Julien Ricaud
email: julien.ricaud@u-cergy.fr

ON UNIQUENESS AND NON-DEGENERACY OF ANISOTROPIC POLARONS

We study the anisotropic Choquard-Pekar equation which describes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium.

Introduction

A polaron describes a quantum electron in a polar crystal. The atoms of the crystal are displaced due to the electrostatic force induced by the charge of the electron, and the resulting deformation is then felt by the electron itself. This coupled system (the electron and its polarization cloud) is a quasi-particle, called a polaron.

When the polaron extends over a domain much larger than the characteristic length of the underlying lattice, the crystal can be approximated by a continuous polarizable medium, leading to the so-called Pekar nonlinear model [START_REF] Pekar | Untersuchungen fiber die Elektronen Theorie der Kristalle[END_REF][START_REF]Research in electron theory of crystals[END_REF]. In this theory, the energy functional is

E V pψq " 1 2 ż R 3 |∇ψpxq| 2 dx ´1 2 ż R 3 ż R 3
|ψpyq| 2 |ψpxq| 2 V px ´yq dy dx, (1.1) where ψ is the wave function of the electron, in units such that the vacuum permittivity, the mass, and the charge of the electron are all normalized to one: 4πε 0 " m e ´" e " 1. While, on the other hand, ´V ‹ |ψ| 2 pxq " ´żR 3 |ψpyq| 2 V px ´yq dy is the mean-field self-trapping potential felt by the electron.

For an isotropic and homogeneous medium, characterized by its relative permittivity (or relative dielectric constant) ε M ě 1, the effective interaction potential is

V pxq " 1 ´ε´1 M |x| . (1.2)
For ε M ą 1, the so-called Choquard-Pekar or Schrödinger-Newton equation ´´∆

2 ´V ‹ |ψ| 2 ¯ψ " ´µψ (1.3)
is obtained by minimizing the energy E V in (1.1) under the constraint ş R 3 |ψ| 2 " 1, with associated Lagrange multiplier µ ą 0. Lieb proved in [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF] the uniqueness of minimizers, up to space translations and multiplication by a phase factor. This ground state Q is positive, smooth, radial decreasing, and has an exponential decay at infinity. That Q is also the unique positive solution to (1.3) was proved in [START_REF] Ma | Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation[END_REF].

In 2009, Lenzmann proved in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF] that Q is nondegenerate. Namely, the linearization of (1.3),

L Q ξ " ´1 2 ∆ξ `µξ ´`V ‹ |Q| 2 ˘ξ ´2Q pV ‹ pQξqq , (1.4)
has the trivial kernel

ker |L 2 pR 3 q L Q " span tB x1 Q, B x2 Q, B x3 Qu (1.5)
which stems from the translation invariance. This nondegeneracy result is an important property which is useful in implicit function type arguments. Uniqueness and nondegeneracy were originally used in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF] to study a pseudo-relativistic model, and then in [START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF][START_REF] Liu | Regularity, symmetry, and uniqueness of some integral type quasilinear equations[END_REF][START_REF] Nodari | Perturbation method for particle-like solutions of the Einstein-Dirac equations[END_REF][START_REF] Stuart | Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system[END_REF][START_REF] Frank | Symmetry of Bipolaron Bound States for Small Coulomb Repulsion[END_REF][START_REF] Sok | Existence of Ground State of an Electron in the BDF Approximation[END_REF] for other models. The purpose of this paper is to study the case of anisotropic media, for which the corresponding potential is

V pxq " 1 |x| ´1 a detpM ´1q|M 1{2 x| , 0 ă M ď 1, (1.6)
where M ´1 ě 1 is the (real and symmetric) static dielectric matrix of the medium. The mathematical expression is simpler in the Fourier domain:

p V pkq " 4π ˆ1 |k| 2 ´1 k T M ´1k ˙.
The form of the potential V in the anisotropic case is well-known in the physics literature and it has recently been derived by Lewin and Rougerie from a microscopic model of quantum crystals in [START_REF] Lewin | Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystal[END_REF].

From a technical point of view, the fact that V in (1.6) is a difference of two Coulomb type potentials complicates the analysis. For this reason, we will also consider a simplified anisotropic model where V is replaced by

V pxq " 1 |p1 ´Sq ´1x| , 0 ď S ă 1, (1.7)
and S is also a real and symmetric matrix. This simplified potential can be seen as an approximation of the potential (1.6) in the weakly anisotropic regime, that is, when M is close to an homothecy. In this paper, we derive several properties of minimizers of E V and of positive solutions to the nonlinear equation (1.3), when V is given by formulas (1.6) and (1.7).

After some preparations in Section 2, we quickly discuss the existence of minimizers and the compactness of minimizing sequences in Section 3. Then, based on the fundamental non degeneracy result by Lenzmann [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF], we prove in Section 4 the uniqueness and non-degeneracy of minimizers in a weakly anisotropic material. In Section 5, considering back general anisotropic materials, we investigate the symmetry properties of minimizers using rearrangement inequalities. Finally we discuss the linearized operator in Section 6. By using Perron-Frobenius type arguments, we are able to prove that for ψ a positive solution of the so-called Choquard-Pekar equation (1.3) sharing the symmetry properties of V , we have ker L ψ " span tB x ψ, B y ψ, B z ψu à ker pL ψ q |L 2 sym pR 3 q .

(1.8)

Where L 2 sym pR 3 q is the subspace of function in L 2 pR 3 q sharing the symmetry properties of V . For instance, in the general case where the three eigenvalues of M (or S) are distinct from each other and V is decreasing in the corresponding directions, L 2 sym pR 3 q is the subspace of functions that are even in these directions. On the other hand, if exactly two eigenvalues are equal, it is the subspace of cylindrical functions that are also even in the directions of the principal axis.

The main difficulty in proving (1.8) is that the operator L ψ is non-local and therefore the ordering of its eigenvalues is not obvious. The next step would be to prove that ker L ψ |L 2 sym pR 3 q " t0u which we only know for now in the weakly anisotropic regime (Theorem 7 below) and in the radial case (see [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF]). We hope to come back to this problem in the future.

Elementary properties

We define the energy E V as in (1.1) and consider, for all λ ą 0, the minimization problem

I V pλq :" min ψPH 1 pR 3 q ||ψ|| 2 2 "λ E V pψq. (2.1)
Let pe 1 , e 2 , e 3 q be the principal axis of the medium, that is, such that each e i P R 3 is a normalized eigenvector associated with the eigenvalue m i of the real symmetric matrix M , where 0 ă m 1 ď m 2 ď m 3 ď 1 (we demand, in addition, that at least m 2 ă 1), or associated with the eigenvalue s i of the real symmetric matrix S where 0 ď s 3 ď s 2 ď s 1 ă 1 in the simplified model.

We define the map M Þ Ñ V as

t0 ă M ď 1 | M symmetric realu Ñ L 2 pR 3 q `L4 pR 3 q M Þ Ñ V pxq " 1 |x| ´1 a detpM ´1q|M 1{2 x| (2.2)
with, in particular, M " Id Þ Ñ V " 0 which corresponds to the vacuum. And, in the simplified model, S Þ Ñ V is defined as

t0 ď M ă 1 | M symmetric realu Ñ L 2 pR 3 q `L4 pR 3 q S Þ Ñ V pxq " |p1 ´Sq ´1x| ´1 (2.3)
with, in particular, S " 0 Þ Ñ V " V 0 . We denote the isotropic potentials by V c pxq " p1 ´cq|x| ´1, for 0 ď c ď 1, and I V c the associated minimization problem. Both maps are well-defined. Indeed, let V as in (2.2) or (2.3) then one can easily show that there exist a ą b ě 0 such that

@x P R 3 ˚, 0 ď b|x| ´1 ď V pxq ď a|x| ´1 ď |x| ´1. (2.4)
Consequently, V P L 2 pR 3 q`L 4 pR 3 q. Moreover, if we restrict ourselves to 0 ă M ă 1 then there exist a ą b ą 0 such that

@x P R 3 ˚, 0 ă b|x| ´1 ď V pxq ď a|x| ´1 ď |x| ´1. (2.5) Lemma 1. Let M Þ Ñ V be defined as in (2.2), S Þ Ñ V as in (2.
3) and let f, g be two functions in H 1 pR 3 q. Then V ‹ pf gq P W 1,8 and, for any 0 ă α ă 1, we have

(1) locally Lipschitzity of

tα ă M ď 1 | M symmetric realu ˆH1 ˆH1 Ñ W 1,8 pM, f, gq Þ Ñ V ‹ pf gq, (2)
uniform Lipschitzity of

t0 ď S ă α | S symmetric realu ˆH1 ˆH1 Ñ W 1,8 pS, f, gq Þ Ñ V ‹ pf gq.
Proof of Lemma 1. First, for any f P L 2 pR 3 q and g P H 1 pR 3 q, by (2.4) together with Hardy inequality, |V ‹ pf gqpxq| ď p| ¨|´1 ‹ |f g|qpxq ď 2||f || 2 ||∇g|| 2 holds. Consequently, for any f, g P H 1 pR 3 q, we have

||V ‹ pf gq|| W 1,8 ď ||V ‹ pf gq|| 8 `||V ‹ pg∇f q|| 8 `||V ‹ pf ∇gq|| 8 ď 2||f || 2 ||∇g|| 2 `4||∇f || 2 ||∇g|| 2 ď 6||f || H 1 ||g|| H 1 . Thus V ‹ pf gq is in W 1,8
. For the rest of the proof, we denote by ||M || the spectral norm of M and fix an α such that 0 ă α ă 1.

For pS, T q P t0 ď M ă α | M symmetric realu 2 , f P L 2 pR 3 q, g P H 1 pR 3 q and x P R 3 , we have

|pV S ´VT q ‹ pf gqpxq| ď ˇˇˇ| p1 ´T q ´1 ¨| ´|p1 ´Sq ´1 ¨| |p1 ´Sq ´1 ¨||p1 ´T q ´1 ¨| ˇˇˇ‹ |f g|pxq ď ˇˇ" p1 ´T q ´1 ´p1 ´Sq ´1‰ ¨| ¨|2 ‹ |f g|pxq ď ˇˇp1 ´Sq ´1pT ´Sqp1 ´T q ´1¨| ¨|2 ‹ |f g|pxq ď ˇˇˇˇp 1 ´Sq ´1ˇˇ| |T ´S|| ˇˇˇˇp 1 ´T q ´1ˇˇ1 | ¨| ‹ |f g|pxq ď 2p1 ´αq ´2||f || 2 ||∇g|| 2 ||S ´T ||.
Thus, for any f, g P H 1 pR 3 q, we have

||pV S ´VT q ‹ pf gq|| W 1,8 ď 6p1 ´αq ´2||f || H 1 ||g|| H 1 ||S ´T ||,
which concludes the proof of (2). For pM, N q P tα ă M ď 1 | M symmetric realu 2 , we have

M 1{2 ´N 1{2 " π ´1ż 8 o ˆM s `M ´N s `N ˙ds ? s " π ´1ż 8 o 1 s `M pM ´N q 1 s `N ?
s ds, which leads to ˇˇˇˇM Moreover, with a similar computation and since det M, det N ą α 3 , we obtain

ˇˇ?det M ´?det N ˇˇď |det M ´det N | 2α 3{2 .
Thus, for f P L 2 pR 3 q, g P H 1 pR 3 q and x P R 3 , we have

|pV M ´VN q ‹ pf gqpxq| ď 1 ? det N ´1 |f g| ‹ ˇˇˇˇM 1{2 ¨ˇ´ˇN 1{2 ¨ˇM 1{2 ¨ˇˇN 1{2 ¨ˇp xq `ˇˇˇ1 ? det N ´1 ´1 ? det M ´1 ˇˇˇ| f g| ‹ ˇˇM 1{2 ¨ˇˇ´1 pxq ď 2 ? det N ˇˇˇˇM ´1ˇˇ1 {2 ˇˇˇˇN ´1ˇˇ1 {2 ||f || 2 ||∇g|| 2 ˇˇˇˇM 1{2 ´N 1{2 ˇˇˇ2 ˇˇˇˇM ´1ˇˇ1 {2 ||f || 2 ||∇g|| 2 ˇˇ?det N ´?det M ˇď ´||M ´N || `α´1{2 |det N ´det M | ¯α´3{2 ||f || 2 ||∇g|| 2 .
Finally, the determinant being locally Lipschitz, we obtain that M Þ Ñ V ‹ pf gq is locally Lipschitz.

Since M ´1 is real and symmetric, there exists R P Op3q such that

R T M R " diagpm 3 , m 2 , m 1 q
and so, for any x P R 3 , after a simple computation, we have

V pRxq " |x| ´1 ´ˇˇd iag ´pm 1 m 2 q ´1{2 , pm 1 m 3 q ´1{2 , pm 2 m 3 q ´1{2 ¯xˇˇˇ´1 , where 0 ă ? m 1 m 2 ď ? m 1 m 3 ď ? m 2 m 3 ă 1 since m 2 ă 1.
Thus, we can consider, without any loss of generality, that

$ & % M " diagpm 1 , m 2 , m 3 q with 0 ă m 3 ď m 2 ď m 1 ă 1, M Þ Ñ V pxq " 1 |x| ´1 |M ´1x| (2.6)
Similarly, for the simplified model, we can also assume that

V pxq " |diagp1 ´s1 , 1 ´s2 , 1 ´s3 q ´1x| ´1, 0 ď s 3 ď s 2 ď s 1 ă 1. (2.7)
For clarity, from now on we denote by E M (resp. E S ) the energy and by I M pλq (resp. I S pλq) the minimization problem since V depends only on the matrix M (resp. on the matrix S). However, for shortness, we will omit the subscripts when no confusion is possible. Lemma 2. Let ψ P H 1 pR 3 q be a solution of the equation (1.3), for V defined as in (2.6) or in (2.7), then px, y, zq Þ Ñ ψp˘x, ˘y, ˘zq are H 1 pR 3 q-solutions to (1.3).

Proof of Lemma 2. This follows from the symmetry properties of V .

Existence of minimizers

We state in this section the existence of minimizers for the minimization problems (see Appendix in [START_REF] Ricaud | Symétrie et brisures de symétrie pour certains systèmes quantiques non linéaires[END_REF] for the proof). We first give some properties of these variational problems. Lemma 3. Let V be defined as in (2.6) or (2.7) and I be defined as in (2.1). Then Ipλq " λ 3 Ip1q ă 0, if λ ą 0.

(3.1)

Consequently, (1) λ Þ Ñ Ipλq is C 8 on R `,
(2) Ipλq ă Ipλ ´λ1 q `Ipλ 1 q, for any λ et λ 1 such that 0 ă λ 1 ă λ, and, in particular:

(3) Ipλq ă Ipλ 1 q, for any 0 ď λ 1 ă λ.

Proof of Lemma 3. Let ψ P H 1 pR 3 q with ||ψ|| 2 L 2 " 1, then ψ λ :" λ 2 ψpλ¨q P H 1 pR 3 q and ||ψ λ || 2 L 2 " λ and, by a direct computation, E pψ λ q " λ 3 E pψq which leads to Ipλq " λ 3 Ip1q. If we now define ψ t " t 3{2 ψpt¨q and use (2.5), we find that

E pψ t q ď 1 2 ||∇ψ t || 2 L 2 ´b 2 ˇˇˇˇ| ψ t | 2 `|ψ t | 2 ‹ | ¨|´1 ˘ˇˇ2 L 2 ď t 2 2 ||∇ψ|| 2 L 2 ´bt 2 ˇˇˇˇ| ψ| 2 `|ψ| 2 ‹ | ¨|´1 ˘ˇˇ2 L 2 ,
and taking t small enough leads to the claimed strict negativity. The rest follows immediately.

Lemma 4. Let V be defined as in (2.6) or (2.7). Let I be as in (2.1) and let λ ą 0.

Then Iptλq ą tIpλq, for all t P p0, 1q.

Proof of Lemma 4. Let t P p0, 1q. By Lemma 3, 0 ą Iptλq " t 3 Ipλq ą tIpλq.

These two lemmas imply the existence of minimizers and the compactness of minimizing sequences, as stated in the following theorem which gives also some properties of these minimizers.

Theorem 5 (Existence of a minimizer). Let V be as in (2.6) or (2.7) and λ ą 0. Then Ipλq has a minimizer and any minimizing sequence strongly converges in H 1 pR 3 q to a minimizer, up to extraction of a subsequence and after an appropriate space translation.

Moreover for any minimizer ψ, we have

(1) ψ is a H 2 pR 3 q-solution of the Choquard-Pekar equation (1.3) with ´µ " d dλ Ipλq ă 0 being the smallest eigenvalue of the self-adjoint operator H ψ :" ´∆{2 ´|ψ| 2 ‹ V and being simple;

(2) µλ " ´λ d dλ Ipλq " ´3λ 3 Ip1q "

3 2 ||∇ψ|| 2 2 " 3 4 @ V ‹ |ψ| 2 , |ψ| 2 D ; (3.2)
(3) |ψ| is a minimizer and |ψ| ą 0;

(4) ψ " z|ψ| for a given |z| " 1.

See Appendix in [START_REF] Ricaud | Symétrie et brisures de symétrie pour certains systèmes quantiques non linéaires[END_REF] for the proof. Note that for the isotropic potentials V c , Lieb proved several of these results in 1977 [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF] using only the fact that |x| ´1 is radially decreasing.

Uniqueness in a weakly anisotropic material

We recall that the uniqueness of the minimizer, up to phases and space translations, in the isotropic case, was proven by Lieb in [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF]. In this section, we extend this result to the case of weakly anisotropic materials, meaning that we consider static dielectric matrices close to an homothecy.

We first prove the continuity of I M pλq, with respect to pM, λq, which we will need in the proof of uniqueness.

Lemma 6 (Minimums' convergence). Let V be defined as in (2.6) or (2.7), I be defined as in (2.1) and pλ, λ 1 q P `R˚˘2 . Then

I M 1 pλ 1 q Ý ÝÝÝÝÝÝÝ Ñ ||M 1 ´M||Ñ0 |λ 1 ´λ|Ñ0 I M pλq.
Thus, the continuity of the corresponding Euler-Lagrange multiplier, ´µM 1 ,λ 1 , holds as well:

µ M 1 ,λ 1 Ý ÝÝÝÝÝÝÝ Ñ ||M 1 ´M||Ñ0 |λ 1 ´λ|Ñ0 µ M,λ .
Proof of Lemma 6. Let ψ (resp. ψ 1 ) be a minimizer of I M pλq (resp. I M 1 pλq) for a given λ ą 0. First, for any ϕ P H 1 pR 3 q, we have

|E M pϕq ´EM 1 pϕq| " 1 2 ˇˇ@ |ϕ| 2 , |ϕ| 2 ‹ pV ´V 1 q Dˇˇď 1 2 ˇˇˇˇ| ϕ| 2 ‹ pV ´V 1 q ˇˇˇˇ8 ||ϕ|| 2 2 .
Thus, by Lemma 1, M Þ Ñ E M pϕq is Lipschitz for any ϕ P H 1 pR 3 q. Moreover

E M pψq ´EM 1 pψq ď I M pλq ´IM 1 pλq ď E M pψ 1 q ´EM 1 pψ 1 q,
which implies that M Þ Ñ I M pλq is Lipschitz for any λ ą 0.

Thanks to Lemma 3, we conclude the proof of the convergence of I since ˇˇI M pλq ´IM 1 pλ 1 q ˇˇÀ |I M p1q| ˇˇλ 3 ´pλ 1 q 3 ˇˇ`ˇˇˇˇM ´M 1 ˇˇˇˇ.

Then, the equality ´µM,λ " 3λ 2 I M p1q gives the convergence of the µ M 1 ,λ 1 's.

We now give our theorem of uniqueness in the weakly anisotropic case.

Theorem 7 (Uniqueness and non-degeneracy in the weakly anisotropic case). Let λ ą 0. i. Let 0 ă s ă 1. There exists ε ą 0 such that, for every real symmetric 3 ˆ3 matrix 0 ă M ă 1 with ||M ´s ¨Id|| ă ε, the minimizer ψ of the minimization problem I M pλq, for V pxq " |x| ´1 ´|M ´1x| ´1 as in (2.6), is unique up to phase and space translations. ii. Let 0 ď s ă 1. There exists ε ą 0 such that, for every real symmetric 3 ˆ3 matrix 0 ď S ă 1 with ||S ´s ¨Id|| ă ε, the minimizer ψ of the minimization problem I S pλq, for V pxq " |p1 ´Sq ´1x| ´1 as in (2.7), is unique up to phase and space translations. Moreover, in both cases, the minimizer is even along each eigenvectors of M and ker L ψ " span tB x ψ, B y ψ, B z ψu, where L ψ is the linearized operator defined in (1.4).

The proof of this theorem is based on a perturbative argument around the isotropic case, using the implicit functions theorem. The fundamental nondegeneracy result in the isotropic case, proved by Lenzmann in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF], is a key ingredient of our proof.

Proof of Theorem 7. The proof of ii being similar to the one of i, we will only give the latter. Let us fix 0 ă s ă 1, define D :" t0 ă M ă 1 | M symmetric realu and denote by Q the unique positive minimizer of the isotropic minimization problem Ipλq :" I s¨Id pλq for V pxq " V s¨Id pxq " p1 ´sq|x| ´1, which is radial and solves (1.3):

´1 2 ∆Q `µQ ´p|Q| 2 ‹ V qQ " 0, with ||Q|| 2 2 " λ.
There λ is fixed hence is µ :" µ s¨Id,λ ą 0 by Lemma 3.

Step 1: Implicit function theorem and local uniqueness. By Proposition 5 in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF], we know that the linearized operator L Q given by

L Q ξ " ´1 2 ∆ξ `µξ ´`V ‹ |Q| 2 ˘ξ ´2Q pV ‹ pQξqq , (4.1)
acting on L 2 pR 3 q with domain H 2 pR 3 q, has the kernel

ker L Q " span tB x1 Q, B x2 Q, B x3 Qu . (4.2)
Let us define u as

H 1 pR 3 , Rq ˆD u Ñ L 2 pR 3 , Rq pψ, M q Þ Ñ ´`|ψ| 2 ‹ V ˘ψ and G as pker L Q q K ˆD G Ñ H 1 pR 3 , Rq pψ, M q Þ Ñ ψ `p´∆{2 `µM q ´1 upψ, M q,
where pker L Q q K is the orthogonal of ker L Q for the scalar product of L 2 pR 3 q, which we endow with the norm of H 1 pR 3 q, and µ M :" µ M,λ " 3λ 2 I M p1q. We emphasize here that we consider real valued functions, meaning that we are constructing a branch of real valued solutions. Moreover, Gpψ, M q " 0 is equivalent to ´1 2 ∆ψ μψ ´p|ψ| 2 ‹ V qψ " 0. Differentiating with respect to x i , for i " 1, 2, 3, we get L ψ B xi ψ " 0, for i " 1, 2, 3, and thus span tB x ψ, B y ψ, B z ψu Ă ker L ψ .

By the Hardy-Littlewood-Sobolev and Sobolev inequalities, u is well defined. Moreover, splitting upψ, M q ´upψ 1 , M 1 q into three pieces and using (2.5) together with the Hardy inequality, one obtains

ˇˇˇˇu pψ, M q ´upψ 1 , M 1 q ˇˇˇˇL 2 ď ˇˇˇˇV ‹ |ψ| 2 ˇˇˇˇL 8 ˇˇˇˇψ ´ψ1 ˇˇˇˇL 2 `ˇˇp V ´V 1 q ‹ |ψ| 2 ˇˇˇˇL 8 ˇˇˇˇψ 1 ˇˇˇˇL 2 `ˇˇV 1 ‹ `|ψ| 2 ´|ψ 1 | 2 ˘ˇˇL 8 ˇˇˇˇψ ´ψ1 ˇˇˇˇL 2 ď ˇˇˇˇV ‹ |ψ| 2 ˇˇˇˇL 8 ˇˇˇˇψ ´ψ1 ˇˇˇˇL 2 `ˇˇp V ´V 1 q ‹ |ψ| 2 ˇˇˇˇL 8 ˇˇˇˇψ 1 ˇˇˇˇL 2 `2 `||ψ|| H 1 `ˇˇψ 1 ˇˇˇˇH 1 ˘ˇˇψ ´ψ1 ˇˇˇˇ2 L 2 .
Therefore, using Lemma 1, u is locally Lipschitz on H 1 pR 3 , Rq ˆD. Then, since p´∆{2 `µM q ´1 maps L 2 pR 3 q onto H 2 pR 3 q Ă H 1 pR 3 q, G is also well defined. Moreover, since ˇˇˇˇˇˇp ´∆ `νq ´1ˇˇˇL 2 ÑH 2 ď maxt1, ν ´1u (for ν ą 0) and p´∆{2 `aq ´1 ´p´∆{2 `bq ´1 " pb ´aq p´∆{2 `aq ´1 p´∆{2 `bq ´1 , for all a, b ą 0, we have

ˇˇˇˇG pψ, M q ´Gpψ 1 , M 1 q ˇˇˇˇH 1 ď ˇˇˇˇψ ´ψ1 ˇˇˇˇH 1 `ˇˇˇˇp ´∆{2 `µM q ´1 `upψ, M q ´upψ 1 , M 1 q ˘ˇˇˇˇH 1 `|µ M 1 ´µM | ˇˇˇˇp´∆{2 `µM q ´1 p´∆{2 `µM 1 q ´1 upψ 1 , M 1 q ˇˇˇˇH 1 À ˇˇˇˇψ ´ψ1 ˇˇˇˇH 1 `maxt2, pµ M q ´1u ˇˇˇˇu pψ, M q ´upψ 1 , M 1 q ˇˇˇˇL 2 `maxt2, pµ M q ´1u maxt2, pµ M 1 q ´1u ˇˇˇˇu pψ 1 , M 1 q ˇˇˇˇL 2 ˇˇˇˇM 1 ´M ˇˇˇˇ,
which proves that G is also locally Lipschitz.

A simple computation shows that

B ψ upψ, M qξ " ´`|ψ| 2 ‹ V ˘ξ ´2ψ ppψξq ‹ V q , (4.3)
acting on ξ P pker L Q q K , and that

B ψ Gpψ, M q " 1 `p´∆{2 `µM q ´1 B ψ upψ, M q. (4.4)
We claim B ψ Gpϕ, M q, defined from pker

L Q q K ˆD into L `pker L Q q K , L 2 pR 3 , Rq ˘, to be continuous. Indeed, ||B ψ upψ, M qξ|| L 2 ď ||ξ|| L 2 ˇˇˇˇV ‹ |ψ| 2 ˇˇˇˇL 8 `2||ψ|| L 2 ||pψξq ‹ V || L 8 ď 3||ψ|| H 1 ||ψ|| L 2 ||ξ|| L 2 ,
thus B ψ upψ, M qξ P L 2 pR 3 , Rq for any pϕ, M, ξq P pker

L Q q K ˆD ˆpker L Q q K .
Splitting again the term into pieces and using (2.5), for ξ P L 2 pR 3 , Rq, one obtains

ˇˇˇˇB ψ upψ, M qξ ´B1 upψ 1 , M 1 qξ ˇˇˇˇL 2 ď ˇˇˇˇV ‹ `|ψ| 2 ´|ψ 1 | 2 ˘ˇˇL 8 ||ξ|| L 2 `ˇˇ`V ´V 1 ˘‹ |ψ 1 | 2 ˇˇˇˇL 8 ||ξ|| L 2 `2||V ‹ pψξq|| L 8 ˇˇˇˇψ ´ψ1 ˇˇˇˇL 2 `2ˇˇˇˇV ‹ ppψ ´ψ1 qξq ˇˇˇˇL 8 ˇˇˇˇψ 1 ˇˇˇˇL 2 `2ˇˇˇˇp V ´V 1 q ‹ pψ 1 ξq ˇˇˇˇL 8 ˇˇˇˇψ 1 ˇˇˇˇL 2 " O ´ˇˇp ψ, M q ´pψ 1 , M 1 q ˇˇˇˇH 1 ˆD ¯||ξ|| L 2 .
Then, since

ˇˇˇˇB ψ Gpψ, M qξ ´Bψ Gpψ 1 , M 1 qξ ˇˇˇˇH 1 À maxt2, pµ M q ´1u ˇˇˇˇB ψ upψ, M q ´Bψ upψ 1 , M 1 q ˇˇˇˇL 2 `maxt2, pµ M q ´1u maxt2, pµ M 1 q ´1u ˇˇˇˇB ψ upψ 1 , M 1 q ˇˇˇˇL 2 ˇˇˇˇM 1 ´M ˇˇˇˇ, we have ˇˇˇˇˇˇB ψ Gpψ, M q ´Bψ Gpψ 1 , M 1 q ˇˇˇˇˇˇÑ 0, if ˇˇˇˇp ψ, M q ´pψ 1 , M 1 q ˇˇˇˇH 1 ˆD Ñ 0.
This concludes the proof of the continuity of

B ψ Gpϕ, M q from pker L Q q K ˆD into L `pker L Q q K , H 1 pR 3 , Rq ˘.
We now apply the implicit function theorem to G. Indeed, by the definition of pker L Q q K , the restriction of L Q to pker L Q q K has a trivial kernel. On the other hand, the operator p´∆{2 `µM q ´1 B ψ upQ, s ¨Idq is compact on L 2 pR 3 q (see Appendix in [START_REF] Ricaud | Symétrie et brisures de symétrie pour certains systèmes quantiques non linéaires[END_REF]), therefore ´1 does not belong to its spectrum. We deduce from this the existence of the inverse operator

pB ψ GpQ, s ¨Idqq ´1 : RanpGq Ă H 1 pR 3 , Rq Ñ pker L Q q K . (4.5)
Then, by the continuity of G and B ψ G, the existence of pB ψ GpQ s , s ¨Idqq ´1 and since GpQ, s ¨Idq " 0, the inverse function theorem 1.2.1 of [START_REF] Chang | Methods in nonlinear analysis[END_REF] implies that there exist δ, ε ą 0 such that there exists a unique ψpM q P pker L Q q K satisfying:

GpψpM q, M q " 0 for ||M ´s ¨Id|| ď ε and ||ψpM q ´Q|| H 1 ď δ.

Moreover, the map M Þ Ñ ψpM q is continuous. Additionally, ker B ψ GpψpM q, M q " t0u, i.e. ker |pker LQq K L ψ " t0u which leads to dim ker pL ψ q ď 3 since dim ker pL Q q " 3 by (4.2).

We now claim that ψpM q is symmetric with respect to the three eigenvectors of M , te i u i"1,2,3 , and consequently that, for i " 1, 2, 3, B xi ψpM q is odd along e i and even along e j for j ‰ i. Indeed V being symmetric, the eight functions px, y, zq Þ Ñ ψpM qp˘x, ˘y, ˘zq, which are in pker L Q q K , are zeros of Gp¨, M q. If ψpM q were not symmetric with respect to each e i , then at least two of the functions px, y, zq Þ Ñ ψpM qp˘x, ˘y, ˘zq would be distinct functions but both verifying (4.6), since Q is symmetric with respect to each e i , which is impossible by local uniqueness.

Thus the B xi ψpM q's are orthogonal and we have dim span tB x ψ, B y ψ, B z ψu " 3. Since span tB x ψ, B y ψ, B z ψu Ă ker L ψ , this leads to dim ker pL ψ q ě 3. Which proves that ker L ψ " span tB x ψ, B y ψ, B z ψu.

Let us emphasize that, at this point, we do not know the masses ||ψpM q|| 2 2 of those ψpM q. Note also that we could prove here that |ψ| ą 0, since ´µM stays the first eigenvalue by continuity and with a Perron-Frobenius type argument, but we do not give the details here since this fact will be a consequence of Step 2.

Step 2: Global uniqueness. Let pM n q n be a sequence of matrices in D such that M n ÝÑ nÑ8 s ¨Id and let pψ Mn q n be a sequence of minimizers of pI Mn pλqq n which we can suppose, up to phase, strictly positive by Theorem 5 and, up to a space translation (for each M n ), in pker L Q q K . Indeed, for any ψ P H 1 pR 3 q, let us define the continuous function f pτ q :" ş ∇Qp¨qψp¨´τ q which is bounded, by the Cauchy-Schwarz inequality. Then ş f pτ q dτ " ş ψpxq ş ∇Qpx ´τ q dτ dx " 0 since ş ∇Q " 0. Thus, f being continuous, there exists τ such that f pτ q " ş ψpx ´τ q∇Qpxq dx " 0, i.e. ψp¨´τ q P pker L

Q q K since ker L Q " span tB x1 Q, B x2 Q, B x3 Qu.
By continuity of pI Mn pλqq n , given by Lemma 6, pψ Mn q n is a minimizing sequence of I s¨Id pλq. So, by Theorem 5, pψ Mn q n strongly converges in H 1 pR 3 q to a minimizer of I s¨Id pλq, up to extraction of a subsequence. But, since the ψ Mn are positive and in pker L Q q K , they converge to a positive minimizer of I s¨Id pλq in pker L Q q K which is Q.

So, there exists ε 1 ď ε such that if ||M ´s ¨Id|| 8 ď ε 1 , then each ψ Mn verifies Gpψ Mn , M n q " 0, by definition of pψ Mn q n , and ||ψ Mn ´Q|| H 1 ď δ i.e. verifies (4.6). So the ψ Mn are unique (up to phases and spaces translation). Which concludes the proof of Theorem 7.

Moreover, we now know that, in fact, the masses ||ψpM n q|| 2 2 of the unique ψpM n q found in the local result were in fact all equal to λ. We also proved incidentally that our choice of translation to obtain pψ Mn q n Ă pker L Q q K was, in fact, unique.

Rearrangements and symmetries

The goal of this section is to prove that minimizers are symmetric and strictly decreasing in the directions along which V is decreasing, without assuming that V is close to the isotropic case as we did in the previous section. More precisely, we will consider here the general anisotropic case m 3 ď m 2 ď m 1 (resp. s 3 ď s 2 ď s 1 ) and, in particular, the two cylindrical cases m 3 " m 2 ă m 1 (resp. s 3 " s 2 ă s 1 ) and m 3 ă m 2 " m 1 (resp. s 3 ă s 2 " s 1 ). Our main result in this section is Theorem 9 below. As a preparation, we first give conditions for V to be its own Steiner symmetrization.

As in [START_REF] Capriani | The Steiner rearrangement in any codimension[END_REF], for f defined on R n " spante 1 , . . . , e n u, we denote: ' by f ˚its Schwarz symmetrization, for n ě 1; ' by St i1,...,i k pf q its Steiner symmetrization (in codimension k) with respect to the subspace spanned by e i1 , . . . , e i k , for n ě 2 and 1 ď k ă n. Let us remark that the Steiner symmetrization St i1,...,i k pf q of f is the Schwarz symmetrization of the function px i1 , ¨¨¨,

x i k q Þ Ñ f px 1 , ¨¨¨, x n q.
Proposition 8 (Criterion for V to be its own Steiner symmetrization).

(1) Let V be given by (2.6), with 0 ă m 3 ď m 2 ď m 1 ă 1. Then V " St 1 pV q (thus V is e 1 -symmetric strictly decreasing). Moreover, for k P t2, 3u, V " St k pV q (thus V is e k -symmetric strictly decreasing) if and only if

m 3 1 ď m 2 k . (5.1 k ) Moreover, i. if m 3 ă m 2 " m 1 , then V " St 1,2 pV q. Thus V is pe 1 , e 2 q-radial strictly decreasing. ii. if m 3 " m 2 ă m 1 , then V " St 2,3 pV q -thus V is pe 2 , e 3 q-radial
strictly decreasing -if and only if

m 3 1 ď m 2 2 " m 2 3 ; (5.2)
(2) Let V be given by (2.7), with 0 ď s 3 ď s 2 ď s 1 ă 1. Then V " St k pV q (thus V is e k -symmetric strictly decreasing) for k " 1, 2, 3. Moreover, i. if s 3 ă s 2 " s 1 , then V " St 1,2 pV q. Thus V is pe 1 , e 2 q-radial strictly decreasing; ii. if s 3 " s 2 ă s 1 , then V " St 2,3 pV q. Thus V is pe 2 , e 3 q-radial strictly decreasing.

Proof of Proposition 8. Suppose V is given by (2.6), then it obviously has the claimed properties of symmetry and, moreover, the cylindrical ones in cases i. and ii.. So the proof that V is equal to its symmetrization is reduced to the proof of decreasing properties.

For any x ‰ 0 and k " 1, 2, 3, we have

B |x k | V px 1 , x 2 , x 3 q " m ´2 k |x k | pm ´2 1 x 2 1 `m´2 2 x 2 2 `m´2 3 x 2 3 q 3{2 ´|x k | px 2 1 `x2 2 `x2 3 q 3{2 . (5.
3) Thus, V " St k pV q and V is radially decreasing with respect to x k if and only if

0 ď pm ´2 1 ´m´4{3 k qx 2 1 `pm ´2 2 ´m´4{3 k qx 2 2 `pm ´2 3 ´m´4{3 k qx 2 3 a.e. on R 3 which is equivalent to m 1 ď m 2{3 k . Consequently, V " St 1 pV q always holds. If m 3 " m 2 ă m 1 , denoting u " |px 2 , x 3 q|, and computing B u V , we obtain that V " St 2,3 pV q if and only if m 1 ď m 2{3 2 " m 2{3 3 , in which case V is pe 2 , e 3 q-radial decreasing. If m 3 ă m 2 " m 1 , denoting u " |px 1 , x 2 q|, and computing B u V , we obtain that V " St 1,2 pV q if and only if m 3 ď m 2{3 2 " m 2{3 1
, which always holds thus V is pe 1 , e 2 q-radial decreasing.

We now need to prove the strict monotonicity. Thanks to (5.3), ∇V " 0 holds only on measure-zero sets (note that we use the computation but do not use any condition on m 1 , m 2 and m 3 except that they are strictly less than 1). Thus |tV " tu| " 0 for any t P R `and then |tV ˚" tu| " 0 for any t P R `. Hence V ˚is radially strictly decreasing. Same results of strict decreasing hold for Steiner symmetrizations since, as noted before, a Steiner symmetrization is a Schwarz symmetrization on a subspace.

The proof for V given by (2.7) is very similar and easier.

We now state our main result about the symmetries of minimizers.

Theorem 9 (Symmetries of minimizers). Let λ ą 0.

(1) Let V be given by (2.6) and ψ M ě 0 be a minimizer of I M pλq. Then, up to a space translation, ψ M is e 1 -symmetric strictly decreasing. If m 3 1 ď m 2 2 as in p5.1 2 q, then ψ M is also e 2 -symmetric strictly decreasing. Finally, if m 3 1 ď m 2 3 as in p5.1 3 q, then ψ M is additionally e 3 -symmetric strictly decreasing. Moreover, i. if m 3 ă m 2 " m 1 , then ψ M is cylindrical strictly decreasing with axis e 3 . Meaning that ψ M is pe 1 , e 2 q-radial strictly decreasing. If additionally p5.1 3 q holds, then ψ M is cylindrical-even strictly decreasing with axis e 3 . This means that ψ M is cylindrical strictly decreasing with axis e 3 and e 3 -symmetric strictly decreasing; ii. if m 3 " m 2 ă m 1 and m 3 1 ď m 2 2 " m 2 3 as in (5.2), then ψ M is cylindrical-even strictly decreasing with axis e 1 .

(2) Let V be given by (2.7) and ψ S ě 0 be a minimizer of I S pλq. Then, up to a space translation, ψ S is e k -symmetric strictly decreasing for k " 1, 2, 3. Moreover, i. if s 3 ă s 2 " s 1 , then ψ S is cylindrical-even strictly decreasing with axis e 3 ; ii. If s 3 " s 2 ă s 1 , then ψ S is cylindrical-even strictly decreasing with axis e 1 .

To prove the symmetry properties of the minimizers, we need symmetrizations of a minimizer to be minimizers, which is proved in the following lemma.

Lemma 10. Suppose that V , given by (2.6) or by (2.7), verifies one of the symmetric strictly decreasing property (resp. radial strictly decreasing property) described in Proposition 8, and define ψ St the symmetrization of ψ corresponding to this symmetric strictly decreasing property of V .

If ψ is a minimizer then ψ St too. Moreover the following equalities hold i.

||∇ψ|| 2 2 " ˇˇˇˇ∇ ψ St ˇˇˇˇ2 2 , ii. @ |ψ| 2 , |ψ| 2 ‹ V D 2 " @ |ψ St | 2 , |ψ St | 2 ‹ V D 2 .
Proof of Lemma 10. On one hand, since the symmetrization conserves the L 2 norm and ψ is a minimizer, we have E pψq ď E pψ St q. On the other hand, given the Riesz inequality (see [START_REF] Burchard | Cases of equality in the Riesz rearrangement inequality[END_REF]), the fact that the kinetic energy is decreasing under symmetrizations (see Theorem 2.1 in [START_REF] Capriani | The Steiner rearrangement in any codimension[END_REF]) and since V " V St by Proposition 8, we have E pψq ě E pψ St q. So finally Ipλq " E pψq " E pψ St q. Consequently, given (3.2) in Theorem 5 and that minimizers ψ and ψ St have the same Lagrange multiplier µ " ´3λ 2 Ip1q, we immediately obtain both equalities.

Using the analycity of minimizers (Lemma 12) we can now prove the strict monotonicity of Steiner symmetrizations of minimizers.

Lemma 11. Let λ ą 0 and ψ be a real minimizer of Ipλq for V given by (2.6) or by (2.7), then ψ ˚is radially strictly decreasing. Moreover, for any permutation ti, j, ku of t1, 2, 3u, we have i. for any x P spante j , e k u, St i pψqpx, ¨q is radially strictly decreasing, ii. for any x P spante i u, St j,k pψqpx, ¨q is radially strictly decreasing.

Proof of Lemma 11. By Theorem 5, ψ is a solution of (1.3) in H 2 pR 3 , Rq with a real Lagrange multiplier µ. Then, by the following lemma (see the Appendix of [START_REF] Ricaud | Symétrie et brisures de symétrie pour certains systèmes quantiques non linéaires[END_REF] for the proof), ψ is real analytic. Lemma 12. Any ψ P H 2 pR 3 , Rq solution of (1.3) for µ P R is analytic.

Thus |tψ " tu| " 0 for any t P R `and this is equivalent to |tψ ˚" tu| " 0 for any t P R `. Hence ψ ˚is radially strictly decreasing.

Given that for any 1 ď k ă 3 and any x P R 3´k , ψpx, ¨q is analytic and since a Steiner symmetrization is a Schwarz symmetrization, we obtain ii. and iii. by the same reasoning to ψpx, ¨q.

Finally, to prove our Theorem 9 on the symmetries of minimizers, we need a result on the case of equality in Riesz' inequality for Steiner's symmetrizations. We emphasize that different Steiner symmetrizations do not commute in general. However, if the Steiner symmetrizations are made with respect to the vectors of an orthogonal basis then the radial strictly decreasing properties are preserved.

For shortness, we write u Stk :" St k puq and, in cylindrical cases, u St1,2 :" St 1,2 puq and u St2,3 :" St 2,3 puq.

Proposition 13 (Steiner symmetrization: case of equality for g strictly decreasing). Let f, g, h be three measurable functions on R 3 such that g ą 0 and f, h ě 0 where 0 ‰ f P L p pR 3 q, with 1 ď p ď `8, and 0 ‰ h P L q pR 3 q, with 1 ď q ď `8. Define Jpf, g, hq "

1 2 ż R 3 ż R 3
f pxqgpx ´yqhpyq dx dy ď 8.

(1) Let pi, j, kq be a permutation of p1, 2, 3q and J `f Sti , g, h Sti ˘ă 8. If for any px j , x k q P R 2 the functions g, f Sti and h Sti are all strictly decreasing with respect to

|x i |, then Jpf, g, hq " J `f St i , g, h St i ˘ô D a P R 3 , # f " f Sti p¨´aq, h " h St i p¨´aq,
a.e. on R 3 .

(2) Let pi, j, kq be a permutation of p1, 2, 3q and J `f St j,k , g, h St j,k ˘ă 8. If for any x i P R the functions g, f St j,k and h St j,k are all radially strictly decreasing with respect to px j , x k q, then Jpf, g, hq

" J `f St j,k , g, h St j,k ˘ô D a P R 3 , # f " f Stj,k p¨´aq, h " h St j,k p¨´aq.
a.e. on R 3 .

(3) Let St and St 1 be two Steiner symmetrizations, acting on two orthogonal directions, T " St 1 ˝St and J `f T , g, h T ˘ă 8. If the functions g, f St , h St are all radially strictly decreasing in the direction (or the plane) of St, and g, f St 1 and h St 1 are all radially strictly decreasing in the direction (or the plane) of St 1 , then

Jpf, g, hq " J `f T , g, h T ˘ô D a P R 3 , # f " f T p¨´aq, h " h T p¨´aq.
a.e. on R 3 .

Proof of Proposition 13. The implications ð all follow from a simple changes of variable. We show the implications ñ and start with (1). Define, for any permutation pi, j, kq of p1, 2, 3q and any px j , x 1 j , x k , x 1 k q P R 4 , the functions

J i pf, g, hqpx j , x 1 j , x k , x 1 k q " 1 2 ż R ż R f pXqgpX ´X1 qhpX 1 q dx i dx 1 i ,
where X " px 1 , x 2 , x 3 q and X 1 " px 1 1 , x 1 2 , x 1 3 q. We claim that for almost all px j , x 1 j , x k , x 1 k q P R 4 , we have J i pf, g, hqpx j , x 1 j , x k , x 1 k q " J i pf St i , g, h St i qpx j , x 1 j , x k , x 1 k q. Indeed, assume that there exists a non-zero measure set E Ă R 2 ˆR2 such that J i pf, g, hqpy, y 1 q ‰ J i pf Sti , g, h Sti qpy, y 1 q for any py, y 1 q P E. Thus, by Riesz inequality on R, J i pf, g, hq ă J i pf St i , g, h St i q necessarily holds on E, since g " g St i , and consequently Jpf, g, hq ă Jpf Sti , g, h Sti q, reaching a contradiction.

We now use the following result of Lieb [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF]:

Lemma 14 ([9, Lemma 3]: Case of equality in Riesz' inequality for g strictly decreasing). Suppose g is a positive spherically symmetric strictly decreasing function on R n , f P L p pR n q and h P L q pR n q are two nonnegative functions, with p, q P r1; `8s, such that Jpf ˚, g, h ˚q ă 8. Then Jpf, g, hq " Jpf ˚, g, h ˚q ñ D a P R n , f " f ˚p¨´aq and h " h ˚p¨´aq a.e..

Thus, for almost all py, y 1 q P R 2 ˆR2 , there exists a i py, y 1 q P R such that f py, x i q " f St i py, x i ´ai py, y 1 qq and hpy 1 , x i q " h St i py 1 , x i ´ai py, y 1 qq, for almost all x i P R. Using now the assumed strict monotonicity of f Sti py, ¨q and h Sti py 1 , ¨q, it follows that a i does not depend on py, y 1 q, and (1) is proved.

The case (2) is very similar, defining this time J j,k pf, g, hqpx i , x 1 i q "

1 2 @ f p¨, x i q, gp¨, x i ´x1 i q ‹ hp¨, x 1 i q D L 2 pR 2 q , @px i , x 1 i q P R 2 . We now prove (3). Let St be one of the Steiner's symmetrization described (1) and (2) and the same for St 1 . We claim that J St pf, g, hq " J St pf St , g, h St q and J St 1 pf, g, hq " J St 1 pf St 1 , g, h St 1 q, a.e.. Indeed, Riesz inequality gives Jpf, g, hq ď Jpf St , g, h St q ď Jpf T , g, h T q. Since first and third terms are equal, the three of them are. From the first equality, there exists a P R ℓ (ℓ " 1, 2) such that f " f St p¨´a, ¨q and h " h St p¨´a, ¨q. Then, since St and St 1 act on orthogonal directions, we have Jpf T , g, h T q " J ´f St 1 p¨`a, ¨q, g, h St 1 p¨`a, ¨q¯" Jpf St 1 , g, h St 1 q and so the second claim holds true too. Then, for almost every y :" px, zq P R 3 , we have

$ & % f T `y ´pa 1 , aq ˘" `f St px ´a, ¨q˘S t 1 pz ´a1 q " f St 1 px, z ´a1 q " f px, zq " f pyq,
h T `y ´pa 1 , aq ˘" `hSt px ´a, ¨q˘S t 1 pz ´a1 q " h St 1 px, z ´a1 q " hpx, zq " hpyq.

We now have all the ingredients to prove Theorem 9.

Proof of Theorem 9. Let ψ be a minimizer and ψ St one (or a composition) of its Steiner symmetrizations with a direction (or a plane) for which V " V St . We take f " h " |ψ| 2 P and g " V . So we have f " h ą 0 (thanks to Theorem 5), g ą 0 (thanks to (2.5)) and Jpf St , V, f St q finite. Indeed by the Hardy-Littlewood-Sobolev inequality and (2.5),

Jpf St , V, f St q À ˇˇˇˇf St ˇˇˇˇ2 6{5 " ||f || 2 6{5
ă `8 since f P H 1 pR 3 q. Moreover, the assumption on the m k 's gives that g " g St is radially strictly decreasing by Proposition 8, and the strict monotonicity of f St " h St is obtained by Lemma 11. Finally, by Lemma 10, ψ St is a minimizer and Jp|ψ| 2 , V, |ψ| 2 q " J `p|ψ| 2 q St , pV q St , p|ψ| 2 q St ˘" J `p|ψ| 2 q St , V, p|ψ| 2 q St ˘.

By Proposition 13, there exists a such that |ψ| 2 " p|ψ| 2 q St p¨´aq " p|ψ| St q 2 p¨´aq holds a.e. thus ψ " ψ St p¨´aq since ψ ě 0. This concludes the proof of Theorem 9.

Study of the linearized operator

In this section we study the linearized operator L Q , on L 2 pR 3 q with domain H 2 pR 3 q, associated with the Euler-Lagrange equation ´∆Q `Q ´p|Q| 2 ‹ V qQ " 0 (1.3), which is given by

L Q ξ " ´∆ξ `ξ ´`V ‹ |Q| 2 ˘ξ ´2Q pV ‹ pQξqq , (6.1)
and we give partial characterization of its kernel. We first consider the true model (2.6) for which, following the scheme in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF], we will use a Perron-Frobenius argument on subspaces adapted to the symmetries of the problem. The main difficulty will stand in dealing with the non-local operator Q pV ‹ pQξqq and, in particular, with proving that this operator is positivity improving. The fundamental use of Newton's theorem in the proof of this property in the isotropic case does not work here, therefore we need a new argument. Our proof will rely on the conditions (5.1 k )'s for which V is e k -symmetric strictly decreasing for each k (see Proposition 8). Then we discuss in a similar way the cylindrical case for the simplified model (2.7), which will need another argument.

6.1. The linearized operator in the symmetric decreasing case. We consider the general case for V , given by (2.6), verifying the three conditions (5.1 k ), for k " 1, 2, 3, and define the subspaces of L 2 pR 3 q L 2 τx,τy,τz :"

$ ' &
' % f P L 2 pR 3 q ˇˇˇˇˇˇf p´x, y, zq " τ x f px, y, zq, f px, ´y, zq " τ y f px, y, zq, f px, y, ´zq " τ z f px, y, zq

, / . / - , (6.2)
obtained by choosing τ x , τ y , τ z P t˘1u. We prove the following theorem which is basically saying that the kernel of the linearized operator around solutions is reduced to the kernel on functions that are even in all three directions.

Theorem 15. Let V , be given by (2.6), verifying (5.1 k ), for all k, and let Q be a positive and symmetric strictly decreasing (with respect to each e k separately) solution of (1.3). Then

ker L Q " span tB x Q, B y Q, B z Qu à ker pL Q q |L 2 `,`,`. (6.3)
For instance, Q could be a minimizer for I M pλq.

The proof of this result is inspired by Lenzmann's proof in [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF] of the fundamental similar result for the linearized operator in the radial case which corresponds to m 1 " m 2 " m 3 . In that case, Lenzmann proved that ker pL Q q |L 2 `,`,`" t0u. Note that by the result of Section 4, we know that this is still true in the weakly anisotropic case. Moreover, a theorem similar to Theorem 15 holds true for the simplified model (2.7) (with no conditions on the matrix S) but we do not state it here for shortness.

The rest of this Section 6.1 being dedicated to the proof of the theorem, let V and Q verify the assumptions of Theorem 15 for the entire Section 6.1.

6.1.1. Direct sum decomposition. First, one can easily verify that L Q stabilizes the spaces L 2 τx,τy,τz . Let us then introduce the direct sum decomposition

L 2 pR 3 q " L 2 x´' L 2 x`" L 2 y´' L 2 y`" L 2 z´' L 2 zẁ here $ ' ' ' ' ' ' & ' ' ' ' ' ' % L 2 x´: " à τy,τz"˘L 2
´,τy,τz , L We claim that those spaces -with corresponding projectors P x´, P x`, P y´, P y`, P z´a nd P z`-reduce the linerized operator L Q (see [START_REF] Teschl | Mathematical Methods in Quantum Mechanics[END_REF] for a definition of reduction), where P x˘ψ pr, ϕ, zq " ψpx, y, zq ˘ψp´x, y, zq 2 and similarly for the other projections. The reduction property is straightforward for ´∆ `1 ´`V ‹ |Q| 2 ˘. Moreover, since Q is even in x, we have

V ‹ pQP x˘ψ q " V ‹ pQϕq ˘V ‹ pQϕp´¨, ¨, ¨qq 2
" V ‹ pQϕq ˘rV ‹ pQϕqsp´¨, ¨, ¨q 2 " P x˘r V ‹ pQψqs
and, Q being also even in y and in z, we obtain the result for the other projections. Thus we can apply [21, Lemma 2.24] which gives us that

L Q " L xQ ' L xQ " L yQ ' L yQ " L zQ ' L zQ ,
with the six operators being self-adjoint operators on the corresponding L 2 pR 3 q spaces with domains P w H 2 pR 3 q, and w P tx´, x`, y´, y`, z´, z`u. Note that P x´H 2 pR 3 q " H 2 x´p R 3 q :" H 2 pR 3 q X L 2 x´p R 3 q and similarly for P y´a nd P z´. Let us then redefine for now on the operator L xQ (resp. L yQ and L zQ ) by restricting it to x-odd (resp. y-odd and z-odd) functions through the isomorphic identifications

L 2 x´p R ˚ˆR 2 q « L 2 x´p R 3 q and H 2 x´p R ˚ˆR 2 q « H 2 x´p R 3 q.
Thus, L xQ , as an operator on L 2 x´p R ˚ˆR 2 q with domain H 2 x´p R ˚ˆR 2 q, can be written

L xQ " ´∆ `1 `Φp´q `Wp´q
where the strictly negative multiplication local operator, on R ˚ˆR 2 , is

Φ p´q px, Y q " ´`V ‹ |Q| 2 ˘px, Y q " ´żR ˚ˆR 2 Q 2 px 1 , Y 1 qrV px ´x1 , Y ´Y 1 q `V px `x1 , Y ´Y 1 qs dY 1 dx 1
and the non-local term W p´q , on R ˚ˆR 2 , is

pW p´q f qpx, Y q " ´2Qpx, Y qż R ˚ˆR 2 Qpx 1 , Y 1 qf px 1 , Y 1 qrV px ´x1 , Y ´Y 1 q ´V px `x1 , Y ´Y 1 qs dY 1 dx 1 .
The same properties hold for L yQ and L zQ .

The key fact to deal with the non-local operator, in order to adapt Lenzmann's proof to anisotropic case, is the positivity improving property of ´Wp´q .

Lemma 16. The operator ´Wp´q is positivity improving on L 2 x´p R ˚ˆR 2 q. Proof of Lemma 16. Since X Þ Ñ V pX, Y q is |X|-strictly decreasing, due to conditions (5.1 k ), and x `x1 ą |x ´x1 | on pR ˚q2 , we obtain, for x, x 1 ą 0 and pY, Y 1 q P `R2 ˘2, that V px ´x1 , Y ´Y 1 q ´V px `x1 , Y ´Y 1 q ą 0. Moreover Q ą 0. Thus, ´Wp´q is positivity improving on L 2 x´p R ˚ˆR 2 q.

6.1.2. Perron-Frobenius property. We can now prove that the three operators L xQ , L yQ and L zQ verify a Perron-Frobenius property.

Proposition 17 (Perron-Frobenius properties). The operators L xQ , L yQ and L zQ are self-adjoint on L 2 ´pR ˚ˆR 2 q with domain H 2 ´pR ˚ˆR 2 q and bounded below. Moreover they have the Perron-Frobenius property: if λ x0 (resp. λ y0 and λ z0 ) denotes the lowest eigenvalue of L xQ (resp. L yQ and L zQ ), then λ x0 (resp. λ y0 and λ z0 ) is simple and the corresponding eigenfunction ψ x0 (resp. ψ y0 and ψ z0 ) is strictly positive.

Proof of Proposition 17. We follow the structure of the proof of [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF]Lemma 8]. Moreover, we only write the proof for L xQ which we denote L Q for simplicity. The argument is the same for the other directions.

Self-adjointness. We have Q P H 2 pR 3 q Ă C 0 pR 3 q X L 2 pR 3 q X L 8 pR 3 q and, by (2.5), V ‹ |Q| 2 is in L 4 pR 3 q X L 8 pR 3 q since V " V 2 `V4 P L 2 pR 3 q `L4 pR 3 q. Defining, for any f P L 2 ´pR ˚ˆR 2 q, f P L 2 ´pR have 2xf, gy L 2 ´pR ˚ˆR 2 q " x f , gy L 2 ´pR 3 q and so Φ p´q `1 is bounded on L 2 ´pR ˚ˆR 2 q.

Moreover, by Young inequalities, for any ξ P L 2 ´pR ˚ˆR

2 q, ||V ‹ pQ ξq|| L 8 ď p||V 4 || L 4 ||Q|| L 4 `||V 2 || L 2 ||Q|| L 8 q || ξ|| L 2
holds. Thus, for p P r2, 8s, we have

ˇˇˇˇW p´q ξ ˇˇˇˇL p pR ˚ˆR 2 q ď 2||Q|| L p pR ˚ˆR 2 q ||V ‹ pQ ξq|| L 8 pR ˚ˆR 2 q ď ||Q|| L p ||V ‹ pQ ξq|| L 8
and W p´q ξ P L 2 ´pR ˚ˆR 2 q X L 8 pR ˚ˆR 2 q. Finally, 1 `Φp´q `Wp´q and, thus, L Q are bounded below on L 2 ´pR ˚ˆR 2 q. Finally, we deduce the self-adjointness of the operator

L Q on L 2 ´pR ˚ˆR 2 q with domain H 2 ´pR ˚ˆR 2 q from the self-adjointness of the operator L Q on L 2 ´pR 3 q with domain H 2 ´pR 3 q.
Positivity improving. We know (see [10] for example) that p´∆ `µq ´1ξpX q " 1 4π

ż R 3
e ´?µ|X´Y | |X ´Y | ξpY q dY , @µ ą 0, @ξ P L 2 pR 3 q.

Consequently, for ξ P L 2 ´pR ˚ˆR 2 q and px, xq P R ˚ˆR 2 , we have

p´∆ `µq ´1ξpx, xq " 1 4π ż R ˚ˆR 2 "
e ´?µ|px´y,x´ỹq| |px ´y, x ´ỹq| ´e´?µ|px`y,x´ỹq| |px `y, x ´ỹq|  ξpy, ỹq dy dỹ.

Thus, with the same arguments as in the proof of Lemma 16, p´∆`µq ´1 is positivity improving on L 2 ´pR ˚ˆR 2 q for all µ ą 0. Moreover, ´pΦ p´q `Wp´q q is positivity improving on L 2 ´pR ˚ˆR 2 q since ´Φp´q is a positive multiplication operator and ´Wp´q is positivity improving by Lemma 16. Then similarly to the proof of [7, Lemma 8], for µ " 1,we have ´LQ `µ¯´1 " p´∆ `µ `1q ´1 ¨`1 `pΦ p´q `Wp´q qp´∆ `µ `1q

´1˘´1

.

Since pΦ p´q `Wp´q q is bounded, we have ˇˇˇˇp Φ p´q `Wp´q qp´∆ `µq ´1ˇˇL 2 ´pR ˚ˆR 2 q ă 1, for µ large enough. This implies, for µ " 1, by Neumann's expansion that ´LQ `µ¯´1 " p´∆ `µ `1q

´1 8 ÿ p"0 " ´pΦ p´q `Wp´q qp´∆ `µ `1q ´1‰ p ,
which is consequently positivity improving on L 2 ´pR ˚ˆR 2 q for µ " 1.

Conclusion.

We choose µ " 1 such that pL Q `µq ´1 is positivity improving and bounded. Then, by [15, Thm XIII.43], the largest eigenvalue sup σppL Q `µq ´1q is simple and the associated eigenfunction

ψ 0 P L 2
´`R ˚ˆR 2 ˘is strictly positive. Since, for any ψ P L 2

´`R ˚ˆR 2 ˘, having ψ being an eigenfunction of L Q for the eigenvalue λ is equivalent to having ψ being an eigenfunction of pL Q `µq ´1 for the eigenvalue pλ `µq ´1, we have proved Proposition 17.

´∆Q `Q ´p|Q| 2 ‹ V qQ " 0 (1.3), we obtain L Q B x Q " 0. Moreover, Q is positive symmetric strictly decreasing, thus B x Q P L 2
x´p R 3 q, and this shows that L xQ B x Q " 0. Then, Q ą 0 being symmetric strictly decreasing, B x Q ă 0 on R ˚ˆR 2 and, by the Perron-Frobenius property, it is (up to sign) the unique eigenvector associated with the lowest eigenvalue of L xQ , namely λ x0 " 0. Since L xQ acts on L 2 x´: " À τy,τz"˘L

2

´,τy,τz , we obtain

# ker pL Q q |L 2
´,`,`p R 3 q " spantB x Qu;

ker pL Q q |L 2
´,´,`p R 3 q " ker pL Q q |L 2 ´,`,´p R 3 q " ker pL Q q |L 2 ´,´,´p R 3 q " t0u. This the exact same arguments for the two other directions we finally obtain that

ker L Q " span tB x Q, B y Q, B z Qu à ker pL Q q |L 2
`,`,`p R 3 q . Which concludes the proof of Theorem 15.

6.2. The linearized operator in the cylindrical case. We now consider the case where the static dielectric matrix has exactly two identical eigenvalues. Obviously, Theorem 15 holds and it tells us that the kernel is reduced to the kernel on functions that are even in the z-direction and even in any direction of the plane orthogonal to z. However, this does not tell us that it is reduced to the kernel on cylindrical functions, which is what we are interested in. Indeed, instead of the kernel of L Q on L 2 `,`,`p R 3 q, we want the remaining term in the direct sum to be the kernel on L 2 rad,`p R 3 q, namely the subset of cylindrical functions that are also even in the direction of their principal axis.

Unfortunately, our method fails to prove it for V given by (2.6) since we are not able to prove a positivity improving property for the non local operator. Therefore, in this section, we will only consider the simplified model where V is given by (2.7).

We use the cylindrical coordinates pr, zq where e z is the vector orthogonal to the plane of symmetry. Namely, e z " e 3 if s 3 ă s 2 " s 1 and e z " e 1 if s 3 " s 2 ă s 1 . We then define the following subspaces L 2 τ pR 3 q :" f P L 2 pR 3 q | f px, y, ´zq " τ f px, y, zq ( , for τ " L2

`pR ˚ˆRq :" f P L 2 pR ˚ˆR, r dr dzq | f pr, ´zq " f pr, zq ( L 2 rad,`p R 3 q :" L 2 `pR ˚ˆRq b 1.

(6.4)

Theorem 18. Let V be given by (2.7) with S having one eigenvalue of multiplicity 2 and let Q be a cylindrical-even decreasing and positive solution of (1.3). Then

ker L Q " span tB x Q, B y Q, B z Qu ' ker pL Q q |L 2 rad,`p R 3 q . (6.5)
For instance, Q could be a minimizer for I S pλq.

Several parts of the proof of this theorem being identical to the ones in the proof of the Theorem 15, we will only give the details for the parts that differ. 6.2.1. Cylindrical decomposition. Since V is cylindrical-even strictly decreasing by Proposition 8 and since minimizers are cylindrical-even strictly decreasing by Proposition 9, L Q commutes with rotation in the plane of symmetry. Let us then introduce the direct sum decomposition

L 2 pR 3 q " L 2 ´pR 3 q ' à ně0,σ"˘L 2 `pR ˚ˆRq b Y σ n , (6.6)
with # Y 0 " p2πq ´1 2 ; Y 0 " 0;

Y ǹ " π ´1 2 cospn¨q; Y ń " π ´1 2 sinpn¨q, for n ě 1. (6.7)
The operator L Q stabilizes L 2 ´pR 3 q and the spaces L 2 `pR ˚ˆRq b Y σ n . Let us immediately decompose the potential V in order to give the fundamental property in the cylindrical case (Proposition 19 below), which is what allows us to adapt the original work of Lenzmann, namely the strict positivity of each zodd terms of the cylindrical decomposition of V . For any r " pr, ϕ, zq P R 3 and r1 " pr 1 , ϕ 1 , z 1 q P R 3 , defining ρ :" pr ´r1 , 0, z ´z1 q and θ :" ϕ ´ϕ1 , we have, as soon as pr 1 , z 1 q ‰ pr, zq:

θ Þ Ñ V pr ´r1 q " 1 b
|p1 ´Sq ´1ρ| 2 `2p1 ´s2 q ´2rr 1 p1 ´cos θq ą 0, (

which is in C 8 pRq, 2π-periodic and even. Thus, for any r ‰ r 1 ,

V pr ´r1 q " 8 ÿ n"0 v n pr, r 1 , z ´z1 qY ǹ pϕ ´ϕ1 q, with v n pr, r 1 , z ´z1 q " ż π ´π V pr ´r1 qY ǹ pθq dθ " 2 ż π 0 V pr ´r1 qY ǹ pθq dθ. (6.9)

Proposition 19. Let V be given by (2.7), the Y ǹ 's by (6.7) and the v n 's by (6.9) for any pn, r, r 1 , z, z 1 q P N ˆR˚ˆR˚ˆR ˆR. Then v n pr, r 1 , z ´z1 q ą 0, @pn, r, r 1 , z, z 1 q P N ˆR˚ˆR˚ˆR ˆR.

Proof of Proposition 19. Defining for r, r 1 ą 0,

m ˘:" d ˆr `r1 1 ´s2 ˙2 `ˆZ 1 ´sz ˙2 ˘dˆr ´r1 1 ´s2 ˙2 `ˆZ 1 ´sz

˙2

" max ϕ´ϕ 1 ˇˇp1 ´Sq ´1pr ´r1 q ˇˇ˘min ϕ´ϕ 1 ˇˇp1 ´Sq ´1pr ´r1 q ˇˇą 0, we note that m `ą m ´and obtain

V pr ´r1 q " 2 m `1 c 1 ´2 m ḿ `cos θ `´m ḿ `¯2
.

We now give the explicit expansion of p1 ´2t cos θ `t2 q ´1{2 in the following lemma.

Lemma 20. For p0, 1q ‰ pθ, tq P R ˆr0, 1s, we have

1 ? 1 ´2t cos θ `t2 " 8 ÿ k"0 β 0,2k t 2k `8 ÿ n"1 8 ÿ k"n β 2n,2k t 2k Y 2n pθq `8 ÿ n"0 8 ÿ k"n β 2n`1,2k`1 t 2k`1 Y 2n`1 pθq. (6.10) with $ ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % β 0,2k " ? 2π `2k k ˘2 2 4k ą 0, 0 ď k; β 2n,2k " 2 ? π `2pk`nq k`n ˘`2pk´nq k´n 24k ą 0, 0 ă n ď k; β 2n`1,2k`1 " 2 ? π `2pk`n`1q k`n`1 ˘`2pk´nq k´n 24k`2 ą 0, 0 ď n ď k.
Proof of Lemma 20. The proof of this lemma is entirely inspired by the original computation of Legendre 1 in his famous mémoire [START_REF] Le Gendre | Recherches sur la figure des planètes[END_REF] where he introduced the polynomials that are nowadays called after him. Let us first rewrite the fraction, for

As proved in the last Section, L 2 ´pR 3 q, with corresponding projectors P ´, reduces L Q . We claim that the spaces L 2

`pR ˚ˆRq b Y σ n , with corresponding projectors P ǹ,σ ψpr, ϕ, zq "

ˆż 2π 0 ψpr, ϕ 1 , zq `ψpr, ϕ 1 , ´zq 2 Y σ n pϕ 1 q dϕ 1 ˙Y σ n pϕq, also reduce L Q . Given that `V ‹ |Q| 2 ˘P L 2 rad,`p R 3 q, d 2 dϕ 2 Y σ n " ´n2 Y σ n and ∆ " B 2 Br 2 `1 r B Br `B2 Bz 2 `1 r 2 B 2 Bϕ 2 , (6.11)
we have

" ´∆ `1 ´`V ‹ |Q| 2 ˘‰ pf Y σ n q " " ´∆pnq `1 ´`V ‹ |Q| 2 ˘‰ pf qY σ n , (6.12)
for any f P L 2 `pR ˚ˆRq, and so belonging to L

V ‹ pQP ǹ1 ,σ 1 c ǹ,σ Y σ n q " P ǹ1 ,σ 1 `V ‹ pQc ǹ,σ Y σ n q ˘,
for any n, n 1 ě 0 and σ, σ 1 " ˘, in order to conclude. We have

" V ‹ pQc ǹ,σ Y σ n q ‰ pr, ϕ, zq " ż R ˚ż π ´π ż R Qpr 1 , z 1 qc ǹ,σ pr 1 , z 1 qY σ n pϕ 1 qV pr ´r1 qr 1 dz 1 dϕ 1 dr 1 " ˜żR ˚żR
Qpr 1 , z 1 qc ǹ,σ pr 1 , z 1 qv n pr, r 1 , z ´z1 qr 1 dz 1 dr 1 ¸Y σ n pϕq.

(6.13)

Then using the parity of v n with respect to its third variable (which is straightforward with (6.9)), we obtain V ‹ pQc ǹ,σ Y σ n q P L 2 `pR ˚ˆRq b Y σ n and the reduction property follows. Thus we can apply [21, Lemma 2.24] which gives us that

L Q " L ´' à ně0,σ"˘L ǹ,σ ,
with L ´" L zQ being the same operator as before and each L ǹ,σ a self-adjoint operator on L 2

`pR ˚ˆRq b Y σ n with domain P ǹ,σ H 2 pR 3 q. For shortness, we omit the Q subscript in the decomposition L Q .

Given (6.12) and (6.13), for any n ě 0 we note L ǹ the operator on L 2

`pR ˚ˆRq such that L ǹ,`p f Y ǹ q " L ǹ pf qY ǹ and L ǹ,´p f Y ń q " L ǹ pf qY ń . This operator is Similarly to the non-cylindrical case, we need to prove that ´Wpnq is positivity improving on L 2 `pR ˚ˆRq and this is where the result of Proposition 19 is needed.

L ǹ " ´∆pnq `1 `Φ `
Lemma 22. For n ě 0, the operator ´Wpnq is positivity improving on L 2 `pR ˚ˆRq.

Proof of Lemma 22. Given the definition (6.14) of ´Wpnq , the fact that the v n 's are strictly positive as soon as r, r 1 ą 0 (by Proposition 19) and that Q ą 0, it follows that ´Wpnq is positivity improving on L 2 `pR ˚ˆRq for any n ě 0.

6.2.2. Perron-Frobenius property. We now prove that the L ǹ 's verify the Perron-Frobenius property.

Proposition 23 (Perron-Frobenius properties). For n ą 0, the L ǹ 's are essentially self-adjointness on C 8 0 pR ˚ˆRq and bounded below. Moreover they have the Perron-Frobenius property: if λ n 0 denotes the lowest eigenvalue of L ǹ , then λ n 0 is simple and the corresponding eigenfunction ψ n 0 is strictly positive.

Proof of Proposition 23. We follow the structure of the proof of [START_REF] Lenzmann | Uniqueness of ground states for pseudorelativistic Hartree equations[END_REF]Lemma 8].

Self-adjointness. We still have V ‹ |Q| 2 P L 4 pR 3 q X L 8 pR 3 q. Moreover, defining f pr, ¨, zq " f pr, zqY ǹ P L 2

`pR ˚ˆRq b Y ǹ Ă L 2 pR 3 q, for any f P L 2 `pR ˚ˆRq, we have xf, gy L 2 `pR ˚ˆRq " x f ,gy L 2 pR 3 q and, consequently, that Φ `1 is a bounded operator on L 2 `pR ˚ˆRq. Then, for any ξ P L 2 `pR ˚ˆRq and p P r2, 8s, we have

ˇˇˇˇW pnq ξ ˇˇˇˇL p pR`ˆRq ď ||Q|| L p ||V ‹ pQ ξq|| L 8 .
Thus W pnq ξ P L 2 `pR ˚ˆRq X L 8 pR ˚ˆRq and, finally, 1 `Φ `Wpnq and, thus, L ǹ are bounded below on L 2 `pR ˚ˆRq. Furthermore, it is known that ´∆pnq is essentially self-adjoint on C 8 0 pR ˚ˆRq provided that n ą 0. Thus, given that 1 `Φ `Wpnq is bounded (so ´∆pnq -bounded of relative bound zero), symmetric (moreover self-ajoint) and that its domain contains the domain of ´∆pnq , we obtain by the Rellich-Kato theorem the essentially self-adjointness of L ǹ on C 8 0 pR ˚ˆRq.

Positivity improving. We claim that e t∆ pnq is positivity improving for all t ą 0 on L 2 pR ˚ˆRq. Indeed we have the explicit formula for the integral kernel e t∆ on R 3 , namely, e t∆ px, yq " p4πtq ´3{2 e ´|x´y| 2 4t " p4πtq ´3{2 e ´r2 `r12 `pz´z 1 q 2 4t e rr 1 2t cospϕ´ϕ 1 q , (6.15) for all x :" pr, ϕ, zq and y :" pr 1 , ϕ 1 , z 1 q. On the other hand we have

e x cos θ " ? 2π 8 ÿ m"0 ? 2 δ tmě1u I m pxqY m pθq, @x P R, (6.16)
where I n pxq " π ´1 ∫ π 0 exppx cos θq cospnθq dθ are the modified Bessel functions of the first kind, that are strictly positive for n ě 0 and x ą 0. From these two relations, we deduce the integral kernel e t∆ pnq acting on L 2 pR `ˆRq and that it is positive, which are the two points of the following lemma.

Lemma 24. Let f P L 2 pR ˚ˆRq, r, r 1 ą 0 and n ě 0. Then the integral kernel e t∆ pnq acting on L 2 pR ˚ˆRq verifies e t∆ pnq ppr, zq, pr 1 , z f pr 1 , z 1 q ˆż π ´π e rr 1

2t cospϕ´ϕ 1 q Y σ n pϕ 1 q dϕ 1 ˙r1 dr 1 dz 1 " ? 2 Which allows to conclude the proof of Lemma 24.

So, for all n ě 0, e t∆ pnq is positivity improving on L 2 pR ˚ˆRq for all t ą 0 and consequently on L 2 `pR ˚ˆRq. Then, by functional calculus, we have `´∆ pnq `µ˘´1 " ż 8 0

e ´tµ e t∆ pnq dt, @ µ ą 0, thus p´∆ pnq `µq ´1 is positivity improving on L 2 `pR ˚ˆRq for all µ ą 0. Moreover we claim that ´pΦ `Wpnq q is positivity improving on L 2 `pR ˚ˆRq since ´Φ is a positive multiplication operator and ´Wpnq is positivity improving on L 2 `pR ˚ˆRq.

The end of the proof uses the exact same arguments as in the proof of the Perron-Frobenius property in the non-cylindrical case (Proposition 17) and, consequently, this ends the proof of Proposition 23. 6.2.3. Proof of Theorem 18. First, using the results of the previous Section, we have ker pL Q q |L 2 ´pR 3 q " ker pL Q q |L 2 ´,`,`p R 3 q " spantB z Qu and

L Q B x Q " L Q B y Q " 0. But now Q is furthermore cylindrical-even, thus B x Q " x r B r Q P L 2
`pR ˚ˆRq b Y 1 and B y Q " y r B r Q P L 2 `pR ˚ˆRq b Y 1 , which implies that L 1 B r Q " 0. Then, Q ą 0 being cylindrical-even strictly decreasing, B r Q ă 0 on R `ˆR and, by the Perron-Frobenius property (Proposition 23), it is (up to sign) the unique eigenvector associated with the lowest eigenvalue of L 1 , namely λ 1 0 " 0. Consequently, B x Q (resp. B y Q) is the unique eigenvector -up, in addition, to rotations in the cylindrical plane -associated with the lowest eigenvalue λ 1,0 " 0 (resp. λ 1,0 " 0) of L 1,`( resp. L 1,´) . To summarize, we know at this point that ker L Q " span tB x Q, B y Q, B z Qu ' ker pL Q q |L 2 rad,`p R 3 q ' à ně2 σ"˘k er pL Q q |L 2 `pR ˚ˆRqbY σ n , and we have to deal with the higher orders. The end of the paper is devoted to the proof that ker L ǹ,σ " t0u, @n ě 2, σ " ˘. (6.18)

For n ě 2, let 0 ă ϕ n P L 2 `pR ˚ˆRq be an eigenvector of L ǹ associated with λ n 0 . Then ϕ n Y ǹ (resp. ϕ n Y ń ) is an eigenvector of L ǹ,`( resp. L ǹ,´) associated to the eigenvalue λ n,0 " λ n 0 (resp. λ n,0 " λ n 0 ). Thus, for n ě 2 and σ " ˘, we have λ rQϕ n spr, zqrQϕ n spr 1 , z 1 q rv n ´v1 s pr, r 1 , z ´z1 qrr 1 dz dz 1 dr dr 1 .

Since Q ą 0 and ϕ n ą 0 (by the Perron-Frobenius property in Proposition 23), in order to prove that λ n,σ 0 ą λ 1,σ 0 , it is sufficient to prove that v n ă v 1 almost everywhere on pR ˚ˆRq 2 . Using the explicit formula (6.8) for V , this is equivalent to prove that T n :" ż π 0 cospnθq ´cos θ a K `2a 2 2 rr 1 p1 ´cos θq dθ ă 0, for a.e. pr, zq, pr 1 , z 1 q P `R˚ˆR ˘2 , where K " |p1 ´Sq ´1pr ´r1 , 0, z ´z1 q| 2 . First, let us remark that the points ! 2 k n´1 π

) kPZ and ! 2 k n`1 π
) kPZ are the zeros of cospn¨q´cosp¨q and that the function g " " K `2p1 ´s2 q ´2rr 1 p1 ´cosp¨qq ‰ ´1{2 is strictly decreasing on s0, πr. Let us define, θ 2tn{2u :" π and, for k an integer in r0, tn{2u ´1s, θ 2k :" 2 k n´1 π and θ 2k`1 :" 2 k`1 n`1 π which are all the zeros of cospn¨q ´cosp¨q in r0, πs, except θ 2tn{2u if n is even. Then, noticing that cospn¨q ´cosp¨q is strictly negative on intervals sθ 2k , θ 2k`1 r, strictly positive on intervals sθ 2k`1 , θ 2k`2 r and that nθ 2k " 2kπ `θ2k , we have Thus we have just proved, for n ě 2 and σ " ˘, that λ n,σ 0 ą λ 1,σ 0 " 0, consequently ker L ǹ,σ " t0u.

This concludes the proof of Theorem 18.

Date:

  December 2, 2014. The author is grateful to M. Lewin for helpful discussions and advices. The author acknowledges financial support from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement MNIQS 258023), and from the French ministry of research (ANR-10-BLAN-0101).

6. 1 . 3 .

 13 Proof of Theorem 15. Differentiating, with respect to x the Euler-Lagrange equation

f pr 1 , z 1 qI n ˆrr 1 2t ˙exp ˆzz 1 2t ˙r1 dr 1

 111 dz 1 Y σ n pϕq.

  L 1 ϕ n D L 2 pR`ˆRq ´@ϕ n , L ǹ ϕ n D L 2 pR`ˆRq ď ´żR`ˆR n 2 ´1 r 2 pϕ n pr, zqq 2 r dr dz `2 ij pR`ˆRq 2

  ´cos θq dθ.If n " 2 or n " 3, this immediately leads to T n ă 0. Otherwise, if n ě 4, we have

  Thus the reduction property follows for ´∆ `1 ´`V ‹ |Q| 2 ˘. Moreover, since V ‹ pQ¨q and P ǹ,σ are linear and using the decomposition

					2 `pR ˚ˆRq b Y σ n , and where
	´∆pnq :"	´B2 Br 2 ´1 r	B Br	´B2 Bz 2	`n2 r 2 .
	ψpr, ϕ, zq " c ´pr, ϕ, zq	`ÿ ně0,σ"˘c	ǹ,σ pr, zqY σ n pϕq,
	we have to prove that

  Wpnqwhere Φ is the strictly negative multiplication local potential, on R ˚ˆR, , z 1 q| 2 v 0 pr, r 1 , z ´z1 qr 1 dz 1 dr 1 ă 0

	Φpr, zq " ´`V ‹ |Q| 2 ˘pr, zq " ´?2π	ż	R ˚ˆR	|Qpr 1

and W pnq is the non-local operator, on R ˚ˆR, pW pnq f qpr, zq " ´2Qpr, zq ż R ˚ˆR Qpr 1 , z 1 qf pr 1 , z 1 qv n pr, r 1 , z ´z1 qr 1 dz 1 dr 1 . (6.

[START_REF]Research in electron theory of crystals[END_REF]

  Proof of Lemma 24. Let f P L 2 pR ˚ˆRq. Using (6.15), for n ě 0, we have `et∆ pf Y σ n q ˘pr, ϕ, zq

	" p4πtq ´3{2	ż	R`ˆR	e	´r2 `r12 `pz´z 1 q 2 4t
					1 qq "	? 4πt 3{2 e 2 ´δ0 n	´r2 `r12 `z2 `z12 4t	I n	ˆrr 1 2t	˙exp	ˆzz 1 2t	˙ą 0.	(6.17)

or Legendre and Laplace, according to a famous paternity controversy.

J. RICAUD p0, 1q ‰ pθ, tq P R ˆr0, 1s:

1 ´2t cos θ `t2 " p1 ´eiθ tq ´1{2 p1 ´e´iθ tq ´1{2 .

Then, since Γp1{2 ´pq " p´4q p p! p2pq! Γp1{2q and using the following expansion

we obtain:

With the definition of the Y ǹ 's, this concludes the proof of Lemma 20.

Defining all the others β p,q 's to be zero, this proves Proposition 19:

for n ě 0, r, r 1 ą 0 and z, z 1 P R. Moreover, for r ‰ r 1 , we have

Remark 21. (The anisotropic potential (2.6)) If we define v n in a similar fashion for the true model based on (2.6), even with the conditions p5.1 k q and p5.2q, the v n 's have no sign for n ě 2, since we have

hich changes sign for n ě 2. This is why our method fails if V is given by (2.6). Note that the strict positivity however holds true for v 0 and for v 1 if r, r 1 ą 0, which is straightforward using (6.9).


Claudio Guidi
email: cguidi@cs.unibo.it

Saverio Giallorenzo

Maurizio Gabbrielli

Towards a Composition-based APIaaS Layer

Keywords: Service-Oriented Computing, Cloud Computing, API, Composition, SaaS, PaaS, APIaaS

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Application Programming Interfaces (APIs) are a standard feature of any application that exposes its functionalities to external invokers. APIs can be composed to create new applications at client or server side. In the former case composition takes the form of a mashup [START_REF] Liu | Towards service composition based on mashup[END_REF], whereas in the latter case it is usually realized within a specific application. Depending on the desired result, developers choose for a server-or client-side composition, although there are some cases in which only a server-side application can comply with the requirements of the project, e.g., : (i) the developer needs to publish the composition as a new API; (ii) the composition has some strategic value, therefore the developer does not want to publish it as a readable script at client-side; (iii) APIs could use API keys and other identification data that must remain private and/or pass only through certified channels like VPNs or SSH; (iv) privacy policies and national agreements might enforce a known, certified, and geographically restricted route for data; (v) API contracts require the user to be part of a specific federated network. For these reasons and since our interest regards API composition and publication, in this work we focus on server-side API composition.

Composition of APIs implies the choice of a set of different technologies that span from the implementation language to the communication and serialisation protocols used by the application. The choice on these technologies it is very important since it strongly af-fects development, deployment, and maintenance of the project. In addition, it is not clear what longterm implications could derive from the choice of a specific set of technologies at design-time. In order to clarify how technology could affect a server-side API composition project, let us consider the example of Fig. 1 which reports the architectural view of a realistic application, dubbed PhotoBlog. PhotoBlog manages the publication of photos and their captions, translated in several languages, on various blogging platforms. PhotoBlog derives from the composition of three APIs: (i) image, for image handling (resize, cut, etc.) and storage; (ii) translation, for the automatic translation of captions; (iii) blogging, for publishing images and captions as posts. Moreover, Pho-toBlog itself exposes its own APIs to client interfaces. Assuming T the technology chosen for the development of the project, the typical implementation of such a composition passes by the following phases:

• documentation analysis: the developer must understand the parameters of interaction of each API with T . Documentation analysis usually consists of reading the companion documentation issued by the API provider;

• API connection: the developer must include, or develop by her own, specific libraries for T to handle communication and serialisation protocols with the composed APIs. Notably, each API could require the support of a different protocol;

• development: the developer implements the composition in T , employing the specific patterns of composition provided by T ;

• deployment: the administrator of PhotoBlog deploys the application on a platform of a provider which supports T . Convenience, diffusion, and support for T is another strong concern which influences its choice;

• maintenance and evolution: the developer has to maintain the project up-to-date wrt new releases of APIs. API updates could require upgrades of T by either the administrator or the provider of the platform, depending on the service-level agreement.

Even in such a simple example as PhotoBlog, composing APIs requires a lot of "boilerplate" work to enable communication with the composed applications. This diverts the focus of the developer from the core composition functionality to the creation of a suitable environment for composition. In addition such an environment requires an exhausting work of maintenance.

What emerges from our brief analysis is that APIs are a commodity of any application, especially when deployed on the cloud, but the heterogeneity of protocols makes interacting with several applications an unaffordable task, if compared to its yield. The costs of setting a suitable environment easily overcome the potential value of the composition and represent an important drawback of programming web and cloud applications. This fact highlights the strong need for tools that enable efficient development of heterogeneous APIs, either in the cloud or on the Internet in general. However, to the best of our knowledge, currently there is no structured programming patterns nor formal theories which deal with API server-side composition in a cloud environment.

In this paper we try to attack this problem by introducing the notion of an API-as-a-Service (API-aaS) layer as a tool which eases the development and deployment of applications based on API compositions. By doing this, we also elicit the desirable general features of such a layer. Moreover, we also provide a proof-of-concept APIaaS prototype based on a service-oriented approach. In our vision, the APIaaS layer is a PaaS layer which provides all the tools for facilitating API composition, in particular by abstracting communication protocols and message formats. Service-oriented orchestration programming could be adopted as a programming reference for abstracting technological details and focusing on composition.

In § 2 we enhance the PhotoBlog example in order to list all the problems related on API composition. In § 3 we list all the features of an APIaaS layer should provide, whereas in § 4 we present our proof-of-concept APIaaS prototype which exploits a service-oriented approach both for the programming and the architectural level. Finally, in § 5 we discuss some related work, drawing our conclusions in § 6.

PROBLEMS WITH API COMPOSITION

In this section we list and discuss the main difficulties related to the development of applications that use and publish APIs. We extend the example of PhotoBlog, assuming T the technology selected for its implementation.

Documentation. Documentation regards the retrieval and integration of documents that describe APIs for the given technology T . In our example such a step must be performed for the three composed APIs, namely image, translation and blogging. Usually the documentation is described for RESTful JSON messages so, for each service, the developer has first to check the documentation in order to extract information on the identifier, the type, and the semantics of the values of request-response operations (if any).

Then the documentation of specific libraries for T must be analysed, when supplied. This process alone can take an important amount of time [START_REF] Erl | Service-Oriented Architecture: A Field Guide to Integrating XML and Web Services[END_REF] when compared to that one of the development. Moreover, when the documentation is partial or missing the developer could end in a timecostly trial-and-error phase to acquire the information on how to interact with an API.

Interoperability. Interoperability concerns the mechanisms and technologies needed for requesting services and exchanging data between different APIs. In our example let us suppose that T natively supports interoperability with the translation and blogging APIs, but needs specific libraries for interacting with the image one. In this situation the developer faces two scenarios:

1. Including libraries. The developer integrates libraries of T which offer ready-to-use functionalities and reduce the workload of the developer.

However library inclusion causes the following drawbacks:

• strong dependencies in the project. Once included, dependencies need to be kept up-todate and changes between versions may lead to major changes in the implementation of the applications; • conflicts among libraries, source of bugs and unwanted side-effects; • limited knowledge of the internal mechanisms of libraries. This constitutes a "grey area" of the project on which the developer has little to no knowledge, adding complexity to the debugging process.

2. Ad-hoc solutions. The developer could spend a significant amount of time in implementing adhoc solutions to support interoperability with the image API in T . In this case, she would need a comprehensive knowledge on the communication and serialisation protocols employed by the image API.

Publication. Published APIs allow interaction with client interfaces and other server-side applications. Publication concerns both APIs deployment and documentation, which represent two concerns:

• the problems addressed for interoperability hold also for API deployment. High levels of interoperability bring strong dependencies on several libraries, whilst supporting only a few protocols reduces the cost of implementation to the detriment of interoperability;

• publishing an API requires also the publication of its companion documentation. Such documentation is produced ad-hoc and requires a steady commitment as it must be up-to-date wrt the implementation it describes.

Security. Security regards secure transition, storage, and retrieval of data and it is a major issue of any application available on any open network. Implementing and administrating security requires specific knowledge, experience, and skills. Indeed the developer can easily include in T cryptographic libraries for secure communication like TSL and SSL and encryption/decryption libraries for data storage/retrieval. However security still requires a tremendous effort to ensure a specific level of reliability: a developer that chooses to implement and deploy the security of data on its own must be aware of the latest best practices [START_REF] Mcgraw | Software security. Security Privacy[END_REF] in security. Moreover, she must invest a lot of effort to check the chain of com-ponents that handle sensible data to prevent security exploits [START_REF] Barber | Hacking techniques: The tools that hackers use, and how they are evolving to become more sophisticated[END_REF].

In our example, PhotoBlog must provide APIs that enable users to set persistent configurations on the service. Users have to register to the service, link one or more blogs to their account, and identify themselves before publishing a post. This implies two APIs concerning security: registration/access of users and registration/access to blogs, depicted in Fig. 2. Monitoring. After PhotoBlog has been deployed, its administrator wants to check the performances of her application. An important step in project development is therefore monitoring. Monitorability [START_REF] Skene | The monitorability of service-level agreements for applicationservice provision[END_REF] is a major concern when Service-Level Agreements (SLAs) define performances and availability of services with legal consequences such as penalties and compensations. As for security, T probably benefits from libraries to administer monitoring.

In PhotoBlog the monitoring function, provided as an API, can serve both clients and its administrator. The administrator can monitor the activities on its application, checking for availability, performances, and resource requirements. Clients can also take advantage of the monitoring feature to assess the availability of the service. In addition, third parties can offer impartial monitoring of services to assess compliance with SLAs.

Portability. Portability is the degree of freedom of an application wrt its dependencies. Internal dependencies enforce the usage of libraries whose changes potentially lead to major rewrites of the application. External dependencies imply the presence of a specific elements in the running environment of the application. It could be the running environment itself, e.g., a specific interpreter, or other applications that must coexist in the same platform to enable a particular behaviour. In the previous paragraphs we extensively addressed the problems linked to internal dependency, however also external dependencies are a major issue which may lead to a locked-in situation. To illus-trate this point let us say that the technology T chosen for PhotoBlog is Java, relying on Spring [START_REF] Johnson | Professional Java Development with the Spring Framework[END_REF] as supporting framework. Once released and deployed by the developer, the administrator of PhotoBlog is forced to choose a running environment that offers Java with Spring. In case the administrator wants to switch to another provider, she must choose one that offers a similar running environment.

Otherwise she could spend a significant amount of time in setting the necessary running environment on a generic platform i.e., installing the JVM, the Spring framework, setting it properly, etc..

FEATURES OF APIAAS

In this section we discuss the basic features which should characterize a composition-based APIaaS layer in a cloud framework. Notably, APIs can be divided into two groups, based on their implementation: core and composed APIs. Core APIs belong to applications that directly implement their functionalities without relying on external APIs (e.g., image, translation, and blogging APIs of PhotoBlog). Composed APIs derive from a behavioural composition of other APIs like the PhotBlog API. In our perspective an APIaaS layer should make no distinction in supporting core or composed APIs.

In this work we propose an APIaaS layer as a Platform-as-a-Service layer tailored to cope with the issues of API composition (3b). Developers can access such layer by means of private and secure credentials which give access to a private area that supports design of new API compositions starting from other existing core APIs. Developers can search existing APIs in the API registry of the APIaaS layer. APIs come with a formal interface which defines the message types and the behaviour of the interaction. They are also enriched with formal SLA documentation which provides all the necessary information for using/buying them. Each developer can decide to publish her APIs as private, inner or public. Private APIs are visible only by the author, inner APIs are visible only within the same APIaaS platform, whilst public APIs are visible from outside. Notably, inner and public visibility does not imply also accessibility and inner and public APIs can be released for free or under payment. Deployment costs and revenues depend on the business model of the APIaaS provider. Finally each deployed API should be monitorable and its performances visible by the administrator of the API and its clients, guaranteeing impartial SLA conformance.

Coherently with this scenario we propose to di- vide the APIaaS layer into two main sub-layers: the Abstract APIAaaS layer and the Deployment APIaaS one, as shown in Fig. 3c. The former sub-layer supports the abstract composition of APIs. The latter provides all the necessary tools for their management. Such a division allows to separate programming issues, considered in the abstract layer, from the architectural ones, addressed by the deployment layer.

In the following, we list and discuss the main desirable features for these two sub-layers.

Abstract APIaaS

• Composability: is the availability of the linguistic tools that provide a suitable abstraction for achieving composition;

• Formal documentability: is the possibility of documenting and describing each API by means of formal and abstract interface specifications. Such specifications can be used to achieve composition abstracting the details of real communication protocols and message formats. Formal specifications enables the automatic extraction of stubs of descriptive documentation, thus helping developers in API documentation. In addition, semiautomatic production of documentation uniforms the overall documentation easing the interpretation of API documentation by clients. This feature is currently missing in RESTful APIs where there exist no formal nor standard language for defining API interfaces.

• Portability: deals with the possibility of moving an API composition from an APIaaS platform to another one without affecting its code. Portability is important to avoid lock-ins on APIaaS providers. Lock-in avoidance is one of the key to guarantee a competitive API market. Moreover, it also allows to separate functionality issues from the one related to deployment and performances.

Deployment APIaaS

• Protection: deals with the possibility of composing APIs within the APIaaS layer for inner usage, without publishing them thought a public Software-as-a-Service layer. This feature allows the creation of inner utility APIs used by more complex APIs.

• Interoperability: allows the APIaaS layer to connect with different applications and APIs by supporting several communication protocols and message formats (e.g., Http/JSON, Http/SOAP, Java RMI, XML RPC, etc.). This feature allows to connect the formal interfaces of the abstract layer to concrete communication protocols and interoperate with external APIs.

• Publishability: deals with the possibility to select and publish a subset of the developed APIs on the public SaaS layer. Such feature allows mashups and applications commonly developed with other technologies to employ the developed APIs.

• Measurability: concerns the possibility to measure the performances of APIs in terms of response time, memory usage, served sessions per time unit, etc.. All these information serve for the definition of SLA contracts both between APIaaS providers and API developers and between API developers and their clients. Since APIs can be internally composed, it should be possible to derive minimal SLAs starting from the composition of the SLAs of the composed APIs. As an example consider the case described in § 2. Let us suppose that the image blurring process takes at maximum 500msec whilst the process of image posting takes at maximum 200msec. Provided these data, a developer can guarantee in PhotoBlog's SLA that the response time of its image posting API has a upper bound of 700msec.

• Sandboxing: relates to the protection feature and extends it. In particular sandboxing of code enables each developer to access a different environment which runs and contains all her code. In the sandboxing environment the developer defines the visibility of its compositions i.e., which should be visible internally or which have public visibility for external usage.

• Security: deals with different aspects of security of the APIaaS framework, i.e., : i) access security; ii) private data consistency; iii) malware prevention. Access security deals with well-known security issues like authentication and access token release to allow clients access APIs; private data consistency deals with issues related to corruption of private data performed by malicious ac-tivities from the outside (e.g., hacking attacks that alter private data); malware prevention deals with issues related to malicious behaviour of the deployed composed APIs.

• Scalability: deals with the possibility to scale the APIaaS architecture depending on the number of subscribed developers and the number of developed APIs.

• Federability: deals with the possibility of federating different APIaaS layers in order to share the inner developed APIs among different providers. Such a feature allows for the creation of a horizontal layer among different cloud infrastructures and provides the abstraction of a unique and common development framework for APIs. Such federation could be useful for the creation of groups of specialised APIaaS providers which maximise variety, efficiency, and security of offered APIs.

As an example, let us consider the case of a Smart City that providers the federation of three different APIaaS to citizens and companies: the municipality APIaaS, the main hospital APIaaS, and the public transportation company APIaaS. In this case, developers can easily access one of the three APIaaS and immediately exploit also all the APIs offered by the other two with the ease, efficiency, and security of the inner composition.

A COMPOSITION-BASED APIAAS PROTOTYPE

In this section we describe our composition-based APIaaS prototype, called JSOA, which is built following a service-oriented paradigm. The prototype represents an important enhancement wrt the PaaSSOA project [START_REF] Guidi | Paassoa: An open paas architecture for service oriented applications[END_REF]. In particular, JSOA introduces formal documentability, a web GUI designer, monitoring enhancements, publishability, and a prototype of the SaaS layer.

Abstract APIaaS

In the Abstract APIaaS layer we follow service orchestration principles to supply a composition language for APIs. In particular we use Jolie [START_REF]Java Orchestration Language Interpreter Engine[END_REF] as orchestration language because it combines all the basic mechanisms of the well-known web services standards WSDL (W3C, 2001) and WS-BPEL (OASIS, 2007) in the same linguistic domain.

In the following we discuss how composability, formal documentability, and portability are achieved by the JSOA Abstract APIaaS layer. Composability JSOA addresses composability in a natural way using the linguistic primitives supplied by an orchestration language (e.g., Jolie) which easily permit to coordinate concurrent service calls, thus focusing the value on composition and not on other aspects of programming. In particular these primitives cover the following aspects: i) workflow behaviour: the behaviour is expressed in terms of a workflow that provides both sequence and parallel operators, therefore no thread programming is required; ii) synchronous invocation primitive: APIs can be invoked by exploiting blocking request-response primitives which can be easily composed within the behaviour; iii) statefulness primitives: orchestration languages provide specific language primitives for addressing statefulness, e.g., such as correlation sets [START_REF] Mauro | An efficient management of correlation sets with broadcast[END_REF] which ease session and state handling at application level; iv) process spawn: process spawn is automatically managed by the orchestration engines, no linguistic primitives are required; v) endpoint abstraction: endpoints allows to invoke APIs abstracting away from communication protocols and message formats. List. 1 reports a snippet of code corresponding to the behavioural composition of PhotoBlog in § 2. Notably, List. 1 does not report the declaration part which we excluded for reasons of presentation. In List. 1 we suppose that getImgWithFaceBlur@ImageAPI (Lines 3-4), getEngTranslation@TranslationAPI (Lines 6-7), and setPost (Line 10) are the names of the API for the image application, the translation application, and PhotoBlog respectively. Moreover, we define the name of the new composed API as setIntoBlog (Line 2). When the API is invoked (Line 2), the layer spawns an instance of its behaviour, executing it. In particular List. 1 composes the image and translation APIs in parallel (Line 5) and when they both return a response it commands the sequential execution of the subsequent instructions (Lines 8-9). Formal documentability In Jolie, each input and output endpoints of a service is joined with an interface which defines all the available APIs for the given port and the message types1 . Each Jolie interface can be enriched with comments and automatically transformed into an human-readable HTML document.

Portability Portability differs if considered within the same technology or across different technologies. In the former case, JSOA Abstract APIaaS layer completely cope with portability issues. Since the Jolie interpreter is the same, compositions developed into the APIaaS can seamlessly execute without any significant modification of the code (except for endpoint rebinding which consists in a mere url location renaming) on any environment that provides said interpreter. On the contrary, in the latter case, portability is not directly achieved but since Jolie has been developed upon a formal semantics [START_REF] Guidi | Sock: A calculus for service oriented computing[END_REF] ad-hoc encoding translators can be developed. • PaaSSOA: the component in charge of providing the PaaS abstraction to the developer. It equips a central registry that lists all running orchestrators in any SOABoot available. Registered orchestrators can be exploited for creating new service APIs. Moreover PaaSSOA manages a central monitoring service which collects all the logs from the running services.

Deployment APIaaS

• WSOA: the component that provides the SaaS abstraction and that is in charge of publishing service APIs outside JSOA.

At the present, JSOA does not respect all the features discussed in § 3. In the following, we briefly discuss the implemented ones whilst we address the missing ones in § 6 as future work.

Protection is inherently provided by JSOA since all deployed orchestrators and core APIs can be easily invoked by new API compositions within the Abstract APIaaS layer.

Interoperability is addressed by managing the deployment of the Jolie endpoints depending on the required protocols. Jolie provides both input and output endpoints used by an orchestrator to receive and send messages. As a test, we developed two core API wrappers that invoke WordReference (WordReference, 2013) and CKAN APIs (CKAN, 2013).

Publishability is implemented by the WSOA component which allows to select a set of already deployed APIs and to export them under different protocols (e.g., Http/JSON, Http/SOAP, Http/POST/XML, etc.). As a test, we developed a plugin that enables the composition of already published APIs within the editor of the SATIN project [START_REF] Satin | Satin project[END_REF].

Measurability is provided by the monitoring architecture. A built-in monitor is attached to each deployed orchestrator. The monitor sends all the logged events to the SOABoot local monitor. In turn SOA-Boot local monitors send collected local data to the central monitor which stores them. The monitoring architecture also provides analysis of data via the Drools engine [START_REF] Drools | Drools -the business logic integration platform[END_REF]. The current implementation monitors start and end of sessions and start and end of operations. Although simple these data allows for the calculation of important statistics like API response time, memory usage, etc.. Scalability is addressed by SOABoot and the registry architecture. Depending on the load of the available computational resources new SOABoots can be added. In the same way, computational resources can be released when the corresponding SOABoots do not execute any orchestrator.

RELATED WORK

Recently the communities behind the main languages used for web development like JavaScript, Ruby, Java, and PHP produced several RESTful [START_REF] Richardson | Restful Web Services. O'Reilly[END_REF] libraries to ease the task of designing, writing, and composing RESTful APIs (some examples are, respectively, (Backbone.js, 2013), (Ruby on Rails, 2013), (Java Community Process, 2013), and [START_REF] Lavarel | The lavarel php framework[END_REF]). RESTful frameworks ease the development of server-side API composition and deployment yet the implementation is part of the basic core of an application which translates into high costs of creation and maintenance.

An inspiring work in the direction of a PaaS middleware for application deployment is [START_REF] Wettinger | Enabling dynamic deployment of cloud applications using a modular and extensible paas environment[END_REF] where the authors address the problem of composition (and dependencies) with a middlewareoriented approach. In particular, we notice the focus on decoupling the application logic from the necessary middleware and infrastructure resources. On the same topic, other two interesting works are [START_REF] Caporuscio | ubiSOAP: A Service Oriented Middleware for Ubiquitous Networking[END_REF][START_REF] Caporuscio | ubirest: A restful service-oriented middleware for ubiquitous networking[END_REF] where the authors introduce the concept of a middleware to leverage integration of heterogeneous services for ubiquitous networking. Such middleware is designed and implemented following a service-oriented approach supporting, respectively, SOAP and RESTful Services. This flexible and dynamic vision is also proposed in [START_REF] Caporuscio | Restful service architectures for pervasive networking environments[END_REF] in which thirdparty applications discovered at run time should be integrated in applications to offer more sophisticated functionalities.

CONCLUSIONS

In this paper we discussed a new proposal for an composition-based APIaaS layer which facilitates API programming and deployment on the cloud by enabling API composition. In our perspective the APIaaS providers should supply large repositories of core APIs which developers could compose. Composed APIs could be available on the same repository of core APIs, enabling the publication of new and more sophisticated services, thus fostering a liquid market for APIs and the reuse of services.

We also discuss a list of basic features which should be supported by an APIaaS layer and present a proof-of-concept prototype, dubbed JSOA, which implements a subset of such features. We based JSOA on a service-oriented paradigm and exploited primitives offered by its implementation language (Jolie) to orchestrate APIs and, more in general, to implement said features.

The aim of this work is to start a critical discussion towards a shared definition of the concepts and the foundational architecture of an open APIaaS layer.

Future work. We intend to continue the development of our prototype. Our experience with JSOA proved that prototyping constitutes a relevant resource for testing and analysing benefits and drawbacks of our approach. Namely, the next steps of JSOA development count the implementation of features like sandboxing, security, and federability.

In addition, we also plan to analyse how different economic models affect a federated market of APIaaS providers (as represented in the example regarding the federability feature in § 3). What we expect is that the features of our composition-based APIaaS layer would avoid technology lock-in, lower dependencies on specific technologies, and enact a beneficial competition among different providers.

Figure 1 :

 1 Figure 1: An example of composition of APIs.

Figure 2 :

 2 PhotoBlogregister registers to service

Figure 3 :

 3 Figure 3: APIaaS layer in the common cloud stack.

Figure 4 :

 4 Figure 4: The JSOA Architecture.

Fig. 4

 4 Fig. 4 reports a sketch of the architecture of JSOA which implements the Deployment APIaaS layer. The main components are: • SOABoot: the basic component which enables service execution in the cloud. It must be available in each computational resource provided by the IaaS and it takes the form of a service itself. SOA-Boot provides three main functionalities: (i) local

  Listing 1: Composition behaviour of PhotoBlog 2 in Jolie.

	1 main {
	2 setIntoBlog ( r )( p ) {
	3	getImgWithFaceBlur @
	4	ImageAPI ( r. image )( s. image )
	5	|
	6	getEngTranslation @
	7	TranslationAPI ( r. text )( s. text );
	8	s . username = r. username ;
	9	s . password = r. password ;
	10	setPost @ BlogAPI ( s )( p )
	11	}
	12 }

Here we do not report the syntax of Jolie interfaces. The interested reader may refer to(Jolie,

2013)