1 -Log u(x) Log (||x||) .
That is to say δ(x) = -Log u(x) Log (||x||) for ||x|| large enough.

Proposition 16. Let A(•) be a matrix belonging to Q(α, β). Let us assume that, for any positive continuous function f with compact support, the solution

u(•) = u f (•) of -T r(AD 2 u f ) = f in R N u f (x) > 0 ∀ x , lim ||x||→+∞ u f (x) = 0
has the following asymptotic behavior :

a(f ) ||x|| δ f (x) u(x) b(f ) ||x|| δ f (x) , ∀ x , ||x|| R f > 0 with 0 < a(f ) b(f ) , b(f ) a(f ) γ , b(f ) ||u|| ∞ 2 Then δ f (•) is independant of f and satisfies N. α β -2 δ f (•) N. β α -2
Proof . We need two steps.

1) First step : asymptotic behavior of the somme of two solutions. Let us consider

(24) -T r(AD 2 u i ) = f i in R N , i = 1, 2 u i (x) > 0 , lim ||x||→+∞ u i (x) = 0 . Let us set δ i (•) = δ fi (•), ω(•) = u 1 (•) + u 2 (•), g = f 1 + f 2 , a i = a(f i ), b i = b i (f ), R i = R fi . For any x, ||x|| sup R 1 , R 2 , R g we have :        a(g) ||x|| δ g ω(x) b(g) ||x|| δ g a i ||x|| δi u i (x) b i ||x|| δi , i = 1, 2 .
Since ω = u 1 + u 2 , we obtain

a i ||x|| δi u i (x) b(g) ||x|| δg . There exists R sup R 1 , R 2 , R g such that δ i (x) δ g (x) ∀ x , ||x|| R .
This entails that we have inf δ 1 (x), δ 2 (x) δ g (x) . As we have

(25) a(g) ||x|| δ g ω = u 1 + u 2 b 1 ||x|| δ 1 + b 2 ||x|| δ 2 b 1 + b 2 ||x|| inf(δ1,δ2)
we obtain δ g (x) inf δ 1 (x), δ 2 (x) for ||x|| sufficiently large. Thus we have

(26) δ g (x) = inf δ 1 (x), δ 2 (x) .
And since operator -T r(AD 2 •) is linear, we have :

(27) δ kf (x) = δ f (x) , ∀ k > 0 .
And thus from (26) and (27) we obtain (28)

δ kf2+f1 (x) = inf δ 1 (x), δ 2 (x)
∀ k > 0 and ||x|| large enough .

2) Second step : a perturbation result.

For any ε > 0 we set g ε = f 1 + εf 2 ; and u ε 2 (x) = εu 2 (x) is the solution of (24) with εf 2 instead of f 2 . From (28) we obtain

δ g ε (x) = inf δ 1 (x), δ 2 (x) = δ g (x). From (25) we can write (29) a(g ε ) ||x|| δ1 a(g ε ) ||x|| δg u 1 (x) + εu 2 (x) b 1 ||x|| δ1 + εb 2 ||x|| δ2 for ||x|| sufficiently large, since δ 1 (x) δ g (x). Consequently as b(g ε ) ||u 1 + εu 2 || ∞ 2 ||u 1 || ∞ -ε||u 2 || ∞ 2 ||u 1 || ∞ 4 ,
for ε sufficiently small, we have

β 0 a(g ε ) b(g ε ) ||u 1 || ∞ 4
and thus a(g ε )

||u 1 || ∞ 4β0
. Let us pass to the limit in (29) as ε goes to zero. We obtain

1 ||x|| δg ||u 1 || ∞ 4β 0 u 1 (x) b 1 ||x|| δ1 .
This entails, for ||x|| sufficiently large, δ g (x) δ 1 (x) . In the same way we can prove that δ g (x) δ 2 (x), and thus we have

(30) δ g (x) sup δ 1 (x), δ 2 (x) .
Finally ( 26) and (30) give the result :

δ g (x) = δ 1 (x) = δ 2 (x) . The inequalities -2 + N α β δ(x) -2 + N β α come from [4]
The behavior at infinity of matrices is important as shown by the following result.

Proposition 17. Let us give two matricies

A 1 (•) and A 2 (•) belonging to Q(α, β) such that A 1 (x) = A 2 (x) for any x, ||x|| R 0 > 0. Then the solutons u i (•), i = 1, 2, of (31) -T r(A i D 2 u i ) = f i in R N u i (x) > 0 , lim ||x||→+∞ u i (x) = 0
where

f i = f , (i = 1, 2)
, is a positive continuous function with compact support, have the same asymptotic behavior, that is to say there exist 0

< a i < b i , i = 1, 2 and x -→ δ(x) ∈ [N α β -2, N β α -2] such that a i ||x|| δ(x) u i (x) b i ||x|| δ(x) i = 1, 2
for ||x|| sufficiently large.

Proof . From (31) it follows

(32) -T r A 1 D 2 (u 1 -u 2 ) = g(x)
where

g(x) = -T r (A 2 -A 1 )D 2 u 2 (x) is a continuous function with compact support in B(O, R 0 ) since A 2 (x)-A 1 (x) = 0, ∀ x, ||x|| > R 0 . Let us set h(x) = |g(x)|. From comparison principle -( cf [1] , [3] )-we have (33) 0 u 1 (x) -u 2 (x) s(x)
where s(•) is the solution of (31) with h(•) and A 1 instead of f i = f and A i , respectively.

From proposition 16 we have (33.1)

       a(h) ||x|| δ 1 (x) s(x) b(h) ||x|| δ 1 (x) ∀ x, ||x|| R R 0 , a i (f ) ||x|| δi(x) u i (x) b i (f ) ||x|| δi(x) ∀ x, ||x|| R R 0 , i = 1, 2 .
From (33) we have u 2 (x) u 1 (x) + s(x) and thus it follows

a 2 (f ) ||x|| δ2(x) b 1 (f ) + b(h) ||x|| δ1(x) .
For ||x|| large enough, this give δ 2 (x) δ 1 (x). In the same way, reversing the role of u 1 and u 2 we get δ 1 (x) = δ 2 (x).

Proposition 18. Consider the two functions

H i (x) = +∞ ||x|| 2 1 s ∧i(x) s 0 σ ∧ i (x) g i (σ) dσ ds i = 1, 2 .
Let us assume that

∧ 1 (x) = ∧ 2 (x) = ∧(x) ∀ x, ||x|| R ,
and g i (•) is positive bounded function with compact support, i = 1, 2. Then the two functions H 1 (•) and H 2 (•) have the same behavior as the function 1 ||x|| 2(∧(x)-1) at infinity.

Proof . Let us recall that we have

0 < +∞ 0 σ ∧i(x) g i (σ) < Γ i < +∞
It is sufficient to prove the result for H 1 (•), for instance. We have

0 H 1 (x) Γ 1 . +∞ ||x|| 2 ds s ∧ 1 (x) = Γ 1 ∧ 1 (x) -1 ||x|| 2(∧1-1)
that is to say

0 H 1 (x) Γ 1 ∧ 1 (x) -1 ||x|| 2(∧(x)-1) ∀ x , ||x|| R .
Now let us prove the reverse inequality. Since g 1 (•) has a compact support, there exists

R 1 R such that +∞ 0 σ ∧ 1 (x) g 1 (σ) dσ = R1 0 σ ∧ 1 (x) g 1 (σ) dσ R1 1 σ ∧1 g 1 (σ) dσ + 1 0 σ ∧1 g 1 (σ) dσ Γ 0 > 0
where ∧1 = inf ||x|| R1 ∧ 1 (x) and ∧1 = sup ||x|| R 1 ∧ 1 (x). This gives the following :

∀ x, ||x|| R 1 , H 1 (x) Γ 0 +∞ ||x|| 2 ds s ∧1(x) = Γ 0 ∧ 1 (x) -1 ||x|| 2(∧(x)-1)
Remark 6.1. The previous result will enable us to prove that the solution u(•) of (12) behaves like a suitable function H associated to ∧(•). This shows that only ∧(•) controls the behaviour of u(•). The following result is easy to obtain. Proposition 19. Let us set H(x) = h g (x, ||x|| 2 ) where r -→ h g (x, r) is defined by ( 19) and (20). Then function H(•) satisfies the following

T r A(x)D 2 H = -4 A(x).x, x .g ||x|| 2 + K 1 (x, ||x|| 2 , g).T r A(x)D 2 ∧ + K 2 (x, ||x|| 2 , g). A(x)∇∧, ∇ ∧ + 4K 3 (x, ||x|| 2 ,

g). A(x)∇∧, x

Proof . For any x ∈ R N , let us consider the solution r -→ h(x, r) of (19) given by (20). To conclude it is sufficient to see that

A r (x)h(x, •) ||x|| 2 = -4 A(x).x, x g ||x|| 2
Let us introduce our essential assumption (H) : there exists R 0 (H) := R 0 > 0 such that for any R > 0 and any positive, bounded function g with compact support in B(O, R), the spectral dispersion ∧(•) satisfies the following :

-K 1 (x, ||x|| 2 , g).T r A(x)D 2 ∧ -K 2 (x, ||x|| 2 , g). A(x)∇∧, ∇ ∧ -4K 3 (x, ||x|| 2 , g). A(x)∇∧, x ∈ - c 0 (R)||g|| ∞ 1 + ||x|| 2(∧(x)+δ(x)) , c 0 (R)||g|| ∞ 1 + ||x|| 2(∧(x)+δ(x)) ∀ x , ||x|| R 0 ,
with δ(x) δ 0 > 0 and ∧ -∧ + δ 0 > 0 , where ∧ and ∧ are defined in (18.4).

Remark 5.1. if ∧ = ∧ = ∧ ∞ then we have lim R→+∞ sup ||x|| R ∧ (x) -∧ ∞ = 0, and the result of [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] can apply.

Comments about the assumption (H) :

The differential inequality (H) is suitable as shown by the following remarks

1) if ∧(•) is constant for ||x|| R, (H) is satisfied.
And in this case our result already follows from [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF]. And from proposition 3 section II this result does not obtained by [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] since in this work [START_REF] Meyyers | The exterior Dirichlet problem for second order elliptic partial differential equations[END_REF] lim ||x||→+∞ A(x) = A 0 , a constant matrix.

2) Let us consider the matrix M (∧) defined by :

M (∧) = K 1 .D 2 ∧ +K 2 .∇ ∧ ⊗∇ ∧ +K 3 . ∧(x) ||x|| 2 .N (∧)
where

K i = K i (x, ||x|| 2 , g) and N (∧) ij = x i . ∂∧ ∂x j + x j . ∂∧ ∂x i ∀ i, j .
If M (∧) satisfies the inequalities (4) for any x, ||x|| R, then from corollary 7 there exists a matrix x -→ A(x) ∈ Q(α, β) such that

(34) K 1 .T r A(x)D 2 ∧ + K 2 A(x)∇∧, ∇ ∧ + K 3 A(x)∇∧, x = 0 , ∀ x, ||x|| R.
And thus (H) is satisfied by A(•) and ∧(•).

3) If in addition ∧(•) is the spectral dispersion of the matrix that we seek, then we use the remark 2 section II with the two following matrices V = M (∧) and R = I -2∧ ||x|| 2 .x⊗x. And we find A(•) ∈ Q(α, β) satisfying (H) and such that ∧(•) is its spectral dispersion.

4)

Let us write the equation (34) in the form

(35) T r A(x)D 2 ∧ + K 2 K 1 . A(x)∇∧, ∇ ∧ + K 3 K 1 A(x)∇∧, x = 0 , ∀ x, ||x|| R.
Then for R > 0 large enough and A = I, the equation ( 35) is "very close" to the radial form :

(36)

   -∧ + 1 2 Log ||x|| 2 .||∇ ∧ || 2 + ∧(x) x.∇∧ ||x|| 2 = 0 . ∀ x , ||x|| R .
The equation ( 36) is obtained by "approximation" using the proposition 14 section IV. The proposition 8 section II shows that the equation (36) has many non constant positive and bounded solutions. This gives a partial answer to the question posed in the remark 3 section II. 5) an elementary example satisfying (H) : Let k be a positive real number such that

ϕ(x) = k + sin 2π 1 + ||x|| 2m ∈ α β N, β α N
for any x ∈ R N . After some elementary calculations we obtain that there exist some positive constant c such that

|||D 2 ϕ(x)||| + ||∇ϕ(x)|| c 1 + ||x|| 2m-1 ∀ x ∈ R N and thus c 1 (x)T rA(x)D 2 ϕ(x) + c 2 (x)a ∇ϕ(x), ∇ϕ(x) + c 3 (x)a ∇ϕ(x), x c 1 + ||x|| 2m-1
for any c 1 , c 2 , c 3 bounded functions in R N , and any matrix A(•) belonging to Q(α, β). The positive real number m is arbitrary.

B1 -Construction of a super-solution of (12) or (18) Thanks to the proposition 16 we can suppose that the function f (•) has its support in the open ball B(O, ρ 1 ). To choose a suitable function r -→ g(r) for which the function H(•) = H g (•) is a super-solution of (12), we introduce the following radial function

f (r) = sup f (x) / x ∈ R N , ||x|| 2 = r .
Regularizing f (•) if necessary, thanks to proposition 16, we can suppose that f (•) is continuous and satisfies f (||x|| 2 ) f (x) for any x. And we can suppose sup f ⊂ [0, ρ 2 1 [. Let k be a positive real number such that (37)

k.α.||x|| 2 χ K ||x|| 2 > f ||x|| 2 where K = x / ||x|| 2 < ρ 2 1 = B(O, ρ 1 ), χ K ||x|| 2 = 1 if x ∈ K, χ K ||x|| 2 = 0 if x / ∈ K. Let us set (38) G(x) := g ||x|| 2 := kχ K r ||x|| 2
where χ Kr ||x|| 2 := χ [0,ρ 2 1 ] ||x|| 2 . We need to introduce the following function σ(•) defined by : (38.1)

   σ (r)/σ(r) = ∧(r) r , r 0 , σ(1) = 1 with ∧(r) = ∧ + η for any r R 0 (H) = R 0 and such that ∧(||x|| 2 ) ∧(x) ∀ x ∈ R N .
We have : η) for θ large enough.

(38.2) σ(r) = exp r 1 ∧(θ) θ dθ . 1) (39)    R 0 > ρ 1 3α 2 . ρ 2 1 0 σ(θ) dθ > c 0 (ρ 1 ). +∞ R 2 0 σ(θ) (1 + θ ∧(η)+δ 0 )θ dθ since we have ∧(η) -∧(η) + δ 0 = ∧ -∧ + δ 0 -2η and σ(θ) ∼ c.θ ∧(
2) there exists

R 1 = R 1 (ε) such that (40)    ρ 1 < R 1 < R 0 3α 2 . ρ 2 1 0 σ ∧+1 dθ > 4 R 2 0 R 2 1 c 1 (ρ 1 )σ ∧(η) (1 + θ (∧-1+ε) )θ dθ
with R 1 close enough to R 0 . Thanks to the proposition 17, we can modify our matrix A(•) in a suitable ball. This procedure does not affect the assumption (H) since this hypothesis is supposed true at infinity. For this, let θ ε = θ > 0, be small enough with respect to R 0 -R 1 and ε. And let us consider the following matrix

Ãε (x) = Ã(x) = A(x) ∀ x , ||x|| R 0 -θ τ.I ∀ x , ||x|| < R 0 -θ
where τ is a real number such that α < τ < β, and I the identity matrix. From the approximation lemma giving the approximation of the matrix Ã, we obtain the regular matrix A 1 (x) which satisfies :

(38.3) A 1 (x) = A(x) ∀ x , ||x|| R 0 τ.I ∀ x , ||x|| R 1
The matrix A 1 (•) depends to ε, R 0 , R 1 , η. From the approximation lemma the matrix A 1 (•) belongs to Q(α, β). This is now the regularized matrix A 1 (•) that we use in the equation ( 18), (or equivalently in (12) ), to construct a super-solution. Hence the equation (18) becomes

-T r A 1 (x)D 2 u 1 = f in R N , u 1 (x) > 0 , lim ||x||→+∞ u 1 (x) = 0 . The spectral dispersion of A 1 (•) is ∧(•) if ||x|| R 0 and N 2 if ||x|| R 1 . Thus A 1 (•) satisfies (H) for x belonging to {x / ||x|| R 0 } ∪ {x / ||x|| R 1 }.
To simplify the notation we denote again ∧(•) the spectral dispersion of A 1 (•).

Theorem 20. Let g(•) be a radical function satisfying (37) and (38). For any x let us consider the function r -→ h(x, r) = h g (x, r), solution of (19) associated to the matrix A 1 (•). Then the function

x -→ H g (x) := H 1 (x) = h g (x, ||x|| 2 ) = h(x, ||x|| 2 ) ,
is a super-solution of (18).

Remark 5. In the sequel we use the notation

H 1 (•) instead of H g (•) because the operator -T r(A 1 D 2 •) is associated to the matrix A 1 (•).
where σ(•) and ∧(•) are given by (38.1). It is easy to see that this function satisfies the condition ω rad (0) = 0 and the differential equation in (45). It remains to prove the others conditions. For this it is sufficient to verify that we have :

(46) ∀ s 0 , 0 s 0 σ(θ) Ψ(θ) dθ +∞ 0 σ(θ) Ψ(θ) dθ = Γ ∞ < +∞ .
We have four cases to distinguish.

First case : for 0 s ρ 2 1 we have

s 0 σ(θ) Ψ(θ) dθ = 3kα 4β . s 0 σ(θ) dθ 0 . Second case : for ρ 2 1 s R 2 1 we have s 0 σ(θ) Ψ(θ) dθ = 3kα 4β . ρ 2 1 0 σ(θ) dθ > 0 . Third case : for R 2 1 s R 2 0 we have s 0 σ(θ) Ψ(θ) dθ 3kα 4β . ρ 2 1 0 σ(θ) dθ - k 4β R 2 0 R 2 1 c 1 (ρ 1 )σ(θ) (1 + θ (∧(η)-1-ε) )θ dθ > 0 .
by the choice of R 1 and R 0 given by (40). And thus ω rad (•) is the solution of (45) and behaves like 1 r ∧(η)-1 as r goes to infinity. From (46) and since ∧(x) ∧(||x|| 2 ) for any x, we have by (45) :

-ω rad (r) -ω rad (r) ∧(x) r -ω rad (r) -ω rad (r) ∧(r) r = Ψ(r) , for any x ∈ R N and r = ||x|| 2 . We obtain :

-4 A 1 (x).x, x ω rad (||x|| 2 ) -4ω rad (||x|| 2 ) ∧ (x). A 1 (x).x, x ||x|| 2 4 A 1 (x).x, x Ψ(||x|| 2 ) that is to say, after some computations and using (44,) the function ω(x) = ω rad (||x|| 2 ) satisfies (47) -T r(A 1 D 2 ω) ϕ(x) in R N , ω(x) > 0 , lim ||x||→+∞ ω(x) = 0 .

Second step : Let us prove that we have :

H 1 (x) -ω(x) ω rad (||x|| 2 ) > 0 , ∀ x ∈ R N .
It follows from (47) that we have

(48) -T r A 1 D 2 (H 1 -ω) + T r(A 1 D 2 ω) ϕ(x) -Ψ(||x|| 2 ) 0 in R N lim ||x||→+∞ ω(x) = lim ||x||→+∞ (H 1 -ω)(x) = 0 .
Since x -→ ϕ(x) -Ψ(||x|| 2 ) is non negative, with compact support we proceed as in [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] to prove that H 1 (x) -ω(x) ω(x) = ω rad (||x|| 2 ) > 0 for any x ∈ R N : we consider (48) in a ball B(O, n) with the Dirichlet boundary condition. Let us denote by H 1,n , ω n , ω rad,n the corresponding approximations of H 1 , ω and ω rad respectively. The maximum principle, [1] - [START_REF] Protter | Principles in Differential Equations[END_REF], gives that we have

(H 1,n -ω n )(x) -ω rad,n (||x|| 2 ) 0 , ∀ x ∈ B(O, n) .
After establishing some suitable estimates, we pass to the limit as n goes to infinity and we conclude that we have :

H 1 (x) -ω(x) ω rad (||x|| 2 ) > 0 , ∀ x ∈ R N .
Third step : Since by construction f (||x|| 2 ) f (x) ∀ x ∈ R N , we prove as previously [START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] that we have ω(x) u 1 (x) ∀ x ∈ R N . And the final result follows :

H 1 (x) u 1 (x) ∀ x ∈ R N
that is to say H 1 (•) is a super-solution of (18) and recalling that u 1 (•) is the solution of (18) associated to the operator -T r(A 1 D 2 •).

B2 -Construction of a sub-solution of (18) or (12). We proceed exactly as before. To avoid repetition we only indicate the key points that differ from the section B1. Before to choose the suitable function g wich allows us to construct a sub-solution H g (•) of ( 18), let us introduce, as previously, the following function

f (r) = inf f (x) / x, ||x|| 2 = r .
Thanks to the proposition 16 we can assume that the non negative and regular function f (•), with compact support, is such that :

Definition 2 .

 2 If, for any matrix A(•) belonging to Q(α, β), we have J ∞ (B ∞ ) = 0, the matrix A(•) is called constant as ||x|| goes to infinity or shortly constant at infinity. Theorem 13. [2] [4]. If the matrix A(•) is constant for any x, ||x|| R, the solution of the problem (12) has the following asymptotic behavior :

Remark 4 .

 4 For any matrix B ∈ Q(α, β), we use the change of variables y = √ B -1 .x. Thus the problem (12) and

  The function γ(•) can play the role of θ, θ or γ(||x|| 2 ) given in the introduction, for instance. Indead the conclusion in the last line of the proof -([4] p 180, line 5 from the top) -is not correct because it is some what fast : the various constants depend to the variable ε. T o overcome this difficulty we need to replace the hypothesis (H3) of[START_REF]Tahraoui Comparison principle for second order elliptic operators and applications[END_REF] by the following : the decreasing function r -→ ϕ(r) = sup ||x|| r ∧ (x) -γ(||x|| 2 ) satisfies exp

Fourth case : for s R 2 c 1 2 0c 0

 2120 (ρ 1 )σ(θ) (1 + θ (∧(η)-1-ε) )θ dθ -k +∞ R (ρ 1 )σ(θ) (1 + θ (∧(η)+δ) )θ dθ > 0 from (39) and (40). Thus we conclude that for any s 0 we have : dθ = Γ ∞ < +∞ .

( 2 1 <ρ<R 2 0 1 ∧

 221 49) ρ -→ f (ρ) ≡ 0 , supportf = {ρ / 0 ρ < ρ 2 1 } . Let us introduce the function (50) G(x) := g(||x|| 2 ) = k.χ Kr ||x|| 2whereΨ(r) := -kχ [ρ / ρ>R 2 0 ] (r).c 0 (ρ 2 ) 1 + r (∧(η)-1-ε) -kχ [ρ / R ] (r).c 1 (ρ 2 ) 1 + r (∧(η)-1-ε) +4kβχ Kr (r).r .As previously we remark that the radial functionω rad (r) = ) Ψ(θ) dθ dsis the solution of the following ordinary differential equation (56)   -ω (r) -∧(r) r ω (r) = Ψ(r) 4βr := Ψ(r) ω (0) = 0 , lim r→+∞ ω(r) = 0 , ω (r) 0 , with σ(r) = exp r (s) s ds .Using (52) and the definition of Ψ(•) we can prove that (57) ∀ s 0 , 0s 0 σ(θ) Ψ(θ) dθ Γ ∞ < +∞and thus ω rad (r) 0 for any r > 0. This entails that ω rad (•) behaves like the functionr -→ +∞ r dθ σ(θ)as r goes to infinity. Now in a second step we acheave the proof of our result.Second step : Let us set ω(x) := ω rad (||x|| 2 ). We can prove that ω(•) satisfies(58) -T r(A 2 D 2 ω) Ψ(||x|| 2 ) ϕ(x) -T r A 2 D 2 (ω -H 2 ) .Since ω rad (•) is non negative, from the Maximum Principle we obtain, using (58)0 ω rad (||x|| 2 ) ω(x) -H 2 (x) ∀ x ∈ R N , that is to say H 2 (x) ω(x) ∀ x ∈ R N .Denoting by u 2 (•) the solution of (18) associated to A 2 (•), it follows ω(x) u 2 (x) for any x, since f (x) f (x) ∀ x. And thus H 2 (x) u 2 (x) ∀ x ∈ R N C -Asymptotic behavior of u(•).

Corollary 22 : 2 ||x|| 2 A

 2222 asymptotic behavior of u. For any non negative bounded function f (•) with compact support, and for any matrixA(•) belonging to Q(α, β) such that (H) is satisfied, the solution u(•) of the equation -T r(AD 2 u) = f in R N u(x) > 0 , lim ||x||→+∞ u(x) = 0From (61) and proposition 17 u(•) behaves like the solution u 0 (•) of-T r A 0 (x)D 2 u 0 (x) = f (x) , x ∈ R N , u 0 (x) > 0 , lim ||x||→+∞ u 0 (x) = 0 .In addition if∧ 0 (x) =T r A 0 (x) 0 (x).x, x satisfies hypothetis (H), then u(•) behaves like x -→ 1 ||x|| 2(∧ 0 (x)-1)

Proof of theorem 20. From (H) and the definition of H 1 (•) given in proposition 18, and using the proposition 19, we obtain :

kc 0 (ρ 1 ).χ [y / ||y|| R0] (x)

1 + ||x|| 2(∧(x)+δ(x))

c 1 (ρ 1 ).k.χ [R 1 <||y||<R 0 ] (x)

1 + ||x|| 2(∧(x)-1-ε) +4 A 1 (x).x, x .G(x) := p(x) in R N with lim ||x||→+∞ H 1 (x) = 0 and where c 1 (ρ 1 ) = c ε .[ρ ∧+1 1 ]. Let us consider the solution ω of the following :

Writing (41) -( 42) we obtain :

In the sequel we propose to estimate from below H 1 -ω by a suitable radial function ω rad (•).

First step : introduction of ω rad (•).

This function will be a solution of an ordinary differential equation. For this let us estimate from below the function ϕ(•) in ( 43) by a radial one. By (37) and (38) we have :

From the definitions (18.4) and (18.5) we have ||x|| 2∧(x) ||x|| 2∧(η) for any x, ||x|| R 1 and it follows that the function ϕ(•) is estimated from below by the following radial function

Then we define ω rad (•) as the solution of the following ordinary differential equation :

Let us set

. The real number k > 0 is choosen such that (51)

There exist two positive real numbers R 0 , R 1 , 0 < R 1 < R 0 , large enough and satisfying (52)

As previously in (38.3), from (52) we construct a regular matrix

We have the following result :

Theorem 21. Let g(•) be a function satisfying (49) to (51). For any x ∈ R N we denote by r -→ h(r, x) = h g (r, x) the solution of (18) associated to A 2 (•). Then the function

Proof . We need several steps. From (H), propositions 18 and 19 we obtain :

where

Let ω(•) be the solution of

Writing (53) + (54), it follows :

Now, our goal is to prove that we have ω(x) -H 2 (x) 0 for any x ∈ R N . To do this we use the same way like the section B1. In a first step we construct a positive radial function ω rad (•) which estimates ω -H 2 from below.

First step : we estimate ϕ(•) from below by a radial function Ψ(•). Using the same idea as in section B1, we obtain

behaves like the function x -→ 1 ||x|| 2(∧(x)-1) , when ||x|| goes to infinity, where

.

Proof . Let us consider the previous functions u 1 (•) and u 2 (•) associated, respectively, to the matrices A 1 (•) and A 2 (•). By the propositions 16 and 17 the three functions u 1 (•), u 2 (•) and u(•) has the same asymptotic behavior since

Applying the proposition 18 we can claim that H 1 (•) and H 2 (•) tend to zero like x -→

Then we can introduce the following functions r -→ q 1 (r) = inf u 1 (y) u 2 (y) / y, ||y|| r which is non decreasing, r -→ q 2 (r) = sup u 1 (y) u 2 (y) / y, ||y|| r which is non increasing.

Consequently there exists r 0 > 0 large enough such that (59) 0 < q 1 (r 0 )

since u 1 (•) and u 2 (•) have the same asymptotic behavior. Using (59), theorems (20) and ( 21) and proposition 12 it follows :

that it to say u 1 (•) goes to zero as x -→ 1 ||x|| 2(∧(x)-1) , thanks the proposition 18. And consequently the final result follows for u 2 (•) and u(•). D -Application. We can apply the previous corollary to find the asymptotic behavior in a nonlinear equation :

where (x, η) → A(x, η) ∈ Q(α, β). This problem is in the form of (12) with B(x) = A x, u(x) ∈ Q(α, β). We assume that there exist two positive real constants R 0 and η 0 such that for any x ∈ R N , ||x|| R 0 , any η ∈ R, |η| η 0 we have :


Abdullah Almaksour
email: abdullah.almaksour@irisa.fr

Eric Anquetil
email: eric.anquetil@irisa.fr

Solen Quiniou
email: solen.quiniou@synchromedia.ca

Mohamed Cheriet
email: mohamed.cheriet@etsmtl.ca

Evolving Fuzzy Classifiers: Application to Incremental Learning of Handwritten Gesture Recognition Systems

Keywords: incremental learning, evolving, handwriting recognition, fuzzy classifier

published or not. The documents may come

L'archive ouverte

I. INTRODUCTION AND BACKGROUND

Classification techniques appear frequently in many application areas, and become the basic tool for almost any pattern recognition task. The main problem in classification is to induce a classifier from a set of data samples. A large amount of samples is needed to set up and evaluate a classification system that can achieve a high accuracy, and it is practically very difficult to have such number of samples covering the expanse of the variability of classes. Therefore, life-long classifier adaptation becomes more and more an essential point. Moreover, in many application contexts, the classifier needs to take into account new unseen classes and to integrate them in the classification process, which increases the need for "evolving" classifiers. One good example of such applications is the use of online handwritten gesture classifiers which aim at facilitating interactions with computers using pen-based interfaces like whiteboards, tablet PCs, PDA...Etc. The main drawback in the current existing systems is that they are trained "offline" on a specific group of gestures and then implemented to operate without changing their structure during the use. This fixed structure does not allow the user to choose his own set of gestures or to add new ones according to his special needs, for example.

In our work, we aim at building an handwriting classifier, on-the-fly, from scratch and using only few data. Thus, the classifier will be incrementally adapted to achieve high recognition rates as soon as possible and to keep the system robust when introducing new unseen classes at any moment in the life-long learning process. An incremental learning algorithm is defined in [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF] by the following criteria: it should be able to learn additional information from new data; it should not require access to the original data (i.e. data used to train the existing classifier); it should preserve previously acquired knowledge (it should not suffer from catastrophic forgetting, i.e. significant loss of original learned knowledge); and it should be able to accommodate new classes that may be introduced with new data. Many of the existing "incremental learning" algorithms are not truly incremental because at least one of the mentioned criteria is violated. These criteria can be briefly expressed by the so-called "plasticity-stability dilemma" [START_REF] Zwickel | New Directions in Human Associative Learning[END_REF]. It says that a system must be able to learn to adapt to a changing environment but that constant change can lead to an unstable system that can learn new information only by forgetting everything it has learned so far. We can distinguish two main types of incremental learning algorithms: algorithms for parameter learning and algorithms for structure learning. The incremental learning of parameters can be considered as an "adaptation" algorithm. The structure in such systems is fixed and initialized at the beginning of the learning process, and the system parameters are learned incrementally according to newly available data. Some examples of these systems are presented in [START_REF] Laviola | A practical approach for writer-dependent symbol recognition using a writer-independent symbol recognizer[END_REF], [START_REF] Jang | Anfis: adaptive-network-based fuzzy inference system[END_REF]. Most of the structure incremental learning algorithms are based on the principle of the ART clustering algorithm [START_REF] Carpenter | The art of adaptive pattern recognition by a self-organizing neural network[END_REF], such as [START_REF] Sadri | A new clustering method for improving plasticity and stability in handwritten character recognition systems[END_REF], [START_REF] Lughofer | Flexfis: A robust incremental learning approach for evolving takagi-sugeno fuzzy models[END_REF]. The main problem of these systems is that they are sensitive to the selection of the vigilance parameter, to the noise level in the training data and to the order in which the training data is presented. A promised incremental clustering approach had been presented in [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF] based on the Mountain Clustering algorithm (originally introduced in [START_REF] Yager | Learning of fuzzy rules by mountain clustering[END_REF]). The main idea of the proposed approach is that of a potential of a given point: it corresponds to a value representing the density in the data space at that point. The potential of a sample can be defined as the inverse of the sum of the distances between that data sample and all the other ones. Samples with high potential are then considered to be candidates to form a cluster.

In this paper, we extend the recursive mountain clustering by combining it with a robust fuzzy adaptation method, and we use this hybrid algorithm to incrementally learn an evolving fuzzy rule-based classifier. Our system is used for the recognition of online handwritten gestures, and must be able to learn new classes of gestures and to evolve, sample after sample, without using all the old data.

A brief description of the system architecture is presented in Section II. Then, we explain the two main elements of the learning algorithm in Section III. IV studies the experimental evaluation results.

II. SYSTEM ARCHITECTURE

Our system is based on a fuzzy rule-based classifier. The fuzzy rules make a link between intrinsic models (premises) and system outputs by consequent functions. For a K classes problem, a rule R i is built for each fuzzy model P i :

R i : IF x is P i T HEN y 1 i AN D . . . AN D y k i ( 1
)
where x is the n-dimensional feature vector, P i is a fuzzy model (prototype) defined by a center µ i and a covariance matrix Q i . The degree of membership of x to P i is given by the Mahalanobis distance:

β i ( x) = 1/(1 + d Qi ( x, µ i )) (2)
For the consequent part, we can distinguish different structures [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]: (i) the zero order Takagi-Sugeno (TS) with binary consequents (y m i = 1 if P i belongs to class m, 0 otherwise), (ii) the zero order TS with constant consequents (y m i ∈ [0, 1] represents the participation of P i in the description of class m), (iii) the first order TS, where the consequents are linear functions y m i = a m i x. Finally, the sum-product inference is used to compute the system output for each class:

y m ( x) = R i=1 β i ( x) y m i (3)

III. INCREMENTAL LEARNING ALGORITHMS

In order to incrementally learn a fuzzy rule-based classifier in an on-line manner, we need, on the one hand, to evolve its structure by adding (or deleting) rules, and, on the other hand, to adjust its parameters (prototypes' centers, covariance matrices and the consequent parameters). The emphasis in this paper is on the learning of the premise part of the system. We aim at extending the recursive mountain clustering by combining it with a robust adaptation method that can constantly re-center the fuzzy prototypes and reshape their influence zones, according to each single data sample.

A. Recursive Mountain Clustering

As mentioned earlier in Section I, a recursive (on-line, one-pass, non-iterative) version of the mountain clustering method was introduced in [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF]. The recursive formula avoids memorizing the whole previous data but keeps -using few variables -the density distribution in the feature space, based on the previous data:

P k (x(k)) = k -1 (k -1)α(k) + γ(k) -2ζ(k) + k -1 (4)
where P k (x(k)) denotes the potential of the k'th data sample and

α(k) = n j=1 x 2 j (k) (5)
γ(k) = γ(k -1) + α(k -1), γ(1) = 0 (6) ζ(k) = n j=1
x j (k)η j (k),

η j (k) = η j (k -1) + x j (k -1), η j (1) = 0 (7)
Introducing a new sample affects the potential values of the centers of the existing clusters, which can be recursively updated by:

P k (µ i ) = (k -1)P k-1 (µ i ) k -2 + P k-1 (µ i ) + P k-1 (µ i ) n j=1 µ i -x 2 j (8)
If the potential of the new sample is higher than the potential of the existing centers, then this sample will be a center of a new cluster (and a new fuzzy rule will be formed in the case of fuzzy rule-based classifier). If the high potential sample is close to an existing center µ i , then this sample will replace µ i and no new cluster will be created.

B. Fuzzy Vector Quantization

As can be noted in section III-A, the condition to have a high potential is a very hard one, and it is inversely proportional to the growing number of data. In this way, we can imagine a cluster center µ i which is not really in the optimal center position (according to the data history), but that remains the center because it still has the highest potential value. Therefore, the incremental clustering process of the premise part of the fuzzy classifier will not be able to take advantage of the data points that do not have a very high potential to move (or reshape) the existing clusters. We enhance the incremental clustering process (described in section III-A) by an adaptation algorithm that allows the modification of all the fuzzy prototypes by re-centering and re-shaping them for each new data point. For this purpose, we use a fuzzy version of the Vector Quantization algorithm [START_REF] Chung | Fuzzy competitive learning[END_REF]. In this method, the farther the normalized activation β i of the premise of the rule i is away from its objective score β * i , the more it must be moved:

∆µ i = λ * (β * i -β i ( x)) * ( x -µ i ) (9)
where the adaptation parameter λ lies between 0 and 1. The objective score β * i is 1 if the prototype P r and x belong to the same class and 0 otherwise. In the same way, a fuzzy recursive formula is given in [START_REF] Backer | Texture segmentation by frequencysensitive elliptical competitive learning[END_REF] to update the inverse of the covariance matrix as follows:

Q -1 i ⇐ Q -1 i 1 -αδ i - αδ i 1 -αδ i • (Q -1 i d) • (Q -1 i d) T 1 + αδ i ( d T Q -1 i d) (10)
δ i = β * i -β i ( x) (11)
where d x -µ r and α lies between 0 and 1.

C. Learning algorithm

The incremental learning algorithm of the consequent parameters depends on the type of the used fuzzy system. If we take the three structures mentioned in Section II: (i) no consequent learning is needed for the simple structure with binary consequents, (ii) for the second structure, an online estimation of the constant consequents is presented in [13] and (iii) the linear consequents learning problem in a first-order TS can be solved by the weighted Recursive Least Square method (wRLS) [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF]. The complete learning algorithm can be summarized by Algorithm 1.

Algorithm 1 Online incremental learning algorithm

for all new sample x do if x is the first sample of a new class then add a new fuzzy prototype centered on x to the system; let its potential be 1 else calculate the potential of x by [START_REF] Jang | Anfis: adaptive-network-based fuzzy inference system[END_REF] update the potentials of the existing prototypes centers using [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF] if

P ( x) > P k ( µ i ) ∀i ∈ [1, R] then if
x is close to a center µ i then let x be the center of the prototype P i else add a new fuzzy prototype centered on x to the system; let its potential be 1 end if else apply premise adaptation according to x by [START_REF] Yager | Learning of fuzzy rules by mountain clustering[END_REF] and [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] end if end if update the consequent parameters; end for

IV. EVALUATION

We will particularly focus in our experiments on the rapidity of the performance improvement in the beginning of the incremental learning process, and on the stability and the recovery speed of the performance when introducing new unseen classes. We led the experiments on the "SIGN" database, which is a database of on-line handwritten gestures. It is composed of 17 different gestures drawn by 17 different writers on Tablet PCs. Each writer has drawn 100 samples of each gesture, i.e. 1,700 gestures in each writerspecific dataset. The dataset (and additional information on the data collection protocol) can be found in [14]. Each gesture is described by a set of 21 features. The presented results are the average of results of 17 different tests for the 17 writers. In order to get the results unbiased by the data order effect, we repeat the experiment for each writer 40 times with different random data orders and the mean results are considered. We used about half of the database for the incremental learning process and the rest is used to estimate the evolution of the performance during the learning process. Two fuzzy incremental learning models are compared in these experiments: (a) ETS: Evolving zero-order TS classifier with binary consequents and recursive mountain clustering learning, (b) ETS+: our extended version of ETS in which we integrate the fuzzy vector quantization algorithm in the incremental learning process. The parameters λ and α are set to 0.005 and 0.001 respectively. In the first experimental protocol, we introduce the 17 gestures in the beginning of the learning process. A new sample from each class is presented to the system between each two consecutive evaluation points. In order to have referential values in the evaluation of the recognition rate on the used dataset, we trained two well-known non-incremental classifiers using the whole training set in batch mode, and we measured their performance on the test dataset. We choose a Multi-Layer Perceptron (MLP) classifier and a K-Nearest Neighbor (K-NN, with K=5) classifier. We note from Figure 1 that integrating the premise adaptation in the model significantly boosts the incremental learning process. By comparing the performance of ETS+ with that of ETS, we see that the recognition error rate decreases by about 40% thanks to premise adaptation. We note that using ETS+, a recognition rate of 90% is reached after only 15 samples per class. It is important to mention as well that the classifier contains, in average, less than 20 fuzzy rules for 17 classes of gesture, which is computationally reasonable. Furthermore, we can note from the same that the presented incremental one-pass learning model (ETS+) can achieve or exceed the performance of some well-known iterative (or batch) classification methods. In the second experiment, we emulate the real application context in which the classifier starts with few classes of gestures, and then the user adds a set of new gestures according to his needs. We aim at studying the ability of the classifier to learn new classes of data without fully destroying the knowledge learned from the old ones. Thus, the learning process starts with 10 classes of gestures, and then, few samples of 7 new unseen gestures and some samples of the already learned ones are introduced between each two consecutive evaluation points. Figure 2 shows how ETS+ resists better when introducing new classes and it is able to re-estimate rapidly all the covariance matrices and to improve rapidly the recognition performance for the old and the new gestures.

V. CONCLUSION

In the context of handwritten gesture recognition systems, we present in this paper an online incremental learning algorithm for fuzzy rule-based classifiers. Using this algorithm, the recognition system can start to learn from scratch and with few learning data. Moreover, the dynamic nature of the presented classifiers allows as well adding new unseen classes without destroying the already learned ones.

Figure 1 :

 1 Figure 1: Evolution of the recognition rate in the beginning of the incremental learning process.

Figure 2 :

 2 Figure 2: Performance recovery when introducing new unseen classes.