Lucio Boccardo
email: boccardo@mat.uniroma1.it

Gisella Croce
email: gisella.croce@univ-lehavre.fr

Luigi Orsina
email: orsina@mat.uniroma1.it

NONLINEAR DEGENERATE ELLIPTIC PROBLEMS WITH W 1,1 0 (Ω) SOLUTIONS

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W 1,1 0 (Ω) distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.

Introduction and statement of the results

In this paper we are going to study the nonlinear elliptic equation

(1.1)      -div a(x, ∇u) (1 + |u|) γ + u = f in Ω, u = 0 on ∂Ω,
under the following assumptions. The set Ω is a bounded, open subset of R N , with N > 2, γ > 0, f belongs to some Lebesgue space, and a : Ω × R N → R N is a Carathéodory function (i.e., a(•, ξ) is measurable on Ω for every ξ in R N , and a(x, •) is continuous on R N for almost every x in Ω) such that

(1.2) a(x, ξ) • ξ ≥ α |ξ| 2 , (1.3) |a(x, ξ)| ≤ β |ξ| , (1.4) [a(x, ξ) -a(x, η)] • (ξ -η) > 0 ,
for almost every x in Ω and for every ξ and η in R N , ξ = η, where α and β are positive constants. We are going to prove that, under suitable assumptions on γ and f , problem (1.1) has a unique distributional solution u obtained by approximation, with u belonging to the (nonreflexive) Sobolev space W 1,1 0 (Ω). Furthermore, we are going to prove that problem (1.1) does not have a solution if γ > 1 and the datum f is a bounded Radon measure concentrated on a set of zero harmonic capacity.

Problems like (1.1) have been extensively studied in the past. In [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity, dedicated to[END_REF] (see also [START_REF] Giachetti | Existence results for some non uniformly elliptic equations with irregular data[END_REF], [START_REF] Giachetti | Elliptic equations with degenerate coercivity: gradient regularity[END_REF], [START_REF] Porretta | Uniqueness and homogenization for a class of noncoercive operators in divergence form, dedicated to Prof. C. Vinti[END_REF]), existence and regularity results were proved, under the assumption that a(x, ξ) = A(x) ξ, with A a uniformly elliptic bounded matrix, and 0 < γ ≤ 1, for the problem (1.5)

     -div A(x) ∇u (1 + |u|) γ = f in Ω, u = 0 on ∂Ω,
where f belongs to L m (Ω) for some m ≥ 1.

The main difficulty in dealing with problem (1.5) (or (1.1)) is that the differential operator, even if well defined between H 1 0 (Ω) and its dual H -1 (Ω), is not coercive on H 1 0 (Ω) due to the fact that if u is large, 1 (1+|u|) γ tends to zero (see [START_REF] Porretta | Uniqueness and homogenization for a class of noncoercive operators in divergence form, dedicated to Prof. C. Vinti[END_REF] for an explicit example). This lack of coercivity implies that the classical methods used in order to prove the existence of a solution for elliptic equations (see [START_REF] Leray | Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder[END_REF]) cannot be applied even if the datum f is regular. However, in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity, dedicated to[END_REF], a whole range of existence results was proved, yielding solutions belonging to some Sobolev space W 1,q 0 (Ω), with q = q(γ, m) ≤ 2, if f is regular enough. Under weaker summability assumptions on f , the gradient of u (and even u itself) may not be in L 1 (Ω): in this case, it is possible to give a meaning to solutions of problem (1.5), using the concept of entropy solutions which has been introduced in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF].

If γ > 1, a non existence result for problem (1.5) was proved in [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF] (where the principal part is nonlinear with respect to the gradient), even for L ∞ (Ω) data f . Therefore, if the operator becomes "too degenerate", existence may be lost even for data expected to give bounded solutions. However, as proved in [START_REF] Boccardo | Some remarks on a class of elliptic equations[END_REF], existence of solutions can be recovered by adding a lower order term of order zero. Indeed, if we consider the problem

(1.6)      -div A(x)∇u (1 + |u|) γ + u = f in Ω, u = 0 on ∂Ω,
with f in L m (Ω), then the following results can be proved in the case γ > 1 (see [START_REF] Boccardo | Some remarks on a class of elliptic equations[END_REF] and [START_REF] Croce | The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity[END_REF]): i) if m > γ N 2 , then there exists a weak solution in

H 1 0 (Ω) ∩L ∞ (Ω); ii) if m ≥ γ+2, then there exists a weak solution in H 1 0 (Ω)∩L m (Ω); iii) if γ+2 2 < m < γ + 2, then there exists a distributional solution in W 1, 2m γ+2 0 (Ω) ∩ L m (Ω); iv) if 1 ≤ m ≤ γ+2
2 , then there exists an entropy solution in L m (Ω) whose gradient belongs to the Marcinkiewicz space M 2m γ+2 (Ω). Note that if γ + 2 ≤ m < γ N 2 and m tends to γ N 2 , the summability result of ii) is not "continuous" with the boundedness result of i), according to the following example (see also Example 3.3 of [START_REF] Boccardo | Some remarks on a class of elliptic equations[END_REF]).

Example 1.1. If 2 γ < σ < N -2, then u(x) = 1 |x| σ -1 is a distributional solution of (1.6) with A(x) ≡ I, and f (x) = σ(N -2+σ(γ-1)) |x| 2-σ(γ-1) + 1 |x| σ -1.
Due to the assumptions on σ, both f and u belong to L m (Ω), with m < γ N 2 . If m tends to γ N 2 , i.e., if σ tends to 2 γ , the solution u does not become bounded.

As stated before, this paper is concerned with two borderline cases connected with point iv) above:

A. if m = γ+2
2 , we will prove in Section 2 the existence of W 1,1 0 (Ω) distributional solutions, and in Section 3 their uniqueness; B. if f is a bounded Radon measure concentrated on a set E of zero harmonic capacity and γ > 1, we will prove in Section 4 non existence of solutions. In the linear case, i.e., for the boundary value problem (1.6), a simple proof of the existence result is given in [START_REF] Boccardo | A semilinear problem with a W 1[END_REF].

Remark 1.2. Let a(x, ξ) = A(x) ξ, with A a bounded and measurable uniformly elliptic matrix, and let u ≥ 0 be a solution of

-div A(x)∇u (1 + u) γ + u = f , with γ > 1 and f ≥ 0. If we define z = 1 γ -1 1 - 1 (1 + u) γ-1 , then z is a solution of -div(A(x)∇z) + 1 (1 -(γ -1)z) 1 γ-1 -1 = f ,
which is an equation whose lower order term becomes singular as z tends to the value 1 γ-1 . For a study of these problems, see [START_REF] Boccardo | On the regularizing effect of strongly increasing lower order terms, dedicated to Philippe Bénilan[END_REF] and [START_REF] Dupaigne | Elliptic equations with vertical asymptotes in the nonlinear term[END_REF]. Remark 1.3. We explicitely state that our existence results can be generalized to equations with differential operators defined on W 1,p 0 (Ω), with p > 1: if γ ≥ (p-2) + p-1 and if m = γ(p-1)+2 p , then it is possible to prove the existence of a distributional solution u in W 1,1 0 (Ω) ∩ L m (Ω) of the boundary value problem

(1.7)      -div a(x, ∇u) (1 + |u|) γ(p-1) + u = f in Ω, u = 0 on ∂Ω,
where a(x, ξ) satisfies (1.2), (1.3) and (1.4) with p instead of 2 (in (1.3), a grows as |ξ| p-1 ).

2. Existence of a W 1,1 0 (Ω) solution In this section we prove the existence of a W 1,1 0 (Ω) solution to problem (1.1). Our result is the following.

Theorem 2.1. Let γ > 0, and let f be a function in L γ+2 2 (Ω). Then there exists a distributional solution

u in W 1,1 0 (Ω) ∩ L γ+2 2 (Ω) of (1.1), that is, (2.1) Ω a(x, ∇u) • ∇ϕ (1 + |u|) γ + Ω u ϕ = Ω f ϕ , ∀ϕ ∈ W 1,∞ 0 (Ω) .
Remark 2.2. The previous result gives existence of a solution u in W 1,1 0 (Ω) to (1.6) for every γ > 0. If 0 < γ ≤ 1 existence results for (1.1) can also be proved by the same techniques of [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity, dedicated to[END_REF]. More precisely, if f belongs to L m (Ω) with m > N N (1-γ)+1+γ then (1.1) has a solution in W 1,q 0 (Ω), with q = N m(1-γ) N -m(1+γ) . Note that when m tends to N N (1-γ)+1+γ , then q tends to 1. We have now two cases:

if γ+2 2 > N N (1-γ)+1+γ , that is, if 0 < γ < 2
N -1 , our result is weaker than the one in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity, dedicated to[END_REF]. On the other hand, if 2 N -1 ≤ γ ≤ 1, then our result, which strongly uses the lower order term of order zero, is better.

Remark 2.3. The same existence result, with the same proof, holds for the following boundary value problem

-div(b(x, u, ∇u)) + u = f in Ω, u = 0 on ∂Ω, with b : Ω × R × R N → R N a Carathéodory function such that α|ξ| 2 (1 + |s|) γ ≤ b(x, s, ξ) • ξ ≤ β|ξ| 2 ,
where α, β, γ are positive constants.

To prove Theorem 2.1 we will work by approximation. First of all, let g be a function in L ∞ (Ω). Then, by the results of [START_REF] Boccardo | Some remarks on a class of elliptic equations[END_REF], there exists a solution v in

H 1 0 (Ω) ∩ L ∞ (Ω) of (2.2)      -div a(x, ∇v) (1 + |v|) γ + v = g in Ω, v = 0 on ∂Ω.
In order for this paper to be self contained, we give here the easy proof of this fact. Let M = g L ∞ (Ω) + 1, and consider the problem

(2.3)      -div a(x, ∇v) (1 + |T M (v)|) γ + v = g in Ω, v = 0 on ∂Ω.
Here and in the following we define T k (s) = max(-k, min(s, k)) for k ≥ 0 and s in R. Since the differential operator is pseudomonotone and coercive thanks to the assumptions on a and to the truncature, by the results of [START_REF] Leray | Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder[END_REF] there exists a weak solution v in H 1 0 (Ω) of (2.3). Choosing (|v|g L ∞ (Ω) ) + sgn(v) as a test function we obtain, dropping the nonnegative first term,

Ω |v| (|v| -g L ∞ (Ω) ) + ≤ Ω g L ∞ (Ω) (|v| -g L ∞ (Ω) ) + .
Thus,

Ω (|v| -g L ∞ (Ω) ) (|v| -g L ∞ (Ω) ) + ≤ 0 , so that |v| ≤ g L ∞ (Ω) < M. Therefore, T M (v) = v,
and v is a bounded weak solution of (2.2).

Let now f n be a sequence of L ∞ (Ω) functions which converges to f in L

     -div a(x, ∇u n ) (1 + |u n |) γ + u n = f n in Ω, u n = 0 on ∂Ω. A solution u n in H 1 0 (Ω) ∩ L ∞ (Ω) exists choosing g = f n in (2.
2). We begin with some a priori estimates on the sequence {u n }.

Lemma 2.4. If u n is a solution to problem (2.4), then, for every k ≥ 0, (2.5)

{|un|≥k} |u n | γ+2 2 ≤ {|un|≥k} |f | γ+2 2 ;
(2.6)

{|un|≥k} |∇u n | 2 (1 + |u n |) γ+2 2 ≤ C {|un|≥k} |f | γ+2 2 2 γ+2 ; (2.7) {|un|≥k} |∇u n | ≤ C {|un|≥k} |f | γ+2 2 1 γ+2 ; (2.8) α Ω |∇T k (u n )| 2 ≤ k (1 + k) γ Ω |f | .
Here, and in the following, C denotes a positive constant depending on α, γ, meas(Ω), and the norm of

f in L γ+2 2 (Ω).
Proof. Let k ≥ 0, h > 0, and let ψ h,k (s) be the function defined by

ψ h,k (s) =        0 if 0 ≤ s ≤ k, h (s -k) if k < s ≤ k + 1 h , 1 if s > k + 1 h , ψ h,k (s) = -ψ h,k (-s) if s < 0. -k k ψ h,k -1 1 -k -1 h k + 1 h Note that lim h→+∞ ψ h,k (s) =    1 if s > k, 0 if |s| ≤ k, -1 if s < -k. Let ε > 0, and choose (ε + |u n |) γ 2 ψ h,k (u n ) as a test function in (2.4); such a test function is admissible since u n belongs to H 1 0 (Ω) ∩ L ∞ (Ω)
and ψ h,k (0) = 0. We obtain (2.9)

γ 2 Ω a(x, ∇u n ) • ∇u n (ε + |u n |) γ 2 -1 (1 + |u n |) γ |ψ h,k (u n )| + Ω a(x, ∇u n ) • ∇u n (1 + |u n |) γ ψ ′ h,k (u n ) (ε + |u n |) γ 2 + Ω u n (ε + |u n |) γ 2 ψ h,k (u n ) = Ω f n (ε + |u n |) γ 2 ψ h,k (u n ) .
By (1.2), and since ψ ′ h,k (s) ≥ 0, the first two terms are nonnegative, so that we obtain, recalling that

|f n | ≤ |f |, Ω u n (ε + |u n |) γ 2 ψ h,k (u n ) ≤ Ω |f | (ε + |u n |) γ 2 |ψ h,k (u n )| .
Letting ε tend to zero and h tend to infinity, we obtain, by Fatou's lemma (on the left hand side) and by Lebesgue's theorem (on the right hand side, recall that

u n belongs to L ∞ (Ω)), {|un|≥k} |u n | γ+2 2 ≤ {|un|≥k} |f | |u n | γ 2 .
Using Hölder's inequality on the right hand side we obtain

{|un|≥k} |u n | γ+2 2 ≤ {|un|≥k} |f | γ+2 2 2 γ+2 {|un|≥k} |u n | γ+2 2 γ γ+2
.

Simplifying equal terms we thus have

{|un|≥k} |u n | γ+2 2 ≤ {|un|≥k} |f | γ+2 2 ,
which is (2.5). Note that from (2.5), written for k = 0, it follows (2.10)

Ω |u n | γ+2 2 ≤ Ω |f | γ+2 2 = f γ+2 2 L γ+2 2 (Ω)
. Now we consider (2.9) written for ε = 1. Dropping the nonnegative second and third terms, and using that

|f n | ≤ |f |, we have γ 2 Ω a(x, ∇u n ) • ∇u n (1 + |u n |) γ+2 2 |ψ h,k (u n )| ≤ Ω |f |(1 + |u n |) γ+2 2 |ψ h,k (u n )| .
Using (1.2), and letting h tend to infinity, we get (using again Fatou's lemma and Lebesgue's theorem)

α γ 2 {|un|≥k} |∇u n | 2 (1 + |u n |) γ+2 2 ≤ {|un|≥k} |f |(1 + |u n |) γ 2 .
Hölder's inequality on the right hand side then gives

α γ 2 {|un|≥k} |∇u n | 2 (1 + |u n |) γ+2 2 ≤ {|un|≥k} |f | γ+2 2 2 γ+2 {|un|≥k} (1 + |u n |) γ+2 2 γ γ+2 ≤ {|un|≥k} |f | γ+2 2 2 γ+2 Ω (1 + |u n |) γ+2 2 γ γ+2
, so that, by (2.10),

α γ 2 {|un|≥k} |∇u n | 2 (1 + |u n |) γ+2 2 ≤ C {|un|≥k} |f | γ+2 2 2 γ+2 , which is (2.6).
Then, again by Hölder's inequality, and by (2.6) and (2.10),

(2.11)

{|un|≥k} |∇u n | = {|un|≥k} |∇u n | (1 + |u n |) γ+2 4 (1 + |u n |) γ+2 4 ≤ {|un|≥k} |∇u n | 2 (1 + |u n |) γ+2 2 1 2 {|un|≥k} (1 + |u n |) γ+2 2 1 2 ≤ C {|un|≥k} |f | γ+2 2 1 γ+2 Ω (1 + |u n |) γ+2 2 1 2 ≤ C {|un|≥k} |f | γ+2 2 1 γ+2
, so that (2.7) is proved.

Finally, choosing T k (u n ) as a test function in (2.4) we get, dropping the nonnegative linear term, and using (1.2),

α Ω |∇T k (u n )| 2 ≤ k(1 + k) γ Ω |f | , which is (2.8).
Lemma 2.5. If {u n } is the sequence of solutions to (2.4), there exists a subsequence, still denoted by {u n }, and a function u in L γ+2 2 (Ω), with T k (u) belonging to H 1 0 (Ω) for every k > 0, such that u n almost everywhere converges to u in Ω, and T k (u n ) weakly converges to T k (u) in H 1 0 (Ω). Proof. Consider (2.6) written for k = 0:

(2.12)

Ω |∇u n | 2 (1 + |u n |) γ+2 2 ≤ C f L γ+2 2 (Ω)
.

Since (if γ = 2)

|∇u n | 2 (1 + |u n |) γ+2 2 = 16 (2 -γ) 2 |∇[(1 + |u n |) 2-γ 4 -1]| 2 , the sequence v n = 4 2-γ [(1 + |u n |) 2-γ 4 -1]sgn(u n ) is bounded in H 1 0 (Ω) by (2.12). If γ = 2 we have |∇u n | 2 (1 + |u n |) 2 = |∇ log(1 + |u n |)| 2 , so that v n = [log(1 + |u n |)]sgn(u n ) is bounded in H 1 0 (Ω).
In both cases, up to a subsequence still denoted by v n , v n converges to some function v weakly in H 1 0 (Ω), strongly in L 2 (Ω), and almost everywhere in Ω. If γ < 2, define

u(x) = 2 -γ 4 |v(x)| + 1 4 2-γ -1 sgn(v(x)) , if γ > 2 define u(x) =          2-γ 4 |v(x)| + 1 4 2-γ -1 sgn(v(x)) if |v(x)| < 4 γ-2 , +∞ if v(x) = 4 γ-2 , -∞ if v(x) = -4 γ-2 , while if γ = 2, define u(x) = [e |v(x)| -1]sgn(v(x)) .
Thus, u n almost everywhere converges, up to a subsequence still denoted by u n , to u. From now on, we will consider this particular subsequence, for which it holds that u n almost everywhere converges to u.

We use now (2.5) written for k = 0:

Ω |u n | γ+2 2 ≤ Ω |f | γ+2 2 ≤ C .
Since u n almost everywhere converges to u, we have from Fatou's lemma that

Ω |u| γ+2 2 ≤ C .
Hence u belongs to L γ+2 2 (Ω), which implies that u is almost everywhere finite (note that if γ > 2 this fact did not follow from the definition of u, since |v| could have assumed the value 4 γ-2 on a set of positive measure).

Let now k > 0; since from (2.8) it follows that the sequence {T k (u n )} is bounded in H 1 0 (Ω), there exists a subsequence T k (u n j ) which weakly converges to some function v k in H 1 0 (Ω). Using the almost everywhere convergence of u n to u, we have that v k = T k (u). Since the limit is independent on the subsequence, then the whole sequence {T k (u n )} weakly converges to T k (u), for every k > 0.

Remark 2.6. Using the fact that T k (u) is in H 1 0 (Ω) for every k > 0, and the results of [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF], we have that there exists a unique measurable function v with values in R N , such that

∇T k (u) = v χ {|u|≤k}
almost everywhere in Ω, for every k > 0.

Following again [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF], we will define ∇u = v, the approximate gradient of u.

Remark 2.7. We emphasize that if γ = 2, then (2.11), written for k = 0, becomes

Ω |∇u n | ≤ Ω |∇u n | 2 (1 + |u n |) 2 1 2 Ω (1 + |u n |) 2 1 2 . Since |∇u n | 2 (1 + |u n |) 2 = |∇ log(1 + |u n |)| 2 ,
a nonlinear interpolation result follows: let A be in R + and let v in L 2 (Ω) be such that log(A + |v|) belongs to H 1 0 (Ω). Then v belongs to W 1,1 0 (Ω), and

Ω |∇v| ≤ log(A + |v|) H 1 0 (Ω) Ω (A + |v|) 2 1 2
.

Our next result deals with the strong convergence of T k (u n ) in H 1 0 (Ω). Proposition 2.8. Let u n and u be the sequence of solutions to problems (2.4) and the function in L γ+2 2 (Ω) given by Lemma 2.5. Then, for every fixed k > 0, T k (u n ) strongly converges to T k (u) in H 1 0 (Ω), as n tends to infinity.

Proof. We follow the proof of [START_REF] Leone | Entropy solutions for nonlinear elliptic equations in L 1[END_REF].

Let h > k and choose

T 2k [u n -T h (u n ) + T k (u n ) -T k (u)] as a test function in (2.4). We have (2.13) Ω a(x, ∇u n ) • ∇T 2k [u n -T h (u n ) + T k (u n ) -T k (u)] (1 + |u n |) γ = - Ω (u n -f n ) T 2k [u n -T h (u n ) + T k (u n ) -T k (u)] .
We observe that the right hand side converges to zero as first n and then h tend to infinity, since u n converges to u almost everywhere in Ω and u n and f n are bounded in L γ+2 2 (Ω). Thus, if we define ε(n, h) as any quantity such that

lim h→+∞ lim n→+∞ ε(n, h) = 0 , then Ω (u n -f n ) T 2k [u n -T h (u n ) + T k (u n ) -T k (u)] = ε(n, h) . Let M = 4k+h. Observing that ∇T 2k [u n -T h (u n )+T k (u n )-T k (u)] = 0 if |u n | ≥ M, by (2.13) we have ε(n, h) = {|un|<k} a(x, ∇T M (u n )) • ∇[u n -T h (u n ) + T k (u n ) -T k (u)] (1 + |u n |) γ + {|un|≥k} a(x, ∇T M (u n )) • ∇[u n -T h (u n ) + T k (u n ) -T k (u)] (1 + |u n |) γ . Since u n -T h (u n ) = 0 in {|u n | ≤ k} and ∇T k (u n ) = 0 in {|u n | ≥ k}, we have, using that a(x, 0) = 0, ε(n, h) = Ω a(x, ∇T k (u n )) • ∇[T k (u n ) -T k (u)] (1 + |u n |) γ + {|un|≥k} a(x, ∇T M (u n )) • ∇[u n -T h (u n )] (1 + |u n |) γ - {|un|≥k} a(x, ∇T M (u n )) • ∇T k (u) (1 + |u n |) γ .
The second term of the right hand side is positive, so that

ε(n, h) ≥ Ω [a(x, ∇T k (u n )) -a(x, ∇T k (u))] • ∇[T k (u n ) -T k (u)] (1 + k) γ + Ω a(x, ∇T k (u)) • ∇[T k (u n ) -T k (u)] (1 + |u n |) γ - {|un|≥k} a(x, ∇T M (u n )) • ∇T k (u) (1 + |u n |) γ = I n + J n -K n .
The last two terms tend to zero as n tends to infinity. Indeed

lim n→+∞ J n = lim n→+∞ Ω a(x, ∇T k (u)) • ∇[T k (u n ) -T k (u)] (1 + |u n |) γ = 0 , since T k (u n ) converges to T k (u)
weakly in H 1 0 (Ω) and a(x,∇T k (u)) (1+|un|) γ is strongly compact in (L 2 (Ω)) N by the growth assumption (1.3) on a.

The last term can be rewritten as

K n = Ω a(x, ∇T M (u n )) • ∇T k (u)χ {|un|≥k} (1 + |u n |) γ .
Since M is fixed with respect to n, then the sequence {a(x, ∇T M (u n ))} is bounded in (L 2 (Ω)) N . Hence, there exists σ in (L 2 (Ω)) N , and a subsequence {a(x, ∇T M (u n j ))}, such that

lim j→+∞ a(x, ∇T M (u n j )) = σ ,
weakly in (L 2 (Ω)) N . On the other hand,

lim n→+∞ ∇T k (u)χ {k≤|un|} (1 + |u n |) γ = ∇T k (u)χ {k≤|u|} (1 + |u|) γ = 0 ,
strongly in (L 2 (Ω)) N , and so lim j→+∞

K n j = lim j→+∞ {|un j |≥k} a(x, ∇T M (u n j )) • ∇T k (u) (1 + |u n j |) γ = 0 .
Since the limit does not depend on the subsequence, we have

lim n→+∞ K n = lim n→+∞ {|un|≥k} a(x, ∇T M (u n )) • ∇T k (u) (1 + |u n |) γ = 0 , as desired. Therefore, ε(n, h) ≥ I n = Ω [a(x, ∇T k (u n )) -a(x, ∇T k (u))] • ∇[T k (u n ) -T k (u)] (1 + k) γ , so that, thanks to (1.4), lim n→+∞ Ω [a(x, ∇T k (u n )) -a(x, ∇T k (u))] • ∇[T k (u n ) -T k (u)] = 0 .
Using this formula, (1.4) and the results of [START_REF] Browder | Existence theorems for nonlinear partial differential equations[END_REF] and [START_REF] Boccardo | Resultats d'existence pour certains problèmes elliptiques quasi linéaires[END_REF], we then conclude that T k (u n ) strongly converges to T k (u) in H 1 0 (Ω), as desired. Corollary 2.9. Let u n and u be as in Proposition 2.8. Then ∇u n converges to ∇u almost everywhere in Ω, where ∇u has been defined in Remark 2.6.

Lemma 2.10. Let u n and u be as in Proposition 2.8. Then ∇u n strongly converges to ∇u in (L 1 (Ω)) N . Moreover u n strongly converges to u in L γ+2 2 (Ω).

Proof. We begin by proving the convergence of ∇u n to ∇u. Let ε > 0, and let k > 0 be sufficiently large such that (2.14)

{|un|≥k} |f | γ+2 2 1 γ+2 < ε ,
uniformly with respect to n. This can be done thanks to (2.10) and to the absolute continuity of the integral. Let E be a measurable set. Writing

E |∇u n | = E |∇T k (u n )| + E∩{|un|≥k} |∇u n |
we have, by (2.7), and by (2.14),

E |∇u n | ≤ E |∇T k (u n )| + Cε .
Using Hölder's inequality and (2.8), we obtain

E |∇u n | ≤ C meas(E) 1 2 k 1 2 (1 + k) γ 2 Ω |f | 1 2 + Cε .
Choosing meas(E) small enough (recall that k is now fixed) we have

E |∇u n | ≤ Cε ,
uniformly with respect to n, where C does not depend on n or ε. Since ∇u n almost everywhere converges to ∇u by Corollary 2.9, we can apply Vitali's theorem to obtain the strong convergence of ∇u n to ∇u in (L 1 (Ω)) N .

As for the second convergence, by (2.5) we have

E |u n | γ+2 2 ≤ E∩{|un|≤k} |u n | γ+2 2 + E∩{|un|≥k} |u n | γ+2 2 ≤ k γ+2 2 meas(E) + {|un|≥k} |f | γ+2 2 .
As before, we first choose k such that the second integral is small, uniformly with respect to n, and then the measure of E small enough such that the first term is small. The almost everywhere convergence of u n to u, and Vitali's theorem, then imply that u n strongly converges to u in L γ+2 2 (Ω).

Remark 2.11. Since we have proved that ∇u n strongly converges to ∇u in (L 1 (Ω)) N , so that u belongs to W 1,1 0 (Ω), then the approximate gradient ∇u of u is nothing but the distributional gradient of u (see [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]).

Proof of Theorem 2.1. Using the previous results, we pass to the limit, as n tends to infinity, in the weak formulation of (2.4). Starting from

Ω a(x, ∇u n ) • ∇ϕ (1 + |u n |) γ + Ω u n ϕ = Ω f n ϕ , ϕ ∈ W 1,∞ 0 (Ω) ,
the limit of the second and the last integral is easy to compute; indeed, recall that by Lemma 2.10, and by definition of f n , the sequences {u n } and {f n } strongly converge to u and f respectively in L γ+2 2 (Ω), hence in L 1 (Ω). For the first integral, we have that a(x, ∇u n ) converges almost everywhere in Ω to a(x, ∇u) thanks to Corollary 2.9, and the continuity assumption on a(x, •); furthermore, (1.3) implies that

|a(x, ∇u n )| ≤ β|∇u n | ,
and the right hand side is compact in L 1 (Ω) by Lemma 2.10. Thus, by Vitali's theorem a(x, ∇u n ) strongly converges to a(x, ∇u) in (L 1 (Ω)) N , so that

lim n→+∞ Ω a(x, ∇u n ) • ∇ϕ (1 + |u n |) γ = Ω a(x, ∇u) • ∇ϕ (1 + |u|) γ ,
where we have also used that u n almost everywhere converges to u, and Lebesgue's theorem. Thus, we have that

Ω a(x, ∇u) • ∇ϕ (1 + |u|) γ + Ω u ϕ = Ω f ϕ , ∀ϕ ∈ W 1,∞ 0 (Ω) , i.e., u satisfies (2.1)
.

Uniqueness of the solution obtained by approximation

Let f ∈ L γ+2 2 (Ω), let f n be a sequence of L ∞ (Ω) functions converging to f in L γ+2 2 (Ω), with |f n | ≤ |f |, and let u n be a solution of (2.4). In Section 2 we proved the existence of a distributional solution u in W 1,1 0 (Ω) ∩ L γ+2 2 (Ω) to (1.1), such that, up to a subsequence,

lim n→+∞ u n -u W 1,1 0 (Ω)∩L γ+2 2 (Ω) = 0 . Now, let g ∈ L γ+2 2 (Ω), let g n be a sequence of L ∞ (Ω) functions con- verging to g in L γ+2 2 (Ω), with |g n | ≤ |g|, and let z n in H 1 0 (Ω) ∩ L ∞ (Ω) be a weak solution of (3.2)      -div a(x, ∇z n ) (1 + |z n |) γ + z n = g n in Ω, z n = 0 on ∂Ω. (3.1)
Then, up to a subsequence, we can assume that

(3.3) lim n→+∞ z n -z W 1,1 0 (Ω)∩L γ+2 2 (Ω) = 0 , where z in W 1,1 0 (Ω) ∩ L γ+2 2 (Ω) is a distributional solution of (3.4)      -div a(x, ∇z) (1 + |z|) γ + z = f in Ω, z = 0 on ∂Ω.
Our result, which will imply the uniqueness of the solution by approximation (see [START_REF] Dall | Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations[END_REF]) of (1.1), is the following.

Theorem 3.1. Assume that u n and z n are solutions of (2.4) and (3.2) respectively, and that (3.1) and (3.3) hold true, with u and z solutions of (1.1) and (3.4) respectively. Then

(3.5) Ω |u -z| ≤ Ω |f -g| . Moreover, (3.6)
f ≤ g a.e. in Ω implies u ≤ z a.e. in Ω.

Proof. Substracting (3.2) from (2.4) we obtain

-div a(x, ∇u n ) (1 + |u n |) γ - a(x, ∇z n ) (1 + |z n |) γ + u n -z n = f n -g n .
Choosing T h (u nz n ) as a test function we have

Ω a(x, ∇u n ) (1 + |u n |) γ - a(x, ∇z n ) (1 + |z n |) γ • ∇T h (u n -z n ) + Ω (u n -z n )T h (u n -z n ) = Ω (f n -g n )T h (u n -z n ) .
This equality can be written in an equivalent way as

Ω [a(x, ∇u n ) -a(x, ∇z n )] • ∇T h (u n -z n ) (1 + |u n |) γ + Ω (u n -z n )T h (u n -z n ) = Ω (f n -g n )T h (u n -z n ) - Ω 1 (1 + |u n |) γ - 1 (1 + |z n |) γ a(x, ∇z n ) • ∇T h (u n -z n ) .
By (1.4), the first term of the left hand side is nonnegative, so that it can be dropped; using Lagrange's theorem on the last term of the right hand side, we therefore have, since the absolute value of the derivative of the function s → 1 (1+|s|) γ is bounded by γ,

Ω (u n -z n )T h (u n -z n ) ≤ Ω (f n -g n )T h (u n -z n ) +γh Ω |a(x, ∇z n )||∇T h (u n -z n )| .
Dividing by h we obtain

Ω (u n -z n ) T h (u n -z n ) h ≤ Ω |f n -g n | |T h (u n -z n )| h +γ Ω |a(x, ∇z n )||∇T h (u n -z n )| .
Since, for every fixed n, u n and z n belong to H 1 0 (Ω), and a(x, ξ) satisfies (1.3), the limit as h tends to zero gives

(3.7) Ω |u n -z n | ≤ Ω |f n -g n | ,
which then yields (3.5) passing to the limit and using the second part of Lemma 2.10.

The use of T h (u nz n ) + as a test function and the same technique as above imply that

Ω (u n -z n ) + ≤ {un≥zn} (f n -g n ) .
Hence, passing to the limit as n tends to infinity, we obtain, if we suppose that f ≤ g almost everywhere in Ω,

Ω (u -z) + ≤ {u≥z} (f -g) ≤ 0 , so that (3.6) is proved.
Thanks to (3.5), we can prove that problem (1.1) has a unique solution obtained by approximation. Corollary 3.2. There exists a unique solution obtained by approximation of (1.1), in the sense that the solution u in W 1,1 0 (Ω) ∩ L γ+2 2 (Ω) obtained as limit of the sequence u n of solutions of (2.4) does not depend on the sequence f n chosen to approximate the datum f in L γ+2 2 (Ω).

Remark 3.3. Note that (3.7) implies the uniqueness of the solution of (2.2), while (3.6) implies that if f ≥ 0, then the solution u of (1.1) is nonnegative.

Remark 3.4. Corollary 3.2, together with estimates (3.5) and (2.5), implies that the map S from L γ+2 2 (Ω) into itself defined by S(f ) = u, where u is the solution of (1.1) with datum f , is well defined and satisfies

S(f ) -S(g) L 1 (Ω) ≤ f -g L 1 (Ω) , S(f ) L γ+2 2 (Ω) ≤ f L γ+2 2 (Ω)
.

A non existence result

As stated in the Introduction, we prove here a non existence result for solutions of (1.1) if the datum is a bounded Radon measure concentrated on a set E of zero harmonic capacity.

Theorem 4.1. Assume that γ > 1, and let µ be a nonnegative Radon measure, concentrated on a set E of zero harmonic capacity. Then there is no solution to

     -div a(x, ∇u) (1 + u) γ + u = µ in Ω, u = 0 on ∂Ω.
More precisely, if {f n } is a sequence of nonnegative L ∞ (Ω) functions which converges to µ in the tight sense of measures, and if u n is the sequence of solutions to (2.4), then u n tends to zero almost everyhwere in Ω and

lim n→+∞ Ω u n ϕ = Ω ϕ dµ ∀ ϕ ∈ W 1,∞ 0 (Ω) .
Remark 4.2. A similar non existence result for the case γ ≤ 1 is much more complicated to obtain. Indeed, if for example a(x, ξ) = ξ, and γ = 1, the change of variables

v = log(1 + u) yields that v is a solution to      -∆v + e v -1 = µ in Ω, u = 0 on ∂Ω.
Existence and non existence of solutions for such a problem has been studied in [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF] (where the concept of "good measure" was introduced) and in [START_REF] Vazquez | On a semilinear equation in R 2 involving bounded measures[END_REF] (if N = 2) and [START_REF] Bartolucci | Semilinear equations with exponential nonlinearity and measure data[END_REF] (if N ≥ 3).

Proof. Let µ be as in the statement. Then (see [START_REF] Maso | Prignet: Renormalized solutions of elliptic equations with general measure data[END_REF]) for every δ > 0 there exists a function

ψ δ in C ∞ 0 (Ω) such that 0 ≤ ψ δ ≤ 1 , Ω |∇ψ δ | 2 ≤ δ , Ω (1 -ψ δ )dµ ≤ δ .
Note that, as a consequence of the estimate on ψ δ in H 1 0 (Ω), and of the fact that 0 ≤ ψ δ ≤ 1, ψ δ tends to zero in the weak * topology of L ∞ (Ω) as δ tends to zero.

If f n is a sequence of nonnegative functions which converges to µ in the tight convergence of measures, that is, if

lim n→+∞ Ω f n ϕ = Ω ϕ dµ , ∀ϕ ∈ C 0 (Ω) , then (4.1) 0 ≤ lim n→+∞ Ω f n (1 -ψ δ ) = Ω (1 -ψ δ ) dµ ≤ δ .
Let u n be the nonnegative solution to the approximating problem (2.4).

If we choose 1 -(1 + u n ) 1-γ as a test function in (2.4), we have, by (1.2), and dropping the nonnegative lower order term,

α(γ -1) Ω ∇u n (1 + u n ) γ 2 ≤ (γ -1) Ω a(x, ∇u n ) • ∇u n (1 + u n ) 2γ ≤ Ω f n .
Recalling (1.3), we thus have

Ω a(x, ∇u n ) (1 + u n ) γ 2 ≤ β Ω ∇u n (1 + u n ) γ 2 ≤ C Ω f n ,
with C depending on α, β and γ. Therefore, up to a subsequence, there exist σ in (L 2 (Ω)) N and ρ in L 2 (Ω) such that

(4.2) lim n→+∞ a(x, ∇u n ) (1 + u n ) γ = σ , lim n→+∞ ∇u n (1 + u n ) γ = ρ , weakly in (L 2 (Ω)) N and L 2 (Ω) respectively. The choice of [1 -(1 + u n ) 1-γ ](1 -ψ δ ) as a test function in (2.4) gives (4.3) (γ -1) Ω a(x, ∇u n ) • ∇u n (1 + u n ) 2γ (1 -ψ δ ) + Ω u n [1 -(1 + u n ) 1-γ ](1 -ψ δ ) = Ω f n [1 -(1 + u n ) 1-γ ](1 -ψ δ ) + Ω a(x, ∇u n ) • ∇ψ δ (1 + u n ) γ [1 -(1 + u n ) 1-γ ] ≤ Ω f n (1 -ψ δ ) + Ω a(x, ∇u n ) • ∇ψ δ (1 + u n ) γ [1 -(1 + u n ) 1-γ ] .
We study the right hand side. For the first term, (4.1) implies that

lim δ→0 + lim n→+∞ Ω f n (1 -ψ δ ) = 0 ,
while for the second one, we have, using (4.2), and the boundedness of

[1 -(1 + u n ) 1-γ ], lim n→+∞ Ω a(x, ∇u n ) • ∇ψ δ (1 + u n ) γ [1 -(1 + u n ) 1-γ ] = Ω σ • ∇ψ δ [1 -(1 + u) 1-γ ] .
Recalling that σ is in (L 2 (Ω)) N , that ψ δ tends to zero in H 1 0 (Ω), and using the boundedness [1 -(1 + u) 1-γ ], we have

lim δ→0 + lim n→+∞ Ω a(x, ∇u n ) • ∇ψ δ (1 + u n ) γ [1 -(1 + u n ) 1-γ ] = 0 .
Therefore, since both terms of the left hand side of (4. (1-ψ δ ) = 0 , which implies that ρ = 0. Thus, since

∇u n (1 + u n ) γ = 1 γ -1 ∇ 1 -(1 + u n ) 1-γ ,
by the second limit of (4.2) the sequence 1-(1+u n ) 1-γ weakly converges to zero in H 1 0 (Ω), and so (up to subsequences) it strongly converges to zero in L 2 (Ω). Therefore u n (up to subsequences) tends to zero almost everywhere in Ω. Since the limit does not depend on the subsequence, the whole sequence u n tends to zero almost everywhere in Ω.

We now have, for Φ in (L 2 (Ω)) N , and by (1.3), (Ω), as desired.

Remark 4.3. With minor technical changes (see [START_REF] Maso | Prignet: Renormalized solutions of elliptic equations with general measure data[END_REF]) one can prove the same result if µ is a signed Radon measure concentrated on a set E of zero harmonic capacity.

γ+2 2 (

 2 Ω), and such that |f n | ≤ |f | almost everywhere in Ω, and consider the approximating problems(2.4)

Ω

  a(x, ∇u n ) (1 + |u n |) γ • Φ ≤ Ω a(x, ∇u n ) (1 + |u n |) γ |Φ| ≤ β Ω |∇u n | (1 + |u n |) γ |Φ| . Thus, by (4.2), Ω σ • Φ = lim n→+∞ Ω a(x, ∇u n ) (1 + |u n |) γ • Φ ≤ β Ω ρ |Φ| = 0 ,which implies that σ = 0. Therefore, passing to the limit in (2.4), that is, inΩ a(x, ∇u n ) • ∇ϕ (1 + u n )we get, since the first term tends to zero,lim n→+∞ Ω u n ϕ = Ω ϕ dµ ,for every ϕ in W 1,∞ 0


Gisella Croce
email: gisella.croce@univ-lehavre.fr

AN ELLIPTIC PROBLEM WITH TWO SINGULARITIES

We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.

Introduction

In this note we study existence of solutions to the following elliptic problem:

(1.1)

   -div a(x)∇u (1 + |u|) p = f |u| γ in Ω, u = 0 on ∂Ω,
where Ω is an open bounded subset of R N , N ≥ 3, p and γ are positive reals, f is an L m (Ω) non-negative function and a : Ω → R is a measurable function such that 0 < α ≤ a(x) ≤ β, for two positive constants α and β.

The operator v → -div a(x)∇v (1+|v|) p is not coercive on H 1 0 (Ω), when v is large (see [START_REF] Porretta | Uniqueness and homogeneization for a class of noncoercive operators in divergence form[END_REF]); moreover, the right-hand side is singular in the variable u. These two difficulties has led us to study problem (1.1) by approximation. More precisely, we will define a sequence of problems (see problems (2.1)), depending on the parameter n ∈ N, in which we "truncate" the degenerate coercivity of the operator term and the singularity of the right hand side. We will prove in Section 2 that these problems admit a bounded H 1 0 (Ω) solution u n , n ∈ N, with the property that for every subset ω ⊂⊂ Ω there exists a positive constant c ω > 0 such that u n ≥ c ω almost everywhere in ω for every n ∈ N. In Section 3 we will get some a priori estimates on u n and, by compactness, a function u to which u n converges. By passing to the limit in the approximating problems (2.1), u will turn out to be a solution to problem (1.1) in the following sense. For every ω ⊂⊂ Ω there exists c ω > 0 such that u ≥ c ω > 0 in ω and

(1.2) Ω a(x) ∇u • ∇ϕ (1 + u) p = Ω f u γ ϕ ∀ ϕ ∈ C 1 0 (Ω) .
We are now going to motivate the study of problem (1.1). The lack of coercivity of the operator term can negatively affect the existence and regularity of solutions to

(1.3)    -div a(x)∇u (1 + |u|) p = f in Ω, u = 0 on ∂Ω.
This was first pointed out in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF], in the case p < 1. In the case where p > 1, the authors of [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF] proved that no solutions exist for some constant sources f . A natural question is then to search for lower order terms which regularize the solutions to problem (1.3). This leads to the study of existence and regularity of solutions to problems of the form

(1.4)    -div a(x)∇u (1 + |u|) p + g(u, ∇u) = f in Ω, u = 0 on ∂Ω,
under various hypotheses on g : Ω × R N → R. In [START_REF] Boccardo | Some remarks on a class of elliptic equations with degenerate coercivity[END_REF] the case g = g(s) = s was examined. This work takes advantage of the fact that, just as for semilinear elliptic coercive problems (cf. [START_REF] Brézis | Semi-linear second-order elliptic equations in L 1[END_REF]), the summability of the solutions is at least that of the source f . In [START_REF] Croce | The regularizing effects of some lower order terms on the solutions in an elliptic equation with degenerate coercivity[END_REF] we analysed two different lower order terms g = g(s). The first, a generalization of the lower order term studied in [START_REF] Boccardo | Some remarks on a class of elliptic equations with degenerate coercivity[END_REF], is g(s) = |s| r-1 s, r > 0. The second is a continuous positive increasing function such that g(s) → +∞, as s → s - 0 , for some s 0 > 0 (see [START_REF] Boccardo | On the regularizing effect of strongly increasing lower order terms[END_REF]). The lower order term of [START_REF] Boccardo | Quasilinear elliptic equations with natural growth terms: the regularizing effects of the lower order terms[END_REF] is, roughly speaking, b(|s|)|ξ| 2 where b is continuous and increasing with respect to |s|. In [START_REF] Croce | An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term[END_REF] we showed the regularizing effects of the lower order term g(s, ξ) = |ξ| 2 s q , q > 0, which grows as a negative power with respect to s and has a quadratic dependence on the gradient variable (see [START_REF] Arcoya | Existence and non-existence of solutions for singular quadratic quasilinear equations[END_REF] and [START_REF] Boccardo | Dirichlet problems with singular and gradient quadratic lower order terms[END_REF] for elliptic coercive problems with the same lower order term).

In all of the above papers a regularizing effect on the solutions to (1.3) was shown under a sign condition on g: either g(s, ξ) ≥ 0 for every (s, ξ) ∈ R × R N and the source is assumed to be positive or g(s, ξ)s ≥ 0 for every (s, ξ) ∈ R × R N .

In this paper we study the effects on the solutions of a different term, f u γ , on the right hand side. Our inspiration is taken from [START_REF] Boccardo | Semilinear elliptic equations with singular nonlinearities[END_REF] where the authors considered the same right hand side, for elliptic semilinear problems whose model is -∆u = f u γ , with zero Dirichlet condition on the boundary. Our main result shows that, despite the singularity in u, this term has some regularizing effects on the solutions to (1.3). The regularity depends on the different values of γp : we distinguish the cases p-1 ≤ γ < p+ 1, γ = p+ 1, and γ > p+ 1. These statements are made more precise in the following

Theorem 1.1. Let γ ≥ p -1. (1) Let γ < p + 1. (a) If f ∈ L m (Ω), with m ≥ 2 * 2 * -p -1 + γ
, there exists a solution u ∈ H 1 0 (Ω)

to (1.1) in the sense of (1.2). If 2 * 2 * -p -1 + γ ≤ m < N 2 , then u belongs to L m * * (γ+1-p) (Ω). (b) If f ∈ L m (Ω), with max 1, 1 * 2 • 1 * -p -1 + γ < m < 2 * 2 * -p -1 + γ , there exists a solution u ∈ W 1,σ 0 (Ω), σ = N m(γ + 1 -p) N -m(p + 1 -γ) , to (1.1) in the sense of (1.2). (2) Let γ = p + 1 and assume that f ∈ L 1 (Ω).
Then there exists a solution u ∈ H 1 0 (Ω) to (1.1) in the sense of (1.2).

(3) Let γ > p + 1 and assume that f ∈ L 1 (Ω). Then there exists a solution u ∈ H 1 loc (Ω) to (1.1) in the sense of (1.2), such that u

γ+1-p 2 ∈ H 1 0 (Ω). (4) Let f ∈ L m (Ω), with m > N 2
. Then the solution found above is bounded.

Let us point out the regularizing effects of the right hand side. It is useful to recall the results obtained in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF] for problem (1.3). Let p < 1 and q

= N m(1 -p) N -m(1 + p) . a) If 1 < m ≤ 2N N + 2 -p(N -2)
, then there exists u ∈ W 1,q 0 (Ω) or

|∇u| s ∈ L 1 (Ω), ∀ s < q. b) If 2N N + 2 -p(N -2) ≤ m < N 2 , then there exists u ∈ H 1 0 (Ω) ∩ L m * * (1-p) (Ω). c) If m > N 2 , then there exists u ∈ H 1 0 (Ω) ∩ L ∞ (Ω)
. We now compare the summabilities obtained in Theorem 1.1 to the previous ones. First of all, we have a solution for every p > 0, if γ ≥ p -1. This is not the case for problem (1.3), as proved in [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF]. Under the same conditions on f , the summability of the solutions to (1.1) is better than or equal to that of the solutions to (1.3), since σ > q and m * * (γ + 1p) > m * * (1p). Moreover, we get H 1 0 (Ω) solutions for less regular sources than in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF].

Indeed, if p -1 ≤ γ < p + 1, one has 2 * 2 * -p -1 + γ < 2N N (1 -p) + 2(p + 1)
; if γ = p + 1 we get a finite energy solution for every L 1 (Ω) source.

Approximating problems

As explained in the Introduction, we will work on the following approximating problems:

(2.1)

     -div a(x)∇u n (1 + |T n (u n )|) p = T n (f ) |u n | + 1 n γ in Ω, u n = 0 on ∂Ω,
where n ∈ N and

(2.2)

T n (s) =      -n, s ≤ -n s, -n ≤ s ≤ n n, s ≥ n .
Observe that we have"truncated" the degenerate coercivity of the operator term and the singularity of the right hand side.

Proposition 2.1. Problems (2.1) are well posed, that is, there exists a non-negative solution

u n ∈ H 1 0 (Ω) ∩ L ∞ (Ω) for every fixed n ∈ N.
Proof. In this proof we will use the same technique as in [START_REF] Boccardo | Semilinear elliptic equations with singular nonlinearities[END_REF]. Let S : L 2 (Ω) → L 2 (Ω) be the map which associates to every v ∈ L 2 (Ω) the solution

w n ∈ H 1 0 (Ω) to (2.3)      -div a(x)∇w n (1 + |T n (w n )|) p = T n (f ) |v| + 1 n γ in Ω, w n = 0 on ∂Ω.
Observe that S is well-defined by the results of [START_REF] Leray | Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder[END_REF] and w n is bounded by the results of [START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF]. Let us choose w n as a test function. Then,

α Ω |∇w n | 2 (1 + n) p ≤ Ω a(x)|∇w n | 2 (1 + T n (w n )) p = Ω T n (f )w n (|v| + 1 n ) γ ≤ n γ+1 Ω |w n | ≤ |Ω| 1/2 n γ+1 w n L 2 (Ω)
by the hypotheses on a and Hölder's inequality on the right hand side. Poincaré's inequality on the left hand side implies

αP w n 2 L 2 (Ω) ≤ |Ω| 1/2 (1 + n) p n γ+1 w n L 2 (Ω) .
Thus there exists an invariant ball for S. Moreover it is easily seen that S is continuous and compact by the H 1 0 (Ω) ֒→ L 2 (Ω) embedding. By Schauder's theorem, S has a fixed point. Therefore there exists a solution u n ∈ H 1 0 (Ω) to problems (2.1). Observe that u n is bounded; by the maximum principle, u n is non-negative since f is non-negative.

Remark 2.2. We remark that the existence of solutions to (2.1) could not be inferred by the results established in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF].

Proposition 2.3. Let u n be the solution to problem (2.1). Then u n ≤ u n+1 a.e. in Ω. Moreover for every ω ⊂⊂ Ω there exists c ω > 0 such that u n ≥ c ω a.e. in ω for every n ∈ N.

Proof. We will use the same technique as in [START_REF] Porretta | Uniqueness and homogeneization for a class of noncoercive operators in divergence form[END_REF] to prove uniqueness of the solutions to problem (1.3). In the proof C will denote a positive constant independent of n (depending on α, β, p and the constant P of Poincaré's inequality). The solution u n to problem (2.1) satisfies

-div a(x)∇u n (1 + T n (u n )) p = T n (f ) u n + 1 n γ ≤ T n+1 (f ) u n + 1 n+1 γ . Therefore -div a(x)∇u n (1 + T n (u n )) p - a(x)∇u n+1 (1 + T n+1 (u n+1 )) p = T n+1 (f )   1 u n + 1 n+1 γ - 1 u n+1 + 1 n+1 γ   ≤ 0 . By choosing T k ((u n -u n+1 ) + ) as a test function we get α Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p ≤ β Ω ∇u n+1 • ∇T k ((u n -u n+1 ) + ) 1 (1 + T n+1 (u n+1 )) p - 1 (1 + T n (u n )) p by the hypotheses on a. In {0 ≤ u n -u n+1 ≤ k} one has 1 (1 + T n+1 (u n+1 )) p - 1 (1 + T n (u n )) p ≤ k 1 (1 + T n+1 (u n+1 )) p + 1 (1 + T n (u n )) p . Therefore Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p ≤ Ck {0≤un-un+1≤k} |∇u n+1 ||∇T k ((u n -u n+1 ) + )| 1 (1 + T n+1 (u n+1 )) p + 1 (1 + T n (u n )) p . For sufficiently small k, one has in {0 ≤ u n -u n+1 ≤ k} 1 2 p 1 (1 + T n (u n )) p ≤ 1 (1 + T n+1 (u n+1 )) p ≤ 2 p (1 + T n (u n )) p .

This implies that

Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p ≤ Ck {0≤un-un+1≤k} |∇u n+1 | (1 + T n+1 (u n+1 )) p/2 |∇T k ((u n -u n+1 ) + )| (1 + T n (u n )) p/2 .
Hölder's inequality on the right hand side gives

Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p ≤ Ck    {0≤un-un+1≤k} |∇u n+1 | 2 (1 + T n+1 (u n+1 )) p    1 2    {0≤un-un+1≤k} |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p    1 2
, and then

(2.4)

Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p ≤ Ck 2 {0≤un-un+1≤k} |∇u n+1 | 2 (1 + T n+1 (u n+1 )) p .
On the other hand, by Poincaré's inequality and (2.4)

k 2 |{0 ≤ u n -u n+1 ≤ k}| ≤ Ω |∇T k ((u n -u n+1 ) + )| 2 (1 + T n (u n )) p (1 + T n (u n )) p ≤ Ck 2 {0≤un-un+1≤k} |∇u n+1 | 2 (1 + n) p (1 + T n+1 (u n+1 )) p , that is, |{0 ≤ u n -u n+1 ≤ k}| ≤ C {0≤un-un+1≤k} |∇u n+1 | 2 (1 + n) p (1 + T n+1 (u n+1 )) p .
The right hand side of the above inequality tends to 0, as k → 0. Therefore |{0 ≤ u nu n+1 ≤ k}| → 0 as k → 0. This implies that u n ≤ u n+1 a.e. in Ω. We remark that u 1 is bounded, that is, |u 1 | ≤ c, for some positive constant c. Setting

h(s) = s 0 dt (1 + T 1 (t)) p , we have -div (a(x)∇(h(u 1 ))) = -div a(x) ∇u 1 (1 + T 1 (u 1 )) p ≥ T 1 (f ) (c + 1) γ .
Let z be the H 1 0 (Ω) solution to -div (a(x)∇z) = T 1 (f ) (c + 1) γ By the strong maximum principle, for every ω ⊂⊂ Ω there exists a positive constant c ω such that z ≥ c ω a.e. in ω. By the comparison principle, we have h(u 1 ) ≥ z a.e. in Ω. The strict monotonicity of h implies the existence of a constant c ω > 0, for every ω ⊂⊂ Ω, such that u 1 ≥ c ω a.e. in ω. Since u n is an increasing sequence, as proved above, u n ≥ c ω a.e. in ω for every n ∈ N.

Existence results

We are going to prove in the following three lemmata some a priori estimates on the solutions u n to problems (2.1). They will allow us to show Theorem 1.1. In the proofs C will denote a positive constant independent of n.

Lemma 3.1. Assume that p -1 ≤ γ < p + 1. (1) Let f ∈ L m (Ω), with m ≥ 2 * 2 * -p -1 + γ
. Then the solutions u n to (2.1) are

uniformly bounded in H 1 0 (Ω). If 2 * 2 * -p -1 + γ ≤ m < N 2
then the solutions u n are uniformly bounded in L m * * (γ+1-p) (Ω).

(

) Let f ∈ L m (Ω), with max 1, 1 * 2 • 1 * -p -1 + γ < m < 2 * 2 * -p -1 + γ . Then the solutions u n to (2.1) are uniformly bounded in W 1,σ 0 (Ω), σ = N m(γ + 1 -p) N + γm -m(p + 1) 2
.

Proof. In case (1) let us choose (1 + u n ) p+1 -1 as a test function; the hypotheses on a imply that

α(p + 1) Ω |∇u n | 2 (1 + T n (u n )) p (1 + u n ) p ≤ C Ω |f |u p+1-γ n .
By Sobolev's inequality on the left hand side and Hölder's inequality with exponent

m = 2 * 2 * -p -1 + γ = 2N N (γ + 1 -p) + 2(p + 1 -γ) (> 1
) on the right one, we have

Sα(p + 1) u n 2 L 2 * (Ω) ≤ α(p + 1) ∇u n 2 L 2 (Ω) ≤ f L m (Ω) Ω |u n | m ′ (p+1-γ) 1 m ′ . We remark that 2 * = m ′ (p + 1 -γ). Moreover 2 2 * ≥ 1 m ′ as γ ≤ p + 1.
Then the above estimate implies that the sequence u n is bounded in L 2 * (Ω) and in H 1 0 (Ω). We are now going to prove that u n is bounded in

L m * * (γ+1-p) (Ω), if m < N 2 . Let us choose (1 + u n ) δ -1 as a test function: by the hypotheses on a, one has 4αδ (-p + δ + 1) 2 Ω |∇[(1 + u n ) -p+δ+1 2 -1]| 2 = αδ Ω |∇u n | 2 (1 + u n )) p-δ+1 ≤ αδ Ω |∇u n | 2 (1 + T n (u n )) p (1+u n ) δ-1 ≤ Ω T n (f ) (u n + 1 n ) γ [(u n +1) δ -1] ≤ C +C Ω |f | (u n + 1) -δ+γ .
By Sobolev's inequality on the left hand side and Hölder's inequality on the right one we have

Ω [(1 + u n ) -p+δ+1 2 -1]| 2 * 2 2 * ≤ C f L m (Ω) Ω |u n + 1| m ′ (δ-γ) 1 m ′ .
Let δ be such that

(1 + δ -p)N N -2 = (δ -γ)m m -1 and 2 2 * ≥ 1 m ′ , that is, δ = (1 -p)N (m -1) + γm(N -2) N -2m and m ≤ N 2 . We observe that (-p + δ + 1) 2 * 2 = m * * (γ + 1 -p) > 1. This implies that u n is bounded in L m * * (γ+1-p) (Ω).
In case (2), let us choose (1

+ u n ) θ -1, θ = (p -1)N (m -1) -γm(N -2) 2m -N
, as a test function. With the same arguments as before, we have

Ω [(1 + u n ) -p+θ+1 2 -1]| 2 * 2 2 * ≤ C Ω |∇u n | 2 (1 + u n )) p-θ+1 ≤ C f L m (Ω) Ω |u n + 1| m ′ (θ-γ) 1 m ′ .
As above, we infer that u n is bounded in L

N (1+θ-p) N -2
(Ω). We observe that pθ + 1 > 0 and

1 < σ = N m(γ + 1 -p) N -m(p + 1 -γ) < 2
, by the assumptions on m. Writing

Ω |∇u n | σ = Ω |∇u n | σ (1 + u n ) σ p-θ+1 2 (1 + u n ) σ p-θ+1
and using Hölder's inequality with exponent 2 σ , we obtain

Ω |∇u n | σ ≤ Ω |∇u n | 2 (1 + u n ) p-θ+1 σ 2 Ω (1 + u n ) σ p-θ+1 2-σ 2-σ 2
.

The above estimates imply that the sequence u n is bounded in

W 1,σ 0 (Ω) if σ p -θ + 1 2 -σ = N (1 + θ -p) N -2 , that is, σ = N m(γ + 1 -p) N -m(p + 1 -γ) .
Lemma 3.2. Assume that γ = p + 1 and f ∈ L 1 (Ω). Then the solutions u n to (2.1) are uniformly bounded in H 1 0 (Ω). Proof. Let us choose (1 + u n ) p+1 -1 as a test function. Using that a(x) ≥ α a.e. in Ω, we have

α(p + 1) Ω |∇u n | 2 (1 + T n (u n )) p (1 + u n ) p ≤ C Ω |f | .
The previous estimate implies that the sequence u n is bounded in H 1 0 (Ω). Lemma 3.3. Assume that γ > p + 1 and f ∈ L 1 (Ω). Then the solutions u n to (2.1)

(γ + 1 -p) 2 Ω ∇(u γ+1-p 2 n ) 2 = αγ Ω |∇u n | 2 u γ-1-p n ≤ Ω |f | .
This proves that the sequence u Let us prove that u n is bounded in H 1 loc (Ω). For ϕ ∈ C 1 0 (Ω) we choose [(u n + 1) p+1 -1]ϕ 2 as a test function in (2.1). Then, if ω = {ϕ = 0}, one has by the hypotheses on a and Lemma 2.3

α(p + 1) Ω |∇u n | 2 ϕ 2 + 2α Ω ∇u n • ∇ϕ ϕ u n ≤ Ω |f | (u n + 1 n ) γ [(u n + 1) p+1 -1]ϕ 2 ≤ Ω C ω |f |ϕ 2 ≤ C ω ϕ 2 L ∞ (Ω) Ω |f | ,
where C ω is a positive constant depending only on ω ⊂⊂ Ω and p. Then

(3.2) (p + 1) Ω |∇u n | 2 ϕ 2 ≤ -2 Ω ∇u n • ∇ϕ ϕ u n + C ω α ϕ 2 L ∞ (Ω) Ω |f | . Young's inequality implies that 2 Ω ∇u n • ∇ϕ ϕ u n ≤ Ω |∇u n | 2 ϕ 2 + Ω |∇ϕ| 2 u 2 n . From (3.2) we infer that p Ω |∇u n | 2 ϕ 2 ≤ Ω |∇ϕ| 2 u 2 n + C ω α ϕ 2 L ∞ (Ω) Ω |f | .
Since u n is uniformly bounded in L 2 (Ω) by (3.1), this proves our result.

We are now able to prove Theorem 1.1.

Proof. We will prove point (1); the second and the third point can be proved in a similar way. Lemma 3.1 gives the existence of a function u ∈ H 1 0 (Ω) such that u n → u weakly in H 1 0 (Ω) and a.e. in Ω, up to a subsequence. We will prove that u is a solution to (1.1) passing to the limit in (2.1). For every ϕ ∈ C 1 0 (Ω),

A k |∇u n | 2 (1 + u n ) p-γ ≤ A k |f | (u n + 1) γ+1 -(k + 1) γ+1 (u n + 1 n ) γ ≤ c(γ) A k |f |(u n -k) ,
where A k = {u n ≥ k} and c(γ) denotes a positive constant depending only on γ.

For pγ ≤ 0, (3.3) is the starting point of of the proof of Theorem 4.1 in [START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF]. For 0 < pγ ≤ 1, (3.3) is the starting point of Lemma 2.2 in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF]. In both cases u n is uniformly bounded in L ∞ (Ω) and therefore (at the a.e. limit) the solutions u found in the previous results are bounded. Remark 3.4. We observe that we have the boundedness of the solution to problem (1.1) for any value of γ ≥ p -1.

γ+1-p 2 n

 2 is bounded in H 1 0 (Ω). Sobolev's inequality on the left hand side applied to u

Ω 1 ( 1 +,

 11 a(x)∇u n • ∇ϕ (1 + T n (u n )) p → Ω a(x)∇u • ∇ϕ (1 + u) p , since T n (u n )) p → 1 (1 + u) p in L r (Ω), for every r ≥ 1. For the limit of the right hand side of (2.1), let ω = {ϕ = 0}. One can use Lebesgue's theorem, sinceT n (f )ϕ u n + 1where c ω is the constant given by Lemma 2.3. In the case where2 * 2 *p -1 + γ ≤ m < N 2 ,since m * * (γ + 1p) > 1, Lemma 3.1 implies that u n converges weakly in L m * * (γ+1-p) (Ω) to some function, which is u by identification.To prove that u is bounded for γ ≥ p -1, let us choose [(u n + 1) γ+1 -(k + 1) γ+1 ] + as a test function in (2.1):(3.3) α(γ + 1)

Acknowledgments

Part of this work was done during a visit to La Sapienza Università di Roma whose hospitality is gratefully acknowledged.


A L Do

O Sename

L Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

B Soualmi

Multi-objective optimization by genetic algorithms in H ∞ /LP V control of semi-active suspension

Keywords: Genetic algorithms, H ∞ control, LPV systems, semi-active suspensions

) by using Genetic Algorithms (GAs) to select the optimal weighting functions for the H ∞ /LP V synthesis. First, a general procedure for the optimization of the weighting functions for the H ∞ /LP V synthesis is proposed and then applied to the semi-active suspension control. Thanks to GAs, the comfort and road holding are handled using a single high level parameter and illustrated via the Pareto optimality. The simulation results performed on a nonlinear vehicle model emphasize the efficiency of the method.

INTRODUCTION

The H ∞ control design approach is an efficient tool to improve the performances of a closed-loop system in predefined frequency ranges. The key step of the H ∞ control design relies on the selection of the weighting functions which depends on the engineer skill and experience. In many real applications, the difficulty in choosing the weighting functions increases because the performance specification is not accurately defined, i.e., it is simply to achieve the best possible performances (optimal design) or to achieve an optimally joint improvement of more than one objective (multi-objective design). So it is still interesting to optimise the selection of the weighting functions to get the desired closed-loop performances, as studied in [START_REF] Beaven | Weighting function selection in the H∞ design process[END_REF], [START_REF] Hu | Systematic H∞ weighting function selection and its application to the real-time control of a vertical take-off aircraft[END_REF] ... They propose to consider each system, no matter how complex it is, as a combination of sub-systems of the first and second order, for which it is easy to find the good weighting functions to be used in the H ∞ control methodology. However, there is no explicit method to find these functions in the general case. The usual way is to proceed by trial-and-error. Recently, the idea to use an optimization tool was proposed in [START_REF] Alfaro-Cid | Optimisation of the weighting functions of an H∞ controller using genetic algorithms and structured genetic algorithms[END_REF]. The choice of GAs seems natural because their formulation is well suited for this type of problematic.

The semi-active control problem has been studied during the recent decades (see [START_REF] Savaresi | Semi-Active Suspension Control for Vehicles[END_REF] and reference therein) and H ∞ control is one of the most investigated approaches. This paper aims at using GAs to obtain optimal weighting functions for the H ∞ /LP V control of semi- active suspension systems. First, from the frequency based industrial performance criteria of suspension systems as defined in [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions: some practical aspects[END_REF], an appropriate cost function is formulated. This cost function is in fact a convex combination of two specifications: passenger comfort and road holding, allowing to tune the suspension system using a single high level parameter. Second, the weighting functions are then tuned by GAs in such a way that the cost function is minimized. Finally, the best weighting functions are obtained are used for the synthesis of H ∞ /LP V controller. For illustration, the performances of the closed-loop semi-active suspension system associated with the controller obtained with GAs will be analyzed to emphasize the efficiency of the approach. It is important to note that this methodology provides a new efficient and generic tool for suspension control design, where the industrial performance objectives are included in the design procedure (and not evaluated a posteriori).

The outline is as follows. In Section 2, a brief introduction about multi-objective optimization using GAs is presented. The procedure for the optimization of the weighting functions for H ∞ /LP V problem using GAs is then introduced in Section 3. In Section 4, the H ∞ /LP V control synthesis for semi-active suspensions, presented in [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF], is recalled. In Section 5, the application of weighting function selection using GAs on the H ∞ /LP V control problem formulated in Section 4 is presented. The results obtained in simulations with a nonlinear quarter car model are discussed in Section 6. Finally, some conclusions and perspectives are given in Section 7.

GENETIC ALGORITHMS AND MULTI-OBJECTIVE OPTIMIZATION

Genetic algorithms

Genetic algorithms have now grown strongly from the first study of [START_REF] Holland | Adaptation in natural and artificial systems, an introductory analysis with application to biology, control and artificial intelligence[END_REF], a popular theory-oriented book of [START_REF] Goldberg | Genetic Algorithms in Searching Optimisation and Machine Learning[END_REF] and an application-oriented book of [START_REF] Davis | Handbook of genetic algorithms[END_REF]. The algorithms are based on the mechanism of the natural selection and have been proven to be very effective in optimization with many real applications such as in finance and investment strategies, robotics, engineering design, telecommunications... They are likely global optimization techniques (despite the high computational expense) (see [START_REF] Marler | Survey of multi-objective optimization mehtods for engineering[END_REF]) using probabilistic, multi-points search, random combination (crossover, mutation) and information of previous iterations to evaluate and improve the population. A great advantage of GAs compared to other searching methods (for example gradient methods) is that they search regardless of the nature of the objective functions and constraints.

GAs initialize with a random population. Through the genetic operation: selection, crossover and mutation, a new population is obtained. By using the selection process, the fittest individuals based on their fitness values will be chosen; crossover and mutation will be then applied to create the new population. The genetic operation on individuals of population continues until the optimization criterion is satisfied or a certain number of generations is reached.

Multi-objective optimization

One well-known application of the GAs is to find the optimal solution for the multi-objective optimization problem involving multiple and conflicting objectives. This is a very popular problem in practice and can be described as follows

min x∈C F (x) =     f 1 (x) f 2 (x)
. . .

f n obj (x)     , n obj 2, ( 1
)
where x is called the decision vector, C the set of possible decision vectors (or the searching space), and F (x) the objective vector. The existence of an ideal solution x * that can minimize simultaneously all objective functions f 1 , f 2 ... f n obj is in fact rarely feasible. Hence, in this case, the concept of Pareto-optimal or Pareto-set is usually used to describe the solution of multi-objective optimization problem (see [START_REF] Marler | Survey of multi-objective optimization mehtods for engineering[END_REF]).

There are many formulations to solve the problem (1) like weighted min-max method, weighted global criterion method, goal programming methods... see [START_REF] Marler | Survey of multi-objective optimization mehtods for engineering[END_REF] and references therein. One of the most popular and simple approaches is the weighted sum method which converts the multi-objective problem into a single objective problem. In this paper, a particular case of the weighted sum method is used, where the multi-objective function F is replaced by the convex combination of objectives

min J = n obj i=1 α i f i (x), s.t x ∈ C (2)
where

n obj i=1 α i = 1. The vector α = (α 1 , α 2 , ...α n obj ) repre-
sents the gradient of function J. By using various sets of α, one can generate several points in the Pareto set.

OPTIMIZATION OF WEIGHTING FUNCTIONS

FOR LP V /H ∞ SYNTHESIS USING GENETIC ALGORITHM 3.1 General H ∞ /LP V problem Consider a general LPV system ẋ = A(ρ(t))x + B 1 (ρ(t))w + B 2 (ρ(t))u z = C 1 (ρ(t))x + D 11 (ρ(t))w + D 12 (ρ(t))u y = C 2 (ρ(t))x + D 21 (ρ(t))w + D 22 (ρ(t))u (3) where x ∈ R n , u ∈ R m , w ∈ R q , z ∈ R r and y ∈ R p are
the state, the input, the disturbance vectors, the control output and the measured output, respectively. ρ(t) ∈ R k is a vector of scheduling parameters and assumed to be known (measured or estimated). From now on, ρ(t) is simply denoted as ρ.

It is well-known that, to satisfy some required closed-loop performance, one must select suitable weighting functions.

While the weights on disturbance inputs are always defined thanks to the knowledge about their bandwidth and magnitude, the selection of weighting functions on control outputs is more difficult and play a key role in the synthesis procedure. Without loss of generality, it is assumed that there is no weight on disturbance inputs. The control outputs are weighted as z w (s) = W (s)z(s) where the weighting function matrix W (s) has the following statespace representation ẋw = A w (ν)

x w + B w (ν)z z w = C w (ν)x w + D w (ν)z ( 4
)
where ν is the vector of the weighting functions parameters which need to be found through optimization using GAs.

From ( 3) and ( 4), the augmented system used for the controller synthesis is written as

ξ = A(ρ, ν)ξ + B 1 (ρ, ν)w + B 2 (ν)u z w = C 1 (ρ, ν)ξ + D 11 (ρ, ν)w + D 12 (ν)u (5) y = C 2 ξ + D 21 w where ξ = x T x w T T A(ρ, ν) = A(ρ) 0 Bw(ν)C 1 (ρ) Aw(ν) , B 1 (ρ, ν) = B 1 (ρ) Bw(ν)D 11 (ρ) , B 2 (ν) = B 2 Bw(ν)D 12 , C 1 (ρ, ν) = Dw(ν)C 1 (ρ) Cw(ν) , C 2 = C 2 D 11 (ρ, ν) = Dw(ν)D 11 (ρ), D 12 (ν) = Dw(ν)D 12 , D 21 = D 21 .
The LP V controller is defined as follows

K(ρ) : ẋc u = A c (ρ) B c (ρ) C c (ρ) D c (ρ) x c y (6)
where x c , y and u are respectively the state, the input and output of the controller, respectively, of the controller associated with the system (5).

A c ∈ R n×n , B c ∈ R n×ny , C c ∈ R nu×n and D c ∈ R nu×ny . H ∞ /LP V problem -
The objective of the synthesis is to find an LPV controller K(ρ) of the form ( 6) such that the closed-loop system is quadratically stable and that, for a given positive real γ, the L 2 -induced norm of the operator mapping w into z w is bounded by γ i.e

z w 2 w 2 ≤ γ (7)
for all possible trajectories of ρ.

In this paper, the polytopic approach with quadratic Lyapunov function is employed for the synthesis of a H ∞ /LP V controller. The following is assumed: ρ ∈ conv{ρ 1 , ...ρ k }, the matrices B 2 , D 12 , C 2 , D 21 are parameter independent and D 22 = 0. For known weighting functions and a suitable pre-defined real positive scalar γ, the sufficient condition that solves the H ∞ /LP V problem is given by Eq. ( 8)-( 9) (see the detail of the solution in [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]). It is worth noting that, unlike the original solution, the weighting function parameters are present in the LMIs problem and will be optimized by genetic algorithm.

  M 11 (ρ i , ν) * * * M 21 (ρ i , ν) M 22 (ρ i , ν) * * M 31 (ρ i , ν) M 32 (ρ i , ν) -γIm * M 41 (ρ i , ν) M 42 (ρ i , ν) M 43 (ρ i , ν) -γIp   ≺ 0 (8) X I I Y ≻ 0 (9) for i = 1 : k where M 11 (ρ i , ν) = A(ρ i , ν)X + XA(ρ i , ν) T + B 2 Ĉ(ρ i ) + Ĉ(ρ i ) T B T 2 M 21 (ρ i , ν) = Â(ρ i ) + A(ρ i , ν) T + C T 2 D(ρ i ) T B T 2 M 22 (ρ i , ν) = Y A(ρ i , ν) + A(ρ i , ν) T Y + B(ρ i )C 2 + C T 2 B(ρ i ) T M 31 (ρ i , ν) = B 1 (ρ i , ν) T + D T 21 D(ρ i ) T B T 2 M 32 (ρ i , ν) = B 1 (ρ i , ν) T Y + D T 21 B(ρ i ) T M 41 (ρ i , ν) = C 1 (ρ i , ν)X + D 12 Ĉ(ρ i ) M 42 (ρ i , ν) = C 1 (ρ i , ν) + D 12 D(ρ i )C 2 M 43 (ρ i , ν) = D 11 (ρ i , ν) + D 12 D(ρ i )D 21
The controller K ci at vertex i is then reconstructed as

Dc(ρ i ) = D(ρ i ) Cc(ρ i ) = Ĉ(ρ i ) -Dc(ρ i )C 2 X M -T Bc(ρ i ) = N -1 B(ρ i ) -Y B 2 Dc(ρ i ) Ac(ρ i ) = N -1 Â(ρ i ) -Y A(ρ i )X -Y B 2 Dc(ρ i )C 2 X M -T -Bc(ρ i )C 2 XM -T -N -1 Y B 2 Cc(ρ i ) (10)
where M , N are defined such that M N T = I n -XY which can be solved through a singular value decomposition and a Cholesky factorization. The global H ∞ /LP V controller is the convex combination of these local controllers.

Optimizing the weighting functions

Decision vector -Certainly ν is one element of the decision vector. Besides, the authors observed that the real positive scalar γ must be taken into account in the decision vector as well. Indeed one aims at optimizing a predefined multiobjective function J (see below) and not γ.

To avoid the infeasibility of the LMIs (8)-( 9) resulting from the bad (i.e too small) value of γ generated by GAs, γ will be split into two positive real elements γ min and γ ga where γ min is the minimal γ satisfying the LMIs (8)-( 9), and γ ga is tuned by GAs. Due to the convexity of the LMIs problem, the existence of γ min will ensure the feasibility of LMIs ( 8)-( 9) with γ = γ min + γ ga for any positive real γ ga . The minimal value γ min can be found by using LMIs toolbox like Yalmip & Sedumi.

Objective functions -In this paper, the frequency-based objective functions are considered. With the remarks above, the performance of the closed-loop system depends on the choice of weighting functions and the real positive scalar γ. Hence, to refer to these implicit dependencies, the objective function can be written as J = J(ν, γ).

Proposed weighting function optimization procedure for H ∞ /LP V synthesis

• Step 1: Initialize with random positive weighting functions ν = ν 0 and random positive real γ ga = γ 0 ga . • Step 2: Solve the minimization problem of γ subject to the LMIs ( 8)-( 9) to compute the minimal real scalar γ min . Solve again the the LMIs ( 8)-( 9) with the couple (ν, γ) where γ = γ min + γ ga . At the end of this step, compute the objective function J(ν, γ).

m s zs = -F spring -F mr m us zus = F spring + F mr -k t (z us -z r ) (11)
where F spring = k s z def is the spring force, z def = z sz us the damper deflection (assumed to be measured or estimated), and żdef = żsżus the deflection velocity (can be directly computed from z def ).

In this paper, the behavior of the semi-active suspension is modeled by the following nonlinear equation, as in [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF]:

F mr = a 2 ( żdef + v 0 x 0 z def ) + a 1 tanh(a 3 ( żdef + v 0 x 0 z def )) (12)
where a 2 , a 3 , v 0 and x 0 are constant parameters and a 1 is varying according to the electrical current in coil (0 < a 1min ≤ a 1 ≤ a 1max ). This model allows fulfilling the dissipativity constraint of the semi-active damper.

The quarter vehicle used in this paper is the "Renault Mégane Coupé" model (see [START_REF] Zin | A nonlinear vehicle bicycle model for suspension and handling control studies[END_REF]) whose specific parameters are: m s = 315 kg, m us = 37.5 kg, k s = 29500 N/m, k t = 210000 N/m. The damper model parameters have been chosen according to the MR damper in [START_REF] Nino-Juarez | Minimizing the frequency in a black box model of a mr damper[END_REF]:

a 2 = 1500 Ns/m, a 3 = 129 s/m, v 0 = 0.788x10 -3 m/s, x 0 = 1.195x10 -3 m, a 1min = 0 N, a 1max = 400 N, F 1 = (a 1max -a 1min )/2 = 200.
After some mathematical transformations as in [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF], a control oriented LPV model with input saturation is obtained with two scheduling parameters ρ * 1 and

ρ * 2 ẋ = A (ρ 1 * , ρ 2 * ) x + Bu c + B 1 w y = Cx (13)
where

x = x s x f T , A (ρ 1 , ρ 2 ) = A s + ρ 2 B s2 C s2 ρ 1 B s C f 0 A f , B = 0 B f , B 1 = B s1 0 , C = C s 0 T ρ 1 * = tanh(C s2 x s ) tanh( C f x f F1 ) F1 C f x f , ρ 2 * = tanh(Cs2xs)

Cs2xs

.

x s , A s , B s , B s1 , C s are state and matrices of a state-space representation of the plant (11); x f , A f , B f , C f are state and matrices of a representation of the following low-pass filter W f ilter = Ω f s+Ω f . This filter is added to the plant to make the control input matrices parameter independent. Notice also that ρ * 1 and ρ * 2 are not independent and the relation between them is presented in Fig. 2. For the polytopic approach, the polytope P 1 P 2 P 3 P 4 that contains the set of (ρ * 1 , ρ * 2 ) will be considered. While P 3 and P 4 are related to the real set of (ρ * 1 , ρ * 2 ), P 1 and P 2 in fact do not belong to the set. Fig. 2. Set of (ρ 1 * , ρ 2 * ) (shaded area + curved line) and set of (ρ 1 , ρ 2 ) (curved line).

Suspension Performance Objectives

The performance objectives for comfort and road holding are obtained following ideas from [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions: some practical aspects[END_REF].

• Comfort in high frequencies ([3 -10] Hz): min

10 3 zs /z r (f )df . • Comfort in low frequencies ([0-4] Hz): min σ(z s /z r )(f ). • Road holding ([0 -20] Hz): min σ(z us /z r (f )) (Re-
bound of wheel) and 20 0 (z usz r )/z r (f )df (Contact force).

SEMI-ACTIVE SUSPENSION CONTROL USING GENETIC ALGORITHMS

The procedure for optimizing the weighting functions in Sec. 3.2 is now applied to the LPV semi-active suspension system (13) with the following specifications.

Parameterizing weighting functions

Along with the bounded real positive scalar γ, the following weighting functions are used for the H ∞ /LP V synthesis

W f ilter = Ω f s + Ω f (14) W zs = k zs s 2 + 2ξ 11 Ω 11 s + Ω 11 2 s 2 + 2ξ 12 Ω 12 s + Ω 12 2 (15)
W zus = k zus s 2 + 2ξ 21 Ω 21 s + Ω 21 2 s 2 + 2ξ 22 Ω 22 s + Ω 22 2 (16)
The decision vector is as follows ν = [ γ Ω f Ω 11,12 ξ 11,12 k zs Ω 21,22 ξ 21,22 k zus ] T (17)

It is worth noting that the structures of these weighting functions are the same as in [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF].

The parameter (i.e. each element in the decision vector) encoding is defined as follows Parameter Value = (a + 0.1b + 0.01c + 0.001d) × 10

Objective and fitness function

Objective function -The objective function is as follows

J = αJ Comfort + (1 -α)J Road Holding ( 19
) where J Comfort and J Road Holding are performance indices corresponding to vertex P 3 (Note that solutions at P 3 and P 4 are identical because of the symmetric property of the polytope (P 1 , P 2 , P 3 , P 4 ) and defined as follows (according to Sec. 4.2) Remark 1 : The feasibility of the new generated elements is not guaranteed because of the possible infeasibility of the LMIs ( 8)-( 9). The problem can be overcome by repeating the crossover or mutation until the feasible solution is obtained. However, a simpler way is to assign a large objective value (for instant J = ∞) to these infeasible solutions and as a consequence, they will be eliminated by the selection procedure after some generations.

J Comfort = 10 1 zs /z r (f )df
Remark 2 : In many cases, to preserve the performance of the closed-loop system with input saturation, a stable stabilizing controller is required. A stable LP V controller can be obtained by eliminating the "unstable solutions" corresponding to at least one unstable local controller during the synthesis.

To sum up, the objective function is chosen as follows.

Algorithm 1: Objective value assignment if (8)-( 9) is feasible then if all local controllers K ci are stable then Calculate J using Eq. 19, 20 and 21

else J = ∞ end else J = ∞ end.
Fitness function -Because the working principle of GAs is the maximization of fitness, the following fitness function is used F fitness = 10 -10 1 -5e -12(J-0.6) 1 + 5e -12(J-0.6) (22)

Genetic operation

The selection method used in this paper is the proportionate selection developed by [START_REF] Holland | Adaptation in natural and artificial systems, an introductory analysis with application to biology, control and artificial intelligence[END_REF]. This method assigns to each individual a probability of selection proportional to its relative fitness. Proportionate selection can be illustrated by a roulette wheel. The crossover happens with a probability of 0.9 and the mutation happens with a very small probability 0.095.

Design results

The result of the optimization procedure using GAs is given in Fig. 3. Some points in the Pareto-Set are depicted.

In the next section, we validate this linear design result on a nonlinear quarter car model.

SIMULATION RESULTS

The nonlinear frequency responses using the "variance gain" algorithm by [START_REF] Savaresi | Acceleration driven damper (ADD): an optimal control algorithm for comfort oriented semi-active suspensions[END_REF] are presented in Fig. 567.

In Fig. 4, the performance indices J Comfort and J Road Holding are computed for a nonlinear quarter car model with the H ∞ /LP V controllers corresponding to each value of α with different values of α in the range of [0.1; 0.99]. As expected, with α = 0.99, the smallest J Comfort is obtained; with α = 0.1, the smallest J Road Holding is obtained.

Three stable H ∞ /LP V controllers C 1 , C 2 , C 3 corresponding to α = 0.99, α = 0.55, α = 0.1 (respectively) synthesized with the optimized weighting functions (whose parameters are presented in Tab. 1) are chosen for the analysis. Their performances are compared with those of the H ∞ /LP V controller obtained in [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF], of the passive soft MR Damper (a 1 = a 1min = 0 N ) and of the passive hard MR Damper (a 1 = a 1max = 400 N ).

• The comfort oriented controller C 1 is the best one for comfort but not good for road holding. • The road holding (rebound of wheel) oriented controller C 3 is the best for preventing the rebound of wheel in high frequency. It can be seen that this controller has the same behavior as that of the hard MR damper. However, a good comfort in high frequency.

CONCLUSION

In this paper, a weighting function optimization using GAs for H ∞ /LP V semi-active control problem has been proposed. The input used for this optimization procedure is a frequency-based objective function, instead of the disturbance attenuation level γ as in conventional H ∞ /LP V design. The simulation results have shown the efficiency of GAs in finding a suitable controller to satisfy some performance objectives. For future work, powerful GAs in multi-objective optimization like SPEA2 or NSGA-II will be considered and compared with the simple weighted sum of objectives method used in this paper.

⋆

  This work was supported by the French national project INOVE/ ANR 2010 BLAN 0308.

  where a, b, c, d, e are natural number and (a, b, c, d, e) ∈ [0; 9] 5 . This encoding covers the range (0; 100).

  s /z r ) P assive (f )df (20)J Road Holding = σ f ∈[10-20] (z us /z r (f )) σ f ∈[10-20] (z us /z r (f )) P assive (21)Note that the criteria with "passive" are computed for the open-loop system (13) with u c = 0.

Fig. 3 .

 3 Fig. 3. Performance indices of the closed-loop linear system corresponding to vertex P 3 (with different values of α).

Fig. 4 .

 4 Fig. 4. Performance indices of the closed-loop nonlinear system (with different values of α).

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Comfort in high frequencies zs /z r .

•

  Step 3: Select the individuals. • Step 4: Apply crossover and mutation to generate a new generation: ν = ν new and γ ga = γ new ga . • Step 5: Evaluate the new generation: If the criteria of interest (for example, reaching the limit number of generation) are not satisfied, go to Step 2 with ν = ν new and γ ga = γ new ga ; else, stop and save the best individual ν opt = ν new and γ opt = γ new ga .

	4. SEMI-ACTIVE SUSPENSION CONTROL
	4.1 System description
	Consider a simple quarter vehicle model (see Fig. 1) made
	m s m us k s up of a sprung mass (m f k t	z s z us z r Semi-active damper
	Fig. 1. Model of quarter vehicle with a semi-active damper.
	vehicle are governed by:

s ) and an unsprung mass (m us ) connected by a spring with the stiffness coefficient k s and a semi-active damper. The wheel tire is modeled by a spring with the stiffness coefficient k t . In this model, z s (respectively z us ) is the vertical position of m s (respectively m us ) and z r is the road profile. It is assumed that the wheel-road contact is ensured. The dynamical equations of a quarter

•

  The intermediate controller C 2 turns out to be the best one. It guarantees a road holding, a good comfort improvement in high frequency and a medium comfort improvement in low frequency.

	To summarize, the overall evaluation of performance based
	on the criteria given in Sec. 4.2 is depicted in Fig. 8.
		1.08	α=0.99
		1.06	α=0.7
			α=0.6
		1.04
	Road Holding	1 1.02
	J	0.98
		0.96	α=0.575
		0.94	α=0.55		α=0.3		α=0.1
		0.92		α=0.5
		0.85	0.9	0.95	1	1.05	1.1	1.15	1.2
						J Comfort

Table 1 .

 1 Parameters for H ∞ /LP V semi-active suspension design.

	Controllers	bounded real scalar Filter			W zs					Wz us
		γ min -γga	Ω f	Ω 11	Ω 12	ξ 11	ξ 12	k zs	Ω 21	Ω 22	ξ 21	ξ 22	kz us
	ACC 2010	363-0	50	125.7	125.7	20	1	1	69.1	69.1	20	1	1
	C 1 (α = 0.99)	456-463	51.1	33.29	28.54	52.32	5.26	1.95	1.29	0.072	4.09	0.136	0.152
	C 2 (α = 0.55)	320-163	8.3	0.48	4.66	37.95	0.76	3.77	0.87	3.3	0.19	1.8	8.1
	C 3 (α = 0.1)	93.9-0.11	1.4	2.373	0.66	0.222	33.87	0.22	24.81	0.49	7.14	5.75	0.343