Jérôme Droniou
email: droniou@math.univ-montp2.fr

Juan-Luis Vázquez
email: juanluis.vazquez@uam.es

Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

Keywords: Mathematics Subject Classification: 35D05, 35B30, 35J25, 35K20, 47B44 Elliptic equation, convection-diffusion, non-coercive equations, Neumann conditions, continuous dependence, accretivity

We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation

Under the assumption that

, we prove that the problem has a solution u ∈ H 1 (Ω) if Ω f dx = 0, and also that the kernel is generated by a function u ∈ H 1 (Ω), unique up to a multiplicative constant, which satisfies u > 0 a.e. on Ω. We also prove that the equation

has a unique solution for all ν > 0 and the map f → u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation

The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

Introduction and main result

We study the existence and uniqueness of solutions for the following convective-diffusive elliptic problem with pure Neumann boundary conditions

-div(D∇u) + div(V u) = f in Ω D ∇u • n = (V • n) u on ∂Ω, (1.1)
under the following basic assumptions:

(H1) Ω is a bounded connected open set of R N , N ≥ 2, with a Lipschitz-continuous boundary, (H2) The matrix valued function D : Ω → M N (R) is bounded and measurable and there exists α > 0 such that D(x) ξ • ξ ≥ α|ξ| 2 for all ξ ∈ R N and a.e. x ∈ Ω,

(H3) V ∈ L p (Ω) N with p = N if N ≥ 3, or p > 2 if N = 2, (H4) f ∈ (H 1 (Ω)) ′ .
Here, n is the unit normal to ∂Ω and dot indicates scalar product in R N . We are mainly interested in the effect of the convection term on the existence of solutions and in finding a natural functional framework where the problem is well-posed. Equations of this form where the convection speed V is the gradient of a potential, V(x) = ∇Φ(x), are known as the stationary version of the Fokker-Planck equation, [START_REF] Risken | The Fokker-Planck equation. Methods of solution and applications[END_REF], but we will not be confined here to such gradient speeds.

We consider weak solutions to (1.1) understood in the standard sense, that is

   u ∈ H 1 (Ω) , ∀ϕ ∈ H 1 (Ω) , Ω D∇u • ∇ϕ dx - Ω uV • ∇ϕ dx = f, ϕ (H 1 ) ′ ,H 1 . (1.2)
Here, •, • is the duality pairing. Under suitable assumptions on the divergence of V and the values of V on ∂Ω, the bilinear form a(u, ϕ) = Ω D∇u • ∇ϕ dx -Ω uV • ∇ϕ dx appearing in (1.2) is coercive on H 1 (Ω), and existence of a solution to this problem is then an immediate consequence of the Lax-Milgram theorem.

However, for general vector fields V, this bilinear form fails to be coercive and existence of a solution is less obvious; it is well known that, in any case, such elliptic problems (for Neumann or other boundary conditions) have finite dimensional kernels and that their solvability requires the right-hand side to satisfy a number of equations (as many equations as the dimension of the kernel of the elliptic problem), see e.g. [START_REF] Brezis | Analyse fonctionnelle: Théorie et applications[END_REF], [START_REF] Evans | Partial Differential Equations[END_REF]. The question is then to determine how many equations are needed and, more precisely, which ones.

For other boundary conditions (Dirichlet, Fourier or mixed conditions), the answer was given in [START_REF] Droniou | Non-coercive Linear Elliptic Problems[END_REF] and is quite simple: no condition on f is required, the convection-diffusion problem always has one and only one solution (it is also the same for some non-linear equations, see [START_REF] Droniou | [END_REF], and for some singular right-hand sides, see [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] and [START_REF] Droniou | Global and local estimates for nonlinear noncoercive elliptic equations with measure data[END_REF]). But, in these references, the proof of the existence of solutions is made via a priori estimates which are not available, at least by direct methods, in the case of pure Neumann boundary conditions. Hence, the question for (1.2) remains open: which necessary and sufficient conditions must be placed on f , and what are the degrees of freedom on the solutions?

The study of non-coercive linear elliptic problems as (1.1) is usually performed simultaneously with the study of the associated dual problem, that is

-div(D T ∇v) -V • ∇v = g in Ω, D T ∇v • n = 0 on ∂Ω, (1.3)
where g ∈ (H 1 (Ω)) ′ and D T is the transpose of D, cf. for instance [START_REF] Droniou | Non-coercive Linear Elliptic Problems[END_REF] for the study with other boundary conditions. The weak formulation of this dual problem is

   v ∈ H 1 (Ω) , ∀ϕ ∈ H 1 (Ω) , Ω D T ∇v • ∇ϕ dx - Ω ϕV • ∇v dx = g, ϕ (H 1 ) ′ ,H 1 . (1.4)
Here too, the bilinear form a in the left-hand side of (1.4) is not coercive without special assumptions on V, so the question is now to understand under which conditions this problem has a solution.

There are however two straightforward facts on these primal and dual problems.

a) In order for (1.2) to have a solution, one must have f,

1 (H 1 ) ′ ,H 1 = 0. b) If v is a solution to (1.4), then so is v + C.
Assertion a) states at least one condition on the right-hand side of (1.2): we need to determine if it is the only one. Moreover, since there are as many conditions as the dimension of the kernel of this problem, such a kernel is non-trivial: can we describe it? How can we select one and only one solution?

The same holds for (1.4): assertion b) shows that the kernel of the dual problem is non trivial; so we ask ourselves, are there any functions other than the constant functions in this kernel? And since this kernel is non-trivial, there must be some condition, that we want to find, on g for (1.4) to have a solution.

Statement of the main result

We use the notation H ⋆ = {u ∈ H 1 (Ω) | Ω u dx = 0}, and for ϕ ∈ H 1 (Ω) we define

H ′ ♯ϕ = {f ∈ (H 1 (Ω)) ′ | f, ϕ (H 1 ) ′ ,H 1 = 0}
. These spaces are endowed with the topologies of H 1 (Ω) and (H 1 (Ω)) ′ respectively. Here is our main result on Problems (1.1) and (1.3), which answers the preceding questions.

Theorem 1.1 Assume that (H1)-(H4) hold. Then, (i) There exists a function u ∈ H 1 (Ω), unique up to a multiplicative constant, which is a solution of Problem (1.2) with f = 0, satisfies u > 0 a.e. on Ω, and moreover (ii) Problem (1.2) has a solution if and only if f ∈ H ′ ♯1 ; in this case, there exists a unique solution u to (1.2) in H ⋆ and the set of all solutions is u + R u = {u + c u : c ∈ R}. Moreover, the application T : f → u is a bounded linear map from H ′ ♯1 into H ⋆ . (iii) Problem (1.4) has a solution if and only if g ∈ H ′ ♯ u ; in this case, there exists a unique solution v to (1.4) in H ⋆ and the set of all solutions is v + R 1. Moreover, the application

T ′ : g → v is a bounded linear map from H ′ ♯ u into H ⋆ .
A simple case: V is a gradient (i) If D is the identity matrix and V is a gradient, say V = ∇Φ with Φ ∈ C 1 (Ω), then we can take u = e Φ in Theorem 1.1. Indeed, in this situation,

-∆e Φ + div(Ve Φ ) = -div(∇e Φ -Ve Φ ) = -div(e Φ ∇Φ -∇Φ e Φ ) = 0 in Ω and ∇e Φ • n -V • ne Φ = e Φ ∇Φ • n -∇Φ • ne Φ = 0 on ∂Ω.
In fact, in this situation, solving (1.1) is best done by the change of unknown w = e -Φ u, which leads to

-div(e Φ ∇w) = f in Ω, e Φ ∇w • n = 0 on ∂Ω, (1.5)
that is to say a classical heterogeneous Neumann problem (notice that, for some β > 0, β ≤ e Φ ≤ β -1 on Ω). The usual choice to select one and only one solution to (1.5) is to impose Ω w dx = 0, which means that the chosen solution u to (1.2) satisfies Ω e -Φ u dx = 0 and not Ω u dx = 0 but, adding a suitable constant to w, we can find back the solution in H ⋆ of (1.2) given by Theorem 1.1.

(ii) In the case where D = Id, the same considerations hold if V = D∇Φ for some Φ ∈ C 1 (Ω). In this case, we still have u = e Φ and w = e -Φ u satisfies -div(e Φ D∇w) = f in Ω and e Φ D∇w • n = 0 on ∂Ω.

Preliminary considerations. Kernels

Let L : H 1 (Ω) → (H 1 (Ω)) ′ be the bounded linear operator defined by the bilinear form a(u, ϕ) appearing in (1.2), that is to say

Lu, ϕ (H 1 ) ′ ,H 1 = Ω D∇u • ∇ϕ dx - Ω uV • ∇ϕ dx.
Thanks to (H1)-(H3) and the Sobolev embeddings, L is well defined. For γ ∈ R we denote L γ = L + γI where I : H 1 (Ω) → (H 1 (Ω)) ′ is the natural embedding through L 2 (Ω), that is to say

L γ u, ϕ (H 1 ) ′ ,H 1 = Ω D∇u • ∇ϕ dx - Ω uV • ∇ϕ dx + γ Ω uϕ dx.
It is well-known that, for γ > 0 large enough, L γ : H 1 (Ω) → (H 1 (Ω)) ′ is an isomorphism (see also Lemma 4.1); from now on, we take such a γ fixed. The restriction of γ(

L γ ) -1 : (H 1 (Ω)) ′ → H 1 (Ω) to L 2 (Ω) is a compact operator K : L 2 (Ω) → L 2 (Ω). Moreover, since K has values in H 1 (Ω), u ∈ L 2 (Ω) and Ku = u ⇐⇒ u ∈ H 1 (Ω) and Ku = u ⇐⇒ u ∈ H 1 (Ω) and γu = L γ u ⇐⇒ u ∈ H 1 (Ω) and Lu = 0 .
In other terms, ker(Id L 2 -K) = ker(L).

(2.1)

The following lemma gives a first description of ker(L), which will be made precise below. Its proof summons up a technique close to the one used in [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires[END_REF] (to prove the uniqueness of a non-monotone elliptic equation with Dirichlet boundary conditions), but in the framework of Neumann boundary conditions.

Lemma 2.1 Assume that (H1)-(H3) hold. If u ∈ ker(L)\{0}, then either u > 0 a.e. on Ω or u < 0 a.e. on Ω. In particular, ker(L) has dimension 0 or 1.

Proof of Lemma 2.1. Since u is not a.e. null, there exists η > 0 such that u ≤ -η or u ≥ η on a set of non-null Lebesgue measure; upon changing u into -u, we can assume that the case

u ≥ η occurs. Let ϕ ε (r) = 0 if r ≥ ε, ϕ ǫ (r) = r -ε if 0 < r < ε and ϕ ε (r) = -ε if r ≤ 0 (see figure 1). ϕ ε -ε ε Figure 1: The function ϕ ε .
The function u satisfies (1.2) with f = 0; using ϕ ε (u) ∈ H 1 (Ω) as a test function in this equation, since ∇(ϕ ε (u)) = 1 {0<u<ε} ∇u (where 1 B stands for the characteristic function of a set B), we obtain

Ω D∇(ϕ ε (u))•∇(ϕ ε (u)) dx = {0<u<ε} uV•∇(ϕ ε (u)) dx ≤ ε||V|| L 2 ({0<u<ε}) N ||∇(ϕ ε (u))|| L 2 (Ω) N .
Since the measure of {0 < u < ε} tends to 0 as ε → 0, we deduce from this inequality and (H2)

that ||∇(ϕ ε (u))|| L 2 (Ω) N ≤ 1 α ε||V|| L 2 ({0<u<ε}) N = εω(ε)
where ω(ε) → 0 as ε → 0. Let E be a set of non-null Lebesgue measure such that u ≥ η on E; for all 0 < ε < η we have ϕ ε (u) = 0 on E and thus, by Lemma 8.1 (see Appendix), there exists

C E not depending on ε ∈]0, η[ such that ||ϕ ε (u)|| L 2 (Ω) ≤ C E ||∇(ϕ ε (u))|| L 2 (Ω) N ≤ C E εω(ε). But |ϕ ε (u)| ≥ ε/2 on {u ≤ ε/2}, which implies ||ϕ ε (u)|| L 2 (Ω) ≥ {u≤ε/2} |ϕ ε (u)| 2 dx 1/2 ≥ ε 2 (meas({u ≤ ε/2})) 1/2
and therefore (meas

({u ≤ ε/2})) 1/2 ≤ 2C E ω(ε); we deduce that meas({u ≤ 0}) = lim ε→0 meas({u ≤ ε/2}) = 0,
that is to say u > 0 a.e. on Ω.

The proof that the dimension of ker(L) cannot be more than 1 is rather simple. We take nonnull functions (u, v) ∈ ker(L) and prove that they are collinear. Since v = 0, then either v > 0 or v < 0 a.e. on Ω; in particular, λ = Ω u dx/ Ω v dx is well defined. The function w = u -λv belongs to ker(L) and, thus, either w > 0 or w < 0 or w = 0 a.e. on Ω; since Ω w dx = 0 by choice of λ, the first two cases cannot occur, and we therefore have w = 0, which means that u = λv and concludes the proof.

The operator L * : H 1 (Ω) → (H 1 (Ω)) ′ defined by the bilinear form a in (1.4), that is to say

L * v, ϕ (H 1 ) ′ ,H 1 = Ω D T ∇v • ∇ϕ dx - Ω ϕV • ∇v dx , is the adjoint of L: it satisfies Lϕ, ψ (H 1 ) ′ ,H 1 = L * ψ, ϕ (H 1 ) ′ ,H 1 for all (ϕ, ψ) ∈ H 1 (Ω).
(2.2)

Hence, for γ > 0 such that L γ is an isomorphism, L * γ = L * + γI : H 1 (Ω) → (H 1 (Ω)) ′ is also an isomorphism. It is easy to see from (2.2) that the restriction of γ(L * γ ) -1 : (H 1 (Ω)) ′ → H 1 (Ω) to L 2 (Ω) is the dual operator K * : L 2 (Ω) → L 2 (Ω) to K. Hence, as (2.1) we have ker(Id L 2 -K * ) = ker(L * ).
(

The properties of compact operators and Lemma 2.1 now allow us to give a precise description of the kernels of L and L * . Proposition 2.2 Assume that (H1)-(H3) hold. Then, 1. ker(L) is one-dimensional, and is spanned by a function u ∈ H 1 (Ω) which satisfies u > 0 a.e. on Ω, 2. ker(L * ) is exactly the set of constant functions. [START_REF] Evans | Partial Differential Equations[END_REF]). Properties (2.1) and (2.3) and the second part of Lemma 2.1 then show that dim(ker(L * )) = dim(ker(L)) ≤ 1. But the constant functions obviously are in ker(L * ), so that ker(L * ) is exactly made out of these functions and dim(ker(L)) = 1. The fact that ker(L) has a generator function which is a.e. positive then follows from the first part of Lemma 2.1.

Proof. Since K : L 2 (Ω) → L 2 (Ω) is compact, we have dim(ker(Id L 2 -K)) = dim(ker(Id L 2 - K * )) (see
Remark 2.3 Note that u is constant only if div(V) = 0 in Ω and V • n = 0 on ∂Ω. In this case, for all ϕ ∈ H 1 (Ω) one has Ω ϕV • ∇ϕ dx = 0 and, by the Poincaré-Wirtinger's inequality, the bilinear forms in (1.2) and (1.4) are coercive on H ⋆ ; Theorem 1.1 (with u = 1) is then a trivial consequence of the Lax-Milgram theorem.

3 Proof of Theorem 1.1

The proof is a direct consequence of the Fredholm alternative and the characterization of the kernels of L and L * . For the sake of completeness, let us quickly recall the reasoning which leads to these results.

Solving (1.2) comes down to finding u ∈ H1 (Ω) such that Lu = f ; this is equivalent to [START_REF] Evans | Partial Differential Equations[END_REF] ( 1 )) and therefore, by (2.3), there exists

L γ u = f + γu and thus u = (L γ ) -1 f + γ(L γ ) -1 u = (L -1 γ )f + Ku since u ∈ L 2 (Ω). Let w = (L γ ) -1 f ∈ H 1 (Ω); since K has values in H 1 (Ω), finding u ∈ H 1 (Ω) such that u = w + Ku is equivalent to finding u ∈ L 2 (Ω) such that u = w + Ku; but, K being compact, we have (Id L 2 -K)(L 2 (Ω)) = (ker(Id L 2 -K * )) ⊥ (see
u ∈ L 2 (Ω) such that u = w + Ku if and only if w ∈ (ker(Id L 2 -K * )) ⊥ = (ker(L * )) ⊥ . But, by definition of w = (L γ ) -1 f , for all ϕ ∈ ker(L * ), since both w and ϕ are in L 2 (Ω), f, ϕ (H 1 ) ′ ,H 1 = Lw + γw, ϕ (H 1 ) ′ ,H 1 = L * ϕ, w (H 1 ) ′ ,H 1 + γ Ω wϕ dx = γ Ω wϕ dx , and therefore w ∈ (ker(L * )) ⊥ if and only if f, ϕ (H 1 ) ′ ,H 1 = 0 for all ϕ ∈ ker(L * ), i.e. if and only if f, 1 (H 1 ) ′ ,H 1 = 0 by Proposition 2.2. Hence, (1.2) has a solution if and only if f ∈ H ′ ♯1 .
All the solutions to (1.2) can then be written u = u 0 + z where u 0 is any fixed solution and z ∈ ker(L), that is to say, using the u given by Proposition 2.2, u = u 0 + λ u for some λ ∈ R.

Since Ω u dx > 0 (because u > 0 a.e. on Ω), we can select one and only one solution by imposing Ω u dx = 0 (this fixes λ = -Ω u 0 dx/ Ω u dx), and all the other solutions lie in u + R u.

We have therefore proved that, for all f ∈ H ′ ♯1 , (1.2) has a unique solution u ∈ H ⋆ . This defines an application T : H ′ ♯1 → H ⋆ which is clearly linear, and the proof of the continuity of T can be made by way of the closed-graph theorem: if {f n } n≥1 is a sequence in H ′ ♯1 that converges to some f in H ′ ♯1 and if {u n = T f n } converges to some u in H ⋆ then, writing (1.2) with u n and f n and passing to the limit n → +∞ we see that u ∈ H ⋆ is a solution to (1.2) with f as right-hand side, i.e. that u = T f ; the graph of T is therefore closed and T is continuous.

The study of (1.4) is completely similar, inverting the roles of L and L * .

Estimates and global continuity

Since the application f ∈ H ′ ♯1 → u ∈ H ⋆ solution to (1.
2) is linear and continuous, there exists C > 0 not depending on f such that ||u|| H 1 (Ω) ≤ C||f || (H 1 (Ω)) ′ ; however, the preceding proof does not allow to estimate C and, in particular, to understand how it depends on D or V (the same holds for the solution to Problem (1.4)). Such estimates can be of importance in the case where one wants to study the behaviour of u as D or V varies. In this section, we intend to make clearer the way C depends on these data, and to prove that the solution and the kernel of the operator are both continuous with respect to these data.

To study this dependence, we need to vary V; in order to emphasize on the dependence of L on V, we therefore write L = L V (and similarly L γ = L V γ ). Moreover, if E is Banach space and r > 0, B(r; E) denotes the closed ball in E of center 0 and radius r.

Let us begin by making precise how γ must be chosen, for a given V, in order for L V γ to be an isomorphism.

= N if N ≥ 3 or p > 2 if N = 2.
Let also s > p and R > 0. Then, there exist ε 0 > 0, M > 0, and γ 0 > 0 such that for all γ ≥ γ 0 and all

V ∈ B(ε 0 ; L p (Ω) N ) + B(R; L s (Ω) N ), L V γ : H 1 (Ω) → (H 1 (Ω))
′ is an isomorphism and we have

||(L V γ ) -1 || L((H 1 ) ′ ,H 1 ) ≤ M. (4.1)
More precisely, we can take any ε 0 < α/S(Ω, p), where S(Ω, p) is the Sobolev constant of the embedding H 1 (Ω) ֒→ L 2p p-2 (Ω), and then γ 0 = γ 0 (Ω, α, p, s, ε 0 , R) and M = M (Ω, α, p, ε 0 , R).

Proof. For all u ∈ H 1 (Ω) and all γ > 0, using Young's inequality, we have

L V γ u, u (H 1 ) ′ ,H 1 = Ω D∇u • ∇u dx - Ω uV • ∇u dx + γ Ω u 2 dx ≥ α||∇u|| 2 L 2 (Ω) N + γ||u|| 2 L 2 (Ω) -||uV|| L 2 (Ω) N ||∇u|| L 2 (Ω) N . (4.2)
We then write

V = V 1 + V 2 with ||V 1 || L p (Ω) N ≤ ε 0 and ||V 2 || L s (Ω) N ≤ R. By the Sobolev embedding H 1 (Ω) ֒→ L 2p p-2 (Ω), we have ||uV|| L 2 (Ω) N ≤ ||uV 1 || L 2 (Ω) N + ||uV 2 || L 2 (Ω) N ≤ ||u|| L 2p p-2 (Ω) ||V 1 || L p (Ω) N + ||u|| L 2s s-2 (Ω) ||V 2 || L s (Ω) N ≤ S(Ω, p)ε 0 ||u|| H 1 (Ω) + R||u|| L 2s s-2 (Ω)
.

But, since 2s s-2 < 2N N -2 , H 1 (Ω) is compactly embedded in L 2s s-2
(Ω) and, therefore, for all ν > 0 there exists C(Ω, s, ν) > 0 such that ||u||

L 2s s-2 (Ω) ≤ ν||∇u|| L 2 (Ω) N + C(Ω, s, ν)||u|| L 2 (Ω) . We deduce that ||uV|| L 2 (Ω) N ≤ (S(Ω, p)ε 0 + Rν)||∇u|| L 2 (Ω) N + (S(Ω, p)ε 0 + RC(Ω, s, ν))||u|| L 2 (Ω)
and, coming back to (4.2) and using Young's inequality, for all η > 0,

L V γ u, u (H 1 ) ′ ,H 1 ≥ (α -S(Ω, p)ε 0 -Rν) ||∇u|| 2 L 2 (Ω) N + γ||u|| 2 L 2 (Ω) -(S(Ω, p)ε 0 + RC(Ω, s, ν))||u|| L 2 (Ω) ||∇u|| L 2 (Ω) N ≥ (α -S(Ω, p)ε 0 -Rν -η) ||∇u|| 2 L 2 (Ω) N + γ - 1 4η (S(Ω, p)ε 0 + RC(Ω, s, ν)) 2 ||u|| 2 L 2 (Ω) .
Fixing ε 0 < α/S(Ω, p), we can choose ν = ν(Ω, α, p, ε 0 , R) > 0 and η = η(Ω, α, p, ε 0 , R) > 0 such that β(Ω, α, p, ε 0 , R) = α -S(Ω, p)ε 0 -Rν -η > 0 and, taking γ 0 (Ω, α, p, s, ε 0 , R) = β(Ω, α, p, ε 0 , R) + 1 4η (S(Ω, p)ε 0 + RC(Ω, s, ν)) 2 we see that, for all γ ≥ γ 0 (Ω, α, p, s, ε 0 , R),

L V γ u, u (H 1 ) ′ ,H 1 ≥ β(Ω, α, p, ε 0 , R)||u|| 2 H 1 (Ω) . (4.3) Lax-Milgram's theorem then shows that L V γ : H 1 (Ω) → (H 1 (Ω)) ′ is an isomorphism and, for all f ∈ (H 1 (Ω)) ′ , applying (4.3) to u = (L V γ ) -1 f , we find ||u|| H 1 (Ω) ≤ β(Ω, α, p, ε 0 , R) -1 ||f || (H 1 (Ω)) ′ , which concludes the proof.
We can now state and prove some more precise estimates on the bounds of the solution to (1.2) in H ⋆ . Theorem 4.2 Assume that (H1) and (H2) hold. Let p = N if N ≥ 3 or p > 2 if N = 2. Let s > p and R > 0, and let S(Ω, p) and ε 0 < α/S(Ω, p) be as in Lemma 4.1. Then, there exists

C(Ω, D, p, s, ε 0 , R) > 0 such that, for all V ∈ B(ε 0 ; L p (Ω) N ) + B(R; L s (Ω) N ) and all f ∈ H ′ ♯1 , if u is the solution to (1.2) in H ⋆ then ||u|| H 1 (Ω) ≤ C(Ω, D, p, s, ε 0 , R)||f || (H 1 (Ω)) ′ .
(4.4)

Remark 4.3 Note that, for all ε 0 > 0 and all s > N ,

L p (Ω) N = B(ε 0 ; L p (Ω) N ) + ∪ R>0 B(R; L s (Ω) N ).
Hence, this theorem gives an estimate for any V satisfying (H3); this estimate however does not only depend on the norm of V in L p (Ω) N but also on the way we can split V as a small function in L p (Ω) N and a function slightly more integrable; it is therefore completely similar to the estimates obtained for non-coercive elliptic problems with other boundary conditions (see [START_REF] Droniou | Non-coercive Linear Elliptic Problems[END_REF]).

Proof of Theorem 4.2. By linearity, it is sufficient to make the proof for ||f || (H 1 (Ω)) ′ = 1. Assume that the result does not hold: then there exists (

V n ) n≥1 ∈ B(ε 0 ; L p (Ω) N ) + B(R; L s (Ω) N ) such that the solution u n ∈ H ⋆ to (1.2) with V = V n satisfies ||u n || H 1 (Ω) → +∞ as n → +∞.
By Lemma 4.1, there exists γ > 0 and M > 0 not depending on n such that, for all n ≥ 1,

L Vn γ is an isomorphism and ||(L Vn γ ) -1 || L((H 1 ) ′ ,H 1 ) ≤ M. (4.5)
As we have seen in the proof of Theorem 1.1, the fact that u n is a solution to (1.2) with V = V n , i.e. that L Vn u n = f , is equivalent to

u n = w n + γ(L Vn γ ) -1 u n , (4.6)
where

(w n ) n≥1 = ((L Vn γ ) -1 f ) n≥1 is bounded in H 1 (Ω) thanks to (4.5). If (u n ) n≥1
were bounded in L 2 (Ω), and therefore in (H 1 (Ω)) ′ , then (4.5) would give a bound on (γ(L Vn γ ) -1 u n ) n≥1 in H 1 (Ω), and, by (4.6), (u n ) n≥1 would be bounded in H 1 (Ω); but we have precisely rejected this from the beginning.

Hence, (u n ) n≥1 is not bounded in L 2 (Ω) and, up to a subsequence,

||u n || L 2 (Ω) → +∞ as n → +∞. Let h n = u n /||u n || L 2 (Ω) ∈ H ⋆ ; from (4.6) we deduce h n = w n ||u n || L 2 (Ω) + γ(L Vn γ ) -1 h n . (4.7)
We have ||h n || L 2 (Ω) = 1 for all n ≥ 1 and thus (4.5) implies that (γ(L Vn γ ) -1 h n ) n≥1 is bounded in H 1 (Ω); by (4.7), (h n ) n≥1 is therefore bounded in H 1 (Ω). Up to a subsequence, (h n ) n≥1 converges to some h weakly in H 1 (Ω) and strongly in L 2 (Ω). Since Ω h n dx = 0 and ||h n || L 2 (Ω) = 1 for all n ≥ 1, the same holds for h; in particular, Ω h dx = 0 and h = 0.

Let ϕ ∈ C ∞ (Ω); by definition of u n and h n = u n /||u n || L 2 (Ω) , we can write

Ω D∇h n • ∇ϕ dx - Ω h n V n • ∇ϕ dx = f ||u n || L 2 (Ω) , ϕ (H 1 ) ′ ,H 1 . (4.8) Since (V n ) n≥1 is bounded in L p (Ω) N
, up to a subsequence we can assume that V n → V weakly in L p (Ω) N , and in particular weakly in L 2 (Ω) N ; the convergence of h n to h weakly in H 1 (Ω) and strongly in L 2 (Ω) and the fact that ||u n || L 2 (Ω) → +∞ then allow to pass to the limit n → +∞ in (4.8) to see that

Ω D∇h • ∇ϕ dx - Ω hV • ∇ϕ dx = 0.
This equation has been proved for ϕ ∈ C ∞ (Ω) but, since h ∈ H 1 (Ω) and V ∈ L p (Ω) N , by density it also holds for all ϕ ∈ H 1 (Ω), which proves that h ∈ ker(L V ). Since Ω h dx = 0 and h = 0, this is in contradiction with Proposition 2.2 which entails, in particular, that the only function in ker(L V ) which has a null mean value is the null function. Hence, the proof is concluded.

Remark 4.4 This proof shows how C depends on V (through p, s, ε 0 and R), but it does not give an explicit bound on this constant or on the way it depends on D, contrary to the direct estimates made in the case of other boundary conditions (see [START_REF] Droniou | Non-coercive Linear Elliptic Problems[END_REF]). In fact, for other boundary conditions, the existence of a solution is deduced from a priori explicit estimates; here, we first proved existence of a solution and then deduced a posteriori non-explicit estimates on this solution. This can be a problem when transferring these estimates to a discrete setting, in order to study numerical schemes on (1.1) (see e.g. [START_REF] Droniou | A finite volume scheme for a noncoercive elliptic equation with measure data[END_REF] for the adaptation of the continuous estimates to the setting of finite volume schemes, in the case of convection-diffusion noncoercive problems with Dirichlet boundary conditions).

Remark 4.5 There are however some situations where we can ensure that the constant C in Theorem 4.2 does not explode as D varies: for example, if (D n ) n≥1 is a bounded sequence of matrix-valued functions which satisfy (H2) (with a uniform α > 0) and which converges a.e. to some D, then a straightforward adaptation of the proof of Theorem 4.2 shows that the corresponding C(Ω, D n , p, s, ε 0 , R) stays bounded as n → ∞. This particular case will be useful in the following.

(1.4) in H ⋆ then, for all F ∈ (H 1 (Ω)) ′ , denoting u the solution to (1.2) in H ⋆ with f = F - 1 meas(Ω) F, 1 (H 1 ) ′ ,H 1 ∈ H ′ ♯1 , since Ω v dx = 0, F, v (H 1 ) ′ ,H 1 = f, v (H 1 ) ′ ,H 1 = L V u, v (H 1 ) ′ ,H 1 = (L V ) * v, u (H 1 ) ′ ,H 1 = g, u (H 1 ) ′ ,H 1 ≤ ||g|| (H 1 (Ω)) ′ ||u|| H 1 (Ω) ≤ C(Ω, D, p, s, ε 0 , R)||g|| (H 1 (Ω)) ′ ||f || (H 1 (Ω)) ′ and it is clear that ||f || (H 1 (Ω)) ′ ≤ 2||F || (H 1 (Ω)) ′ so that, taking the supremum on F ∈ (H 1 (Ω)) ′ having norm smaller than 1, we obtain ||v|| H 1 (Ω) ≤ 2C(Ω, D, p, s, ε 0 , R)||g|| (H 1 (Ω)) ′ .
Let us now study the continuity of the kernel of L and the solution to (1.2) with respect to all the data. We normalize the choices of functions u in Theorem 1.1 by imposing || u|| H 1 = 1. Theorem 4.7 Let (H1) hold and take p = N if N ≥ 3 or p > 2 if N = 2. Assume that (D n ) n≥1 is a bounded sequence of uniformly elliptic matrix-valued functions which converges a.e. to some D, and let [START_REF] Crandall | Generation of semi-groups of nonlinear transformations on general Banach spaces[END_REF] We could of course also state a result on the continuity of the solution to (1.4).

V n → V in L p (Ω) N . Then, i) The (normalized) function u n from Theorem 1.1 corresponding to (D n , V n ) converges in H 1 (Ω) to the (normalized) function u corresponding to (D, V), ii) If f n → f in H ′ ♯1 , then the solution u n ∈ H ⋆ of (1.2) for (D n , V n , f n ) converges in H 1 (Ω) to the solution u ∈ H ⋆ of (1.2) for (D, V, f ). Remark 4.
Proof of Theorem 4.7. The proofs of i) and ii) are completely similar, so we only consider ii).

Since (V n ) n≥1 converges in L p (Ω) N , it is easy to see that, for all ε 0 > 0, there exists R > 0 such that, for all n ≥ 1,

V n ∈ B(ε 0 ; L p (Ω) N )+B(R; L ∞ (Ω) N ) (write V n = (V n -T M (V n ))+T M (V n )
for M large enough, where T M is the truncation of each component at level M ). Hence, by Theorem 4.2 and Remark 4.5, (u n ) n≥1 is bounded in H 1 (Ω), and converges up to a subsequence weakly in H 1 (Ω) and strongly in L 2 (Ω) to some u ∈ H ⋆ . The strong convergence of the data and the weak convergence of the solutions allow to pass to the limit in the equations (1.2) with (D n , V n , f n ) statisfied by u n to see that u is the (unique in H ⋆ ) solution to this problem with (D, V, f ) (and thus that the whole sequence u n converges to u, not only a subsequence). It remains to prove that the convergence of u n to u is strong in H 1 (Ω).

In order to do so, we first notice that

Ω D n ∇u n • ∇u n = Ω u n V n • ∇u n + f n , u n (H 1 ) ′ ,H 1 = Ω V n • ∇ u 2 n 2 + f n , u n (H 1 ) ′ ,H 1 . (4.9) Since u n is bounded in H 1 (Ω), ∇(u 2 n /2) = u n ∇u n is bounded in L p ′ (Ω)
N by our choice of p ′ and the Sobolev embeddings, and thus it weakly converges in this space to some U; but u 2 n → u 2 strongly in L 1 (Ω) and, therefore, ∇(u 2 n /2) → ∇(u 2 /2) in the sense of distributions; this shows that U = ∇(u 2 /2). By weak convergence of u n and ∇(u 2 n /2) and strong convergence of f n and V n , we can then pass to the limit in the right-hand side of (4.9) and we deduce, since u is a solution to (1.2),

lim n→∞ Ω D n ∇u n • ∇u n = Ω V • ∇ u 2 2 + f, u (H 1 ) ′ ,H 1 = Ω uV • ∇u + f, u (H 1 ) ′ ,H 1 = Ω D∇u • ∇u. (4.10)
The rest is quite classical: we have

α||∇u n -∇u|| 2 L 2 (Ω) N ≤ Ω D n ∇(u n -u) • ∇(u n -u) = Ω D n ∇u n • ∇u n - Ω D n ∇u n • ∇u - Ω D n ∇u • ∇u n + Ω D n ∇u • ∇u
and, by strong convergence of D n , weak convergence of ∇u n and (4.10), we can pass to the limit to see that ||∇u n -∇u|| L 2 (Ω) N → 0. Combined with the strong convergence of u n in L 2 (Ω), this concludes the proof.

As a consequence we can state the continuity of the normalized map S : V → u, where u is the unique normalized function of Theorem 1.1 and V satisfies (H3). It is a nonlinear map into H ⋆ , and we know that for V = 0 we get a positive constant:

S(0) = |Ω| -1/2 . Corollary 4.9 S : V → u is continuous from L N if N ≥ 3 (resp. L p with p > 2 if N = 2) to H ⋆ .

Spectral analysis. Optimal solvability

There is an immediate generalization of (2.1) to other possible eigenvalues of K (the restriction to L 2 (Ω) of γ(L γ ) -1 ):

∀α = 0 , ker(αId L 2 -K) = ker L γ(1-1 α ) . (5.1)
This allows in particular to prove that there is no necessity to take γ large in order for L γ to be invertible. Proposition 5.1 Assume that (H1)-(H3) hold. Then, L ν : H 1 (Ω) → (H 1 (Ω)) ′ is an isomorphism for every ν > 0.

Proof. Let us first prove that L ν is one-to-one. Let u ∈ ker(L ν ) and define T ε (r) = min(ε, max(r, -ε)) the truncature function at level ε. Then ∇(T ε (u)) = 1 {-ε<u<ε} ∇u and thus

0 = L ν u, T ε (u) (H 1 ) ′ ,H 1 = Ω D∇u • ∇(T ε (u)) dx - Ω uV • ∇(T ε (u)) dx + ν Ω uT ε (u) dx ≥ α||∇(T ε (u))|| 2 L 2 (Ω) -ε||V|| L 2 (Ω) N ||∇(T ε (u))|| L 2 (Ω) + ν Ω uT ε (u) dx.
Hence, using Young's inequality,

ν Ω uT ε (u) dx ≤ ε 2 ||V|| 2 L 2 (Ω) N 4α ,
so that Ω u T ε (u)/ε dx ≤ Cε with C not depending on ε. But, as ε → 0, T ε (u)/ε → sgn(u) while staying bounded by 1 and, passing to the limit thanks to the dominated convergence theorem, we deduce Ω |u| dx ≤ 0, that is to say u = 0. This proves that ker(L ν ) = {0}.

To prove that L ν is onto, let µ ≫ 1 and denote by K the restriction to L2 (Ω) of (ν+µ)(L ν+µ ) -1 . As at the beginning of the the proof of Theorem 1.1, we notice that, for all f ∈ (H

1 (Ω)) ′ , finding u ∈ H 1 (Ω) such that L ν u = f is equivalent to finding u ∈ L 2 (Ω) such that u = w + µ ν+µ Ku (where w = (L ν+µ ) -1 f ∈ H 1 (Ω)), that is to say ν+µ µ u -Ku = ν+µ µ w. But (5.1) applied to γ = ν + µ and α = ν+µ µ shows that ker ν + µ µ Id L 2 -K = ker(L ν ) = {0}.
Since K is compact, this implies that ν+µ µ Id L 2 -K is onto (see [START_REF] Evans | Partial Differential Equations[END_REF]) and therefore that there exists u ∈ L 2 (Ω) such that ν+µ µ u -Ku = ν+µ µ w, which concludes the proof.

Remark 5.2 It is in fact possible to prove this proposition (which comes down to solving (1.1) with an additional term +νu in the left-hand side of the PDE) using the techniques in [START_REF] Droniou | Global and local estimates for nonlinear noncoercive elliptic equations with measure data[END_REF]. The proof we give above is however much shorter, although it does not give estimates on the norm of (L ν ) -1 .

From this proposition and (5.1), we deduce that all the eigenvalues of K are in ]0, 1] (it is quite clear that 0 is not an eigenvalue of K). The spectral analysis of compact operators then tells that the eigenvalues of K form a decreasing sequence which tends to 0. In terms of L, this means that the eigenvalues of L form a sequence of increasing nonnegative numbers which tends to +∞. Moreover, using Proposition 2.2, we see that the first eigenvalue of L is 0, is of multiplicity 1 and is associated with an a.e. positive eigenfunction ( 2 ). This property of positivity of the first eigenfunction (and therefore its uniqueness up to multiplication by a scalar) is exactly what is stated in the strong forms of the Krein-Rutman theorem. However, since the data D and V we consider are irregular, we had to conduct the study of (1.1) in Sobolev spaces in which the interior of the cone of nonnegative functions is empty; hence, on the contrary to what can be done for smooth solutions (see [START_REF] Du | Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol 1: Maximum Principles and Applications[END_REF]), it does not seem possible to easily deduce the positivity of the first eigenfunction from a strong Krein-Rutman theorem (see [START_REF] Deimling | Nonlinear functional analysis[END_REF] in the case of Dirichlet boundary conditions and regular data).

6 An application: uniqueness for nonlinear problems with measure data

One of the main applications of noncoercive convective-diffusive problems (i.e. without any assumption on the divergence of the convection) is the obtention of uniqueness results for nonlinear elliptic equations with measure data, as shown in [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] in the case of Dirichlet, Fourier or mixed boundary conditions. Now that we have shown how to solve convective-diffusive problems with pure Neumann boundary conditions, we can extend the uniqueness result of [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] to this setting. This is however not completely trivial, because of the lack of explicit estimates on the solutions to (1.2) and (1.4) as D varies (see Remark 4.4); therefore, the adaptation of [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] to Neumann boundary conditions demands a few more arguments than those used for other boundary conditions.

Consider the following nonlinear elliptic equation

-div(a(x, w, ∇w)) = µ in Ω, a(x, w, ∇w) • n = 0 on ∂Ω (6.1)
where

• µ is a bounded signed measure on Ω such that µ(Ω) = 0, (

• a : Ω × R × R N → R N is a Caratheodory function satisfying: ∃γ > 0 , ∃Θ ∈ L 1 (Ω) such that a(x, s, ξ) • ξ ≥ γ|ξ| 2 -Θ(x) , ∃β > 0 , ∃h ∈ L 2 (Ω) such that |a(x, s, ξ)| ≤ h(x) + β|s| + β|ξ| , (6.3) • ∃α > 0 such that (a(x, s, ξ) -a(x, s, η)) • (ξ -η) ≥ α|ξ -η| 2 , 6.2)
• ∃Λ > 0 such that |a(x, s, ξ) -a(x, s, η)| ≤ Λ|ξ -η| , (6.4)
• ∃δ > 0 , ∃ω ∈ [0, 1 N -2 [ , ∃χ ∈ [0, 1 N -1 [ such that |a(x, s, ξ) -a(x, t, ξ)| ≤ δ|s -t|(1 + |s| ω + |t| ω + |ξ| χ ). (6.5)
We refer to [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] for a discussion on these assumptions -notice that Assumption (6.6) is slightly more general here.

The existence of a weak solution to (6.1) in the case of homogeneous Dirichlet boundary conditions is now quite classical, see e.g. [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]. One way to prove this existence is to approximate µ in the weak-* topology by a sequence of data (µ n ) n≥1 ∈ L 1 (Ω) ∩ H -1 (Ω), to consider a weak solution w n ∈ H 1 0 (Ω) to -div(a(x, w n , ∇w n )) = µ n , to prove that (w n ) n≥1 is bounded in W 1,q 0 (Ω) for all q < N N -1 , and to show the strong convergence in these spaces of this sequence towards a weak solution to -div(a(x, w, ∇w)) = µ; the solution thus obtained belongs to ∩ q< N N-1 W 1,q 0 (Ω). With our assumptions on a, this proof of existence of a solution can be easily adapted to the case of Neumann boundary conditions, using an approximation µ n ∈ L 1 (Ω) ∩ (H 1 (Ω)) ′ with null mean value and selecting a solution w n ∈ H 1 (Ω) to (6.1) with µ = µ n which also has a null mean value; the proof of the a priori estimates on (w n ) n≥1 can be made using the same test-functions and methods as in [START_REF] Boccardo | Nonlinear elliptic equations in R N without grow restrictions on the data[END_REF] (3 ) and, to prove the strong convergence of (w n ) n≥1 , the method from [START_REF] Bénilan | An L 1 -Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations[END_REF] works fine. This gives a solution to (6.1) in ∩ q< N N-1 W 1,q (Ω) with null mean value. A solution constructed this way is called a limit solution, or a SOLA: Solution Obtained as the Limit of Approximations (see [START_REF] Dall'aglio | Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations[END_REF]).

The next quaestion concerns the uniqueness of the solution, see e.g. [START_REF] Bénilan | An L 1 -Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations[END_REF], [START_REF] Boccardo | Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data[END_REF], [START_REF] Maso | Renormalized solutions for elliptic equations with general measure data, in preparation, and Definition and existence of renormalized solutions of elliptic equations with general measure data[END_REF], [START_REF] Porretta | Uniqueness of solutions for some nonlinear Dirichlet problems[END_REF] for some results on this subject (for Dirichlet boundary conditions). In the framework of SOLAs and with Neumann conditions, being able to solve linear convective-diffusive problems with non-smooth data D and V (Theorem 1.1) allows to prove the following uniqueness result. Theorem 6.1 Under assumptions (6.2)-(6.6), Problem (6.1) has a unique SOLA. Remark 6.2 We could also state and prove a stability result for the SOLA to (6.1), as it is done in [START_REF] Droniou | A uniqueness result for quasilinear elliptic equations with measures as data[END_REF] for other boundary conditions.

Proof of Theorem 6.1. Let (w, w) be two SOLA of (6.1). There exists thus (µ n

) n≥1 ∈ L 1 (Ω) ∩ H ′ ♯1 and ( µ n ) n≥1 ∈ L 1 (Ω) ∩ H ′ ♯1
which converge weakly- * to µ and such that, for some w n ∈ H ⋆ and w n ∈ H ⋆ weak solutions to (6.1) with respectively µ n and µ n instead of µ, we have w n → w and w n → w in W 1,q (Ω) for all q < N N -1 . Thanks to the assumptions on a and the fact that (w n ) n≥1 and ( w n ) n≥1 strongly converge in W 1,1 (Ω), the functions a n (x) = a(x, w n (x), ∇w n (x)) -a(x, w n (x), ∇ w n (x)) and b n (x) = ∇w n (x) -∇ w n (x) satisfy, up to a subsequence, the assumptions of Lemma 8.2 in the appendix; we still note (a n ) n≥1 and (b n ) n≥1 the subsequences involved and we let (D n ) n≥1 be the matrix-valued functions given by this lemma. These matrices are thus bounded and uniformly elliptic (independently of n) and there exists (ε n ) n≥1 which tends to 0 as n → ∞ such that, on

{|∇w n (x) -∇ w n (x)| ≥ ε n }, a(x, w n (x), ∇w n (x)) -a(x, w n (x), ∇ w n (x)) = D n (x)(∇w n (x) -∇ w n (x)). (6.7)
Hence, subtracting the equations satisfied by w n and w n , and denoting

V n (x) = - a(x, w n (x), ∇ w n (x)) -a(x, w n (x), ∇ w n (x)) w n (x) -w n (x) , we see that Γ n = w n -w n ∈ H ⋆ satisfies, for all ϕ ∈ H 1 (Ω), Ω D n ∇Γ n • ∇ϕ dx - Ω Γ n V n • ∇ϕ dx = µ n -µ n , ϕ (H 1 ) ′ ,H 1 - {|∇Γn|<εn} (a(x, w n , ∇w n ) -a(x, w n , ∇ w n ) -D n ∇Γ n ) • ∇ϕ dx. (6.8)
Thanks to Assumption (6.6) we see that

|V n | ≤ δ(1+|w n | ω +| w n | ω +|∇ w n | χ
) and, since (w n ) n≥1 and ( w n ) n≥1 are bounded in W 1,q (Ω) for all q < N N -1 , we infer that (V n ) n≥1 is bounded in L s (Ω) N for some s > N .

Let h ∈ L ∞ (Ω) and denote by u n ∈ H 1 (Ω) a positive function given by Theorem 1.1 for D = D n and V = V n . Defining t n = Ω h un dx Ω un dx , we see that h -t n ∈ H ′ ♯ un and we can therefore consider the solution ψ n to (1.4) with D = D n , V = V n and g = h -t n , that is to say:

ψ n ∈ H ⋆ and ∀ϕ ∈ H 1 (Ω) , Ω D T n ∇ψ n •∇ϕ dx- Ω ϕV n •∇ψ n dx = h-t n , ϕ (H 1 ) ′ ,H 1 = Ω (h-t n )ϕ dx. (6.9)
We notice that |t n | ≤ ||h|| ∞ and therefore (h -t n ) n≥1 is bounded in L ∞ (Ω) (and thus also in (H 1 (Ω)) ′ ); since (V n ) n≥1 is bounded in L s (Ω) N for some s > N and (D n ) n≥1 is bounded uniformly elliptic and converges a.e. on Ω, Remark 4.5 (see also Remark 4.6) shows that (ψ n ) n≥1 is bounded in H 1 (Ω). (6.10)

Moreover, using again the bound on (h

-t n ) n≥1 in L ∞ (Ω), Theorem A.1 in [11, Appendix A]
shows that there exists κ > 0 such that (ψ n ) n≥1 is bounded in the Hölder space C 0,κ (Ω). (6.11) In order to apply these ideas to our problem we have to select the space X. The choice that better suits our convection-diffusion operator is X = L 1 (Ω). Then, we have to adapt the operator to this framework. To be precise, we define D(A 1 ) as the set of weak solutions u of Problem (1.1) with right-hand f such that u ∈ H 1 (Ω) ⊂ L 1 (Ω) and f ∈ (H 1 (Ω)) ′ ∩ L 1 (Ω). If u ∈ D(A 1 ) we define A 1 (u) = f , i.e., A 1 is a restriction of the differential operator plus boundary conditions used up to now in this paper.

Theorem 7.1 Under assumptions (H1), (H2) and (H3), A 1 defined in this way is a T -accretive operator in L 1 (Ω). Moreover, for every h > 0 we have R(I

+ hA 1 ) = (H 1 (Ω)) ′ ∩ L 1 (Ω) so that it is dense in L 1 (Ω). Finally, D(A 1 ) ⊂ R(I + hA 1 ) and D(A 1 ) is dense in L 1 (Ω).
Proof. (i) Write A instead of A 1 for brevity. Checking the T -accretivity in L 1 (Ω) means therefore that when u ∈ H 1 (Ω) is the weak solution of the problem

-h div(D∇u -u V) + u = f in Ω (D∇u -u V) • n = 0 on ∂Ω, (7.1)
with f ∈ (H 1 (Ω)) ′ ∩ L 1 (Ω), then u + 1 ≤ f + 1 .
There is a standard trick in this theory that consists in using as test functions expressions of the form p n (u) where p n are nondecreasing piecewise C 1 (with bounded derivative) functions which approximate the positive sign function sign + . We do that and get

- Ω div(D∇u -u V)p n (u) dx = Ω p ′ n (u)D∇u • ∇u dx - Ω u V • p ′ n (u)∇u dx.
The first term in the right-hand side is nonnegative. Choosing for instance p n (s) = 0 on ] -∞, 0], p n (s) = ns on [0, 1/n] and p n (s) = 1 on [1/n, ∞[, the second term in the righthand side is bounded from above by Ω 1 {0<u<1/n} |V| |∇u| which tends to 0 as n → ∞ (by the dominated convergence theorem). Hence, passing to the limit we have

- Ω div(D∇u -u V) sign + (u) dx ≥ 0.
Using the equation, this means that

u + 1 = Ω u sign + (u) dx ≤ Ω f sign + (u) dx ≤ f + 1 .
(ii) Let us now prove that R(I + hA) = (H 1 (Ω)) ′ ∩ L 1 (Ω). Indeed, by Proposition 5.1 we can find a weak solution of the equation with h > 0 for every

f ∈ (H 1 (Ω)) ′ . If f ∈ L 1 (Ω) then we write -div(D∇u) + div(Vu) = 1 h (f -u) ∈ (H 1 (Ω)) ′ ∩ L 1 (Ω),
to conclude that u ∈ D(A) in the sense of our definition of operator A, so that hA(u) + u = f and (H 1 (Ω)) ′ ∩ L 1 (Ω) ⊂ R(I + hA) (the other inclusion is trivial). We also derive from this statement that R(I + hA) is dense in L 1 (Ω).

We now examine the definition of D(A) and see that in particular u ∈ H 1 (Ω) ⊂ (H 1 (Ω)) ′ ∩ L 1 (Ω). We conclude that D(A) ⊂ R(I + hA).

(iii) Finally, we prove that D(A 1 ) is dense in L 1 (Ω). Let u ∈ H 1 (Ω) and define f ∈ (H 1 (Ω)) ′ by: for all ϕ ∈ H 1 (Ω),

f, ϕ (H 1 ) ′ ,H 1 = Ω D∇u • ∇ϕ - Ω uV • ∇ϕ + Ω uϕ,
i.e. f = L 1 u with the notation of Section 2. We can approximate f in (H 1 (Ω)) ′ by functions

f n ∈ (H 1 (Ω)) ′ ∩ L 1 (Ω) (simply approximate D∇u, uV and u in L 2 (Ω) by some functions in C ∞ c (Ω)); by Proposition 5.1, u n = (L 1 ) -1 f n converges to u = (L 1 ) -1 f in H 1 (Ω); moreover, L 1 u n = f n implies in particular Lu n = f n -u n ∈ (H 1 (Ω)) ′ ∩ L 1 (Ω)
, and thus u n ∈ D(A 1 ). Hence, D(A 1 ) is dense in H 1 (Ω) and, since H 1 (Ω) is dense in L 1 (Ω) and the topology of L 1 (Ω) is weaker than the topology of H 1 (Ω), this concludes the proof.

The L 2 setting

In Hilbert spaces the concept of accretivity coincides with the better known concept of monotonicity and the theory has better properties. We take as functional space X = L 2 (Ω) and define the operator A 2 in L 2 (Ω) by further restriction of the domain and range so that A 2 goes from D(A 2 ) ⊂ L 2 (Ω) into L 2 (Ω). In that case the corresponding estimate amounts to multiply the equation by u instead of p(u). We easily get

h A 2 u, u + u 2 2 ≤ f 2 u 2 Now, A 2 u, u = Ω D(x)∇u • ∇u dx - Ω uV • ∇u dx,
and the last term equals

- 1 2 Ω V • ∇(u 2 ) dx = 1 2 Ω (∇ • V) u 2 dx - 1 2 ∂Ω (V • n)u 2 dS.
Therefore, the two conditions : (i) ∇ • V ≥ 0 in Ω, and (ii) V • n ≤ 0 on ∂Ω, imply that the operator A 2 is accretive (i. e., monotone) in L 2 (Ω). The theory can still be done if the first condition is replaced by ∇ • V ≥ -2ω, with ω constant, then A 2 u, u ≥ -ω u 2 , and the operator is only ω-monotone instead of monotone. All this is classical theory that is only mentioned as a reminder.

It is quite easy in that case to prove that R(I + hA 2 ) = L 2 (Ω) ⊂ (H 1 (Ω)) ′ , and also that D(A) is contained in R(I + hA 2 ) and is dense in L 2 (Ω).

Parabolic equation. Semigroup

We now want to solve the evolution problem consisting of the equation

∂ t u = -div(D∇u) + div(Vu) for (x, t) ∈ Q = Ω × (0, ∞),
supplied with boundary conditions

D∇u • n -u (V • n) = 0 on ∂Ω × [0, ∞),
Proof. Let γ > 0 and x ∈ Ω such that b n (x) = 0; define the matrix M an(x),bn(x) by: M an(x),bn(x) b n (x) = a n (x) and M an(x),bn(x) = γId on the orthogonal space of Rb n (x), i.e.

M an(x),bn(x) ξ = ξ • b n (x) |b n (x)| a n (x) |b n (x)| + γ ξ -ξ • b n (x) |b n (x)| b n (x) |b n (x)| .
Let ξ ∈ R N ; decomposing ξ on Rb n (x) and (Rb n (x)) ⊥ as ξ = rb n (x) + y, we have, using Young's inequality,

M an(x),bn(x) ξ • ξ = (ra n (x) + γy) • (rb n (x) + y) = r 2 a n (x) • b n (x) + ra n (x) • y + γ|y| 2 ≥ αr 2 |b n (x)| 2 -r|b n (x)| Λ|y| + γ|y| 2 ≥ α 2 r 2 |b n (x)| 2 + γ - Λ 2 2α |y| 2 . Fixing γ = Λ 2 2α + α 2 , we obtain M an(x),bn(x) ξ • ξ ≥ α 2 r 2 |b n (x)| 2 + |y| 2 = α 2 |ξ| 2 . (8.1) Moreover, |M an(x),bn(x) ξ| ≤ |ξ| |an(x)| |bn(x)| + γ , that is to say ||M an(x),bn(x) || ≤ Λ + γ = Λ + Λ 2 2α + α 2 . (8.2)
It is also clear that, if b n (x) → b(x) = 0 and a n (x) → a(x), then M an(x),bn(x) → M a(x),b(x) . Hence, if b n (x) is nowhere null and nowhere converges to 0, the choice D n = M an(x),bn(x) concludes the proof.

In order to take into account the possibility that b n (x) → 0, we take h l : [0, +∞[→ [0, 1] a continuous function which is null on [0, 1/(2l)] and equal to 1 on [1/l, +∞[ and we define Since R l,n and R l are uniformly bounded, these convergences also hold in L 1 (Ω) N ×N . We can thus take ϕ : N → N increasing such that, for all l ≥ 1, ||R l,ϕ(l) -R l || L 1 ≤ 1/l and we have ||R l,ϕ(l) -R|| L 1 ≤ 1 l + ||R l -R|| L 1 → 0 as l → +∞. Hence, there is a subsequence D k = R l k ,ϕ(l k ) which converges a.e. on Ω and the proof is concluded by denoting n k = ϕ(l k ) and ε k = 1/l k .

Maximum principle for other boundary conditions

The maximum principle states that if the right-hand side of the equation is nonnegative, then so is the solution. Of course, it is irrelevant to pure Neumann boundary conditions, because the necessary condition f ∈ H ′ ♯1 prevents any admissible non-trivial right-hand side to be nonnegative. But the question of this principle is relevant for other boundary conditions and, although the existence and uniqueness of solutions to non-coercive problems is known since [START_REF] Droniou | Non-coercive Linear Elliptic Problems[END_REF], there does not seem to be any proof of this maximum principle, at least not in the case of irregular data without assumption on div(V ) (for regular data, this principle is known for (1.3) and can therefore be deduced by duality for (1.1) and, for bounded convection V such that div(V ) ≥ 0, it is proved in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). It could be deduced for irregular data from the case of regular data by approximation; however, slightly modifying the test function used in the proof of Lemma 2.1, it is in fact possible to give a direct and very simple proof of this principle for non-coercive elliptic equations with Dirichlet, Fourier or mixed boundary conditions with irregular data, as shown in the following proposition (stated in the case of pure Dirichlet boundary conditions for convenience). Passing to the limit ε → 0, this gives meas({u < 0}) = 0 and concludes the proof.

Lemma 4 . 1

 41 Assume that (H1) and (H2) hold. Let p

Remark 4 . 6 A

 46 result similar to Theorem 4.2 holds for the solution to (1.4) in H ⋆ . It can be deduced from Theorem 4.2 using the following simple reasoning: if g ∈ H ′ ♯ u , and v is the solution to

R

  l,n (x) = h l (|b n (x)|)M an (x),bn(x) + (1 -h l (|b n (x)|)) α 2 Id (with the obvious choice R l,n (x) = α 2 Id if b n (x) = 0). Since, at each point, R l,n (x) is either α 2 Id or a convex combinaison between this function and M an(x),bn(x) , we deduce from (8.1) and (8.2) that R l,n is uniformly elliptic and bounded independently of l and n. Moreover, if|b n (x)| ≥ 1/l then R l,n (x)b n (x) = M an(x),bn(x) b n (x) = a n (x). Denoting a and b the respective a.e. limits of a n and b n , we see that, for a.e. x,R l,n (x) → R l (x) = h l (|b(x)|)M a(x),b(x) + (1 -h l (|b(x)|)) α2Id as n → ∞ (whatever the definition of M a(x),b(x) if b(x) = 0, since h l (0) = 0). It is also clear that, for all x, R l (x) → R(x) = 1 {|b(x)| =0} M a(x),b(x) + 1 {|b(x)|=0} α 2 Id as l → ∞.

Proposition 8 . 3 . 3 )

 833 Assume that (H1)-(H3) hold, take f ∈ H -1 (Ω) and let u ∈ H 1 0 (Ω) be the weak solution to-div(D∇u) + div(Vu) = f in Ω , u = 0 on ∂Ω. (8If f ≥ 0 (in the sense that f, ϕ H -1 ,H 1 0 ≥ 0 for any ϕ ≥ 0), then u ≥ 0 a.e. on Ω. Proof. Let ε > 0 and φ ε (r) = ϕ ε (r + ε) where ϕ ε is given in Figure 1, i.e. φ ε (r) = 0 if r ≥ 0, φ ε (r) = r if -ε < r < 0 and φ ε (r) = -ε if r ≤ -ε.Then we can use φ ε (u) ∈ H 1 0 (Ω) as a test function in the weak formulation satisfied by u and, since φ ε (u) ≤ 0 on Ω,Ω D∇u • ∇(φ ε (u)) dx -Ω uV • ∇(φ ε (u)) dx = f, φ ε (u) H -1 ,H 1 0 ≤ 0.But ∇(φ ε (u)) = 1 {-ε<u<0} ∇u and we deduce||∇(φ ε (u))|| 2 L 2 (Ω) N ≤ 1 α ε||V|| L 2 ({-ε<u<0}) N ||∇(φ ε (u))|| L 2 (Ω) N = εω(ε)||∇(φ ε (u))|| L 2 (Ω) Nwith ω(ε) → 0 as ε → 0. Using Poincaré's inequality in H 1 0 (Ω), and since|φ ε (u)| ≥ ε 2 on {u ≤ -ε/2}, we obtain (meas({u ≤ -ε/2})) 1/2 ≤ 2 ε ||φ ε (u)|| L 2 (Ω) ≤ 2 ε diam(Ω)||∇(φ ε (u))|| L 2 (Ω) N ≤ 2diam(Ω)ω(ε).

Here and in the following, the orthogonal sets are always taken with respect to the scalar product in L

(Ω).

The same holds for L * , by the properties of the spectrums of compact operators and their adjoint.

It is not clear that the other techniques used in the literature to prove a priori estimates in the case of Dirichlet boundary conditions can be adapted to Neumann boundary conditions.

Acknowledgment. Second author funded by Project MTM2005-08760-C02 (Spain) and European Science Foundation Programme "Global And Geometric Aspects of Nonlinear Partial Differential Equations". Work initiated during a visit of the first author to the UAM, partially funded by the aforementioned project.

Pluging ϕ = Γ n in (6.9) and ϕ = ψ n in (6.8) we deduce, since Γ n has a null mean value,

(6.12) Assumption (6.5) implies 1 {|∇Γn|<εn} (a(x, w n , ∇w n ) -a(x, w n , ∇ w n ) -D n ∇Γ n ) ≤ Cε n with C not depending on n and (6.10) therefore shows that the last term of (6.12) tends to 0 as n → ∞. Moreover, by (6.11), (ψ n ) n≥1 is relatively compact in C(Ω) and, since µ n -µ n → 0 weakly- * in the space of bounded measures on Ω, we deduce that

, we finally obtain, by letting n → +∞ in (6.12), Ω h(w -w) dx = 0. This equality being true for all h ∈ L ∞ (Ω), we infer that w = w and the proof is concluded.

Accretivity and monotonicity. Parabolic equation

We examine here a different perspective for the solution of equation (1.1) with the given boundary conditions. This consists in investigating the properties of the solution map in the L pspaces. In view of the application of these properties to the solution of parabolic problems and the generation of continuous semigroups using the Hille-Yosida-Phillips theorem, a fundamental assumption is accretivity. This concept can be defined as follows: we consider a Banach space X and a (possibly nonlinear) operator A defined in a subset D(A) ⊂ X with values in X. The operator is then called accretive if for every constant h > 0 and every u 1 , u 2 ∈ D(A) we have

In other words, the map I + hA : D(A) → X is one-to-one and its inverse is a contraction from the range R(I + hA) into D(A), if measured with the X-norm. This inverse is usually denoted as J h (A) and is called the h-resolvent. When the operator is linear we can replace the differences by simply u X ≤ (I + hA)u X for every u ∈ D(A) and h > 0. Actually, a stronger variant of this concept appears in the practice when X is a space of real-valued functions, and is called T -accretivity. It consists in replacing • X by (•) + X in the above formulas. Here, (•) + denotes positive part (taken pointwise).

Another important ingredient in the construction of solutions of evolution problems with accretive operators is the so-called range condition. In its stronger version we require R(I + hA) = X for all h > 0 and the operator is called m-accretive. But, in fact, the generation of a semigroup happens under the milder condition D(A) ⊂ R(I + hA) for all h > 0.

The L 1 setting and initial data u(x, 0) = u 0 (x) for x ∈ Ω.

We proceed as follows: using the Hille-Yosida Theorem, [START_REF] Brezis | Analyse fonctionnelle: Théorie et applications[END_REF] for linear operators, or more generally, the Crandall-Liggett theorem [START_REF] Crandall | Generation of semi-groups of nonlinear transformations on general Banach spaces[END_REF] in the nonlinear case, for every accretive operator A (linear or nonlinear) acting in a Banach space X and satisfying the range condition D(A) ⊂ R(I + hA) for all small h > 0, we can construct a continuous semigroup S t that associates to every initial datum u 0 ∈ D(A) a trajectory u(t) ∈ C([0, ∞); X) that solves the abstract problem

in the mild sense. Moreover, the accretivity property implies contractivity of the semigroup: for every two solutions u 1 , u 2 we have

In the case of T -accretivity we get the stronger property

Finally, in the case of ω-monotonicity we get

8 Appendix

Technical results

The proof of the following lemma can be made in a very classical manner, by way of contradiction; we leave it as an exercise to the reader.

Lemma 8.1 Assume that (H1) holds and let E a subset of Ω with positive Lebesgue measure. Then there exists a constant C E > 0 such that, for all v ∈ H 1 (Ω) satisfying v = 0 a.e. on E,

The next lemma is used in the proof of Theorem 6.1.


Keywords: Signal analysis, Image processing, Semi-classical analysis, Schrödinger Operator. MOTS-CLÉS : Analyse des signaux, Traitement d'images, Analyse semi-classique, Opérateur de Schrödinger

. The main idea in this method is to interpret a signal as a potential of a Schrödinger operator and then to use the discrete spectrum of this operator to analyze the signal. In this paper, we are interested in extending this method from one dimension to two dimensions for image processing applications. The efficiency of this approach is shown through some numerical examples. Moreover, the influence of the design parameters in this method is numerically studied.

RÉSUMÉ.

. L'idée principale de cette méthode consiste à interpréter le signal comme un potentiel de l'opérateur de Schrödinger et à l'analyser en utilisant le spectre discret de cet opérateur. Notre intérêt porte particulièrement sur l'extention de cette méthode au cas bidimensionnel pour des applications en traitement d'images. Nous montrons l'efficacité de la méthode proposée à travers quelques exemples numériques . Par ailleurs, l'influence de certains paramètres, dont dépend cette méthode, est numériquement étudiée.

Introduction

Recently, a new signal analysis method based on a semi-classical approach has been proposed (see [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF], [START_REF] Laleg-Kirati | Semi-classical signal analysis[END_REF]). We refer to this method as SCSA for Semi-Classical Signal Analysis. The main idea of this method is to interpret a signal as a potential of a Schrödinger operator depending on a semi-classical parameter [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. Then, the signal can be characterized using the discrete spectrum of the Schödinger operator. Promising results have been obtained when applying the SCSA method to the analysis of arterial blood pressure signals [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF], [START_REF] Laleg-Kirati | Validation of a semi-classical Signal analysis method for Stroke volume variation assessment: a comparison with the PiCCO technique[END_REF] and to the analysis of the proformance of turbomachinery. Moreover, it has been shown that the SCSA method can cope with noisy signals, making this method a potential tool for denoising. The aim of this paper is to extend the SCSA method from one dimensional case to two dimensional case.

Preliminary

In this section, we propose to recall the idea behind of the SCSA method [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF], [START_REF] Laleg-Kirati | Semi-classical signal analysis[END_REF]. Let us consider the following Schrödinger operator:

H 1,h (V 1 ) := -h 2 d 2 dx 2 -V 1 , (1)
where h ∈ R * + is the semi-classical parameter [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF], and V 1 is a positive real valued function belonging to C ∞ (Ω 1 ) where Ω 1 ⊂ R is a compact. Then, the potential V 1 can be estimated using the following proposition.

Proposition 2.1 [2] Let V 1 ∈ C ∞ (Ω 1 ) be positive real valued function, where Ω 1 ⊂ R is a compact. Then, V 1 can be estimated using the following formula: ∀ x ∈ Ω 1 , V 1,h,γ,λ (x) := -λ +   h L cl 1,γ N λ h ∑ n=1 (λ -λ n,h ) γ ψ 2 n,h (x)   2 2γ+1 , ( 2
)
where h ∈ R * + , γ ∈ R + , λ ∈ R * -, and L cl 1,γ is the universal semi-classical constant given by: L cl 1,γ = 1 2 √ π Γ(γ+1) Γ(γ+ 3
2 ) , where Γ is the gamma function. Moreover, λ n,h are the negative eigenvalues of

H 1,h (V 1 ) with λ 1,h < • • • < λ N λ h ,h < λ, N λ
h is the number of negative eigenvalues smaller than λ, and ψ n,h are the associated L 2 -normalized eigenfunctions such that:

H 1,h (V 1 ) ψ n,h = λ n,h ψ n,h .
The efficiency of the proposed signal estimation method and the influence of the design parameters λ, γ and h have been studied in [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF]. In particular, as it is described in [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF] and [START_REF] Laleg-Kirati | Semi-classical signal analysis[END_REF], the semi-classical parameter h plays a key role in this approach. In fact, when h decreases, the estimation V h,γ,λ improves. Since the study of the Schrödinger operator in the case where h tends to 0 is referred to the semi-classical analysis [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF], this justifies the name Semi-Classical Signal Analysis that we give to this approach [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF], [START_REF] Laleg-Kirati | Semi-classical signal analysis[END_REF].

Let us point out that the formula given in ( 2) is still applicable in the case where λ = 0, and it gives good results even if λ is out of the range of definition. It has been used for example in the analysis of the arterial blood pressure signal in [START_REF] Laleg-Kirati | Validation of a semi-classical Signal analysis method for Stroke volume variation assessment: a comparison with the PiCCO technique[END_REF].

Generalization of the SCSA method to two dimensions

In this section, we propose to generalize formula (2) from one dimension to two dimensions. From now on, we consider the following 2D-Schrödinger operator associated to a potential V 2 :

H 2,h (V 2 ) := -h 2 ∆ -V 2 , ( 3
)
where

∆ := d 2 dx 2 + d 2 dy 2 is the 2D-Laplacien operator, h ∈ R * + is the semi-classical parameter [1], and V 2 is a positive real valued function belonging to C ∞ (Ω 2 ) where Ω 2 ⊂ R 2 is a compact.
Let us recall that, as described in the previous section, the main idea behind the SCSA method in one dimension is to use the negative eigenvalues and the associated eigenfunctions of the operator H 1,h (V 1 ) (equation ( 1)) to estimate the potential V 1 . In order to estimate the potential V 2 by using a similar approach, we focus on the following spectral problem:

H 2,h (V 2 )ψ k,h = λ x,y k,h ψ k,h , (4)
where λ x,y k,h and ψ k,h , for k = 1, • • • , K λ h , refer to the negative eigenvalues and the associated L 2 -normalized eigenfunctions respectively.

We propose to use the separation of variables to calculate the eigenfunctions ψ k,h . Hence, by taking

ψ k,h (x, y) = φ n,h (x)ϕ m,h (y) in (4), for n = 1, • • • , N λ h and m = 1, • • • , M λ h with K λ h = N λ h M λ h , we obtain: ∀ (x, y) ∈ Ω 2 , -h 2 ∆ {φ n,h (x)ϕ m,h (y)} -V 2 (x, y)φ n,h (x)ϕ m,h (y) = λ x,y k,h φ n,h (x)ϕ m,h (y). (5)
Then, it implies that:

∀ (x, y) ∈ Ω 2 , λ x,y k,h φ n,h (x)ϕ m,h (y) = ( -h 2 d 2 dx 2 φ n,h (x) - 1 2 V 2 (x, y)φ n,h (x) ) ϕ m,h (y) + ( -h 2 d 2 dy 2 ϕ m,h (y) - 1 2 V 2 (x, y)ϕ m,h (y) ) φ n,h (x). (6)
Let us consider the following eigenvalues problem:

∀ (x, y) ∈ Ω 2 , H x 1,h ( 1 2 V 2 (•, y) ) φ n,h (x) = λ x n,h φ n,h (x), (7)
H y 1,h ( 1 2 V 2 (x, •) ) ϕ m,h (y) = λ y m,h ϕ m,h (y), (8)
where λ x n,h (resp. λ y m,h ) are the negative eigenvalues of the operator

H x 1,h ( 1 2 V 2 (•, y)) (resp. H y 1,h ( 1 2 V 2 (x, •))), n = 1, • • • , N λ h , (resp. m = 1, • • • , M λ h )
, and φ n,h (resp. ϕ m,h ) are the associated L 2 -normalized eigenfunctions. Then, by using ( 7) and ( 8) in (6), we get:

(λ x n,h + λ y m,h ) φ n,h (x)ϕ m,h (y) = λ x,y k,h φ n,h (x)ϕ m,h (y), ∀ (x, y) ∈ Ω 2 .
Consequently, we obtain the following relation

λ x,y k,h = λ x n,h + λ y m,h , (9)
for n = 1, • • • , N λ h , m = 1, • • • , M λ h and k = 1, • • • , N λ h M λ h . Finally, let us recall that the coefficient h L cl 1,γ
given in Proposition 2.1 is related to some Riesz means connected to a Lieb-Thirring's conjecture (see [START_REF] Helffer | Riesz means of bound states and semiclassical limit connected with a Lieb-Thirring's conjecture I[END_REF]). Then we refer to the n dimensional formulation of the Riesz means to find the coefficient of the sum in two dimensions. In particular, we show that this coefficient is given by

h 2 L cl 2,γ
, where the suitable universal semi-classical constant L cl 2,γ is given by

L cl 2,γ = 1 2 2 π Γ(γ+1)
Γ(γ+2) . Consequently, inspired from the idea of tensor product and by using a similar way for obtaining Proposition 2.1, we can obtain the following proposition. Proposition 3.1 Let V 2 be a positive real valued function belonging to C ∞ (Ω 2 ) where Ω 2 ⊂ R 2 is a compact. Then, V 2 can be estimated by the following formula:

∀ (x, y) ∈ Ω 2 , V 2,h,γ,λ (x, y) := -λ +   h 2 L cl 2,γ M λ h ∑ m=1 N λ h ∑ n=1 ( λ -(λ x n,h + λ y m,h ) ) γ φ 2 n,h (x)ϕ 2 m,h (y)   2 2γ+1 , (10) where h ∈ R * + , γ ∈ R + , λ ∈ R -, and L cl 2,γ is the suitable universal semi-classical constant. Moreover, λ x n,h (resp. λ y m,h ) are the negative eigenvalues of the operator H x 1,h ( 1 2 V 2 (•, y)) (resp. H y 1,h ( 1 2 V 2 (x, •)) with λ x 1,h < • • • < λ x N λ h ,h < λ (resp. λ y 1,h < • • • < λ y M λ h ,1 < λ), N λ h (resp. M λ h
) is the number of the negative eigenvalues smaller than λ, and ψ n,h (resp. ϕ m,h ) are the associated L 2 -normalized eigenfunctions.

Some numerical examples

In order to validate formula (10), we have performed numerical tests on two examples. Note that, in this section, we will take λ = 0 in formula (10).

Example 1. In this example, we consider the following function:

V 2 (x, y) = sin( 1 2 x 2 + 1 4 y 2 + 3) cos(2x + 1 -e y ) + 1. (11)
Then, we assume that V 2 is given in a discrete case where x i = iT s and y j = jT s for i, j = -50, • • • , 150 with T s = 0.02. We can remark that since the number of negative eigenvalues increases while h decreases, in practice we can not take h very small as described in [START_REF] Helffer | On semi-classical questions related to signal analysis[END_REF] and [START_REF] Laleg-Kirati | Semi-classical signal analysis[END_REF]. So, before estimating V 2 , we study the influence of the design parameters h and γ. By taking different values of h and γ, by estimating the variation of the mean square errors between V 2 and the estimation V 2,h,γ,0 , we found that there is a minimum at h = 6 10 3 , and γ = 3.5. Then, we can estimate V 2 by using V 2,h,γ,0 with these optimal parameter values (see Figure 1(a)). In particular, we show in Figure 1(b) the original signal V 2 (-0.62, y j ) and the estimation V 2,h,γ,0 (-0.62, y j ) with y j = -1, -0.98, • • • , 3 and in Figure1(c) we show the relative error between the function and its estimation.

Conclusion

In this paper, we have generalized a recent semi-classical signal analysis method from one dimensional case to two dimensional case for image analysis. The efficiency of the proposed method has been shown through some numerical examples. Moreover, the influence of design parameters in the method has been numerically studied. Accurate error analysis will be given in a future work.

Figure 1 .

 1 Figure 1. Example 1: V2(x, y) = sin( 1 2 x 2 + 1 4 y 2 +3) cos(2x+1-e y )+1. (a) V 2,h,γ,0 (xi, yj). (b) V2(-0.62, yj), V 2,h,γ,0 (-0.62, yj) with yj = -1, -0.98, • • • , 3. (c) The relative error between the real function and its estimation.

Example 2 .

 2 In this example, we consider a 440 × 440 pixels image, see Figure2(a). One can note the good reconstruction of this image in Figure2(b), for h = 0.3 and γ = 3.5, and the associated relative error in Figure2(c).

Figure 2 .

 2 Figure 2. Example 2: The beacon in KAUST. (a) Original image. (b) Reconstructed image. (c) The relative error between the original and reconstructed images.

This work was conducted when the first author was visiting the Estimation, Modeling and ANalysis Group at the Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) devision at King Abdullah University of Science and Technology (KAUST). She would like to thank KAUST for its support and generous hospitality.

1

TAMTAM -Alger -2013


J Boudeile
email: justine.boudeile@institutoptique.fr

J Didierjean

F Druon

F Balembois

P Georges

J L Doualan

P Camy

A Benayad

V Ménard

R Moncorgé

Simultaneous measurement of thermal lens and temperature map in ytterbium-doped fluoride crystals

Keywords:

We report on the simultaneous characterization of temperature map and thermal lensing in Yb 3+ :CaF 2 and Yb 3+ :SrF 2 crystals under high-power pumping with/without laser operation. This in situ measurement would allow proper designs of high-power cavities.

1-Introduction

Interest in Yb 3+ -doped materials is still growing, owing to the increasing availability of high power InGaAs laser diodes emitting around 980nm. Ytterbium has very attractive properties including a weak quantum defect, and no problem of concentration quenching. Fluoride crystals, having the cubic Calcium fluoride structure, are well-known to be transparent in a large wavelength region (from IR to VUV) and mix the advantages of the oxide crystals and glass. In particular, they feature broad absorption and emission bands. Fluoride ytterbium-doped crystals are therefore very promising for the development of solid-state lasers and amplifiers [START_REF] Siebold | High-peak-power tunable laser operation of Yb:SrF 2[END_REF][START_REF] Petit | CW and tunable operation of Yb 3+ doped CaF 2[END_REF]. Here, we present the experimental assessment of both the temperature map and the thermal lens (TL) properties of Yb 3+ :CaF 2 and Yb 3+ :SrF 2 under high power pumping, with or without laser operation.

2-Experimental set up

The experimental setup is shown in Fig. 1. The 5 mm 3-at % Yb 3+ :CaF 2 or 3-at % Yb 3+ :SrF2 is directly pumped with a 120 W 400 μm core fiber-coupled laser diode (LIMO) emitting at 980 nm. The probe beam, generated by a fibercoupled LED emitting at 660 nm, is collimated and superposed with the pump beam. The crystal is imaged through a magnifying system composed of two lenses to a Shack-Hartmann wave-front sensor (HASO64, Imagine Optics). A bandpass filter at 660 nm is placed in front of the analyzer to remove the parasitic beams. With pump off, a reference wavefront is recorded to cancel all static aberrations from the optical elements and the cold crystal. The reference is subtracted from the measured wavefront when the pump is on. Simultaneously, an 8μm-12μm high resolution thermal camera with a germanium objective is used to measure the temperature map of one facet of the crystal.

3-Experimental results

The TL dioptric power versus absorbed pump power is plotted in Fig. 2 for CaF 2 and SrF 2 crystals with or without laser operation. We notice that all TL dioptric powers (D th ) are negative. Since the thermal lens focal length (f th ) is inversely proportional to D th , the measured thermal lenses are divergent for both crystals. This result can be understood as follows. The TL dioptric power can be expressed as [START_REF] Chenais | on thermal effects in solid-state lasers: The case of ytterbium-doped materials[END_REF]:

c p abs h r T T c p abs h th th w P C n n dT dn w P f D κ π χ η α α ν κ π η 2 3 2 2 2 ) 1 )( 1 ( 2 1 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + - + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = (1)
where η h is the fractional thermal load, P abs the absorbed pump power, w p the pump beam waist radius, κ c the thermal conductivity, n the refractive index, ν the Poisson's ratio, α T the coefficient of thermal expansion, C r the photo-elastic constant, χ the thermo-optic coefficient. The absolute values of the (n-1)(1+ν)α T and 2n 3 α T C r terms are similar and compensate each other. So, the last term of χ determines the sign of D th , all the other terms being positive. Since the values of (dn/dT) for the CaF 2 and SrF 2 are equal to -10.6 10 -6 K -1 and -12 10 -6 K -1 respectively, divergent thermal lenses are expected. The values of TL focal lengths for CaF 2 :Yb 3+ and SrF 2 :Yb 3+ crystals are equal to -119 cm for an absorbed power of 40 W and -113 cm for an absorbed power of 18 W respectively (fig. 2). Note that the CaF 2 :Yb 3+ crystal needs a higher absorbed pump power to measure the same TL focal length. This difference could be explained by the lower values of the thermal conductivity κ c and the coefficient of thermal expansion α T for the SrF 2 :Yb 3+ crystal: the SrF 2 crystal is not as efficient as CaF 2 to dissipate the heat. Therefore, the temperature gradient is higher in SrF 2 and TL focal length increases more quickly with absorbed power.

Using the expression of the TL dioptric power (1), and taking into account the divergence of the pump beam in the expression of w p , we are able to fit the measured TL dioptric power with laser effect versus incident power (fig. 3). The values of the parameters for the fits are summarized in Table 1.

Crystals

CaF 2 :Yb 3+ (3%) SrF 2 :Yb 3+ (3%) κ c (W.m -1 K -1 ) 6.0 3.3 η h 0.07 0.07 Table 1. Parameters used for the fit of the TL dioptric power.

When operating in laser condition, the fractional thermal load η h was set to a value slightly higher than the quantum defect 1-(λ p /λ l ) where λ p is the pump wavelength and λ l the laser wavelength (quantum defects for CaF 2 and SrF 2 are equal to 0.067 and 0.069 respectively). Indeed, the intracavity power is high enough to consider that the laser extraction efficiency is equal to one over the whole lasing crystal volume. Given this assumption, the good agreement with the experimental data indicates that the heat generated in these crystals is mostly due to the quantum defect, and that the contribution of nonradiative effects is negligible. The evolution of the temperature gradient in the crystal (Delta T) versus incident pump power is shown in Fig. 4 with or without laser operation. A weak temperature gradient is observed for both crystals when the laser is off. This result corroborates the weak measured TL dioptric power without laser operation.

4-Conclusion

To our best knowledge, this is the first combined measurement of both thermal lens properties and temperature map in ytterbium-doped CaF 2 and SrF 2 crystals. These in situ measurements indicate different interesting quantitative results: first the presence of a small negative thermal lenses resulting from various competing (thermo-optics and thermomechanical) effects; second, strong differences of behaviour between lasing and not lasing conditions which is important since it may imply high thermal shocks; third, that the thermal loads are mainly due to the quantum defect implying very few other parasitic non-radiative processes.

5-References

Fig. 1 .

 1 Fig. 1. (a) Experimental setup and (b) zoom of the laser cavity design. PM = plane mirror and IR = interferential filter.

Fig 2 .

 2 Fig 2. TL dioptric power versus absorbed pump power for CaF 2 :Yb 3+ and SrF 2 :Yb 3+ crystals with and without laser operation.

Fig. 3 .

 3 Fig. 3. The TL dioptric power versus incident power fitted for (a) CaF 2 :Yb 3+ and (b) SrF 2 :Yb 3+ crystals with laser effect or not.

Fig. 4 .

 4 Fig. 4. Temperature gradient versus incident pump power for (a) CaF 2 :Yb 3+ and (b) SrF 2 :Yb 3+ crystals with or without laser operation