Dominique Bakry

Olfa Zribi

Curvature dimension bounds on the deltoid model

The deltoid curve in R 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvectors of diffusion operators. As such, they may be considered as a two dimensional extension of the classical Jacobi operators. They belong to one of the 11 families of such bounded domains in R 2 . We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet forms

Curvature dimension inequalities

, in a specific context adapted to our setting. For a given probability space (X, X , µ), we suppose given an algebra A of functions such that A ⊂ ∩ 1≤p<∞ L p (µ),

Introduction

The deltoid curve (also called Steiner's hypocycloid), see figure page 7, is one of the bounded domains in R 2 on which there exist a probability measure µ and a symmetric diffusion process, the eigenvectors of it being orthogonal polynomials for µ. These orthogonal polynomials have been introduced in [START_REF] Koornwinder | Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. i[END_REF][START_REF]Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators[END_REF] and appear in the classification of [START_REF] Krall | Orthogonal polynomials in two variables[END_REF]. They appear in [START_REF] Bakry | Orthogonal polynomials and diffusion operators[END_REF] as one of the eleven models in dimension two for which such polynomials exist. Moreover, it seems that is one of the most difficult models to analyse, since there does not exist many geometric interpretation for it. Beyond this, it is also interesting since it is deeply linked with the analysis of the A 2 root system and of the spectral analysis of SU (3) matrices.

This deltoid model and the associated generators provide an interesting object to check various properties of diffusion operators, since one knows explicitly the eigenvalues, and has many informations on the eigenvectors. For example, they have satisfy recurrence formulas which allows for explicit computations, and in some cases generating functions, see [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF].

Since the associated measures and operators depend on a real parameter λ > 0 (see equation (3.11)), one may try to understand how functional inequalities and curvature properties depend on this parameter λ, and hence on geometric properties of the model. It turns out that for the specific cases λ = 1 and λ = 4, one may produce simple geometric interpretations : in the first case from the Euclidean Laplace operator through the symmetries of the triangular lattice, in the second case from the Casimir operator on SU (3) acting on spectral measures. The SU (3) model provides curvature-dimension inequalities for the generic model for λ ≥ 1. It is not clear however that these inequalities are optimal. It turns out that they indeed are. Quite unexpectedly, the careful investigation of the CD(ρ, n) inequality for this model does not produce better results than the direct consequence of the SU (3) inequality. In comparison with the classical case of Jacobi polynomials, which are orthogonal with respect to the measure C a,b (1 -x) a (1 + x) b dx on (-1, 1), this situation is similar to the symmetric case a = b, but differs from the dissymmetric case (see [START_REF] Bakry | Remarques sur les semi-groupes de Jacobi[END_REF]).

It seems worth to point out at least two interesting features of the computations of curvature-dimension inequalities for this model. The first aspect concerns the existence of an optimal dimension in the inequality. When one looks for curvaturedimension inequalities on a compact Riemannian manifold with dimension n 0 , but for a reversible measure which is not the Riemann measure, there is no optimal one. For any n > n 0 , one may find some constant ρ(n) such that a CD(ρ(n), n) inequality holds. In general, ρ(n) goes to -∞ when n → n 0 . It is only for Laplace operators that one may expect some CD(ρ, n 0 ) inequality. This is not the case here. For any λ > 1, there is a bound n(λ) = 2λ such that no CD(ρ, n) inequality may hold for n < n(λ). However, for this limiting value n = 2λ, the CD( 34 (λ -1), 2λ) holds. Of course, this phenomenon is due to the singularity of the density of the measure at the boundaries of it's support.

The second aspect concerns the use of appropriate coordinate systems. Since the underlying metric is a flat metric in two dimensions, the curvature-dimension inequality amounts to check for which values a and b one has an inequality of the form -∇∇W ≥ aId + b∇W ⊗ ∇W,

where W is the logarithm of the density measure with respect to the Riemann measure. It turns out that a proper choice of the coordinates leads to very simple formulas, which is not the case if the computation is made through the use of the naive usual coordinates in the Euclidean plane. This comes from the fact that we have indeed at disposal a polynomial structure, expressed through the choice of these coordinates, and the function W satisfies nice relations with respect to this, namely a "boundary equation" described in (4.31). This is a good indication that if one wants to study such inequalities for higher dimensional models, these "polynomial coordinates" should be used instead of the usual ones. Then, the curvature-dimension inequalities provide through Sobolev inequalities (2.6) various uniform bounds on the orthogonal polynomials, which turn out, up to some change in the parameters, to be equivalent to the Sobolev inequality itself. They also provide bounds on kernels constructed from other spectral decompositions, that is for operators which do not necessary commute with our starting operator.

Orthogonal polynomials on the interior of the deltoid curve belong to the large family of Heckman-Opdam polynomials associated with root systems [START_REF] Heckman | Root systems and hypergeometric functions[END_REF][START_REF] Heckman | Root systems and hypergeometric functions. I[END_REF], and in the even larger class of MacDonald's polynomials [START_REF]Orthogonal polynomials associated with root systems[END_REF][START_REF] Macdonald | Symmetric functions and orthogonal polynomials[END_REF][START_REF]Affine Hecke algebras and orthogonal polynomials[END_REF]. As such, they may serve as a guide for more general models of diffusions associated with orthogonal polynomials. One may find some extensive presentation of these pluri-dimensional families of orthogonal polynomials, for example in [START_REF] Dunkl | Orthogonal polynomials of several variables[END_REF][START_REF]Affine Hecke algebras and orthogonal polynomials[END_REF].

The associated diffusion generator is associated with some reflection group in dimension 2. Operators associated with reflection groups in R d , known as Dunkl operators, are extensively studied in the literature, from the points of view of special functions related to Lie group analysis or Hecke algebras, as well as from the point of view of the associated heat equations, or in probability and statistics [START_REF] Dunkl | Differential-difference operators associated to reflection groups[END_REF]15,[START_REF] Heckmann | A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups[END_REF][START_REF]Dunkl operators: theory and applications[END_REF][START_REF] Rösler | Generalized hermite polynomials and the heat equation for dunkl operators[END_REF].

Most of the language and notations related to diffusion operators, and in particular the links between curvature-dimension inequalities and Sobolev inequalities, together with the bounds one may deduce for eigenvectors, are borrowed from [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]. The fact that Sobolev inequalities are equivalent to bounds on the heat kernel goes back to [START_REF] Davies | Heat kernels and spectral theory[END_REF][START_REF] Th | Analysis and geometry on groups[END_REF], and the relations between curvature dimension inequalities go back to [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF] and are exposed in [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF], among others. It is classical that one may deduce from them bounds on the spectral projectors. However, the fact that these bounds may in turn provide Sobolev inequalities, which is the content of Theorem 5.4, seems quite new, at least to our knowledge, although a similar result concerning logarithmic Sobolev inequalities is exposed in [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]. Notice that this recovers a Sobolev inequality with a weaker exponent, not to talk about the optimal constants, which are always out of reach with this kind of techniques.

Many of the properties concerning the spectral decomposition of the operator on the deltoid, recurrence formulas for the associated orthogonal polynomials, generating functions, etc., may be found in [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF]. We shall not use the results of this paper here, apart the representations coming from symmetry groups in R 2 and from SU (3), that we recall briefly in Section 3.

The paper is organized as follows. In Section 2, we present the general curvaturedimension inequalities and the associated Sobolev inequalities. We show how this provides bounds on the eigenvectors. Section 3 is a short introduction to the model associated with the deltoid curve, where we explain the two geometric specific cases. Section 4 provides the associated curvature-dimension inequalities, first from the SU (3)-model, then from direct computation. We provide two approaches for this general case, first in subsection 4.2 using the naive system of coordinates, and then in subsection 4.3 with the adapted system of coordinates. The choice to present these two approaches, and the striking difference in the complexity of the computation, aims at underlining the efficiency of the good "polynomial coordinates". Finally, in Section 5, we give the various bounds on polynomials and operators we are looking for.

A is dense in L 2 (µ), and which is stable under composition with smooth functions Φ. In our case, A may be chosen as the class of restrictions to the domain of smooth functions defined in a neighborhood of it, without any boundary condition. This particular choice for A is made possible thanks to a special property of the operator, which satisfies a "boundary equation", which is our model has the specific form (4.31). It is valid as soon one deals with operators having polynomial eigenvectors on a bounded domain, see [START_REF] Bakry | Orthogonal polynomials and diffusions operators[END_REF]. In most of the cases, we may as well restrict our attention to polynomials, although this would not be appropriate for Sobolev inequalities. A bilinear application Γ :

A × A → A is given such that, ∀f ∈ A, Γ(f, f ) ≥ 0, which satisfies Γ(Φ(f 1 , • • • , f k ), g) = i ∂ i ΦΓ(f i , g) for any smooth function Φ. A linear operator L is defined through (2.1) X f L(g)dµ = - X Γ(f, g)dµ
and we assume that L maps A into A. In this context, all the properties of the model are described by Γ and µ, and the model is then entirely described by the triple (X, Γ, µ) (see [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]). We then extend L into a self adjoint operator and we suppose that A is dense in the domain of L.

Then, for f = (f 1 , • • • , f k ) and for any smooth function Φ

(2.2) L(Φ(f )) = k 1 ∂ i Φ(f )L(f i ) + k i,j=1 ∂ 2 ij Φ(f )Γ(f i , f j ).
We have

Γ(f, g) = 1 2 L(f g) -f L(g) -gL(f ) .
We moreover define the Γ 2 operator as

(2.3) Γ 2 (f, g) = 1 2 LΓ(f, g) -Γ(f, Lg) -Γ(g, Lf ) .
Then, for any parameters ρ ∈ R and n ∈ [1, ∞], we say that L satisfies a curvature-dimension inequality CD(ρ, n) if and only if

∀f ∈ A, Γ 2 (f, f ) ≥ ρΓ(f, f ) + 1 n (Lf ) 2 .
It is worth to observe that the CD(ρ, n) inequality is local. For a general elliptic operator on a smooth manifold M with dimension n 0 , one may always decompose L into ∆ g + ∇ log V , where ∆ g is the Laplace operator associated with the co-metric (g) and V is the density of µ with respect to the Riemann measure. In which case, the operator Γ 2 may be decomposed as

(2.4) Γ 2 (f, f ) = |∇∇f | 2 + (Ric g -∇∇ log V )(∇f, ∇f ),
where Ric g denotes the Ricci curvature computed for the Riemannian metric associated with g, ∇∇ log V is the Hessian of log V , also computed in this metric, and |∇∇f | 2 is the Hilbert-Schmidt norm of the Hessian of f .

In this case, the CD(ρ, n) inequality holds if and only if n ≥ n 0 and, when V is not constant, when n > n 0 and (2.5)

Ric g -∇∇ log V ≥ ρg + 1 n -n 0 ∇ log V ⊗ ∇ log V.
Of course, when L = ∆ g , this amounts to n ≥ n 0 and Ric g ≥ ρg. In this case, there exists a best choice for both ρ and n, namely for n the dimension of the manifold and for ρ a lower bound on the Ricci tensor, that is the infimum over M of the lowest eigenvalue of this tensor.

In this paper, we are mainly mainly interested the case where L = ∆ + ∇ log(V ), where ∆ is the Euclidean Laplace operator in some open set of R n 0 . In which case the measure µ is V dx, and the CD(ρ, n) inequality holds if and only if n ≥ n 0 and

-∇∇ log(V ) ≥ ρ + 1 n -n 0 ∇ log(V ) ⊗ ∇ log(V ).
In order for it to be satisfied, we may look at local inequalities CD(ρ(x), n(x)) and try to find such a pair (ρ(x), n(x)) for which ρ(x) is bounded below and n(x) bounded above. For a generic function V , there is no "best" local inequality in general. The CD(ρ, n) inequality requires that the symmetric tensor -∇∇ log V is bounded below by ρId. If ρ 0 is the best real number such that -∇∇ log V ≥ ρ 0 Id (that is ρ 0 is the lowest eigenvalue at the point x of -∇∇ log V ), then the inequality holds as soon as ρ ≥ ρ 0 and (ρ -

ρ 0 )(n -n 0 ) ≥ |∇ log V | 2 .
But in our case, as already mentioned in the introduction, we are not in this situation. There is a lower bound on the admissible dimension, which is strictly bigger than n 0 . To understand this phenomenon, one may analyse a bit further this CD(ρ, n) inequality at a given point on the manifold.

It may happen that, at some point x, the eigenvector of -∇∇ log V corresponding to some eigenvalue ρ 1 (x) > ρ 0 (x) is parallel to ∇ log V . Let then in this case, there is a best choice for both ρ(x) and n(x), which is

ρ(x) = ρ 0 (x), n(x) = n 0 + 1 ρ 1 (x)-ρ 0 (x) |∇ log V | 2 .
In the model that we shall consider later, we shall see that this happens asymptotically on the boundary of the set we are working on, and the constants n and ρ computed at this boundary points are valid for all other points x.

When some CD(ρ, n) inequality holds, with ρ and n constant, and whenever ρ > 0, and 2 < n < ∞, then (X, Γ, µ) satisfies a tight Sobolev inequality. For p = 2n n-2 , and for any f ∈ A, we have (2.6)

X f p dµ 2/p ≤ X f 2 dµ + 4 n(n -2) n -1 ρ X Γ(f, f )dµ.
More generally, an n-dimensional Sobolev inequality is an inequality of the form

(2.7) f 2 2n/(n-2) ≤ A f 2 2 + C Γ(f, f ) dµ.
When µ is a probability measure, we say that the inequality is tight when the constant A is 1, and provided some Sobolev inequality holds, tightness is equivalent to the fact that a Poincaré inequality occurs, which is automatic in our case since the spectrum is discrete (see [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]). When a Sobolev inequality (2.7) holds, then the associated semigroup

P t = exp(tL) is ultracontractive, that is, for any q ∈ [2, ∞] (2.8) P t f q ≤ C 1 t n 2 ( 1 2 -1 q ) f 2 , 0 < t ≤ 1,
with

C 1 = Cn 2 + 2A n/2
. This last constant C 1 is not sharp however. The bound is valid for q = ∞ and indeed, the result for any q ∈ (2, ∞) is a consequence of the case q = ∞ through an interpolation argument. It turns out that his last ultra contractive bound (for some given q, but for any t ∈ (0, 1)) is in turn equivalent to the Sobolev inequality.

There is another equivalent version (2.9)

P t f ∞ ≤ Ct -n/2 f 1 , 0 < t ≤ 1,
for which one deduces immediately that the semigroup P t has a density which is bounded above by Ct -n/2 . When 1 ≤ n ≤ 2, one may replace Sobolev inequalities by Nash inequalities, which play the same rôle, see remark 5.6. However, the best constants that one may deduce from curvature-dimension inequalities for Nash inequalities are not known (see [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF]).

As a consequence of ultracontractive bounds, whenever f is an eigenvector for L with eigenvalue -λ, and provided that f 2 dµ = 1, one has (2.10)

f q ≤ C 1 inf t>0 e λt t n 2 ( 1 2 -1 q ) = C 1 C n,q λ n 2 ( 1 2 -1 q ) , with C n,q = inf s>0 e s s -n 2 ( 1 2 -1 q )
, which follows immediately from the fact that P t f = e -λt f . This applies in particular for q = ∞, and produces uniform bounds on the eigenvectors from the knowledge of their L 2 (µ) norms.

Diffusion processes on the interior of the deltoid curve

We describe first the operator associated with the deltoid curve associated with a family of orthogonal polynomials. Most of the details may be found in [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF]. The deltoid curve is a degree 4 algebraic plane curve which may be parametrized as The connected component of the complementary of the curve which contains 0 is a bounded open set, that we refer to as the deltoid domain D. Indeed, in what follows, we shall work on this domain scaled by the factor 1/3, which will produce much more convenient formulas. It turns out that there exist on this domain a one parameter family L (λ) of symmetric diffusion operator which may be diagonalized in a basis of orthogonal polynomials. It was introduced in [START_REF] Koornwinder | Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. i[END_REF][START_REF]Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators[END_REF], and further studied in [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF]. This is one of the 11 families of sets in dimension 2 carrying such diffusion operators, as described in [START_REF] Bakry | Orthogonal polynomials and diffusion operators[END_REF].

x(t) = 2 cos t + cos 2t, y(t) = 2 sin t -sin 2t
In order to describe the operator, and thanks to the diffusion property (2.2), it is enough to describe Γ(x, x), Γ(x, y), Γ(y, y), L (λ) (x) and L (λ) (y) (the Γ operator does not depend on λ here).

The symmetric matrix Γ(x, x) Γ(x, y) Γ(y, x) Γ(y, y) is referred to in what follows as the metric associated with the operator, although properly speaking it is in fact a cometric. It is indeed easier to use the complex structure of R 2 ≃ C, and use the complex variables Z = x + iy, Z = x -iy, with

L(Z) = (x) + iL(y), L( Z) = L(x) -iL(y), and
Γ(Z, Z) = Γ(x, x) -Γ(y, y) + 2iΓ(x, y), Γ(Z, Z) = Γ(x, y) + Γ(y, y).
The formulas are much simpler with these variables, and L (λ) is then described as

(3.11)            Γ(Z, Z) = Z -Z 2 , Γ( Z, Z) = Z -Z2 , Γ( Z, Z) = 1/2(1 -Z Z), L(Z) = -λZ, L( Z) = -λ Z,
where λ > 0 is a real parameter. The boundary of this domain turns out to the curve with equation (3.12)

P (Z, Z) = Γ(Z, Z) 2 -Γ(Z, Z)Γ( Z, Z) = 0,
and inside the domain, the associated metric is positive definite, so that it corresponds to some elliptic operator on it. The reversible measure associated with it has density C α P (Z, Z) α with respect to the Lebsegue measure, where λ = (6α + 5)/2, and is a probability measure exactly when λ > 0 (we refer to [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF] for more details). There are two particular cases which are worth understanding, namely λ = 1 and λ = 4, corresponding to the parameters α = ±1/2. We briefly present those two models, referring to [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF] for more details, since we shall make a strong use of them in what follows.

In the first case, one sees that this operator is nothing else that the image of the Euclidean Laplace operator on R 2 acting on the functions which are invariant under the symmetries around the lines of a regular triangular lattice.

For the first one, one considers the three unit third roots of identity in C, say (e 1 , e 2 , e 3 ) = (1, j, j). Then, consider the functions Z and z k : C → C which are defined as

(3.13) z k (z) = e iℜ(zē k ) , Z = 1 3 (z 1 + z 2 + z 3 ) They satisfy |z k | = 1 and z 1 z 2 z 3 = 1.
It is easily seen that, for the Euclidean Laplace operator on R 2 , Z and Z satisfy the relations (3.11) with λ = 1. Moreover, the function Z : C → C is a diffeomorphism between the interior T of the triangle T and the deltoid domain D, where T is one of the equilateral triangles with containing the two edges 0 and 4π/3. The functions which are invariant under the symmetries of the triangular lattice generated by this triangle T are exactly functions of Z. Therefore, the image of L (1) through Z -1 : D → T is nothing else that the Laplace operator on T . We may as well look at the image of the operator L (λ) and it is then simply

(3.14) L (λ) = ∆(f ) + (α + 1/2)∇ log W ∇f = ∆(f ) + λ -1 3 ∇ log W ∇f
where ∆ is the usual Laplace operator in R 2 and the function W is defined form the functions z j described in equation (3.13) as

(3.15) W = -(z 1 -z 2 ) 2 (z 2 -z 3 ) 2 (z 3 -z 1 ) 2 .
One should be aware here that thanks to the properties of the functions z j , -(z 1 -z 2 ) 2 (z 2 -z 3 ) 2 (z 3 -z 1 ) 2 is a real valued function taking values in (0, ∞) (and vanishes only at the boundaries of T ).

This representation provides a way of computing CD(ρ, n) inequalities for L (λ) , following the description of Section 2.

The second description follows from the Casimir operator on SU (3). This latter group is a semi-simple compact Lie group, and as such as a canonical Laplace (Casimir) operator which commutes (both from left and right) to the group action [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF][START_REF] Faraut | Analyse sur les groupes de Lie[END_REF] . Namely, in any such semi simple compact Lie group G, one considers it's Lie algebra G, naturally endowed with a Lie algebra structure

G × G → G, (X, Y ) → [X, Y ].
The Lie algebra structure provides on G a natural quadratic form K (the Killing form) as follows : for any element X ∈ G, one considers

ad(X) : G → G, Y → [X, Y ],
and K(X, Y ) = -trace ad(X)ad(Y ) . It turns out that this quadratic form is positive definite exactly when the group is compact and semi-simple. If one considers, for this Killing form, any orthonormal basis (X i ) in the Lie algebra, the quantity i X 2 i , computed in the enveloping algebra, does not depend on the choice of the basis, and commutes with any element on the Lie algebra itself (this means that this commutation property depends only on the Lie-algebra structure and not on the way that the elements of this Lie algebra are effectively represented as linear operators). Now, to any X ∈ G is associated a first order operator D X on G defined as follows

(3.16) D X (f )(g) = ∂ t f (ge tX ) | t=0 .
The application X → D X is a representation of the Lie algebra into the linear space of vector fields

([D X , D Y ] = D [X,Y ]
), and any identity in the Lie algebra (on on it's enveloping algebra) translates to an identity on those differential operators. We work here with the right action g → ge tX but we could as well work with the left action g → e tX g. For any orthonormal basis X i of G for the Killing form K, one defines the Casimir operator

(3.17) L = i D 2 X i .
It does not depend of the choice of the basis and commutes with the group action, that is [L, D X ] = 0 for any X ∈ G.

This Killing form provides an Euclidean quadratic form in the tangent plane at identity in G (the Lie algebra G), which may be translated to the tangent plane at any point g ∈ G through the group action, and endows G with a natural Riemanian structure. It turns out that the Casimir operator L is also the Laplace operator for this structure. For the group SU (d) that we are interested in, one wants to precisely describe the action of this Casimir operator on the entries of the matrix g = (z ij ) in SU (d). That is, writing the entries z pq = x pq + iy pq , we consider x pq and y pq as functions G → R, and, for any i, j, k, l, we want to compute

L SU (d) (x ij ), L SU (d) (y ij ), Γ SU (d) (x ij , x k,l ), Γ SU (d) (x ij , y k,l ), Γ SU (d) (y ij , x k,l ),
where Γ SU (d) is the square field operator associated with L SU (d) . In order to get simpler formulae, it is once again better to work with the complex valued functions z pq , writing for such a function z = x + iy,

L SU (d) (z) = L SU (d) (x) + iL SU (d) (y), Γ SU (d) (z, z) = Γ(x, x) -Γ(y, y) + 2iΓ(x, y), Γ SU (d) (z, z) = Γ(x, x) + Γ(y, y).
If one denotes by E p,q the matrix with entries (E p,q ) i,j = δ ip δ jq , a base of the Lie algebra of SU (d) is given by

R k,l = (E k,l -E l,k ) k<l S k,l = i(E k,l + E l,k ) k<l D 1,l = i(E 1,1 -E l,l ) 1<l .
In order to describe the Casimir operator in a compact form, it is better to work with D k,l = i(E k,k -E l,l ) k<l , and one may write in this way (up to a factor 2 that will play no rôle in the future)

(3.18) L SU (d) = k<l (D 2 R k,l + D 2 S k,l + 2 n D 2 D k,l ).
One may compute then the associated vector fields following formula (3.16), and we get

D Rpq = k z kq ∂ kp -z kp ∂ kq + zkq ∂kp -zkp ∂kq D Spq = i k z kq ∂ kp + z kp ∂ kq -zkq ∂kp -zkp ∂kq . D Dpq = i k z kp ∂ kp -z kq ∂ kq -zkp ∂kp + zkq ∂kq .
With these relations, one may directly compute the action of L SU (d) on the entries of the matrix, and we get

(3.19)            L SU (d) (z pq ) = -2(d 2 -1) d z pq L SU (d) (z pq ) = -2(d 2 -1) d zpq Γ SU (d) (z kl , z rq ) = -2z kq z rl + 2 d z kl z rq , Γ SU (d) (z kl , zrq ) = 2(δ kr δ lq -1
d z kl zrq ). Now, let us consider the special case d = 3 and consider the function Z : G → C defined by Z(g) = 1 3 trace (g) = 1 3 (z 11 + z 22 + z 33 ). Any function on G depending only on the spectrum of g is a function of (Z, Z), since, denoting by λ 1 , λ 2 , λ 3 the eigenvalues of g, with

|λ i | = 1, λ 1 λ 2 λ 3 = 1, such a spectral function is a function of (3Z = λ 1 + λ 2 + λ 3 , λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 )
, but thanks to the properties of the eigenvalues,

λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 = 3 Z.
Then, the characteristic polynomial of g, that is P (X) = det(X -g), may be written as

P (X) = X 3 -3ZX 2 + 3 ZX -1.
It turns out that formulae (3.19) produce an easy way to compute the action of L SU (d) on the entries of the characteristic polynomial. For this, we use the following formulae, valid for any square matrix M with entries (m ij )

(3.20) ∂ m ij log det(M ) = M -1 ji , ∂ 2 m ij m kl log det(M ) = -M -1 jk M -1 li .
Using the change of variable formula (2.2), when applying equation (3.20) to XId-M together with formulae (3.19), one gets, on SU (d) and with P (X) = det(XId -M ),

Γ SU (d) (log P (X), log P (Y )) = (d - XP ′ (X) P (X) )(d - Y P ′ (Y ) P (Y ) ) -d d + X + Y X -Y ( Y P ′ (Y ) P (Y ) - XP ′ (X) P (X) ) - XY X -Y ( P ′ (Y ) P (Y ) - P ′ (X) P (X) ) = XY P ′ (X) P (X) P ′ (Y ) P (Y ) + d X -Y ( P ′ (X) P (X) - P ′ (Y ) P (Y ) ) .
which in turn gives One may now compare this for d = 3 with the formula given by (3.11) for λ = 4, to observe that indeed, with L = 3 4 L SU (3) , they give the same result. In the end, we see that L (4) is nothing else than 3 4 L SU (3) when acting on functions of Z = T /3, where T is the trace of the matrix.

4 Curvature dimension for the deltoid model

Curvature-dimension inequalities for SU (3).

It is well known that the Casimir operator of any compact semi simple Lie group has a constant Ricci curvature (see for example [START_REF] Gallot | Riemannian geometry[END_REF], prop. 3.17, or [START_REF] Saloff-Coste | On the convergence to equilibrium of brownian motion on compact simple lie groups[END_REF]). However, the explicit constant is not straightforward to compute, and for the sake of completeness, we provide for this an easy way through the use of the Γ 2 operator. It relies on the following observation, which may be used in other similar situations. Let E i be some elements of the Lie algebra and X i be the associated vector fields. We do not require that the E i form an orthonormal basis, since we shall use the representation (3.18) for SU (3) which is not given in such a basis (the elements D ij are not orthogonal and are not linearly independent). Suppose that L is given as

L = i X 2 i , [L, X i ] = 0, Γ(f, f ) = i X i (f )X i (f ).
Then, using formula (2.3), one immediately gets

Γ 2 (f, f ) = i,j (X i X j f ) 2 .
We decompose X i X j (f ) into it's symmetric and antisymmetric part

X i X j = 1 2 (X i X j f + X j X i f ) + 1 2 [X i , X j ](f ) = H ij + 1 2 [X i , X j ](f ).
Then

Γ 2 (f, f ) = ij (H ij f ) 2 + 1 4 ij ([X i , X j ](f )) 2 = ij (H ij f ) 2 + 1 2 i<j ([X i , X j ](f )) 2 .
It turns out that this decomposition coincides exactly with the one provided by (2.4) into the second order part and the first order part of the Γ 2 tensor. Therefore, on has

Ric(f, f ) = 1 2 i<j ([X i , X j ](f )) 2 ,
and we are bound to compute the commutators of the elements in the basis. For example, on SU (3), the commutator table is the following, with Di,j = 2/3D i,j , and a = 2/3:

0 R 1,2 R 1,3 R 2,3 S 1,2 S 1,3 S 2,3 D1,2 D1,3 D2,3 R 1,2 0 -R 2,3 R 1,3 2/a D1,2 -S 2,3 S 1,3 -2aS 1,2 -aS 1,2 aS 1,2 R 1,3 0 -R 1,2 -S 2,3 2/a D1,3 S 1,2 -aS 2,3 -2aS 1,3 -aS 1,3 R 2,3 0 -S 1,3 S 1,2 2/a D2,3 aS 2,3 -aS 2,3 -2aS 2,3 S 1,2 0 -R 2,3 -R 1,3 2aR 1,2 aR 1,2 -aR 1,2 S 1,3 0 -R 1,2 aR 1,3 2aR 1,3 aR 1,3 S 2,3 0 -aR 2,3 aR 2,3 2aR 2,3 D1,2 0 0 0 D1,3 0 0 D2,3
0 We just have to add half the sum of all the squares of the element appearing in the table. We get

(1 + 3a 2 )(R 2 2,3 + R 2 1,3 + R 2 1,2 + S 2 1,2 + S 2 1,3 + S 2 2,3 ) + 2 a 2 ( D2 1,2 + D2 2,3 + D2 2,3 ).
Since 1 + 3a 2 = 2/a 2 = 3, we see that the Ricci curvature is constant and it is equal 3, more precisely Ric(f, f ) = 3Γ(f, f ).

Remembering that the dimension of SU (3) is 8, we get From the definition of the Γ 2 operator and of the curvature-dimension inequality, it is immediate that if if L satisfies CD(ρ, n), then cL satisfies CD(cρ, n). We therefore see that 3 4 L SU (3) satisfies a CD( 9 4 , 8) inequality. Then, L (4) being the image of 3 4 L SU (3) through the map g → Z = 1 3 (z 11 + z 22 + z 33 ), applying the CD(ρ, n) inequality on SU (3) on function of (Z, Z) provided directly the Corollary 4.2. L (4) satisfies the CD( 94 , 8) inequality.

Curvature dimension for the general deltoid model

We may now come back to the general deltoid model. Let us write, in the triangle representation, the decomposition (3.14)

L (λ) = ∆ + λ -1 3 ∇ log W,
where W (1-λ)/3 is the density the invariant measure of L (λ) with respect to the Lebesgue measure, and W is given in formula (3.15)s. We also know from the general formulation of the CD(ρ, n) inequality (2.5) that the CD( 94 , 8) for λ = 4 translates into the following

(4.23) -∇∇ log W ≥ 9 4 Id + 1 6 ∇ log W ⊗ ∇ log W.
For λ > 1, multiplying the previous inequality by λ-1 3 , this in turns gives for the general case provides Corollary 4.3. For any λ ≥ 1, L (λ) satisfies a CD( 3(λ-1) 4 , 2λ) inequality.

Observe that indeed the limiting case λ = 1 corresponds to the Laplace operator on R 2 which satisfies a CD(0, 2) inequality.

It is not clear however that this inequality is sharp. Indeed, going from the CD inequality on SU (3) to the same CD inequality for L (4) , we may as well have lost information. In general, as we already mentioned, on a smooth compact manifold (with no boundaries) with dimension n 0 , there is no optimal CD(ρ, n) inequality, and for any n > n 0 one may find some ρ(n 0 ) such that the operator ∆ + ∇ log V satisfies a CD(ρ, n) inequality. Moreover, this does not tell us anything about the case where λ < 0.

As mentioned in the introduction, the optimal computation for this inequality for a generic λ is not elementary. In this section, we shall perform directly the computations of ∇∇ log W and ∇ log W ⊗ ∇ log W to observe that the SU (3) case indeed gives the optimal answer, which is quite surprising. Of course, on SU (3) the CD(ρ, n) inequality is optimal at every point x ∈ SU (3), while on the projected model, it is optimal only at some point on the boundary of the deltoid domain D.

We shall show the following Proposition 4.4.

1. For λ < 1, the operator L (λ) does not satisfy any CD(ρ, ∞) inequality.

2. For λ > 1, the operator satisfy no CD(ρ, n) inequality for any n < 2λ. Moreover, the best constant ρ in the CD(ρ, 2λ) inequality is ρ = 3(λ-1)

4

.

Proof. -Everything boils down, for λ > 1, to check inequalities of the form

-∇∇ log W ≥ c 1 Id + c 2 ∇ log W ⊗ ∇ log W,
and for λ < 1 to check the the tensor ∇∇ log W is not bounded below.

To perform the computations, we shall use the triangle model, that is move back everything of T through the map Z -1 , since on T the metric is the identity and the Hessian is computed in the usual way.

In what follows, we shall use the functions log(z) for a complex variable z = 0 without any precaution about which determination of the argument we chose for the logarithm, since indeed we are only concerned with the one form d log(z) and it's derivative.

Let us recall that

W = -(z 1 -z 2 ) 2 (z 2 -z 3 ) 2 (z 3 -z 1 ) 2 , with z k = e i(E k .z)
, where (4.24)

E 1 = 1 0 , E 2 = -1 2 √ 3 2 , E 3 = -1 2 - √ 3 2
.

Then, setting σ = log(z 1 -z 2 ) + log(z 2 -z 3 ) + log(z 3 -z 1 ), log W = 2σ, up to some additive (eventually complex) constant, and we are looking for (4.25)

-∇∇σ ≥ bI 2 + a∇σ ⊗ ∇σ.

What we want to show is first that ∇∇σ is not bounded above, and then that the former inequality (4.25) may not hold if a > 1/3. Moreover, we want to check that for a = 1/3, the best lower bound for b is b = 9/8. It turns out that this is quite technical.

We shall need a few intermediate steps to check this inequality.

Lemma 4.5. We have

∇z k = iz k E k , ∇ log(z p -z q ) = i z p E p -z q E q z p -z q , ∇∇ log(z p -z q ) = z p z q (z p -z q ) 2 (E p -E q ) ⊙ 2 . Proof. -(Of Lemma 4.5)
The two first identities are immediate. For the last one, we write

∇∇ log(z p -z q ) = i∇( 1 z p -z q (z p E p -z q E q )) = - 1 z p -z q (z p E ⊗ 2 p -z q E ⊗ 2 q ) + 1 (z p -z q ) 2 (z p E p -z q E q ) ⊗ 2
As a consequence, one has Corollary 4.6.

(4.26) ∇∇ log σ = [ z 1 z 2 (z 1 -z 2 ) 2 V ⊙ 2 1 + z 2 z 3 (z 2 -z 3 ) 2 V ⊙ 2 2 + z 1 z 3 (z 3 -z 1 ) 2 V ⊙ 2 3 ]. (4.27) ∇ log σ ⊗ ∇ log σ = -[ z 1 z 1 -z 2 V 1 + z 2 z 2 -z 3 V 2 + z 3 z 3 -z 1 V 3 ] ⊙ 2 .
where

V 1 = E 1 -E 2 , V 2 = E 2 -E 3 , V 3 = E 3 -E 1
and E i , i = 1, 2, 3 are defined in (4.24).

Remark 4.7. Observe that |V i | 2 = 3 and that the complex valued vector

U = z 1 z 1 -z 2 V 1 + z 2 z 2 -z 3 V 2 + z 3 z 3 -z 1 V 3 ,
has purely imaginary components: thanks to the fact that V 1 + V 2 + V 3 = 0, and using zi = z -1 i , one sees that Ū = -U . Therefore -U ⊗ U ≥ 0. On the other hand, the tensor

z 1 z 2 (z 1 -z 2 ) 2 V ⊙ 2 1 + z 2 z 3 (z 2 -z 3 ) 2 V ⊙ 2 2 + z 1 z 3 (z 3 -z 1 ) 2 V ⊙ 2 3 is real.
Proof

∇σ ⊗ ∇σ = -[ z 1 E 1 -z 2 E 2 z 1 -z 2 + z 2 E 2 -z 3 E 3 z 2 -z 3 + z 3 E 3 -z 1 E 1 z 3 -z 1 ] ⊗ 2 = -[E 1 + z 2 z 1 -z 2 (E 1 -E 2 ) + E 2 + z 3 z 2 -z 3 (E 2 -E 3 ) + E 3 + z 1 z 3 -z 1 (E 3 -E 1 )] ⊗ 2
and the fact that

E 1 + E 2 + E 3 = 0.
First, observe that when z 2 = e iφ z 1 with φ → 0, with z 1 = 1, j, j, the tensor ∇∇σ is equivalent to

-4 sin 2 φ/2 V 1 ⊗ V 1 ,
and therefore is not bounded below. This shows that there cannot exist any CD(ρ, ∞) inequality for λ < 1.

From now on, the parameter a ∈ R being fixed, let us call b(a) the best constant b in inequality (4.25) at a given point in the interior of the triangle. It is obtained the lowest eigenvalue of the symmetric tensor ∇∇σ -a∇σ ⊗∇σ. We want to understand when this function is bounded below.

It is then better to change coordinates and consider a basis

W 1 = V 1 / √ 3 and W 2 which is orthogonal to W 1 and norm 1, such that V 2 = √ 3(- 1 2 W 1 + √ 3 2 W 2 ), V 3 = √ 3(- 1 2 W 1 - √ 3 2 W 2 ).
Moreover, we shall set z 2 = z 1 u, |u| = 1, so that z 3 = 1/(z 2 1 u), and z 3 1 = z. Observe that the image of (x, y) → (z, u), is S 2 1 , where S 1 is the set complex numbers with modulus 1.

With those notations, inequality (4.25) becomes

AW ⊙ 2 1 + BW ⊙ 2 2 + CW 1 ⊙ W 2 4(1 -u) 2 (1 -zu) 2 (1 -zu 2 ) 2 ≥ b(W ⊙ 2 1 + W ⊙ 2 2 ),
where

A = 3 a(u 2 + 1) + (2a -4)u (1 + u 6 z 4 ) -uz(u + 1) 4a(u 2 + 1) + u 2 -10u + 1 (1 + z 2 u 3 ) +2u 2 z 2 2a(u 4 + 1) + a(u 3 + u) + (6a -12)u 2 B = 9(u -1) 2 a(u 6 z 4 + 1) -(u 5 z 3 + uz) -(u 4 z 3 + u 2 z) + (-2a + 4)u 3 z 2 C = 6 √ 3(u -1)(1 -z 2 u 3 ) a(u 4 z 2 + 1) + a(u 3 z 2 + u) + (1 -2a)(u 3 z + uz) -2u 2 z
Setting z = e iθ and u = e iφ we have

A = 12u 4 z 2 2 cos(2θ + 3φ)(a cos 2 φ/2 -1) -2 cos(φ/2) cos(θ + 3/2φ)((4a + 1) cos φ -5) +2a cos(2φ) + a cos(φ) + 3(a -2) B = -72u 4 z 2 (sin 2 (φ/2)) a cos(2θ + 3φ) -cos(θ + 2φ) -cos(θ + φ) + (2 -a) C = 48 √ 3z 2 u 4 sin(φ/2) sin(θ + 3/2φ) a cos(θ + 2φ) + a cos(θ + φ) + (1 -2a) cos(φ) -1
while the denominator may be written as

-4 4 z 2 u 4 sin 2 (φ/2) sin 2 (θ/2 + φ/2) sin 2 (θ/2 + φ).
Simplfying everything by u 4 z 2 , and letting

A 1 = -Az -2 u -4 , B 1 = -Bz -2 u -4 , C 1 = Cz -2 u -4 , and N = 4 4 sin 2 (φ/2) sin 2 (θ/2 + φ/2) sin 2 (θ/2 + φ), we see that the best constant b(a), at some point (z, u) is then b(a) = A 1 + B 1 -(A 1 -B 1 ) 2 + C 2 1 2N .
We now may prove the following Let us concentrate on the case a < 1. It is enough to observe the asymptotics of b(a) around θ = φ = 0. The result is obtained when choosing φ = λθ 2 . Then, one has

     A 1 ≃ 12(1 -a)θ 4 = αθ 4 B 1 ≃ 18λ 2 (1 -2a)θ 6 = βθ 6 C 1 ≃ -24 √ 3λaθ 5 = γθ 5
Then, b(a) ≃ c/θ 2 , where the constant c has the sign of 4βα -γ 2 , that is of (1 -3a). When a > 1/3, this converges to -∞ when θ → 0. Remark 4.9. When choosing in the previous argument φ = λθ, one sees that b(a) is unbounded below as soon as a > 1/2. This asymptotics is not enough to capture the optimal bound.

We now concentrate on the case a = 1/3. We are able to compute explicitly the lower bound for a = 1/3, which corresponds and fits with the SU (3) computation, although the explicit computation of the lower bound is not explicit (and not really of interest) for the other values of a < 1/3.

We will study the function b(a) in case a =

b(1/3) = 1 2 1 (u -1) 2 (zu 2 -1) 2 (zu -1) 2 P (z, u) -Q(z, u)
where

P (z, u) = (u 2 -4u + 1)(1 + u 6 z 4 ) -4zu(u + 1)(u 2 -3u + 1)(1 + z 2 u 3 ) +u 2 z 2 (u 4 + 8u 3 -30u 2 + 8u + 1). Q(z, u) = [(z 2 u 4 -zu 3 -zu 2 + u 2 -u + 1)(z 2 u 2 -z 2 u 3 + z 2 u 4 -zu -zu 2 + 1) (z 2 u 4 + z 2 u 3 + zu 3 -6zu 2 + zu + u + 1) 2 ].
Using the same notations than in Lemma 4.8, b(1/3) can be written also in (θ, φ) in this form

(4.28) b(1/3) = 1 2 N (θ, φ) D(θ, φ)
where

N (θ, φ) = 2 2 cos(2θ + 3φ)(cos(φ) -2) -8 cos(φ/2) cos(θ + 3/2φ)(2 cos(φ) -3) + cos(2φ) + 8 cos(φ) -15-| T || 2 cos(φ/2) cos(θ + 3/2φ) + cos(φ) -3 | D(θ, φ) = -2 6 sin 2 (φ/2) sin 2 (θ/2 + φ) sin 2 (θ/2 + φ/2)
where

T = z 2 u 3 -2 cos( φ 2 )zu 3 2 + 2 cos(φ) -1 and | T |= √ T T
Setting x = cos(φ/2), y = cos(θ + 3/2φ), and y = x + w we may rewrite this as

(4.29) b(1/3) = 1 4 (2(1 -x 2 ) -xw) 2 + 3w 2 (1 -x 2 ) -(2(1 -x 2 ) -xw) (2(1 -x 2 ) -xw) 2 + 3w 2 (x 2 -1) (1 -x 2 )w 2
To see this, we just replacing in (4.28)

cos(φ) = 2x 2 -1, cos(2φ) = 8x 4 -8x 2 + 1, cos(2θ + 3φ) = 2y 2 -1 sin 2 ( φ 2 ) = 1 -x 2 , sin(φ + θ 2 ) sin( θ + φ 2 ) = 1 4 (x -y)
we obtain

N (x, y) = 8 2x 4 -8x 3 y + 2x 2 y 2 + x 2 + 10xy -3y 2 -4 - | x 2 + xy -2 | 4x 4 + 4x 2 y 2 -4x 3 y -3y 2 -7x 2 + 2xy + 4 D(x, y) = -2 4 (1 -x 2 )(x -y) 2
then if we set y = x + w we have the result Now, in formula (4.29), we set t =| 2(1-x 2 )-xw)

w √ 1-x 2 |, and then b(1/3) becomes b(1/3) = 1 4 (t 2 + 3 -t t 2 -3), t ≥ √ 3.
It is an easy exercise to check that the lower bound of this last function of t is 9/8.

We now collect the results of Lemmas 4.8 and 4.10 to get Proposition 4.4.

A simpler proof of the curvature-dimension inequality

As mentioned in the introduction, we shall show that the use of the complex coordinates (Z, Z) provide a much simpler proof of Proposition 4.4. Everything relies on the boundary equation (4.31), which is nothing else than a particular case of a general equation which is valid as soon as orthogonal polynomials come into play (see [START_REF] Bakry | Orthogonal polynomials and diffusions operators[END_REF]). In the coordinates (Z, Z) ∈ D, the operator L (λ) takes a simpler form, even if the metric looks more complicated. This illustrates the use of the appropriate coordinates whenever one has a polynomial structure such as this deltoid model. Once again, our aim is to compute the Hessian of the function log P , where P is defined in equation (3.12).

Following [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF], the Hessian of f , applied to dh, dk, that is in a local system of coordinates ∇∇ ij (f )∂ i h∂ j k, may be defined as

(4.30) H[f ](h, k) = 1 2 Γ(h, Γ(f, k)) + Γ(k, Γ(f, h)) -Γ(f, Γ(k, h) .
We want to apply this with f = log P and h, k = Z, Z. For this, one may use the boundary equation which takes in this context the particular form From the previous remarks, we get

H 11 = -3Γ(Z, Z) + 3 2 D(Γ(Z, Z)). H 12 = -3Γ(Z, Z) + 3 2 D(Γ(Z, Z)),
and

H 22 = -3Γ(Z, Z) + 3 2 D(Γ(Z, Z)).
In other words, with the obvious notations, H = -3Γ + 3 2 DΓ In the same way, the tensor ∇ log P ⊗ ∇ log P may be computed in this system of coordinates as

M = 9 Z 2 ZZ ZZ Z 2 .
In the end, the inequality (4.32) -∇∇ log P ≥ b 1 Γ + a 1 ∇ log P ⊗ ∇ log P amounts to

(3 -b 1 )Γ - 3 2 DΓ -9a 1 M ≥ 0,
where Γ denotes the matrix

Γ(Z, Z) Γ(Z, Z) Γ(Z, Z) Γ(Z, Z) .
For such a tensor (R ij ) in complex coordinates, to represent a non negative real tensor amounts to ask that

R 12 ≥ 0 and (R 12 ) 2 ≥ R 11 R 22 .
R 12 ≥ 0 reads 3 -b 1 + (b 1 -18a 1 )ZZ ≥ 0, and for this to be true on Ω amounts to ask b 1 ≤ 3, a 1 ≤ 1/6, since ZZ varies from 0 to 1 on Ω.

The second one writes (4.33)

[(3-b 1 )/2+b 1 /2-9a 1 ZZ] 2 ≥ (3/2-b 1 ) 2 ZZ+(b 1 -9a 1 ) 2 Z 2 Z 2 +(Z 3 +Z 3 )(b 1 -9a 1 )(3/2-b 1 ).
Writing everything in polar coordinates Z = ρe iθ , this writes as

P 2 (ρ 2 ) ≥ 2ρ 3 cos(3θ)(b 1 -9a 1 )(3/2 -b 1 ),
where P 2 is a degree 2 polynomial.

Observe that this requires to be true for any ρ ∈ [0, 1] when cos(3θ) = 0 (which corresponds to the cusps of the deltoid curve).

But, with the explicit computation of P 2 , one gets

P 2 (ρ 2 )-2ρ 3 (b-9a)(3/2-b) = 1 4 (1-ρ)(3-b 1 +b 1 ρ) 3-b 1 +ρ(3-2b 1 )+3ρ 2 (b 1 -12a 1 ) .
For the maximal value a 1 = 1/6,

P 2 (ρ 2 ) -2ρ 3 (b -9a)(3/2 -b) = (1 -ρ) 2 (3 -b 1 + 3ρ(2 -b 1 )),
and we get a bound b 1 ≤ 9/4. For these values a 1 = 1/6 and b 1 = 9/4, equation (4.33) writes

4(1 -ρ 2 ) 2 ≥ ρ 2 + ρ 4 -2ρ 3 cos(3θ),
while the condition Γ(Z, Z) 2 ≥ Γ(Z, Z)Γ(Z, Z), which characterizes the points in D,

writes 1 4 (1 -ρ 2 ) 2 ≥ ρ 2 + ρ 4 -2ρ 3 cos(3θ),
so the the inequality is satisfied everywhere in D. Observe that the critical points for the curvature-dimension inequality for the critical values are attained at the cusps.

where P S 1 (τ ) t (x, dx ′ ) is the semigroup of the torus with radius τ , that is the semigroup of the one-dimensional Brownian motion acing on τ -periodic functions. Since both semigroups have a density which is bounded above by C/ √ t for some constant C and for 0 < t ≤ 1, it turns out that P (1) t has a density which is bounded above by C ′ /t. This is enough to get the bound on the associated eigenvectors. In the end, we get Proposition 5.1. For any λ ≥ 1, there exists a constant C(λ) depending on λ only, such that for any polynomial P eigenvector of L (λ) with eigenvalue µ = 0, one has

(5.35) P ∞ ≤ C(λ)µ λ/2 f 2 .
Remark 5.2. Looking at the constants, whenever λ → 1, the constant C(λ) in (5.35) goes to ∞, and there is an unexpected discontinuity in the constants. Indeed, our computations are not the best possible. One may sharpen them with the help of spectral gaps, that is the knowledge of the lowest non 0 eigenvalue, which here is λ. More precisely, one may reinforce the constants in the ultracontractive bounds under a CD(ρ, n) inequality and the knowledge of this lowest eigenvalue (see remark page 313 in [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF]). But the argument in this modified estimate produces a Sobolev inequality with any dimensional parameter m > n , and a constant which is not improved when m → n. There is therefore a balance in the optimal bound on P between the value of µ (for µ large one wants m to be the lowest possible), and for λ → 1 (when λ → 1, one wants C(λ) not too big). We could such have produced a better bound. But indeed, the remark in [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF] as it stands is not really valid for ρ = 0 which corresponds in our case to = 1. One would have to sharpen this estimate, both for the case ρ = 0 and for the value of m. It is indeed true that one may obtain a n-dimensional Sobolev inequality (under it's entropic form) under a estimate on the lowest eigenvalue and some CD(ρ, n) inequality, even for ρ < 0, but the argument in [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF] is clearly not sufficient for that and requires further analysis.

From the point of view of the invariant measure µ (λ) of L (λ) , what is relevant is the decomposition of L 2 (µ (λ) ) into spaces of orthogonal polynomials. More precisely, when denoting P k the space of polynomials with total degree less than or equal to k, one considers the subspace H k of P k which is orthogonal to P k , such that one has the orthogonal decomposition

L 2 µ (λ) = ⊕ ∞ k=1 H k ,
where H 0 is the space of constant functions. One has Proposition 5.3. There exists a constant C 1 (λ) such that, for any k ≥ 1 and any P ∈ H k (5.36)

P ∞ ≤ C 1 (λ)k λ+1/2 P 2 .
Proof. -One may decompose H k into the eigenspaces associated to L (λ) . There are r k = [k/2] + 1 such eigenspaces, and all the eigenvalues belong to the interval

[k(λ + k -5/4), k(λ + k -1)], or, when k ≥ 1, in the interval [3/4k 2 , λk 2 ].
Writing P ∈ H k as P = r k i=1 a i P i where P i is an eigenvector with P i 2 = 1 and P 2 = r k 1 a 2 i , one has from (5.35) and the bound on the eigenvalues in H k

P ∞ ≤ r k 1 |a i | P i ∞ ≤ C(λ)λ (1+λ/2) r k 1 |a i |k λ ≤ C(λ)λ (1+λ/2) k λ r 1/2 k r k 1 a 2 i ,
from which the bound follows immediately.

One may wonder how far inequality (

P (λ) t P ∞ ≤ C 1 (λ) exp(-3/4tk 2 )k λ+1/2 P 2 .
Observe that this relies only on the bound (5.35) together with the knowledge of the eigenvalues.

Theorem 5.4. Let P t be a symmetric Markov semigroup with reversible probability measure µ and generator L. Assume that L satisfies a Poincaré inequality and that one has a decomposition into orthogonal spaces L 2 (µ) = ⊕ k H k , where H k a linear space, with the property that, for some real number a > 0 and for any f ∈ H k ,

P t f ∞ ≤ Ck p e -atk 2 f 2 .
Then, L satisfies a tight Sobolev inequality with dimension m = 2p + 1.

Proof. -Following [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF], and from the existence of a Poincaré inequality, it is enough to prove that, for t ∈ (0, 1] and for some constant C, P t f ∞ ≤ Ct -m/4 f 2 . We may restrict to the case where

f 2 = 1. For f ∈ L 2 (µ), let us write f = k f k , where f k ∈ H k and k f k 2 2 = 1. P t f ∞ ≤ k P t f k ∞ ≤ k k p e -atk 2 f k 2 ≤ k k 2p e -2atk 2 1/2 .
One may compare the sum k k 2p e -2atk 2 with ∞ 0 x 2p exp(-2atx 2 ) dx, where the function x 2p exp(-2atx 2 ) is increasing on (0, p/(2at) and decreasing on ( p/(2at), ∞), and we see that, for 0 < t ≤ 1, k k 2p e -2atk 2 ≤ C(a, p)t -(p+1)/2 , 0 < t ≤ 1.

Therefore, following the results exposed in Section 2, we get a Sobolev inequality with dimension m = 2(p + 1). The existence of a Poincaré inequality (that is of a strictly positive first non zero eigenvalue for -L) insures that we may get a tight Sobolev inequality (2.7). This gives the result.

Looking at the values for the deltoid model, we see that the provides a Sobolev inequality with dimension m = 2λ + 3, whereas we started from a Sobolev inequality with dimension 2λ. One may wonder if this lost in dimension (from n to n + 3) is due to too crude estimates on both the eigenvalues and the summation in the series, or from the fact that the spaces H k are k dimensional. Indeed, even in the case or one dimensional Jacobi operators, where the eigenspaces are one dimensional, where the eigenvalues for the associated operator are k(k + c) for polynomials with degree k, one would pass with the same method from a Sobolev inequality with dimension n to a Sobolev inequality with dimension n + 1. This is in big contrast with the case of logarithmic Sobolev inequalities, where estimates on the L p bounds on the eigenvectors are indeed equivalent to logarithmic Sobolev inequalities (see [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]).

Finally, we directly get from this a criterium for a symmetric operator constructed from orthogonal polynomial would have a bounded density. Proposition 5.5. Let K be a symmetric operator in L 2 (µ (λ) ) which maps H k into H k and is such that, for any P ∈ H k , K(P ) 2 ≤ ν k P 2 . If A = k ν 2 k k 2λ+1 < ∞, then K 2 may be represented by a bounded kernel, K 2 (f )(x) = f (y)k(x, y) dµ(y), where |k| ≤ A.

Proof. -Arguing as in the proof of Theorem (5.4), we may write f = k f k with f k ∈ H k . Then,

K(f ) ∞ ≤ k K(f k ) ∞ ≤ k k λ/1/2 K(f ) 2 ≤ k ν k k λ/1/2 f k 2 ≤ ( k ν k k λ/1/2 ) 1/2 f 2 .
Therefore, the operator K is bounded from L 2 into L ∞ with norm A 1/2 . By symmetry and duality, the same is true from L 1 into L 2 , and by composition, K 2 is bounded from L 1 into L ∞ with norm A. It therefore may be represented by a kernel k bounded by A.

Remark 5.6. The method presented here says nothing about the case where 0 < λ < 1. Indeed, in this case, one may expect to have a two-dimensional behavior for the heat kernel, that is P 1 2,∞ ≤ Ct -1/2 , 0 < t ≤ 1. In this context, it is better to replace Sobolev inequalities by Nash inequalities, that is inequalities of the form

f 2 2 ≤ f 2θ 1 f 2 2 + C Γ(f, f )dµ 1-θ
, where θ = 2 n+2 is a dimensional parameter. When n > 2, this is equivalent to a Sobolev inequality with dimension n, but for n ∈ (1, 2), this is still equivalent to a bound P t 2,∞ ≤ C ′ t -n/4 (see [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF]). As mentioned, we may expect when 0 < λ < 1 some Nash inequality with dimensional parameter n = 2. We cannot expect smaller value for n since, applied to any function compactly supported in the interior of D, this would contradict the classical two dimensional Nash inequality in an open domain of R 2 . However, the singularity of the measure at the cusps of the deltoid make things a bit hard to analyze. Following the method developed in [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Operators[END_REF], pages 370-371, we are able to prove Nash inequalities with dimension n = 5/2, with however a constant C(λ) which goes to infinity when λ → 0. This is not satisfactory for many points of view. First, because of the bad dimension, and for the lack of continuity in these inequalities when λ → 1. Secondly, when λ → 0, the measure converges to the uniform measure on the three cusps 1 3 (δ 1/3 + δ j/3 + δj /3 ), and the associated polynomials converge to the corresponding polynomials on three point (when one would have to replace Z Z by 1/9 and Z 3 by Z/3 3 ). It is therefore a challenging question to produce for small λ some functional inequality which recaptures this particular structure in the limit.

Figure 1 :

 1 Figure 1: The deltoid domain.

(3. 21 )

 21 Γ(P (X), P (Y )) = XY P ′ (X)P ′ (Y ) + d P ′ (X)P (Y ) -P ′ (Y )P (X) X -Y , and (3.22) L SU (d) (P ) = (1 -d 2 )XP ′ + (1 + d)X 2 P ′′ .

Proposition 4 . 1 .

 41 The Casimir operator L SU (3) defined by equation (3.18) satisfies the optimal CD(3, 8) inequality.

Lemma 4 . 8 .

 48 The function b(a) is unbounded below on the set |z| = |u| = 1 if a > 1/3. Proof. -(Of Lemma 4.8) We shall see in Lemma 4.10 that the function b(a) is bounded below for a = 1/3. This of course shows that it is also bounded below for any a < 1/3. What we have to prove then is that the function b(a) is unbounded below when a > 1/3.

1 3 . 4 . 10 .

 3410 Lemma The function b(1/3) is bounded below and it's lower bound is 9/8 Proof. -(Of Lemma 4.10) In the case a = 1/3 the function b(1/3) have the following form in (z,u):

( 4 .

 4 31) Γ(Z, log(P )) = -3Z, Γ(Z, log P ) = -3Z, and is easily checked from formulae (3.11) From this, we deduce that, for any function G(Z, Z), Γ(log P, G) = -3D(G), where D is the Euler operator Z∂ Z + Z∂ Z Let us write H 11 = H[log P ](Z, Z), H 12 = H[log P ](Z, Z), H 22 = H[log P ](Z, Z).

  5.36) is from the Sobolev inequality we started from. Observe first that, for the heat kernel P P 2 ≤ exp(-3/4tk 2 ) P 2 since all eigenvalues of P t on H k are bounded below by exp(-3/4tk 2 ).Therefore, we have, for any P ∈ H k

	one has	P	(λ)	(λ) t , for any function P ∈ H k ,

t

 Remark 4.11.Observe that the values b 1 = 9/4 and a 1 = 1/6 in inequality (4.32) are once again exactly the bounds obtained in equation (4.23). Moreover, we know that even with a 1 = 0 (corresponding to a CD(ρ, ∞) inequality), if we look for the optimal value for b 1 , it is clear from this method that the best constant b 1 is b max < 3, so that whatever the constant a 1 ∈ [0, 1/6], the optimal value for b 1 lies in the interval [9/4, 3). (The optimal constant b 1 (a 1 ) may be explicitly computed but has no real interest.)

Sobolev inequalities and bounds on the eigenvectors

As described in Section 2, from the curvature dimension inequality, we may obtain bounds on the supremum of the associated eigenvectors. More precisely, whenever a CD(ρ, n) inequality holds with ρ > 0 and n < ∞, there exists a constant C such that for any eigenvector P satisfying L(P ) = -µP , then

where the L 2 norm is computed with respect to the invariant measure of the operator L.

Turning to the case of the operator L (λ) on the deltoid, we recall from [START_REF] Zribi | Orthogonal polynomials associated with the deltoid curve[END_REF] that the associated eigenvectors which are polynomials with total degree n have eigenvalues µ p,q = (λ -1)(p + q) + p 2 + q 2 + pq, with p + q = n. More precisely, of any n ≥ 1 such that p + q = n, when p = q, there is a dimension 2 associated eigenspace. In complex variables, for such value µ p,q , there is a unique degree n polynomial P p,q (Z, Z) with highest degree term Z p Zq and another one which is Pp,q (Z, Z) = P p,q ( Z, Z) = P q,p (Z, Z) eigenvector (the polynomial P p,q having real coefficients). For p = q however, the associated eigenspace is one dimensional. The real forms are S p,q = 1 2 (P p,q + P q,p ) and A p,q = -i 2 (P p,q -P q,p ), which form a real basis for this eigenspace.

When λ > 1, for L (λ) , for any µ p,q and of any polynomial P in the associated eigenspace, one gets from the CD( 3(λ-1) 4 , 2λ) inequality (5.34)

Looking at the constants, this does not produce any estimates for λ = 1 or 0 < λ < 1. However, for λ = 1, one may consider the following. The operator L (1) is nothing else than the usual Laplace operator acting on functions f (Z), where the function Z is given in (3.13). As functions of (x, y) in the real plane, those functions are periodic in x with period 4π and in y with period 4π/ √ 3. As such, the associated semigroup P

(1) t is an image of the product semigroup of the associated 1 dimensional torus (that is the semigroup on the real line acting on periodic functions). More precisely, when considering a function on the deltoid as a function of (x, y) ∈ R 2 , P

t ((x, y, dx ′ , dy ′ ) = P S 1 (4π) t (x, dx ′ )P S 1 (4π/ √ 3) t (y, dy ′ ),


Léo Girardin

Grégoire Nadin

TRAVELLING WAVES FOR DIFFUSIVE AND STRONGLY COMPETITIVE SYSTEMS: RELATIVE MOTILITY AND INVASION SPEED

Our interest here is to nd the invader in a two species, diusive and competitive Lotka Volterra system in the particular case of travelling wave solutions. We investigate the role of diusion in homogeneous domains. We might expect a priori two dierent cases: strong interspecic competition and weak interspecic competition. In this paper, we study the rst one and obtain a clear conclusion: the invading species is, up to a xed multiplicative constant, the more diusive one.

Introduction

Competitive reaction diusion systems have been widely studied in the last few years. This mathematical model is motivated by numerous applications: ecology, chemistry, genetics, etc. In general, the mathematical formulation of this problem is, for some spatial domain Ω (non-necessarily bounded), some n ∈ N and some positive constants (d i , r i , a i , k i,j ) i,j∈{1,...,n} :

(1.1) ∀i ∈ {1, . . . , n}

∂ t u i = d i ∆ x u i + u i   r i -a i u i - j =i k i,j u j   in Ω × (0, +∞)
One tough question is how their solutions and, when it exists, the long-time steady state, depend on the diusion rates (d i ) i∈{1,...,n} . Asymptotically, how do the species (if we see this as continuous approximation of a population dynamics problem) represented by the densities (u i ) i∈{1,...,n} share the domain Ω? Basically, in the neighbourhood of any spatial point x, two cases may occur: either only one species persists (exclusion case) or two or more persist (coexistence case). In the exclusion case, the only persistent species is called invading species. A priori, all the parameters participate in the determination of this invader: number of species n, heterogeneity of Ω, boundedness of Ω, boundary conditions, intrinsic growth rates (r i ) i∈{1,...,n} , interspecic competition rates (k i,j ) i,j∈{1,...,n} , intraspecic competition rates (a i ) i∈{1,...,n} and of course diusion rates (d i ) i∈{1,...,n} .

The dependency on diusion rates is a very open general problem. Previous works show clearly that a very general result is for the moment inachievable and that we are due to consider in each study a specic case for the other parameters of the problem. A key work in this area is the paper by Dockery et al. [START_REF] Dockery | The evolution of slow dispersal rates: a reaction diusion model[END_REF]. They proved that, when Ω is bounded, heterogeneous, with Neumann boundary conditions and when k i,j = 1 for all i, j ∈ {1, . . . , n}, the less motile species that is the one with the lower diusion rate is the invading species. Their result relies deeply on the heterogeneity, the basic idea being that each species loses the individuals trying to invade unfavorable areas while, in favorable areas, the competition helps the more concentrated one, that is the less diusive one.

We leave the extension of Dockery's result for dierent (k i,j ) i,j∈{1,...,n} to others and wonder if a similar result can be obtained in homogeneous domains (bounded or not).

Actually, it is quite tough to guess heuristically what could happen in homogeneous domains. Indeed, on one hand, the more diusive species might be able to ignore its competitors long enough and invade the whole territory while eliminating the competitors slowly. On the other hand, the more concentrated species that is the less diusive one might benet from the maxim unity is strength and eliminate slowly the dispersed competitors and, asymptotically, invade the domain.

It is well-known that diusion tends to bring unexpected results. In any case, if something can revert the invasion, we expect it to be the competition. With this in mind, we decide to focus rst on the innite competition limit which should amplify the eects of competition.

Many papers limit their study to the case n = 2 (and so will we) because then the system becomes monotonic and is therefore much simpler to study than the general case. We will not use the monotonicity explicitly but it will be the underlying mechanism behind many results.

When n = 2, the PDE system can be rewritten:

∂ t u = d 1 ∆ x u + u (r 1 -a 1 u -k 1 v) in Ω × (0, +∞) ∂ t v = d 2 ∆ x v + v (r 2 -a 2 v -k 2 u) in Ω × (0, +∞)
When there is no diusion at all, this system becomes an ODE system. Then, the steady state (u, v) = (0, 1) (resp. (u, v) = (1, 0)) is stable when k1r2 r1a2 > 1 (resp. k2r1 r2a1 > 1), unstable when k1r2 r1a2 < 1 (resp. k2r1 r2a1 > 1). Our interest lies in the bistable case and more precisely in the so-called weak competition case where k1r2 r1a2 and k2r1 r2a1 are larger than 1 but close to 1. In the monostable case, only one species is a strong competitor.

The innite competition limit (k 1 → +∞ and k1 k2 constant) has been studied by Dancer et al. in 1999 in the case of bounded domains with Neumann boundary conditions [START_REF] Dancer | Spatial segregation limit of a competition-diusion system[END_REF] (they also investigated Dirichlet conditions ve years later [START_REF] Crooks | Spatial segregation limit of a competition-diusion system with Dirichlet boundary conditions[END_REF]). They obtained a free boundary Stefan problem and, under regularity assumptions, a spatial segregation with an explicit condition on the interface. In 2007, Nakashima and Wakasa [START_REF] Nakashima | Generation of interfaces for LotkaVolterra competitiondiusion system with large interaction rates[END_REF] studied the generation of interfaces for such systems and obtained a similar free boundary condition.

It is worth mentioning that the spatial segregation in multi-dimensional domains for elliptic PDE yields highly non-trivial issues. It can be either approached as a free boundary problem (Dancer [5], Quitalo [START_REF] Quitalo | A free boundary problem arising from segregation of populations with high competition[END_REF]) or as an optimal partition problem (Conti [START_REF] Conti | A regularity theory for optimal partition problems[END_REF][START_REF] Conti | A variational problem for the spatial segregation of reaction-diusion systems[END_REF]), but in both cases it is really a problem in itself, which requires additional assumptions on the initial conditions and a lot of work. Therefore, our interest goes to unbounded homogeneous domains. Reaction diusion studies in such domains usually conjecture the existence of propagation fronts and, when their existence can be rigorously proved, derive from them some information on the dynamics of the system and the long-time steady state. Here, it is important to recall that the main underlying assumption with propagation fronts is that, when the initial conditions are well-chosen, the solutions of the PDE asymptotically behave like the travelling wave solution. We refer to Gardner [START_REF] Gardner | Existence and stability of travelling wave solutions of competition models: a degree theoretic approach[END_REF] for such results for nite k. We will not treat this aspect of the problem in this paper but will indeed investigate travelling wave solutions.

A straightforward consequence of the travelling wave approach is that it reduces the multi-dimensional Ω × R + to R. The problem becomes one-dimensional, that is an ODE problem, and thus all the free boundary issues vanish. Our hope is to nd a similar spatial segregation limit, with an explicit condition on the interface connecting the invasion speed of the travelling wave to the diusion rates. We know from Gardner [START_REF] Gardner | Existence and stability of travelling wave solutions of competition models: a degree theoretic approach[END_REF] and Kan-On [START_REF] Kan-On | Parameter dependence of propagation speed of travelling waves for competition diusion equations[END_REF] that the invasion speed is constant and bounded by the Fisher KPP's speeds [START_REF] Kolmogorov | Étude de l'équation de la diusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] of the species. Can we use the innite competition limit to derive its sign and therefore know which species invades the other? Will unity be strength?

It is important to remark that the invasion speed is not linearly determined here.

Actually, a linearization near (0, 1) or [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]0) yields no condition at all on the invasion speed and the linearized speed cannot be dened as usual. As far as we know, the linear determinacy for competition diusion systems is useful only with a specic class of monostable problems (Huang [START_REF] Huang | Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model[END_REF], Lewis [START_REF] Lewis | Spreading speed and linear determinacy for twospecies competition models[END_REF]).

In the next section, we fully expose the problem, enunciate our nal result and recall that the problem is well-posed. The third and main section is dedicated to a compactness result and the convergence to a limit problem which is similar in many ways to the one Dancer et al. obtained. Eventually, the last section explicits the relation between the speed and the diusion rates.

Formulation of the problem and main theorem

In this rst section, we present the PDE problem studied in this article, give its ecological interpretation and enunciate our main result. We also check quickly that the problem is well-posed.

2.1. Model.

2.1.1. Reaction diusion system. We rst consider the following one-dimensional Lotka Volterra competition diusion problem:

∂ t µ = d 1 ∂ xx µ + µ (r 1 -a 1 µ -k 1 ρ) in R × (0, +∞) ∂ t ρ = d 2 ∂ xx ρ + ρ (r 2 -a 2 ρ -k 2 µ) in R × (0, +∞) where d 1 , d 2 , r 1 , r 2 , a 1 , a 2 , k 1 , k 2 are
positive constants with ecological meaning (diffusion rates, intrinsic growth rates, intraspecic competition rates, interspecic competition rates). We assume, without loss of generality, that k2a2 r

2 2 ≥ k1a1 r 2 1 . Let k = k1r2 a2r1 > 0, α = k2a2r1 k1a1r2 > 0, d = d2 d1 > 0, r = r2 r1 > 0 and (u k , v k ) : (x, t) → a 1 r 1 µ d 1 r 1 x, 1 r 1 t , a 2 r 2 ρ d 1 r 1 x, 1 r 1 t
We get:

∂ t u k = ∂ xx u k + u k (1 -u k ) -ku k v k in R × (0, +∞) ∂ t v k = d∂ xx v k + rv k (1 -v k ) -αku k v k in R × (0, +∞)
As soon as k > 1 (which will always be assumed thereafter), αk r > 1, that is the system is bistable. Indeed, the free assumption k2a2 r

2 2 ≥ k1a1 r 2 1
we made earlier ensures that α r ≥ 1. A priori, the parameters k, α, d and r can take any positive value. Let P (k, α, d, r) denote this generic PDE problem. Our interest lies in the limit, as k → +∞, of the set of problems {P (k, α, d, r)} k>1 (associated to a given (α, d, r)) (hence the notations u k and v k ).

Moreover, going back to the initial parameters, this means that we actually consider a larger class of ecological problems than just k 1 → +∞ and k1 k2 constant. Indeed, the only restrictions are that d2 d1 , r2 r1 and k2a2 k1a1 are xed along the whole class. For example, the limit k → +∞ may correspond to:

• k 2 proportional (with a xed constant along the whole class) to k 1 and k 1 → +∞ with a 1 and a 2 xed (along the whole class); • k 1 → +∞ and a 1 proportional to 1 k1 with a 2 and k 2 xed; • a 2 proportional to a 1 and a 1 → 0 with k 1 and k 2 xed.

2.1.2. Travelling wave system. Searching for a travelling wave of the variable ξ = x -c k t, where c k ∈ R is the unknown invasion speed, the problem rewrites eventually:

(2.1)

               -u k -c k u k = u k (1 -u k ) -ku k v k in R -dv k -c k v k = rv k (1 -v k ) -αku k v k in R u k (-∞) = 1, u k (+∞) = 0 v k (-∞) = 0, v k (+∞) = 1 u k < 0 in R v k > 0 in R
It is well-known that natural selection tends to dierentiate the niches of competing species. The travelling wave solution corresponds to the case where u k lives essentially in the left half-space while v k lives essentially in the right half-space.

In such a situation, it seems obvious that one species might chase the other and invade the abandoned territory. The whole point of this article is to determine this species, or equivalently, the sign of the invasion speed. Indeed, [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]

c k > 0 i u k chases v k ; (2) c k < 0 i v k chases u k ;
Of course, we aim to nd a result depending on the value of d. Thus in the following pages, when we focus on the dependency of c k on d, we write c k,d ; otherwise, when d is xed, we simply write c k . 2.2. Unity is not strength theorem. Our main result follows.

(1) c ∞,d = 0 if d = α 2 r ; (2) c ∞,d ∈ (0, 2) if d ∈ 0, α 2 r ; (3) c ∞,d ∈ -2 √ rd, 0 if d > α 2 r . Remark 2.
c k ∈ -2 √ rd, 2 , u k ∈ C ∞ (R) and v k ∈ C ∞ (R).
We can moreover assume exactly one of the following normalization hypotheses:

• u k (0) = v k (0), • u k (0) = 1 2 , • v k (0) = 1
2 , and if we do so, u k and v k are unique.

Proof. The well-posedness and the bounds for c k are proven by Gardner in [START_REF] Gardner | Existence and stability of travelling wave solutions of competition models: a degree theoretic approach[END_REF] and also by Kan-On in [START_REF] Kan-On | Parameter dependence of propagation speed of travelling waves for competition diusion equations[END_REF] (actually, Gardner only showed c k ∈ -2 √ rd, 2 but Kan-On showed indeed c k ∈ -2 √ rd, 2 which will be important in the end). It is worth mentioning that their papers actually proved that the problem is well-posed without any monotonicity condition and that the monotonicity is indeed enforced.

Since u k , v k ∈ L ∞ (R) and u k , v k ∈ L 1 (R)
, the regularity just follows from W k,p -estimates and Sobolev's injections.

Remark 2.4. The extremal speeds -2 √ rd and 2 are the invasion speeds of respectively v k when u k = 0 and u k when v k = 0. This is a well-known result from Fisher, Kolmogorov, Petrovsky and Piscounov [START_REF] Kolmogorov | Étude de l'équation de la diusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF].

Limit problem

Here we show that (u k ), (v k ) and (c k ) converge when k → +∞ and formulate the limit problem.

3.1. Existence of limit points. First, (c k ) is relatively compact and therefore, by the Bolzano Weierstrass theorem, has a limit point c ∈ -2 √ rd, 2 .

If c ≤ 0, we x for any k > 1 the normalization u k (0) = 1 2 . On the contrary, if c > 0, we x for any k > 1 v k (0) = 1 2 . This choice will be explained later on. In either case, this implies that the functions k → u k and k → v k are well-dened.

Proposition 3.1. For any i ≥ 1, let

K i = [-i, i]. (u k ) and (v k ) are relatively compact in C (K i ).
Proof. Our aim here is to use Ascoli's theorem. To that end, let us show that each u k is Hölder-continuous with a constant independent of k.

There exists a positive function χ

∈ D (R) such that χ (x) = 0 if x / ∈ [-i -1, i + 1] and χ (x) = 1 if x ∈ [-i, i].
For any k > 1, if we multiply the equation dening u k by u k χ and then integrate, we get:

ˆ(-u k u k χ -c k u k u k χ) = ˆu2 k χ -ˆu2 k (u k + kv k ) χ
The third term is obviously negative. An integration by parts yields:

ˆu 2 k χ - ˆu2 k 2 χ + c k ˆu2 k 2 χ ≤ ˆu2 k χ Finally, since ´u 2 k χ ≥ ´i -i u 2 k and u k L ∞ ≤ 1, we have: u k 2 L 2 (Ki) ≤ ˆ χ + |c k | 2 |χ | + 1 2 |χ |
Then we use Ascoli's theorem: the family (u k ) is bounded in L ∞ (K i ) and uniformly equicontinuous in K i therefore it is relatively compact in C (K i ). The exact same proof works for (v k ).

It is now clear, by a standard diagonal extraction argument, that there exists a subsequence of (u k ) (resp. (v k )) which converges locally uniformly to a limit point u (resp. v).

3.2. Properties of the limit points. c, u and v are actually unique and true limits as it will be proven later on. For the moment, let us just consider extracted convergent subsequences, still denoted (c k ), (u k ) and (v k ).

k ˆuk v k ϕ ≤ ˆuk (1 -u k ) |ϕ| + |c k | ˆuk |ϕ | + ˆuk |ϕ | ≤ C ϕ W 2,1 (R) Hence u k v k → 0 in D (R).
Since u k v k → uv locally uniformly, we get indeed uv = 0.

Remark 3.3. This kind of result is usually referred to as a segregation property.

There is a lot of similar results in the literature. Lemma 3.4. We have Lemma 3.6. u and v have nite limits at ±∞. Besides,

-αu + dv -αcu + cv = αu (1 -u) -rv (1 -v) in D (R).
0 ≤ lim +∞ u ≤ lim -∞ u ≤ 1 and 0 ≤ lim -∞ v ≤ lim +∞ v ≤ 1
Proof. By locally uniform convergence, u and v are monotone, respectively nonincreasing and non-decreasing, and satisfy 0 ≤ u, v ≤ 1.

Lemma 3.7. u and v cannot vanish simultaneously on a non-empty compact set.

Proof. Once again, we consider a non-empty compact set [a, b].

By monotonicity, if u |[a,b] = 0, then u |[a,+∞) = 0. Similarly, v |(-∞,b] = 0. It yields, in D ((-∞, a)), -u -cu = u (1 -u)
and αu -dv = αu . Therefore u is continuous and, using -u -cu = u (1 -u), u is also continuous and the previous dierential equation is satised pointwise. Now, we get by induction that u is C ∞ in (-∞, a). Since it does not explode on the left of a, it is the restriction of a solution on a strictly larger interval. Since u is regular, u (a) = 0 and by Cauchy Lipschitz's theorem, u is identically null. By the same reasoning, v is also identically null.

To prevent u and v from being both null on the whole real line, either one of the two normalization sequences (u k (0)) k>1 = 1 2 and (v k (0)) = 1 2 combined with locally uniform convergence suces. Lemma 3.9. Neither u nor v can be positive everywhere.

Proof. If c ≤ 0, the normalization sequence is (u k (0)) = 1 2 . It ensures that u is not null. We dene

ξ u = sup {ξ ∈ R | u (ξ) > 0} ∈ (-∞, +∞].
If ξ u = +∞ (that is, u positive everywhere), v is null.

In such a case, we have u decreasing, bounded between 0 and 1, with limits at innity, non-constant by normalization, and -u -cu = u (1 -u) everywhere with u ∈ C ∞ (R).

This yields that lim -∞ u = 1 and lim +∞ u = 0. To that end, we use L'Hospital's rule.

Let l = u (-∞), G : ξ → exp (cξ) and F = Gu so that F = G (u + cu ) = -Gu (1 -u). F and G are dierentiable in R, G = 0 and G → +∞ as ξ → -∞; besides, F G → -l(1-l) c . By L'Hospital's rule, F G → -l(1-l) c , that is u (-∞) = -l(1-l)
c . In the end, necessarily, l ∈ {0, 1}. At +∞, we use the other version of L'Hospital's rule, noticing that u is bounded in R + (easy to prove) and checking that F and G go to 0. Eventually, by monotonicity, the limits are 1 at -∞ and 0 at +∞.

Thus u is a travelling wave for the Fisher KPP equation with speed c ≤ 0 < √ 2, hence the contradiction [START_REF] Kolmogorov | Étude de l'équation de la diusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF].

If c > 0, we just apply this reasoning to v with normalization (v k (0)) = 1 2 .

Corollary 3.10. The two quantities

sup {ξ ∈ R | u (ξ) > 0} and inf {ξ ∈ R | v (ξ) > 0}
are real and equal. Up to translation, we can assume it to be 0. By continuity of u and v, u (0) = v (0) = 0.

(8) -u -cu = u (1 -u) in (-∞, 0); (9) -dv -cv = rv (1 -v) in (0, +∞); (10) αu (0) = -dv (0).
The behaviour of these limits is illustrated with the following gure.

v (0) u (0) = -d α v (0) 0 1 0 u v 3.4.
Uniqueness of the limit points.

Theorem 3.16. For any c > -2, the problem -y c -cy c = y c (1 -y c ) in (0, +∞) y c (0) = 0 admits a unique positive solution.

It satises y c > 0 in R + and lim ξ→+∞ y (ξ) = 1. Besides, γ : c → y c (0) is increasing and continuous.

Proof. This result was proved by Du and Lin in [START_REF] Du | Spreading-vanishing dichotomy in the diusive logistic model with a free boundary[END_REF] (prop. 4.1) but wrongly stated. Indeed, the requirement in their theorem should be c < 2, not c ≥ 0 as stated in [START_REF] Du | Spreading-vanishing dichotomy in the diusive logistic model with a free boundary[END_REF]. (Moreover, be aware that our statement is written with -cy c whereas their statement is written with +cy c ; so the requirement c < 2 becomes here c > -2; besides, this also changes the monotonicity of γ.)

Let us clear all doubts by lling the gap in their proof, that is the beginning where they construct their subsolution.

• Case |c| < 2: For all > 0, let y the positive solution of -y -cy = y(1 -y) in (0, ), y(0) = y( ) = 0.

According to Berestycki [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF] (theorem 4), such a solution exists if and only if the Dirichlet principal eigenvalue of the operator -L on (0, ) is negative: λ 1 -L, (0, ) < 0, where L is the operator associated with the linearized equation near y = 0:

Lφ := φ + cφ + φ. It is easy to compute: λ 1 -L, (0, ) = -1 + c 2 /4 + π 2 / 2 .
Hence, when |c| < 2, one has λ 1 -L, (0, ) < 0 when is large enough and thus we can construct y . Vice-versa, if |c| ≥ 2, λ 1 -L, (0, ) > 0 and the solution does not exist (whereas Du and Lin claim it does for all c ≤ 0). • Case c ≥ 2: It suces to remark that, for example, if y 1 is a solution of the previous Dirichlet problem for some c 1 ∈ (-2, 2), then y 1 is a subsolution for the Dirichlet problem with any speed c > c 1 .

In either case, the subsolution is now properly constructed and we can continue the proof as in [START_REF] Du | Spreading-vanishing dichotomy in the diusive logistic model with a free boundary[END_REF] and conclude.

Remark 3.17. We need to change a bit u and v before pursuing in this direction.

Let us consider ũ : ξ → u (-ξ) and ṽ :

ξ → v d r ξ . ũ is a solution of the problem -ũ + cũ = ũ (1 -ũ) in (0, +∞) ũ (0) = 0
ṽ is a solution of the problem -ṽ -c √ rd ṽ = ṽ (1 -ṽ) in (0, +∞) ṽ (0) = 0

Besides, c ∈ -2 √ rd, 2 so -c > -2 and c √ rd > -2, therefore we can apply the theorem.

Corollary 3.18. For any d > 0, there exists a unique (u, v, c) satisfying the limit problem (and may thereafter be called

(u ∞,d , v ∞,d , c ∞,d )).
Proof. The uniqueness of c implies by the previous theorem the uniqueness of u and v.

Corollary 3.19. The sequences (c k ), (u k ) and (v k ) have a unique limit point each. Hence the pointwise convergence of (c k ) and locally uniform convergence of (u k ) and (v k ) are fully proved and there is no need to consider extracted subsequences anymore.

Proof. Recall that, in any metric space, a sequence whose image is relatively compact and which has a unique limit point converges to this limit point.

Dependency of the invasion speed on the diffusion rates

This last section is where we derive from the limit problem the result: how does the invasion speed c depend on the diusion rate d? Thanks to the convergence of (c k ) to c, we will then be able to extend it to c k (for k large enough).

Theorem 4.1. We have: Proof. This could follow from the continuity of each d → c k,d and the locally uniform convergence, but the continuity of d → c k,d is actually a more dicult problem (and is not solved by Kan-On [START_REF] Kan-On | Parameter dependence of propagation speed of travelling waves for competition diusion equations[END_REF]). Therefore, we prove the continuity of d → c ∞,d directly. Our proof being basically a repetition of the whole previous section of this article, we give only a sketch of it.

• if d = α 2 r , c ∞,d = 0; • if d > α 2 r , c ∞,d ∈ -2 √ rd, 0 ; • if d < α 2 r , c ∞,d ∈ (0, 2
First, let 0 < D 1 < D 2 . We have:

{c ∞,d | d ∈ [D 1 , D 2 ]} ⊂ c ∞,d | d ∈ [D 1 , D 2 ] ∩ α 2 r , +∞ ∪ {0} ∪ c ∞,d | d ∈ ∩ 0, α 2 r ⊂    d∈[D1,D2]∩ α 2 r ,+∞ -2 √ rd, 0    ∪ [0, 2] ⊂ -2 rD 2 , 2 Thus, {c ∞,d | d ∈ [D 1 , D 2 ]} is a relatively compact subset of R. Now, let δ ∈ [D 1 , D 2 ] and (δ n ) n∈N ∈ [D 1 , D 2 ]
N a positive sequence which converges to δ. Up to extraction, (c ∞,δn ) converges to a limit point C.

If C ≤ 0, we translate each couple (u ∞,δn , v ∞,δn ) so that (u ∞,δn (0)) = 1

. If

C > 0, we translate each couple (u ∞,δn , v ∞,δn ) so that (v ∞,δn (0)) = 1 • U and V are continuous, αU -δV is C 1 .

• U and V are positive and have nite limits at innity.

• U V = 0.

• If C ≤ 0, U is not identically null by normalization and V cannot be identically null since if it was, U would be a travelling wave for the Fisher KPP equation with a speed smaller than 2. The same reasoning applies for C > 0 and nally, neither U nor V can be identically null.

• U and V cannot be both null on a compact subset by continuity of (αU -δV ) and a Cauchy Lipschitz's argument. Now we translate back so that sup {ξ ∈ R | U (ξ) > 0} = inf {ξ ∈ R | V (ξ) > 0} = 0 It yields U |R+ = 0, V |R-= 0, -U -CU = U (1 -U ) in (-∞, 0), -δV -CV = rV (1 -V ) in (0, +∞) and αU (0) = -δV (0). Basically, C, U and V verify the exact same problem than c ∞,δ , u ∞,δ and v ∞,δ . By uniqueness, C = c ∞,δ , that is c ∞,δ is the unique limit point of (c ∞,δn ) and eventually c ∞,δn → c ∞,δ . Therefore, d → c ∞,d is indeed continuous.

Conclusion

We have proved our Unity is not strength theorem. Some remaining questions concern the shape of the asymptotic speed: what are the limits when d → 0 or d → +∞? Are there optimal diusion rates so that the invasion of one species or the other is the fastest? And eventually, how fast is the convergence to this asymptotic limit and, for example, is it monotone?

These could be adressed with the knowledge of the derivatives of the speed as a function of k or d. These might be determined analytically thanks to Kan-On formulas [START_REF] Kan-On | Parameter dependence of propagation speed of travelling waves for competition diusion equations[END_REF]. However, we did not manage to compute the sign of these derivatives, that is, the monotonicity of the speed with respect to k or d. We leave it as an open problem.

Theorem 2 . 1 .

 21 (d → c k,d ) k>1 converges locally uniformly in (0, +∞) to a continuous function d → c ∞,d which satises:

Lemma 3 . 2 .

 32 uv = 0.Proof. Multiplying by a test function ϕ ∈ D (R) and integrating the equation for u k yields:

  Proof. Multiply the equation for u k by α and substract to it the one for v k . The lefthandside converges trivially in D (R). The right-handside converges by dominated convergence. Lemma 3.5. u, v ∈ C (R) and αu -dv ∈ C 1 (R).

  Proof. The continuity of u and v is immediate thanks to the continuity of each u k and v k and the locally uniform convergence. Let a, b ∈ R such that a < b and I a : C ([a, b]) → C ([a, b]) dened by I a (f ) : x → ´x a f . By continuity of u and v, it is quite obvious that the function αcu -cv + I a (αu (1 -u) -rv (1 -v)) -(αcu (a) -cv (a)) is continuous. But, thanks to the previous lemma, it is also equal in D ((a, b)) to -αu + dv up to an additive constant. Therefore -αu + dv is a well-dened function of C ([a, b]).

Remark 3 . 8 .

 38 We already knew that uv = 0 everywhere. Thus the previous lemma ensures that, for any a < b, u |[a,b] = v |[a,b] = 0 is not possible; one of the two densities has to be positive whereas the other has to be null.

  The equality -αu (0) = dv (0) rewrites αγ (-c) = √ rdγ c √ rd . Now we consider the two functions x → αγ (-x) and x → √ rdγ x √ rd . They necessarily have an intersection point since c exists. But as they are respectively decreasing and increasing, this intersection point is unique.

Remark 3 .

 3 20. It is now clear that the sum up theorem of the previous section gives sucient but far from necessary conditions for uniqueness. For any c, u and v are unique i they are positive and satisfy points 4, 5, 8 and 9 and then the uniqueness of c is just a consequence of point 10.Proposition 3.21. The convergence of(d → c k,d ) k>1 to d → c ∞,d is locally uniform.Proof. Actually, one can see easily that the whole proof of pointwise convergence of (d → c k,d ) k>1 holds if we do not x a priori d. It suces to have d ∈ [D 1 , D 2 ], with D 2 > D 1 > 0 xed, so that we can replace bounds like -2 √ rd by -2 √ rD 2 .

γ

  ). Proof. The sign of c ∞,d is actually a simple consequence of the relation αγ (-c) = √ rdγ c √ rd . Indeed, let us prove that rd < α 2 implies c ∞,d > 0. Indeed, if rd < α 2 , then √ rd α < 1 and as γ c √ rd > 0, we get is increasing, c √ rd > -c, which clearly implies that c > 0. The case rd > α 2 is similar. If rd = α 2 , the relation becomes γ (-c) = γ c √ rd . An obvious zero of x → γ (-x) -γ x √ rd is 0, and by monotonicity it is unique, hence c = 0. Proposition 4.2. The function d → c ∞,d is continuous in (0, +∞).

2 .

 2 In either case, {u ∞,d | d ∈ [D 1 , D 2 ]} and {v ∞,d | d ∈ [D 1 , D 2 ]} are relatively compact in each C (K i) by Ascoli's theorem, and, up to extraction, (u ∞,δn ) and (v ∞,δn ) converge locally uniformly. Let U and V be their limits.• We have -αU + δV -αCU + CV = αU (1 -U ) -rV (1 -V ) in D (R).

  2. This result is profoundly unexpected! It does not suce to compare d to 1 or α to 1. v can loose even if r is large and u can loose even if α is large, for example. This should yield interesting insight into ecological applications.2.3. Well-posedness and regularity of the problem. Theorem 2.3. For any k > 1, there exists a unique c k such that there exist solutions u k and v k of the problem (2.1). It is enforced that

 Lemma 3.11.We have:

• u ∈ C ∞ ((-∞, 0) ∪ (0, +∞)),

• v ∈ C ∞ ((-∞, 0) ∪ (0, +∞)), Besides, we can extend u and v by continuity on the left and on the right respectively and obtain u (0) = lim ξ→0,ξ<0 u (ξ) and v (ξ v ) = lim ξ→0,ξ>0 v (ξ) which are nite and satisfy -αu (0) = dv (0) > 0.

Proof. u is identically zero on (0, ∞) so u |(0,+∞) is trivially C ∞ . In (-∞, 0), it is a weak, and then regular (same routine), solution of u + cu + u (1 -u) = 0.

Eventually, just recall that αu -dv ∈ C 1 (R). If its derivative at 0 is zero, by the same kind of Cauchy Lipschitz reasoning, u = v = 0 everywhere. Remark 3.12. The relation αu (0) + dv (0) = 0 is essentially the free boundary condition obtained by Nakashima and Wakasa in [START_REF] Nakashima | Generation of interfaces for LotkaVolterra competitiondiusion system with large interaction rates[END_REF]. Lemma 3.13. lim -∞ u = 1 and lim +∞ v = 1.

Proof. Same as before. Lemma 3.14

We know from Fisher and KPP [START_REF] Kolmogorov | Étude de l'équation de la diusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] that c = -2 √ rd is the maximal speed for wich there exists a travelling wave v KP P positive, going from 0 at -∞ to 1 at +∞, which satises -dv KP P -cv KP P = rv KP P (1 -v KP P )

We normalize by xing v KP P (ξ ) = 1 2 . Let f = v KP P -v. First, we can easily check that f is in C (R)∩C ∞ ((-∞, 0) ∪ (0, +∞)) and satises

We can therefore apply the maximum principle to the operator

Since lim +∞ f = 0, it gives us that f (ξ) ≤ 0 for any ξ ∈ (ξ , +∞). But we can also apply the minimum principle to the same operator, and we eventually get that f is identically zero in (ξ , +∞). This way, f (ξ ) = 0, hence f is identically zero in (0, +∞), which is impossible since f (0) > 0 and f is continuous in R. Theorem 3.15. There exist locally uniform limits u and v of (u k ) and (v k ) respectively. They satisfy:

(1) u, v ∈ C (R) ∩ C ∞ ((-∞, 0) ∪ (0, +∞));

(2) lim ξ→-∞ u (ξ) = 1;

(3) lim ξ→+∞ v (ξ) = 1;

(4) u |R+ = 0;

(5) v |R-= 0; (6) u ≤ 0 in R -with u (0) dened by left-continuity; [START_REF] Du | Spreading-vanishing dichotomy in the diusive logistic model with a free boundary[END_REF] v ≥ 0 in R + with v (0) dened by right-continuity;