Omar El-Fallah
email: elfallah@fsr.ac.ma

Karim Kellay
email: kellay@cmi.univ-mrs.fr

Thomas Ransford
email: ransford@mat.ulaval.ca

Cantor sets and cyclicity in weighted Dirichlet spaces

Keywords: 2000 Mathematics Subject Classification. 30H05, 46E20, 47A15 Dirichlet space, weight, cyclic vector, α-capacity, Cantor set

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Introduction

In this paper we study the weighted Dirichlet spaces D α (0 ≤ α ≤ 1), defined by

D α := f ∈ hol(D) : D α (f ) := 1 π D |f ′ (z)| 2 (1 -|z| 2 ) α dA(z) < ∞ .
Here D denotes the open unit disk, and dA is area measure on D. Clearly D α is a Hilbert space with respect to the norm • α given by f 2 α := |f (0)| 2 + D α (f ). A classical calculation shows that, if f (z) = n≥0 a n z n , then

f 2 α ≍ n≥0 (n + 1) 1-α |a n | 2 .
Note that D 1 = H 2 is the usual Hardy space, and D 0 is the classical Dirichlet space (thus our labelling convention follows [START_REF] Aleman | Hilbert spaces of analytic functions between the Hardy and the Dirichlet space[END_REF] rather than [START_REF] Brown | Cyclic vectors in the Dirichlet space[END_REF]).

An invariant subspace of D α is a closed subspace M of D α such that zM ⊂ M . Given f ∈ D α , we denote by [f ] Dα the smallest invariant subspace of D α containing f , namely the closure in D α of {pf : p a polynomial}. We say that f is cyclic for D α if [f ] Dα = D α . The survey article [START_REF] El-Fallah | Invariant subspaces of the Dirichlet space[END_REF] gives a brief history of invariant subspaces and cyclic functions in the classical case α = 0.

Our goal is to characterize the cyclic functions of D α . In order to state our results, we introduce the notion of α-capacity. For α ∈ [0, 1), we define the kernel function k α : R + → R ∪ {∞} by k α (t) := 1/t α , 0 < α < 1, log(1/t), α = 0.

The α-energy of a (Borel) probability measure µ on T is defined by

I α (µ) := k α (|ζ -ζ ′ |) dµ(ζ) dµ(ζ ′ ).
A standard calculation gives

I α (µ) ≍ n≥0 | µ(n)| 2 (1 + n) 1-α .
The α-capacity of a Borel subset E of T is defined by C α (E) := 1/ inf{I α (µ) : P(E)}, where P(E) denotes the set of all probability measures supported on compact subsets of E. In particular, C α (E) > 0 if and only if there exists a probability measure µ supported on a compact subset of E and having finite α-energy. If α = 0, then C 0 is the classical logarithmic capacity.

We recall a result due to Beurling and Salem-Zygmund [5, §V, Theorem 3] about radial limits of functions in the weighted Dirichlet spaces.

If f ∈ D α , then f * (ζ) := lim r→1- f (rζ) exists for all ζ ∈ T outside a set of α-capacity zero.
The following theorem gives two necessary conditions for cyclicity in D α .

Theorem 1.1. Let α ∈ [0, 1). If f is cyclic in D α , then • f is an outer function, • {ζ ∈ T : f * (ζ) = 0} is a set of α-capacity zero.
The first part is [2, Corollary 1]. For α = 0, the second part is [2, Theorem 5], and for general α the proof is similar, the only difference being that the logarithmic kernel k 0 is replaced by k α . We omit the details.

Our main result is a partial converse to this theorem. To state it, we need to define the notion of a generalized Cantor set.

Let (a n ) n≥0 be a positive sequence such that a 0 ≤ 2π and

sup n≥0 a n+1 a n < 1 2 .
The generalized Cantor set E associated to (a n ) is constructed as follows.

Start with a closed arc of length a 0 on the unit circle T. Remove an open arc from the middle, to leave two closed arcs each of length a 1 . Then remove two open arcs from their middles to leave four closed arcs each of length a 2 .

After n steps, we obtain E n , the union of 2 n closed arcs each of length a n . Finally, the generalized Cantor set is

E := ∩ n E n . Theorem 1.2. Let α ∈ [0, 1) and let f ∈ D α . Suppose that: • f is an outer function, • |f | extends continuously to D, • {ζ ∈ T : |f (ζ)| = 0} is contained in a generalized Cantor set E of α-capacity zero.
Then f is cyclic for D α .

Functions f satisfying the hypotheses exist in abundance. Indeed, any generalized Cantor set is a so-called Carleson set, and is thus the zero set of some outer function f such that f and all its derivatives extend continuously to D. Moreover, it is very easy to determine which generalized Cantor sets have α-capacity zero. More details will be given in §4.

To prove Theorem 1.2, we adopt the following strategy. In §2, using a technique due to Korenblum, we show that [f ] Dα contains at least those functions g ∈ D α satisfying |g(z)| ≤ dist(z, E) 4 . The idea is then to take one simple such g, and gradually transform it into the constant function 1 while staying inside [f ] Dα , thereby proving that 1 ∈ [f ] Dα . This requires three tools: a general estimate for weighted Dirichlet integrals of outer functions, some properties of generalized Cantor sets, and a regularization theorem. These tools are developed in § §3,4,5 respectively, and all the pieces are finally assembled in §6, to complete the proof of Theorem 1.2.

Theorem 1.2 was established for the classical Dirichlet space, α = 0, in [7, Corollary 1.2]. The proof there followed the same general strategy, but in several places key use was made of a formula of Carleson [START_REF] Carleson | A representation formula for the Dirichlet integral[END_REF] expressing the Dirichlet integral of an outer function f in terms of the values of |f * | on the unit circle. No analogue of Carleson's formula is known in the case 0 < α < 1, and one of the main points of this note is to show how this difficulty may be overcome.

Throughout the paper, we use the notation C(x 1 , . . . , x n ) to denote a constant that depends only on x 1 , . . . , x n , where the x j may be numbers, functions or sets. The constant may change from one line to the next.

Korenblum's method

Our aim in this section is to prove the following theorem.

F := {ζ ∈ T : |f (ζ)| = 0}. If g ∈ D α and |g(z)| ≤ dist(z, F ) 4 (z ∈ D), then g ∈ [f ] Dα .
This theorem is a D α -analogue of [7, Theorem 3.1], which was proved using a technique of Korenblum. We shall use the same basic technique here. However, the proof in [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF] proceeded via a so-called fusion lemma, which, being based on Carleson's formula for the Dirichlet integral, is no longer available to us here. Its place is taken by Corollary 2.3 below.

We need to introduce some notation. Given an outer function f and a Borel subset Γ of T, we define

f Γ (z) := exp 1 2π Γ ζ + z ζ -z log |f * (ζ)| |dζ| (z ∈ D).
We write ∂Γ and Γ c for the boundary and complement of Γ in T respectively.

Lemma 2.2. Let f be a bounded outer function. For every Borel set Γ ⊂ T,

|f ′ Γ (z)| ≤ C(f )(|f ′ (z)| + dist(z, ∂Γ) -4 ) (z ∈ D).
Proof. Without loss of generality, we may suppose that f ∞ ≤ 1. Note that then f Γ ∞ ≤ 1 for all Γ. Also, obviously, log |f * | ≤ 0 a.e. on T, which will help simplify some of the calculations below.

We begin by observing that

f ′ Γ (z) f Γ (z) = 1 2π Γ 2ζ (ζ -z) 2 log |f * (ζ)| |dζ| (z ∈ D), from which it follows easily that |f ′ Γ (z)| ≤ 2 log(1/|f (0)|) dist(z, Γ) 2 (z ∈ D). (1)
Our aim now is to prove a similar inequality, but with ∂Γ in place of Γ.

Set G := {z ∈ D : dist(z, Γ) ≥ dist(z, Γ c ) 2 }. Clearly dist(z, Γ) ≥ dist(z, ∂Γ) 2 for all z ∈ G, so (1) implies |f ′ Γ (z)| ≤ 2 log(1/|f (0)|) dist(z, ∂Γ) 4 (z ∈ G). (2)
Now suppose that z ∈ D \ G. Then dist(z, Γ c ) 2 > dist(z, Γ) ≥ (1 -|z| 2 )/2, and hence

|f Γ c (z)| = exp 1 2π Γ c 1 -|z| 2 |ζ -z| 2 log |f * (ζ)| |dζ| ≥ |f (0)| 2 .
Since obviously

f Γ = f /f Γ c , it follows that, for all z ∈ D \ G, |f ′ Γ (z)| ≤ |f ′ (z)| |f Γ c (z)| + |f (z)| |f Γ c (z)| 2 |f ′ Γ c (z)| ≤ |f ′ (z)| |f (0)| 2 + 1 |f (0)| 4 2 log(1/|f (0)|) dist(z, Γ c ) 2 ,
where once again we have used (1), this time with Γ replaced by Γ c . Noting that dist(z, Γ c ) ≥ dist(z, ∂Γ) for all z ∈ D \ G, we deduce that

|f ′ Γ (z)| ≤ |f ′ (z)| |f (0)| 2 + 1 |f (0)| 4 2 log(1/|f (0)|) dist(z, ∂Γ) 2 (z ∈ D \ G). (3)
The inequalities ( 2) and (3) between them give the result.

Corollary 2.3. Let α ∈ [0, 1) and f ∈ D α ∩H ∞ be an outer function. Then, for every Borel set Γ ⊂ T and every g ∈ D α satisfying |g(z)| ≤ dist(z, ∂Γ) 4 , we have

f Γ g α ≤ C(α, f )(1 + g α ).
Proof. Using Lemma 2.2, we have

|(f Γ g) ′ | ≤ |f ′ Γ ||g| + |f Γ ||g ′ | ≤ C(f )(|f ′ | + 1 + |g ′ |).
The conclusion follows easily from this.

Proof of Theorem 2.1. Let I be a connected component of T \ F , say I = (e ia , e ib ). Let ρ > 1, and define

ψ ρ (z) := (z -1) 4 /(z -ρ) 4 , φ ρ (z) := ψ ρ (e -ia z)ψ ρ (e -ib z). The first step is to show that φ ρ f T\I ∈ [f ] Dα .
Let ǫ > 0 and set I ǫ := (e i(a+ǫ) , e i(b-ǫ) ) and

φ ρ,ǫ (z) := ψ ρ (e -i(a+ǫ) z)ψ ρ (e -i(b-ǫ) z).
By Corollary 2.3,

φ ρ,ǫ f T\Iǫ α ≤ C(f ) φ ρ,ǫ α ≤ C(f, ρ). (4)
φ ρ,ǫ f T\Iǫ ∈ [f ] Dα .
Using (4), we see that φ ρ,ǫ f T\Iǫ converges weakly in

D α to φ ρ f T\I as ǫ → 0. Hence φ ρ f T\I ∈ [f ] Dα , as claimed.
Next, we multiply by g. As g ∈ D α ∩ H ∞ , Aleman's theorem immediately yields φ ρ f T\I g ∈ [f ] Dα . Using the fact that |g(z)| ≤ dist(z, F ) 4 , it is easy to check that φ ρ g α remains bounded as ρ → 1. By Corollary 2.3 again, φ ρ f T\I g α is uniformly bounded, and φ ρ f T\I g converges weakly to f T\I g. Hence f T\I g ∈ [f ] Dα . Now let (I j ) j≥1 be the complete set of components of T \ F , and set J n := ∪ n 1 I j . An argument similar to that above gives f T\Jn g ∈ [f ] Dα for all n. Moreover, f T\Jn g α is uniformly bounded. Thus f T\Jn g converges weakly to g, and so finally g ∈ [f ] Dα .

Estimates for weighted Dirichlet integrals

The following result will act as a partial substitute for Carleson's formula. Theorem 3.1. Let α ∈ [0, 1), and let h : T → R be a positive measurable function such that, for every arc I ⊂ T,

1 |I| I h(ζ) |dζ| ≥ |I| α . ( 5
)
If f is an outer function, then

D α (f ) ≤ 1 π T 2 (|f * (ζ)| 2 -|f * (ζ ′ )| 2 )(log |f * (ζ)| -log |f * (ζ ′ )|) |ζ -ζ ′ | 2 (h(ζ)+h(ζ ′ ))|dζ||dζ ′ |.
Proof. Given z ∈ D, let I z be the arc of T with midpoint z/|z| and arclength

|I z | = 2(1 -|z| 2 ). For ζ ∈ I z , we have |ζ -z| ≤ 2(1 -|z| 2
), and so

P (z, ζ) := 1 -|z| 2 |ζ -z| 2 ≥ 1 4(1 -|z| 2 ) = 1 2|I z | .
Hence, using (5), we have

T P (z, ζ)h(ζ) |dζ| ≥ 1 2|I z | Iz h(ζ) |dζ| ≥ 1 2 |I z | α ≥ 1 2 (1 -|z| 2 ) α .
Therefore, by Fubini's theorem,

D α (f ) := 1 π D |f ′ (z)| 2 (1 -|z| 2 ) α dA(z) ≤ 2 T D ζ (f )h(ζ) |dζ|,
where

D ζ (f ) := 1 π D |f ′ (z)| 2 P (z, ζ) dA(z) (ζ ∈ T). Now D ζ (f )
is the so-called local Dirichlet of integral of f at ζ, which was studied in detail by Richter and Sundberg in [START_REF] Richter | A formula for the local Dirichlet integral[END_REF]. In particular, they showed that, if f is an outer function, then

D ζ (f ) = 1 2π T |f * (ζ)| 2 -|f * (ζ ′ )| 2 -2|f * (ζ ′ )| 2 log |f * (ζ)/f * (ζ ′ )| |ζ -ζ ′ | 2 |dζ ′ |.
Substituting this into the preceding estimate for D α (f ), and noting the obvious fact that h

(ζ) ≤ h(ζ) + h(ζ ′ ), we deduce that D α (f ) is majorized by 1 π T |f * (ζ)| 2 -|f * (ζ ′ )| 2 -2|f * (ζ ′ )| 2 log |f * (ζ)/f * (ζ ′ )| |ζ -ζ ′ | 2 (h(ζ)+h(ζ ′ )) |dζ ′ | |dζ|.
Exchanging the roles of ζ and ζ ′ , we see that D α (f ) is likewise majorized by

1 π T |f * (ζ ′ )| 2 -|f * (ζ)| 2 -2|f * (ζ)| 2 log |f * (ζ ′ )/f * (ζ)| |ζ ′ -ζ| 2 (h(ζ ′ )+h(ζ)) |dζ| |dζ ′ |.
Taking the average of these last two estimates, we obtain the inequality in the statement of the theorem.

We are going to apply this result with h(ζ) := Cd(ζ, E) α , where C is a constant, d denotes arclength distance on T, and E is a closed subset of T. Condition (5) thus becomes

1 |I| I d(ζ, E) α |dζ| ≥ C -1 |I| α for all arcs I ⊂ T. (6)
A set E which satisfies this condition for some α, C is called a K-set (after Kotochigov). K-sets arise as the interpolation sets for certain function spaces, and have several other interesting properties. We refer to [3, §1] and [6, §3] for more details. In particular, if E satisfies [START_REF] Dynkin | The pseudo-analytic extension[END_REF], then it has measure zero and log d(ζ, E) ∈ L 1 (T).

Theorem 3.2. Let α ∈ (0, 1), let E be a closed subset of T satisfying [START_REF] Dynkin | The pseudo-analytic extension[END_REF], and let w : [0, 2π] → R + be an increasing function such that t → ω(t γ ) is concave for some γ > 2/(1 -α). Let f w be the outer function satisfying

|f * w (ζ)| = w(d(ζ, E))
a.e on T.

Then

D α (f w ) ≤ C(α, γ, E) T w ′ (d(ζ, E)) 2 d(ζ, E) 1+α |dζ|. ( 7
)
In particular f w ∈ D α if the last integral is finite.

Proof. The proof is largely similar to that of [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF]Theorem 4.1], so we give just a sketch, concentrating on those parts where the two proofs differ. We begin by remarking that the concavity condition on w easily implies that so log w(d(ζ,E)) ∈ L 1 (T) and the definition of f w makes sense.

| log w(d(ζ, E))| ≤ C(w)| log d(ζ, E)|,
By Theorem 3.1, we have

D α (f w ) ≤ C(α, E) T 2 (w 2 (δ) -w 2 (δ ′ ))(log w(δ) -log w(δ ′ )) |ζ -ζ ′ | 2 (δ α +δ ′α ) |dζ| |dζ ′ |,
where we have written

δ := d(ζ, E) and δ ′ := d(ζ ′ , E).
Let (I j ) be the connected components of T \ E, and set

N E (t) := 2 j 1 {|Ij|>2t} (0 < t ≤ π).
Then, for every measurable function Ω : [0, π] → R + , we have

T Ω(d(ζ, E)) |dζ| = π 0 Ω(t)N E (t) dt.
In particular, as in [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF], it follows that

T 2 (w 2 (δ) -w 2 (δ ′ ))(log w(δ) -log w(δ ′ )) |ζ -ζ ′ | 2 (δ α + δ ′α ) |dζ| |dζ ′ | ≤ C(α) π 0 π 0 (w 2 (s + t) -w 2 (t))(log w(s + t) -log w(t)) s 2 (s + t) α N E (t) ds dt.
The concavity assumption on w implies that t → t 1-1/γ w ′ (t) is decreasing, and thus, as in [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF],

w 2 (t + s) -w 2 (t) ≤ 2γw(t + s)w ′ (t)t (1 + s/t) 1/γ -1 , log w(t + s) -log w(t) ≤ tw ′ (t) (1 + s/t) 1/γ w(t + s) log(1 + s/t).
Combining these estimates, we obtain

π 0 π 0 (w 2 (t + s) -w 2 (t))(log w(t + s) -log w(t)) s 2 (s + t) α dsN E (t) dt ≤ π 0 π 0 2γw ′ (t) 2 t 2+α (1 + s/t) 1/γ -1 (1 + s/t) 1/γ+α log(1 + s/t) ds s 2 N E (t) dt = π 0 2γw ′ (t) 2 t 1+α π/t 0 2γ (1 + x) 1/γ -1 (1 + x) 1/γ+α log(1 + x) dx x 2 N E (t) dt ≤ C(α, γ) π 0 w ′ (t) 2 t 1+α N E (t) dt.
In the last inequality we used the fact that γ > 2/(1 -α).

Generalized Cantor sets

The notion of the generalized Cantor set E associated to a sequence (a n ) was defined in §1. In this section we briefly describe some pertinent properties of these sets. We shall write

λ E := sup n≥0 a n+1 a n .
Recall that, by hypothesis, λ E < 1/2. Our first result shows that generalized Cantor sets satisfy [START_REF] Carleson | Selected Problems on Exceptional Sets[END_REF], and hence that Theorem 3.2 is applicable to such sets. Proposition 4.1. Let E be a generalized Cantor set and let α ∈ [0, 1). Then, for each arc I ⊂ T,

1 |I| I d(ζ, E) α |dζ| ≥ C(α, λ E )|I| α .
Proof. Let I be an arc with |I| ≤ 2a 0 , and choose n so that 2a n < |I| ≤ 2a n-1 . Recall that the n-th approximation to E consists of 2 n arcs, each of length a n , and that the distance between these arcs is at least a n-1 -2a n .

If I meets at least two of these arcs, then I \ E contains an arc J of length a n-1 -2a n , and if I meets at most one of these arcs, then I \ E contains an arc J of length (|I| -a n )/2. Thus I \ E always contains an arc J such that

|J|/|I| ≥ min{1/2 -λ E , 1/4}. Consequently 1 |I| I d(ζ, E) α |dζ| ≥ 1 |I| J d(ζ, E) α |dζ| ≥ 1 |I| |J| α+1 α + 1 ≥ C(α, λ E )|I| α .
In Proof. Given t ∈ (0, a 0 ], choose n so that a n < t ≤ a n-1 . Then clearly

|E t | ≤ 2 n (a n +2t) ≤ 3.2 n t. Also 2 n-1 = (1/λ n-1 E ) log 2/ log(1/λ E ) ≤ (a 0 /t) log 2/ log(1/λ E ) . Hence |E t | ≤ C(a 0 , λ E )t 1-log 2/ log(1/λ E ) .
In particular, every generalized Cantor set E is a Carleson set, that is, π 0 (|E t |/t) dt < ∞. Taylor and Williams [START_REF] Taylor | Ideals in rings of analytic functions with smooth boundary values[END_REF] showed that Carleson sets are zero sets of outer functions in A ∞ (D). This justifies a remark made in §1.

The final property that we need concerns the α-capacity, C α , which was defined in §1.

Theorem 4.3. Let E be a generalized Cantor set and let α ∈ [0, 1). Then

C α (E) = 0 ⇐⇒ π 0 dt t α |E t | = ∞.
Proof. This follows easily from [5, §IV, Theorems 2 and 3].

Regularization

We shall need the following regularization result. The proof is the same, with minor modifications, as that of [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF]Theorem 5.1].

Theorem 5.1. Let α ∈ [0, 1), let σ ∈ (0, 1) and let a > 0. Let φ : (0, a] → R + be a function such that

• φ(t)/t is decreasing, • 0 < φ(t) ≤ t σ for all t ∈ (0, a],

• a 0 dt t α φ(t) = ∞.
Then, given ρ ∈ (0, σ), there exists a function ψ : (0, a] → R + such that • ψ(t)/t ρ is increasing,

• φ(t) ≤ ψ(t) ≤ t σ for all t ∈ (0, a],

• a 0 dt t α ψ(t) = ∞.

Completion of the proof of Theorem 1.2

For α = 0, this theorem was proved in [7, §6]. The proof for α ∈ (0, 1) will follow the same general lines, and once again we shall concentrate mainly on the places where the proofs differ.

Let f be the function in the theorem. Our goal is to show that 1 ∈ [f ] Dα .

Let E be as in the theorem, and let g be the outer function such that We shall construct a family of functions w δ : [0, π] → R + for 0 < δ < 1 such that the associated outer functions f w δ belong to [g] Dα and satisfy:

(i) |f * w δ | → 1 a.e. on T as δ → 0, (ii) |f w δ (0)| → 1 as δ → 0, (iii) lim inf δ→0 f w δ α < ∞.
If such a family exists, then a subsequence of the f w δ converges weakly to 1 in D α , and since they all belong to [g] Dα , it follows that 1 ∈ [g] Dα , as desired.

By Proposition 4.2, there exists µ > 0 such that

|E t | = O(t µ ) as t → 0. Fix ρ, σ satisfying 1 -α 2 < ρ < σ < min 1 -α, 1 -α + µ 2 . Define φ : (0, π] → R + by φ(t) := max min{|E t |, t σ }, t 1-α (t ∈ (0, π]).
Clearly φ(t)/t is increasing and 0 ≤ φ(t) ≤ t σ for all t. We claim also that

π 0 dt t α φ(t) = ∞. (8)
To see this, note that Consequently (8) holds, as claimed.

We have now shown that φ satisfies all the hypotheses of the regularization theorem, Theorem 5.1. Therefore there exists a function ψ : (0, π] → R + satisfying the conclusions of that theorem, namely: ψ(t)/t ρ is increasing, t 1-α ≤ φ(t) ≤ ψ(t) ≤ t σ for all t, and π 0 dt/t α ψ(t) = ∞.

For 0 < δ < 1, we define w δ : [0, π] → R + by

w δ (t) :=            δ ρ ψ(δ) t 1-α-ρ 0 ≤ t ≤ δ A δ -log π t ds s α ψ(s) δ < t ≤ η δ 1 η δ < t ≤ π,
where A δ and η δ are constants chosen to make w δ continuous.

Let us show that f w δ ∈ [g] Dα . Note first that w δ (t)/t 1-α-ρ is a bounded function. Therefore f w δ /g (1-α-ρ)/4 is bounded. Using Theorem 3.2, we have g It remains to check that the functions f w δ satisfy properties (i)-(iii) above. The verifications run along the same lines as those in [7, §6], using the properties of ψ above, and Theorem 3.2 in place of [START_REF] El-Fallah | On the Brown-Shields conjecture for cyclicity in the Dirichlet space[END_REF]Theorem 4.1].

Theorem 2 . 1 .

 21 Let f ∈ D α be an outer function such that |f | extends continuously to D, and let

  Note that |fT\Iǫ | = |f | in a neighborhood of T \ I. Since |f | does not vanish inside I, it follows that |φ ρ,ǫ f T\Iǫ |/|f | is bounded on T. Also f is an outer function. Therefore, by a theorem of Aleman [1, Lemma 3.1],

  |g * (ζ)| = d(ζ, E) 4 a.e. on T.Then |g(z)| ≤ (π/2) 4 dist(z, E) 4 (z ∈ D): indeed, for every ζ 0 ∈ E, we have log |g(z)| ≤ 1 2π T 1 -|z| 2 |z -ζ| 2 4 log (π/2)|ζ -ζ 0 | |dζ| = 4 log (π/2)|z -ζ 0 | .By assumption, the zero setF := {ζ ∈ T : |f (ζ)| = 0} in contained in E, so |g(z)| ≤ (π/2) 4 dist(z, F ) 4 (z ∈ D). Theorem2.1 therefore applies, and we can infer that g ∈ [f ] Dα . It thus suffices to prove that 1 ∈ [g] Dα .

  (1-α-ρ)/4 ∈ D α . Consequently, by a theorem of Aleman [1, Lemma 3.1], f w δ ∈ [g (1-α-ρ)/4 ] Dα . Using another result of Aleman [1, Theorem 2.1], we have g (1-α-ρ)/4 ∈ [g] Dα . Hence f w δ ∈ [g] Dα , as claimed.

  the next two results, we write E t := {ζ ∈ T : d(ζ, E) ≤ t}.

	Proposition 4.2. If E is a generalized Cantor set, then |E t | = O(t µ ) as
	t → 0, where µ := 1 -log 2/ log(1/λ E ).

  Since E is a generalized Cantor set of α-capacity zero, Theorem 4.3 shows that

		t	π	ds s α |E s |	≥	t |E t |	t	π	ds s α+1 = C(α)	t 1-α |E t |	,
	whence
	ǫ	π	dt t α φ(t)	≥	ǫ	π	dt max{t, t α |E t |}
					π ≥ C(α) log ≥ C(α) ǫ	ds π t ds/s α |E s |) t α |E t | t α |E t |( π ǫ dt .
							0	π	dt t α |E t |	= ∞.

1. Research partially supported by a grant from Egide Volubilis (MA09209). 2. Research partially supported by grants from Egide Volubilis (MA09209) and ANR Dynop. 3. Research partially supported by grants from NSERC (Canada), FQRNT (Québec) and the Canada Research Chairs program.


Juan Campos Serrano

Manuel Del Pino

Juan Campos
email: campos@ceremade.dauphine.fr

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

M Del Pino
email: delpino@dim.uchile.cl

Relative equilibria in continuous stellar dynamics

Keywords: Stellar dynamics -gravitation -mass -rotation -angular velocity -solutions with compact support -radial solutions -symmetry breakingorbital stability -elliptic equations - Mathematical subject classification (2000). Primary: 35J20; Secondary: 35B20, 35B38, 35J60, 35J70, 35Q35, 76P05, 76S05, 82B40

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Introduction and statement of the main results

We consider the Vlasov-Poisson system

     ∂ t f + v • ∇ x f -∇ x φ • ∇ v f = 0 φ = - 1 4π | • | * ρ , ρ := R 3 f dv (1)
which models the dynamics of a cloud of particles moving under the action of a mean field gravitational potential φ solving the Poisson equation: ∆φ = ρ. Kinetic models like system [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF] are typically used to describe gaseous stars or globular clusters. Here f = f (t, x, v) is the so-called distribution function, a nonnegative function in L ∞ (R, L 1 (R 3 × R 3 )) depending on time t ∈ R, position x ∈ R 3 and velocity v ∈ R 3 , which represents a density of particles in the phase space, R 3 × R 3 . The function ρ is the spatial density function and depends only on t and x. The total mass is conserved and hence

R 3 ×R 3 f (t, x, v) dx dv = R 3 ρ(t, x) dx = M
does not depend on t.

The first equation in [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF] is the Vlasov equation, also known as the collisionless Boltzmann equation in the astrophysical literature; see [START_REF] Binney | Galactic dynamics[END_REF]. It is obtained by writing that the mass is transported by the flow of Newton's equations, when the gravitational field is computed as a mean field potential. Reciprocally, the dynamics of discrete particle systems can be formally recovered by considering empirical distributions, namely measure valued solutions made of a sum of Dirac masses, and neglecting the self-consistent gravitational terms associated to the interaction of each Dirac mass with itself.

It is also possible to relate [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF] with discrete systems as follows. Consider the case of N gaseous spheres, far away one to each other, in such a way that they weakly interact through gravitation. In terms of system [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF], such a solution should be represented by a distribution function f , whose space density ρ is compactly supported, with several nearly spherical components. At large scale, the location of these spheres is governed at leading order by the N -body gravitational problem.

The purpose of this paper is to unveil this link by constructing a special class of solutions: we will build time-periodic, non radially symmetric solutions, which generalize to kinetic equations the notion of relative equilibria for the discrete N -body problem. Such solutions have a planar solid motion of rotation around an axis which contains the center of gravity of the system, so that the centrifugal force counter-balances the attraction due to gravitation. Let us give some details.

Consider N point particles with masses m j , located at points x j (t) ∈ R 3 and assume that their dynamics is governed by Newton's gravitational equations

m j d 2 x j dt 2 = N j =k=1 m j m k 4π x k -x j |x k -x j | 3 , j = 1, . . . N . (2)
Let us write x ∈ R 3 as x = (x ′ , x 3 ) ∈ R 2 × R ≈ C × R where, using complex notations, x ′ = (x 1 , x 2 ) ≈ x 1 +i x 2 and rewrite system [START_REF]An a priori estimate in the theory of hydrodynamic stability[END_REF] in coordinates relative to a reference frame rotating at a constant velocity ω > 0 around the x 3 -axis. This amounts to carry out the change of variables

x = (e i ω t z ′ , z 3 ), z ′ = z 1 + i z 2 .

In terms of the coordinates (z ′ , z 3 ), system (2) then reads

d 2 z j dt 2 = N j =k=1 m k 4π z k -z j |z k -z j | 3 + ω 2 (z ′ j , 0) + 2 ω i dz ′ j dt , 0 , j = 1, . . . N . (3)
We consider solutions which are stationary in the rotating frame, namely constant solutions (z 1 , . . . z N ) of system [START_REF] Batt | Stationary spherically symmetric models in stellar dynamics[END_REF]. Clearly all z j 's have their third component with the same value, which we assume zero. Hence, we have that

z k = (ξ k , 0) , ξ k ∈ C ,
where the ξ k 's are constants and satisfy the system of equations

N k =j=1 m k 4π ξ k -ξ j |ξ k -ξ j | 3 + ω 2 ξ j , j = 1, . . . N . (4)
In the original reference frame, the solution of (2) obeys to a rigid motion of rotation around the center of mass, with constant angular velocity ω. This solution is known as a relative equilibrium, thus taking the form

x ω j (t) = (e i ω t ξ j , 0) , ξ j ∈ C , j = 1, . . . N .
System (4) has a variational formulation. In fact a vector (ξ 1 , . . . ξ N ) solves ( 4) if and only if it is a critical point of the function

V ω m (ξ 1 , . . . ξ N ) := 1 8π N j =k=1 m j m k |ξ k -ξ j | + ω 2 2 N j=1 m j |ξ j | 2 .
Here m denotes (m j ) N j=1 . A further simplification is achieved by considering the scaling

ξ j = ω -2/3 ζ j , V ω m (ξ 1 , . . . ξ N ) = ω 2/3 V m (ζ 1 , . . . ζ N ) (5)
where

V m (ζ 1 , . . . ζ N ) := 1 8π N j =k=1 m j m k |ζ k -ζ j | + 1 2 N j=1 m j |ζ j | 2 .
This function has in general many critical points, which are all relative equilibria. For instance, V m clearly has a global minimum point.

Our aim is to construct solutions of gravitational models in continuum mechanics based on the theory of relative equilibria. We have the following result.

Theorem 1. Given masses m j , j = 1, . . . N , and any sufficiently small ω > 0, there exists a solution f ω (t, x, v) of equation (1) which is 2π ω -periodic in time and whose spatial density takes the form

ρ(t, x) := R 3 f ω dv = N i=1 ρ j (x -x ω j (t)) + o(1) .
Here o(1) means that the remainder term uniformly converges to 0 as ω → 0 + and identically vanishes away from ∪ N j=1 B R (x ω j (t)), for some R > 0, independent of ω. The functions ρ j (y) are non-negative, radially symmetric, non-increasing, compactly supported functions, independent of ω, with R 3 ρ j (y) dy = m j and the points x ω j (t) are such that

x ω j (t) = ω -2/3 (e i ω t ζ ω j , 0) , ζ ω j ∈ C , j = 1, . . . N and lim ω→0+ V m (ζ ω 1 , . . . ζ ω N ) = min C N V m , lim ω→0+ ∇V m (ζ ω 1 , . . . ζ ω N ) = 0 .
The solution of Theorem 1 has a spatial density which is nearly spherically symmetric on each component of its support and these ball-like components rotate at constant, very small, angular velocity around the x 3 -axis. The radii of these balls are very small compared with their distance to the axis. We shall call such a solution a relative equilibrium of (1), by extension of the discrete notion.

The construction provides much more accurate informations on the solution.

In particular, the building blocks ρ j are obtained as minimizers of an explicit reduced free energy functional, under suitable mass constraints.

It is also natural to consider other discrete relative equilibria, namely critical points of the energy V m that may or may not be globally minimizing, and ask whether associated relative equilibria of system (1) exist. There are plenty of relative equilibria of the N -body problem. For instance, if all masses m j are equal to some m * > 0, a critical point is found by locating the ζ j 's at the vertices of a regular polygon:

ζ j = r e 2 i π(j-1)/N , j = 1, . . . N , (6)
where r is such that

d dr a N 4π m * r + 1 2 r 2 = 0 with a N := 1 √ 2 N -1 j=1 1 1 -cos (2πj/N ) , i.e. r = (a N m * /(4π)) 1/3
. This configuration is called the Lagrange solution, see [START_REF] Perko | Regular polygon solutions of the N -body problem[END_REF]. The counterpart in terms of continuum mechanics goes as follows.

Theorem 2. Let (ζ 1 , . . . ζ N ) be a regular polygon, namely with ζ j given by [START_REF] Dancer | On the superlinear Lazer-McKenna conjecture[END_REF], and assume that all masses are equal. Then there exists a solution f ω exactly as in Theorem 1, but with lim ω→0+ (ζ ω 1 , . . .

ζ ω N ) = (ζ 1 , . . . ζ N )
. Further examples of relative equilibria in the N -body problem can be obtained for instance by setting N -1 points particles of the same mass at the vertices of a regular polygon centered at the origin, then adding one more point particle at the center (not necessarily with the same mass), and finally adjusting the radius. Another family of solutions, known as the Euler-Moulton solutions is constituted by arrays of aligned points.

Critical points of the functional V m are always degenerate because of their invariance under rotations: for any α ∈ R we have

V m (ζ 1 , . . . ζ N ) = V m (e i α ζ 1 , . . . e i α ζ N ) .
Let ζ = ( ζ1 , . . . ζN ) be a critical point of V m with ζℓ = 0. After a uniquely defined rotation, we may assume that ζℓ2 = 0. Moreover, we have a critical point of the function of 2N -1 real variables

Ṽm (ζ 1 , . . . ζ ℓ1 , . . . ζ N ) := V m (ζ 1 , . . . (ζ ℓ1 , 0), . . . ζ N ) .
We shall say that a critical point of V m is non-degenerate up to rotations if the matrix D 2

Ṽm ( ζ1 , . . . ζℓ1 , . . . ζN ) is non-singular. This property is clearly independent of the choice of ℓ.

Palmore in [START_REF] Palmore | Classifying relative equilibria. II[END_REF][START_REF]Classifying relative equilibria. III[END_REF][START_REF]New relative equilibria of the n-body problem[END_REF][START_REF]Minimally classifying relative equilibria[END_REF][START_REF]Measure of degenerate relative equilibria. I[END_REF] has obtained classification results for the relative equilibria. In particular, it turns out that for almost every choice of masses m j , all critical points of the functional V m are non-degenerate up to rotations. Moreover, in such a case there exist at least [2 N -1 (N -2) + 1] (N -2) ! such distinct critical points. Many other results on relative equilibria are available in the literature. We have collected some of them in Appendix A with a list of relevant references. These results have a counterpart in terms of relative equilibria of system (1). ). This paper is organized as follows. In the next section, we explain how the search for relative equilibria for the Vlasov-Poisson system can be reduced to the study of critical points of a functional acting on the gravitational potential. The construction of these critical points is detailed in Section 3. Sections 4 and 5 are respectively devoted to the linearization of the problem around a superposition of solutions of the problem with zero angular velocity, and to the existence of a solution of a nonlinear problem with appropriate orthogonality constraints depending on parameters (ξ j ) N j=1 related to the location of the N components of the support of the spatial density. Solving the original problem amounts to make all corresponding Lagrange multipliers equal to zero, which is equivalent to find a critical point of a function depending on (ξ j ) N j=1 : this is the variational reduction described in Section 6. The proof of Theorems 1, 2 and 3 is given in Section 7 while known results on relative equilibria for the N -body, discrete problem are summarized in Appendix A.

The setup

Guided by the representation (3) of the N -body problem in a rotating frame, we change variables in equation [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF], replacing x = (x ′ , x 3 ) and v = (v ′ , v 3 ) respectively by (e i ω t x ′ , x 3 ) and (i ω x ′ + e i ω t v ′ , v 3 ) .

Written in these new coordinates, Problem (1) becomes

       ∂f ∂t + v • ∇ x f -∇ x U • ∇ v f -ω 2 x ′ • ∇ v ′ f + 2 ω i v ′ • ∇ v ′ f = 0 , U = - 1 4π | • | * ρ , ρ := R 3 f dv . (7)
The last two terms in the equation take into account the centrifugal and Coriolis force effects. System (7) can be regarded as the continuous version of problem (3). Accordingly, a relative equilibrium of System (1) will simply correspond to a stationary state of [START_REF]On the superlinear Lazer-McKenna conjecture[END_REF].

Such stationary solutions of ( 7) can be found by considering for instance critical points of the free energy functional

F [f ] := R 3 ×R 3 β(f ) dx dv + 1 2 R 3 ×R 3 |v| 2 -ω 2 |x ′ | 2 f dx dv - 1 2 R 3 |∇U | 2 dx
for some arbitrary convex function β, under the mass constraint

R 3 ×R 3 f dx dv = M .
A typical example of such a function is

β(f ) = 1 q κ q-1 q f q (8)
for some q ∈ (1, ∞) and some positive constant κ q , to be fixed later. The corresponding solution is known as the solution of the polytropic gas model , see [START_REF] Batt | Stationary spherically symmetric models in stellar dynamics[END_REF][START_REF] Batt | On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation[END_REF][START_REF] Binney | Galactic dynamics[END_REF][START_REF] Sánchez | Orbital stability for polytropic galaxies[END_REF][START_REF] Wolansky | On nonlinear stability of polytropic galaxies[END_REF].

When dealing with stationary solutions, it is not very difficult to rewrite the problem in terms of the potential. A critical point of F under the mass constraint

R 3 ×R 3 f dx dv = M is indeed given in terms of U by f (x, v) = γ λ + 1 2 |v| 2 + U (x) -1 2 ω 2 |x ′ | 2 (9)
where γ is, up to a sign, an appropriate generalized inverse of β ′ . In case (8), γ(s) = κ -1 q (-s)

1/(q-1) +

, where s + = (s + |s|)/2 denotes the positive part of s. The parameter λ stands for the Lagrange multiplier associated to the mass constraint, at least if f has a single connected component. At this point, one should mention that the analysis is not exactly as simple as written above. Identity [START_REF] Dolbeault | Localized minimizers of flat rotating gravitational systems[END_REF] indeed holds only component by component of the support of the solution, if this support has more than one connected component, and the Lagrange multipliers have to be defined for each component. The fact that

U (x) ∼ |x|→∞ - M 4π |x| is dominated by -1 2 ω 2 |x ′ | 2 as |x ′ | → ∞ is
also a serious cause of trouble, which clearly discards the possibility that the free energy functional can be bounded from below if ω = 0. This issue has been studied in [START_REF] Dolbeault | Localized minimizers of flat rotating gravitational systems[END_REF], in the case of the so-called flat systems.

Finding a stationary solution in the rotating frame amounts to solving a non-linear Poisson equation, namely

∆U = g λ + U (x) -1 2 ω 2 |x ′ | 2 if x ∈ supp(ρ) (10)
and ∆U = 0 otherwise, where g is defined by

g(µ) := R 3 γ µ + 1 2 |v| 2 dv .
Hence, the problem can also be reduced to look for a critical point of the functional

J [U ] := 1 2 R 3 |∇U | 2 dx + R 3 G λ + U (x) -1 2 ω 2 |x ′ | 2 dx - R 3 λ ρ dx ,
where λ = λ[x, U ] is now a functional which is constant with respect to x, with value λ i , on each connected component

K i of the support of ρ(x) = g λ[x, U ] + U (x) -1 2 ω 2 |x ′ | 2 ,

and implicitly determined by the condition

Ki

g λ i + U (x) -1 2 ω 2 |x ′ | 2 dx = m i .
By G, we denote a primitive of g and the total mass is M = N i=1 m i . Hence we can rewrite J as

J [U ] = 1 2 R 3 |∇U | 2 dx + N i=1 Ki G λ i + U (x) -1 2 ω 2 |x ′ | 2 dx -m i λ i .
We may also observe that critical points of F correspond to critical points of the reduced free energy functional

G[ρ] := R 3 h(ρ) - 1 2 ω 2 |x ′ | 2 ρ dx - 1 2 R 3 |∇U | 2 dx
acting on the spatial densities if h(ρ) = ρ 0 g -1 (-s) ds. Also notice that, using the same function γ as in [START_REF] Dolbeault | Localized minimizers of flat rotating gravitational systems[END_REF], to each distribution function f , we can associate a local equilibrium, or local Gibbs state,

G f (x, v) = γ µ(ρ(x)) + 1 2 |v| 2
where µ is such that g(µ) = ρ. This identity defines µ = µ(ρ) = g -1 (ρ) as a function of ρ. Furthermore, by convexity, it follows that

F [f ] ≥ F [G f ] = G[ρ] if ρ(x) = R 3 f (x, v
) dv, with equality if f is a local Gibbs state. See [START_REF] Dolbeault | Non linear diffusions as limit of kinetic equations with relaxation collision kernels[END_REF] for more details.

Summarizing, the heuristics are now as follows. The various components K i of the support of the spatial density ρ of a critical point are assumed to be far away from each other so that the dynamics of their center of mass is described by the N -body point particles system, at first order. On each component K i , the solution is a perturbation of an isolated minimizer of the free energy functional F (without angular rotation) under the constraint that the mass is equal to m i . In the spatial density picture, on K i , the solution is a perturbation of a minimizer of the reduced free energy functional G.

To further simplify the presentation of our results, we shall focus on the model of polytropic gases corresponding to [START_REF] Wei | Layered solutions for a semilinear elliptic system in a ball[END_REF]. In such a case, with p :

= 1 q-1 + 3 2 , g is given by g(µ) = (-µ) p + if the constant κ q is fixed so that κ q = 4π √ 2 +∞ 0 √ t (1+t) 1 q-1 dt = (2π) 3 2 Γ ( q q-1 ) Γ ( 3 2 + q q-1 )
.

Free energy functionals have been very much studied over the last years, not only to characterize special stationary states, but also because they provide a framework to deal with orbital stability, which is a fundamental issue in the mechanics of gravitation. The use of a free energy functional, whose entropy part, R 3 ×R 3 β(f ) dx dv is sometimes also called the Casimir energy functional, goes back to the work of V.I. Arnold (see [START_REF] Arnol ′ D | On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid[END_REF][START_REF]An a priori estimate in the theory of hydrodynamic stability[END_REF][START_REF] Wolansky | An extension of Arnol'd's second stability theorem for the Euler equations[END_REF]). The variational characterization of special stationary solutions and their orbital stability have been studied by Y. Guo and G. Rein in a series of papers [START_REF] Guo | Existence and stability of Camm type steady states in galactic dynamics[END_REF][START_REF]Stable steady states in stellar dynamics[END_REF][START_REF]Isotropic steady states in galactic dynamics[END_REF][START_REF]Stable models of elliptical galaxies[END_REF][START_REF] Rein | Flat steady states in stellar dynamics -Existence and stability[END_REF][START_REF]Reduction and a concentration-compactness principle for energy-Casimir functionals[END_REF][START_REF]Non-linear stability of gaseous stars[END_REF][START_REF]Nonlinear stability of newtonian galaxies and stars from a mathematical perspective[END_REF] and by many other authors, see for instance [START_REF] Dolbeault | Localized minimizers of flat rotating gravitational systems[END_REF][START_REF] Dolbeault | Stability for the gravitational Vlasov-Poisson system in dimension two[END_REF][START_REF] Lemou | Orbital stability and singularity formation for Vlasov-Poisson systems[END_REF][START_REF]The orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system[END_REF][START_REF]Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system[END_REF][START_REF]Structure of the linearized gravitational Vlasov-Poisson system close to a polytropic ground state[END_REF][START_REF] Sánchez | Orbital stability for polytropic galaxies[END_REF][START_REF] Schaeffer | Steady states in galactic dynamics[END_REF][START_REF] Wolansky | On nonlinear stability of polytropic galaxies[END_REF].

The main drawback of such approaches is that stationary solutions which are characterized by these techniques are in some sense trivial: radial, with a single simply connected component support. Here we use a different approach to construct the solutions, which goes back to [START_REF] Floer | Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential[END_REF] in the context of Schrödinger equations. We are not aware of attempts to use dimensional reduction coupled to power-law non-linearities and Poisson force fields except in the similar case of a nonlinear Schrödinger equation with power law nonlinearity and repulsive Coulomb forces (see [START_REF] Wei | Layered solutions for a semilinear elliptic system in a ball[END_REF]), or in the case of an attractive Hartree-Fock model (see [START_REF] Krieger | Two soliton solutions to the three dimensional gravitational Hartree equation[END_REF]). Technically, our results turn out to be closely related to the ones in [START_REF] Dancer | On the superlinear Lazer-McKenna conjecture[END_REF][START_REF]On the superlinear Lazer-McKenna conjecture[END_REF].

Compared to previous results on gravitational systems, the main interest of our approach is to provide a much richer set of solutions, which is definitely of interest in astrophysics for describing complex patterns like binary gaseous stars or even more complex objects. The need of such an improvement was pointed for instance in [START_REF]A non-variational approach to nonlinear stability in stellar dynamics applied to the King model[END_REF]. An earlier attempt in this direction has been done in the framework of Wasserstein's distance and mass transport theory in [START_REF] Mccann | Stable rotating binary stars and fluid in a tube[END_REF]. The point of this paper is that we can take advantage of the knowledge of special solutions of the N -body problem to produce solutions of the corresponding problem in continuum mechanics, which are still reminiscent of the discrete system.

Construction of relative equilibria

3.1. Some notations. We denote by x = (x ′ , x 3 ) ∈ R 2 × R a generic point in R 3 . We may reformulate Problem [START_REF] Dolbeault | Stability for the gravitational Vlasov-Poisson system in dimension two[END_REF] in terms of the potential u = -U as follows. Given N positive numbers λ 1 , . . . λ N and a small positive parameter ω, we consider the problem of finding N non-empty, compact, disjoint, connected subsets K i of R 3 , i = 1, 2 . . . N , and a positive solution u of the problem

-∆u = N i=1 ρ i in R 3 , ρ i := u -λ i + 1 2 ω 2 |x ′ | 2 p + χ i , (11)
lim |x|→∞ u(x) = 0 , (12)
where χ i denotes the characteristic function of K i . We define the mass and the center of mass associated to each component by

m ω i := R 3 ρ i dx and x ω i := 1 m i R 3 x ρ i dx .
We shall find a solution of (11) as a critical point of J [u] = J [-u] in case of ( 8), with the notations of Section 2, that is when G(-s) = 1 p+1 s p+1 + for any s ∈ R.

The functional J reads

J [u] = 1 2 R 3 |∇u| 2 dx - 1 p + 1 N i=1 R 3 u -λ i + 1 2 ω 2 |x ′ | 2 p+1 + χ i dx . (13)
Heuristically, our method goes as follows. We first consider the so-called basic cell problem: we characterize the solution with a single component support, when ω = 0 and then build an ansatz by considering approximate solutions made of the superposition of basic cell solutions located close to relative equilibrium points, when they are far apart from each other. This can be done using the scaling invariance, in the low angular velocity limit ω → 0 + . The proof of our main results will be given in Sections 4-7. It relies on a dimensional reduction of the variational problem: we shall prove that for a well chosen u,

J [u] = N i=1 λ 5-p i e * -V ω m (ξ 1 , . . . ξ N ) + o(1)
for some constant e * , up to o(1) terms, which are uniform in ω > 0, small. Hence finding a critical point for J will be reduced to look for a critical point of V ω m as a function of (ξ 1 , . . . ξ N ).

The basic cell problem.

Let us consider the following problem

-∆w = (w -1) p + in R 3 . ( 14
)
Lemma 1. Under the condition lim |x|→∞ w(x) = 0, Equation ( 14) has a unique solution, up to translations, which is positive and radially symmetric.

Proof. Since p is subcritical, it is well known that the problem

-∆Z = Z p in B 1 (0)
with homogeneous Dirichlet boundary conditions, Z = 0, on ∂B 1 (0), has a unique positive solution, which is also radially symmetric (see [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF]). For any R > 0, the function Z R (x) := R -2/(p-1) Z(x/R) is the unique radial, positive solution of -∆Z R = Z p R in B R (0) with homogeneous Dirichlet boundary conditions on ∂B R (0). According to [START_REF] Flucher | Asymptotic shape and location of small cores in elliptic freeboundary problems[END_REF][START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF], any positive solution of ( 14) is radially symmetric, up to translations. Finding such a solution w of ( 14) is equivalent to finding numbers R > 0 and m * > 0 such that the function, defined by pieces as w = Z R + 1 in B R and w(x) = m * /(4π |x|) for any x ∈ R 3 such that |x| > R, is of class C 1 . These numbers are therefore uniquely determined by

w(R -) = 1 = m * 4π R = w(R + ) , w ′ (R -) = R -2 p-1 -1 Z ′ (1) = - m * 4π R 2 = w ′ (R + ) ,
which uniquely determines the solution of [START_REF] Flucher | Asymptotic shape and location of small cores in elliptic freeboundary problems[END_REF]. Now let us consider the slightly more general problem

-∆w λ = (w λ -λ) p + in R 3
with lim |x|→∞ w λ (x) = 0. For any λ > 0, it is straightforward to check that it has a unique radial solution given by

w λ (x) = λ w λ (p-1)/2 x ∀ x ∈ R 3 .
Let us observe, for later reference, that

R 3 (w λ -λ) p + dx = λ (3-p)/2 R 3 (w -1) p + dx =: λ (3-p)/2 m * . (15)
Moreover, w λ is given by

w λ (x) = m * 4π |x| λ (3-p)/2 ∀ x ∈ R 3 such that |x| > R λ -(p-1)/2 .
3.3. The ansatz. We consider now a first approximation of a solution of ( 11)-( 12), built as a superposition of the radially symmetric functions w λi translated to points ξ i , i = 1, . . . N in R 2 × {0}, far away from each other:

w i (x) := w λi (x -ξ i ) , W ξ := N i=1 w i .
Recall that we are given the masses m 1 , . . . m N . We choose, according to formula ( 15), the positive numbers λ i so that

R 3 (w i -λ i ) p + dx = m i for all i = 1, . . . N .
By ξ we denote the array (ξ 1 , ξ 2 , . . . ξ N ). We shall assume in what follows that the points ξ i are such that for a large, fixed µ > 0, and all small ω > 0 we have

|ξ i | < µ ω -2/3 , |ξ i -ξ j | > µ -1 ω -2/3 . (16)
Equivalently,

|ζ i | < µ , |ζ i -ζ j | > µ -1 where ζ i := ω -2/3 ξ i . (17)
We look for a solution of (11) of the form

u = W ξ + φ
for a convenient choice of the points ξ i , where φ is globally uniformly small when compared with W ξ . For this purpose, we consider a fixed number R > 1 such that supp (w λi -λ

i ) + ⊂ B R-1 (0) ∀ i = 1, 2 . . . N
and define the functions

χ(x) =    1 if |x| < R 0 if |x| ≥ R and χ i (x) = χ(x -ξ i ) .
Thus we want to find a solution to the problem

∆(W ξ + φ) + N i=1 W ξ -λ i + φ + 1 2 ω 2 |x ′ | 2 p + χ i = 0 in R 3
with lim |x|→∞ φ(x) = 0, that is we want to solve the problem

∆φ + N i=1 p W ξ -λ i + 1 2 ω 2 |x ′ | 2 p-1 + χ i φ = -E -N[φ] (18)
where

E := ∆W ξ + N i=1 W ξ -λ i + 1 2 ω 2 |x ′ | 2 p + χ i , N[φ] := N i=1 W ξ -λ i + 1 2 ω 2 |x ′ | 2 + φ p + -W ξ -λ i + 1 2 ω 2 |x ′ | 2 p + -p W ξ -λ i + 1 2 ω 2 |x ′ | 2 p-1 + φ χ i .

A linear theory

The purpose of this section is to develop a solvability theory for the operator

L[φ] = ∆φ + N i=1 p W ξ -λ i + 1 2 ω 2 |x ′ | 2 p-1 + χ i φ .
To this end we introduce the norms

φ * = sup x∈R 3 N i=1 |x -ξ i | + 1 |φ(x)| , h * * = sup x∈R 3 N i=1 |x -ξ i | 4 + 1 |h(x)| .
We want to solve problems of the form L[φ] = h with h and φ having the above norms finite. Rather than solving this problem directly, we consider a projected problem of the form

L[φ] = h + N i=1 3 j=1 c ij Z ij χ i , (19)
lim |x|→∞ φ(x) = 0 , (20)
where

Z ij := ∂ xj w i , subject to orthogonality conditions R 3 φ Z ij χ i dx = 0 ∀ i = 1, 2 . . . N, j = 1, 2, 3 . (21)
Equation ( 19) involves the coefficients c ij as Lagrange multipliers associated to the constraints [START_REF] Krieger | Two soliton solutions to the three dimensional gravitational Hartree equation[END_REF]. If we can solve the equations L [START_REF]Stable models of elliptical galaxies[END_REF] and satisfies [START_REF] Krieger | Two soliton solutions to the three dimensional gravitational Hartree equation[END_REF]. However, for existence, we will rather reformulate the question as a constrained variational problem; see Equation ( 23) below.

[ψ] = h and L[Y ij ] = Z ij , and if we define c ij such that R 3 ψ Z ij dx + c ij R 3 Y ij Z ij dx = 0, then we observe that φ = ψ + i, j c ij Y ij solves
Lemma 2. Assume that (16) holds. Given h with h * * < +∞, Problem (19)-( 21) has a unique solution φ =: T[h] and there exists a positive constant C, which is independent of ξ such that, for ω > 0 small enough,

φ * ≤ C h * * . (22)
Proof. In order to solve ( 19)-( 21), we first establish [START_REF] Lemou | Orbital stability and singularity formation for Vlasov-Poisson systems[END_REF] as an a priori estimate.

Assume by contradiction the existence of sequences ω n → 0, ξ n i satisfying ( 16) for ω = ω n , of functions φ n , h n and of constants c n ij for which

φ n * = 1 , lim n→∞ h n * * = 0 , R 3 φ n Z ij χ i dx = 0 ∀ i , j and L[φ n ] = h n + N i=1 3 j=1 c n ij Z ij χ i .
Testing the equation against Z kl , we obtain, after an integration by parts,

O(ω 2/3 n ) φ n * = R 3 h n Z kl dx + c n kl R 3 |Z kl | 2 χ k dx + O(ω 4/3 n ) (i,j) =(k, l) |c n ij | .
From these relations we then get c n kl = O(ω

2/3 n ) + O( h n * * ) → 0 for all k, l. Let us prove that lim n→∞ φ n L ∞ (R 3 ) = 0.
If not, since φ n * = 1, we may assume that there is an index i and a sufficiently large number R > 0 for which lim inf n→∞ φ n L ∞ (BR(ξi)) > 0 .

Using elliptic estimates, and defining ψ n (x) = φ n (ξ n i +x), we may assume that ψ n uniformly converges in C 1 sense over compact subsets of R 3 to a bounded, nontrivial solution ψ of the equation ∆ψ + p w λi -λ i p-1

+ ψ = 0 , R 3 ψ ∂ xj w λi χ dx = 0 ∀ j = 1, 2, 3 .
According to [START_REF] Flucher | Asymptotic shape and location of small cores in elliptic freeboundary problems[END_REF]Lemma 5], ψ must be a linear combination of the functions ∂ xj w λi , j = 1, 2, 3. The latter orthogonality conditions yield ψ ≡ 0. This is a contradiction and the claim is proven. Finally, let

hn := h n + N i=1 3 j=1 c n ij Z ij χ i .
Then we have that

| hn (x)| ≤ O(ω 2/3 n ) + O( h n * * ) k i=1 1 1 + |x -ξ n i | 4
and hence φn , the unique solution in R 3 of

∆ φn = hn , lim |x|→∞ φn (x) = 0 , satisfies | φn (x)| ≤ O(ω 2/3 n ) + O( h n * * ) k i=1 1 |x -ξ n i | . Now, since φ n -φn is harmonic in R 3 \∪ i B R (ξ n i )
, it tends to zero as |x| → ∞ and gets uniformly small on the boundary of this set. By the maximum principle, we get the estimate

|φ n (x)| ≤ O(ω 2/3 n ) + O( h n * * ) k i=1 1 |x -ξ n i | on R 3 \ ∪ i B R (ξ n i ) .
This shows that lim n→∞ φ n * = 0, a contradiction with φ n * = 1, and ( 22) follows.

Now, for existence issues, we observe that problem ( 19)-( 21) can be set up in variational form in the Hilbert space

H = φ ∈ D 1,2 (R 3 ) : R 3 φ Z ij χ i dx = 0 ∀ i = 1, 2 . . . N, j = 1, 2, 3 endowed with the inner product φ, ψ = R 3 ∇ψ • ∇φ dx, as R 3 ∇φ • ∇ψ dx - R 3 N i=1 p W ξ -λ i + 1 2 ω 2 |x ′ | 2 p-1 + χ i φ ψ dx + R 3
ψ h dx = 0 (23) for all ψ ∈ H. Since the potential defined by the second term of the above equality is compactly supported and h decays sufficiently fast, this equation takes the form φ + K[φ] = h where K is a compact linear operator of H. The equation for h = 0 has just the trivial solution in view of estimate [START_REF] Lemou | Orbital stability and singularity formation for Vlasov-Poisson systems[END_REF]. Fredholm's alternative thus applies to yield existence. This concludes the proof of Lemma 2.

We conclude this section with some considerations on the differentiability of the solution with respect to the parameter ξ. Let us assume that h = h(•, ξ) defines a continuous operator into the space of functions with finite • * * -norm. We also assume that ∂ ξ h(•, ξ) * * < +∞. Let φ = φ(•, ξ) be the unique solution of Problem ( 19)- [START_REF] Krieger | Two soliton solutions to the three dimensional gravitational Hartree equation[END_REF] for that right hand side, with corresponding constants c ij (ξ). Then φ is differentiable in ξ. Moreover ∂ ξ φ can be decomposed as

∂ ξ φ = ψ + ij d ij Z ij χ j where ψ solves L[ψ] = ∂ ξ h - N i=1 p ∂ ξ W ξ -λ i + 1 2 ω 2 |x ′ | 2 p-1 + χ i φ + N i=1 3 j=1 c ij ∂ ξ (Z ij χ i ) + b ij Z ij χ i , lim |x|→∞ ψ(x) = 0 , R 3 ψ Z ij χ i dx = 0 ∀ i = 1, 2 . . . N, j = 1, 2, 3 ,
and the constants d ij are chosen so that η :=

N i=1 3
j=1 d ij Z ij satisfies R 3 χ ij Z ij η dx = - R 3 ∂ ξ (χ ij Z ij ) φ dx ∀ i = 1, 2 . . . N, j = 1, 2, 3 .
Lemma 3. With the same notations and conditions as in Lemma 2, we have

∂ ξ φ(•, ξ) * ≤ C h(•, ξ) * * + ∂ ξ h(•, ξ) * * .

The projected nonlinear problem

Next we want to solve a projected version of the nonlinear problem ( 18) using Lemma 2. Thus we consider the problem of finding φ with φ * < +∞, solution of

L[φ] = -E -N[φ] + N i=1 3 j=1 c ij Z ij χ i (24) lim |x|→+∞ φ(x) = 0 ( 25
)
where the coefficients c ij are Lagrange multipliers associated to the orthogonality conditions

R 3 φ Z ij χ i dx = 0 ∀ i = 1, 2 . . . N, j = 1, 2, 3 . (26)
In other words, we look for a critical point of the functional J defined by [START_REF] Floer | Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential[END_REF] under the constraints [START_REF] Macmillan | Permanent configurations in the problem of four bodies[END_REF].

For this purpose, we first have to measure the error E. We recall that

E = ∆W ξ + N i=1 W ξ -λ i + 1 2 ω 2 |x ′ | 2 p + χ i = N i=1 w i + j =i w j -λ i + 1 2 ω 2 |x ′ | 2 p + -(w i -λ i ) p + χ i = N i=1 p w i -λ i + t j =i w j + 1 2 ω 2 |x ′ | 2 p-1 + j =i w j + 1 2 ω 2 |x ′ | 2 χ i
for some function t taking values in (0, 1). It follows that

|E| ≤ C N i=1 j =i 1 |ξ i -ξ j | + 1 2 ω 2 |ξ i | 2 χ i ≤ C ω 2/3 N i=1 χ i ,
from which we deduce the estimate

E * * ≤ C ω 2/3 .
As for the operator N[φ], we easily check that for φ * ≤ 1,

N[φ] ≤ C N i=1 |φ| γ χ i with γ := min{p, 2} , which implies N[φ] * * ≤ C φ γ * .
Let T be the linear operator defined in Lemma 2. Equation ( 24) can be rewritten as

φ = A[φ] := -T E + N[φ] .
Clearly the operator A applies the region

B = φ ∈ L ∞ (R 3 ) : φ * ≤ K ω 2/3 into itself if the constant K is fixed, large enough. It is straightforward to check that N[φ] satisfies in this region a Lipschitz property of the form N[φ 1 ] -N[φ 2 ] * * ≤ κ ω φ 1 -φ 2 *
for some positive κ ω such that lim ω→0 κ ω = 0, and hence existence of a unique fixed point φ of A in B immediately follows for ω small enough. We have then solved the projected nonlinear problem.

Since the error E is even with respect to the variable x 3 , uniqueness of the solution of ( 24)- [START_REF] Macmillan | Permanent configurations in the problem of four bodies[END_REF] implies that this symmetry is also valid for φ itself, and besides, the numbers c i3 are automatically all zero. Summarizing, we have proven the following result. Lemma 4. Assume that ξ = (ξ 1 , ξ 2 , . . . ξ N ) ∈ R 2N is given and satisfies [START_REF] Guo | Existence and stability of Camm type steady states in galactic dynamics[END_REF]. Then Problem (24)-( 26) has a unique solution φ ξ which depends continuously on ξ and ω for the * -norm and satisfies φ ξ * ≤ C ω 2/3 for some positive C, which is independent of ω, small enough. Besides, the numbers c i3 are all equal to zero for i = 1, 2 . . . N .

It is important to mention that φ ξ also defines a continuously differentiable operator in its parameter. Indeed, combining its fixed point characterization with the implicit function theorem and the result of Lemma 3, we find in fact that ∂ ξ φ ξ * ≤ C ω 2/3 . We leave the details to the reader.

With the complex notation of Section 1, let us consider the rotation e i α of an angle α around the x 3 -axis and let e i α ξ = (e i α ξ 1 , . . . e i α ξ N ). By construction, there is a rotational symmetry around the x 3 -axis, which is reflected at the level of Problem ( 24)-(26) as follows.

Lemma 5. Consider the solution φ found in Lemma 4. For any α ∈ R and any (x ′ , x 3 ) ∈ C × R, we have that

φ e i α ξ (x ′ , x 3 ) = φ ξ (e -i α x ′ , x 3 ) .
The proof is a direct consequence of uniqueness and rotation invariance of the equation satisfied by φ ξ .

The variational reduction

We consider the functional J defined in [START_REF] Floer | Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential[END_REF]. Our goal is to find a critical point satisfying [START_REF] Macmillan | Permanent configurations in the problem of four bodies[END_REF], of the form u = W ξ + φ ξ . We estimate J [W ξ ] by computing first

R 3 |∇W ξ | 2 dx = R 3 N i=1 ∇w i 2 dx = N i=1 R 3 |∇w i | 2 dx+ i =j R 3 ∇w i • ∇w j dx .
The last term of the right hand side can be estimated by

R 3 ∇w i • ∇w j dx = - R 3 ∆w i w j dx = R 3 (w i -λ i ) p + w j dx = R 3 w λi -λ i p + w λj (x + ξ i -ξ j ) dx = R 3 w λi -λ i p + m j 4π |x + ξ i -ξ j | dx = m i m j 4π |ξ i -ξ j | + O(ω 4/3 )
where

m i = R 3 w λi -λ i p + dx = m * λ (3-p)/2 i
. Next we find that

R 3 w i + j =i w j -λ i + 1 2 ω 2 |x ′ | 2 p+1 + χ i dx = R 3 (w i -λ i ) p+1 + dx + (p + 1) R 3 (w i -λ i ) p + j =i w j + 1 2 ω 2 |x ′ | 2 dx + O(ω 4/3 ) = R 3 (w i -λ i ) p+1 + dx + (p + 1) j =i m i m j 4π |ξ i -ξ j | + 1 2 ω 2 m i |ξ i | 2 + O(ω 4/3 ) .
Let us define

e * := 1 2 R 3 |∇w| 2 dx - 1 p + 1 R 3 (w -1) p+1 + dx .
Combining the above estimates, we obtain that

J [W ξ ] = N i=1 λ 5-p i e * -V ω m (ξ 1 , . . . ξ N ) + O(ω 4/3 ) , (27)
where

V ω m (ξ) = i =j mi mj 8π |ξi-ξj | + 1 2 ω 2 N i=1 m i |ξ i | 2
has been introduced in Section 1. Here the O(ω 4/3 ) term is uniform as ω → 0 on the set of ξ satisfying the constraints [START_REF] Guo | Existence and stability of Camm type steady states in galactic dynamics[END_REF]. This approximation is also uniform in the C 1 sense. Indeed, we directly check that

∇ ξ J [W ξ ] = -∇ ξ V ω m (ξ) + O(ω 4/3 ) . ( 28
)
According to [START_REF] Binney | Galactic dynamics[END_REF], we have V ω m (ξ) = ω 2/3 V m (ζ) for ζ = ω 2/3 ξ. We get a solution of Problem ( 11)- [START_REF] Elmabsout | Sur l'existence de certaines configurations d'équilibre relatif dans le problème des n corps[END_REF] as soon as all constants c ij are equal to zero in [START_REF]Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system[END_REF]. Lemma 6. With the above notations, c ij = 0 for all i = 1, 2 . . . N, j = 1, 2, 3 if and only if ξ is a critical point of the functional ξ → Λ(ξ)

:= J [W ξ + φ ξ ].
Proof. We have already noticed in Lemma 4 that the numbers c i3 are all equal to zero. On the other hand, we have that

∂ ξij Λ = DJ [W ξ + φ ξ ] • ∂ ξij (W ξ + φ ξ ) = k, ℓ c kl R 3 ∂ ξij (W ξ + φ ξ ) Z kℓ χ ℓ dx = c ij R 3 |Z ij | 2 χ i dx + (k, ℓ) =(i,j) c kℓ O(ω 2/3 ) . (29)
From here the assertion of the lemma readily follows, provided that ω is sufficiently small. Remark 1. An important observation that follows from the rotation invariance of the equation is the following. Assume that the point ξ is such that ξ ℓ = (ξ ℓ1 , 0) = (0, 0) for some ℓ ∈ {1, . . . N }. Then if ∂ ξ kj Λ(ξ) = 0 for all k = 1, . . . N , j = 1, 2, (k, j) = (ℓ, 2) , it follows that actually ξ is a critical point of Λ. Indeed, differentiating in α the relation Λ(e i α ξ) = Λ(ξ) we get

0 = N k=1 ∂ ξ k Λ(ξ) • i ξ k = ∂ ξ ℓ Λ(ξ) • i ξ ℓ = -ξ ℓ1 ∂ ξ ℓ2 Λ(ξ) ,
and the result follows.

Proofs of Theorems 1-3

Let us consider the solution φ ξ of ( 24)-( 26), i.e. of the problem

L[φ ξ ] = -E -N[φ ξ ] + N i=1 3 j=1 c ij (ξ) Z ij χ i lim |x|→+∞ φ ξ (x) = 0
given by Lemma 4. We will then get a solution of Problem ( 11)-( 12), of the desired form u = W ξ + φ ξ , inducing the ones for Theorems 1 and 3, if we can adjust ξ in such a way that c ij (ξ) = 0 for all i = 1, 2 . . . N, j = 1, 2, 3 .

According to Lemma 6, this is equivalent to finding a critical point of the functional

Λ(ξ) := J [W ξ + φ ξ ] .
We expand this functional as follows:

J [W ξ ] = J [W ξ + φ ξ ] -DJ [W ξ + φ ξ ] • φ ξ + 1 2 1 0 D 2 J [W ξ + (1 -t) φ ξ ] • (φ ξ , φ ξ ) dt .
By definition of φ ξ we have that DJ [W ξ + φ ξ ] • φ ξ = 0. On the other hand, using Lemma 4, we check directly, out of the definition of φ ξ , that

D 2 J [W ξ + (1 -t) φ ξ ] • (φ ξ , φ ξ ) = O(ω 4/3 )
uniformly on points ξ i satisfying constraints [START_REF] Guo | Existence and stability of Camm type steady states in galactic dynamics[END_REF]. Hence, from expansion [START_REF] Mccann | Stable rotating binary stars and fluid in a tube[END_REF] we obtain that

Λ(ξ) = N i=1 λ 5-p i e * -V ω m (ξ) + O(ω 4/3 ) .
We claim that this expansion also holds in the C 1 sense. Let us first observe that

R 3 E ∂ ξ W ξ dx = ∇ ξ J [W ξ ] and ∂ ξij W ξ = Z ij .
Then, testing equation ( 24) against Z ij , we see that

R 3 N[φ] Z ij + L[Z ij ] φ dx = - R 3 E Z ij dx + kl c kl R 3 Z ij Z kl χ i dx .
Next we observe that

L[Z ij ] * * = O(ω 2/3 ) and R 3 Z ij Z kl χ i dx = O(ω 2/3 ) if (i, j) = (k, l) .
By lemma 4, φ ξ * = O(ω 2/3 ), and so we get

1 + O(ω 2/3 ) c ij = O(ω 4/3 ) + ∂ ξij J [W ξ ] .
Hence, according to relation [START_REF] Moulton | The straight line solutions of the problem of n bodies[END_REF], we obtain

I 3N + O(ω 2/3 ) ∇ ξ Λ(ξ) = ∇ ξ J [W ξ ] + O(ω 4/3 )
where ∇ ξ J [W ξ ] has been computed in [START_REF] Meyer | Introduction to Hamiltonian Dynamical Systems and the N-Body Problem[END_REF]. Summarizing, we have found that Fixing such a µ, we get from Proposition 1 that, for all sufficiently small ω,

∇ ξ Λ(ξ) = -∇ ξ V ω m (ξ 1 , . . . ξ N ) + O(ω 2/3 ) .
sup Bµ Γ > sup ∂Bµ Γ .
so that the functional Λ has a maximum value somewhere in ω 2/3 B µ , which is close to a maximum value of V ω m . This value is achieved at critical point of Λ, and hence a solution with the desired features exists. The construction is concluded. 7.2. Proof of Theorem 2. When (ζ 1 , . . . ζ N ) is a regular polygon with ζ j given by ( 6) and all masses are equal, the system is invariant under the rotation defined by

x = (x 1 , x 2 , x 3 ) ∈R 3 ≈ ((x 1 + i x 2 ), x 3 ) ∈C×R → e 2 i π/N (x 1 + i x 2 ), x 3 =: R N x .
We can therefore pass to the quotient with respect to this group of invariance and look for solutions u which are invariant under then action of R N and moreover symmetric symmetric with respect to the reflections (x 1 , x 2 , x 3 ) → (x 1 , -x 2 , x 3 ) and (x 1 , x 2 , x 3 ) → (x 1 , x 2 , -x 3 ). Here we assume that (ζ 1 , . . . ζ N ) is contained in the plane {x 3 = 0} and ζ 1 = (r, 0, 0). Altogether this amounts to look for critical points of the functional

J 1 [u] = 1 2 Ω1 |∇u| 2 dx - 1 p + 1 Ω1 u -λ 1 + 1 2 ω 2 |x ′ | 2 p+1 + χ 1 dx
where

Ω 1 = {x = (x ′ , x 3 ) ∈ R 3 ≈ C × R : x ′ = r e i θ s.t. -π N < θ < π N } and u ∈ H 1 (Ω) is invariant under the two above reflections and such that ∇u • n = 0 on ∂Ω \ {0} × R and χ 1 is the characteristic function of the support of ρ 1 = u -λ 1 + 1 2 ω 2 |x ′ | 2 p+1 + χ 1 in Ω 1 .
Here n = n(x) denotes the unit outgoing normal vector at x ∈ ∂Ω 1 . With J defined by [START_REF] Floer | Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential[END_REF], it is straightforward to see on S m is equivalent to that of relative equilibria; see for instance [START_REF] Elmabsout | Sur l'existence de certaines configurations d'équilibre relatif dans le problème des n corps[END_REF]. Let us give some details. Let q be a critical point of U m on S m . Then by definition, there are Lagrange multipliers λ ∈ R and µ ∈ R 2 for which

- 1 8π N i =j m i m j |q i -qj | 3 (q i -qj ) = λ m j qj + m j µ ∀ j = 1, . . . N .
First, adding in j the above relations and using that M = N j=1 m j > 0 we obtain that µ = 0. Second, taking the scalar product of R 2 against qj and then adding in j, we easily obtain that U m (q) = λ. From here it follows that the point ζ = λ 1/3 q is a critical point of the functional V m , hence a relative equilibrium.

With the reparametrization of R 2N given by

ζ(α, p, q) = (ζ 1 , . . . ζ N ) = (α q 1 + p, . . . α q N + p) , (α, p, q) ∈ R × R 2 × S m ,
the Hessian matrix of V m at the critical point ζ = ζ(λ 1/3 , 0, q) found above is represented as the block matrix

D 2 V m ( ζ) =   2 M I 2 λ -1/3 D 2 Sm U m (q)  
where I 2 is the 2 × 2 identity matrix and D 2 Sm represents the second covariant derivative on S m . Reciprocally, we check that a critical point ζ = (ζ j ) N j=1 of V m necessarily satisfies N j=1 m j ζ j = 0. Defining q = 1 2 N j=1 m j |ζ j | 2 -1/2 ζ, we readily check that q is a critical point of U m in S m .

Any rotation e i α q of a critical point q of U m on S m is also a critical point. We say that two such critical points are equivalent in S m . Let us denote by S m the quotient manifold of S m by this equivalence relation. On S m , critical points of the potential U m yield critical points of U m on S m and hence equivalence classes of critical points e i α ζ for V m using the reparametrization.

A critical point q of U m on S m is said to be non-degenerate if the second variation of U m at q is non-singular. Let us assume that qℓ = 0, with either ℓ = 1, or ℓ = 2 if q1 = 0. Then there is a unique representative q of this class of equivalence for which qℓ2 = 0. It is a routine verification to check that q is then a critical point of U m on the (2N -4)-dimensional manifold S m := q ∈ S m : q ℓ = 0 as above , q ℓ2 = 0 .

Moreover, the second derivative of U m on S m at q is non-degenerate if and only if D 2

Sm U m (q) is non-singular. Because of the expression of D 2 V m ( ζ), we see that ζ is non-degenerate as a critical point of V m on the space of ζ ∈ R 2N with q ℓ2 = 0, which is the notion of non-degeneracy up to rotations of a relative equilibrium that we have used in this paper. Finally we define the index of a non-degenerate relative equilibrium ζ as the number of negative eigenvalues of D 2 Sm U m (q). Some results on classification of relative equilibria. For simplicity, we will assume that masses are all different: for any i, j = 1, . . . N , if m i = m j , then i = j. This is the generic case.

The cases N = 2, 3 are well known; see for instance [START_REF] Meyer | Introduction to Hamiltonian Dynamical Systems and the N-Body Problem[END_REF]. For N = 2, the only class of critical points is such that For N = 3, there are two types of solutions, the Lagrange and the Euler solutions. The Lagrange solutions are such that their center of mass is fixed at the origin, the masses are located at the vertices of an equilateral triangle, and the distance between each point is (M/(4π)) 1/3 with M = m 1 + m 2 + m 3 . They give rise to two classes of solutions corresponding to the two orientations of the triangle when labeled by the masses. The Euler solutions are made of aligned points and provide three classes of critical points, one for each ordering of the masses on the line.

|ζ 1 -ζ 2 | = M
In the case N ≥ 4, the classes of solutions for which all points are collinear still exist (see [START_REF] Moulton | The straight line solutions of the problem of n bodies[END_REF]) and are known as the Moulton solutions. But the configuration of relative equilibria where all particles are located at the vertices of a regular N -polygon exists if and only if all masses are equal; see [START_REF] Macmillan | Permanent configurations in the problem of four bodies[END_REF][START_REF] Williams | Permanent configurations in the problem of five bodies[END_REF][START_REF] Perko | Regular polygon solutions of the N -body problem[END_REF][START_REF] Elmabsout | Sur l'existence de certaines configurations d'équilibre relatif dans le problème des n corps[END_REF][START_REF] Xie | A simpler proof of regular polygon solutions of the N -body problem[END_REF]. Various classification results which have been obtained by Palmore are summarized below.

Theorem 4 ( [START_REF] Palmore | Classifying relative equilibria. II[END_REF][START_REF]Classifying relative equilibria. III[END_REF][START_REF]New relative equilibria of the n-body problem[END_REF][START_REF]Minimally classifying relative equilibria[END_REF][START_REF]Measure of degenerate relative equilibria. I[END_REF]). We have the following multiplicity results.

Theorem 3 .

 3 Let (ζ 1 , . . . ζ N ) be a non-degenerate critical point of V m up to rotations. Then there exists a solution f ω as in Theorem 1, which satisfies, as in Theorem 2, lim ω→0+ (ζ ω 1 , . . . ζ ω N ) = (ζ 1 , . . . ζ N

Therefore, setting ξ = ω 2 / 3 ζProposition 1 . 7 . 1 .

 23171 with ζ = (ζ 1 , . . . ζ N ) and defining Γ (ζ) := Λ(ξ) on B µ := ζ ∈ R 2N : (17) holds , we have shown the following result. With the above notations, we have that ω 2/3 V m (ζ) + O(ω 4/3 ) ∇Γ (ζ) = -∇V m (ζ) + O(ω 2/3 ) uniformly on ζ satisfying (17). Here the terms O(•) are continuous functions of ζ defined on B µ . Proof of Theorem 1. If µ > 0 is fixed large enough, we have that inf Bµ V m < inf ∂Bµ V m .

4π 1 / 3 and m 1 ζ 1 +

 131 m 2 ζ 2 = 0 with M = m 1 + m 2 .

( a ) 2 .

 a2 For N ≥ 3, the index of a relative equilibrium is always greater or equal than N -This bound is achieved by Moulton's solutions. (b) For N ≥ 3, there are at least µ i (N ) := N i (N -1 -i) (N -2) ! distinct relative equilibria in S m of index 2N -4 -i if U m is a Morse function. As a consequence, there are at least N -2 i=0 µ i (N ) = [2 N -1 (N -2) + 1] (N -2) ! distinct relative equilibria in S m if U m is aMorse function. (c) For every N ≥ 3 and for almost all masses m ∈ R N + , U m is a Morse function. (d) There are only finitely many classes of relative equilibria for every N ≥ 3 and for almost all masses m = (m i ) N i=1 ∈ R N + .

c 2010 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Acknowledgments. This project is part of the scientific program of the MathAmSud network NAPDE and of the ANR funded research projectCBDif-Fr. It has also been supported by grants Fondecyt 1070389, Fondo Basal CMM and Anillo ACT 125.

that J [u] = N J 1 [u] if u is extended to R 3 by assuming that u(R N x) = u(x). With these notations, we find that

The proof goes as for Theorem 1. Because of the symmetry assumptions, c 1j = 0 if j = 2 or 3. Details are left to the reader.

Similarly, we denote by Γ the restriction of

By assumption, ζ is a non-degenerate critical point of Ṽm , i.e. an isolated zero of ∇ Ṽm . Besides, its local degree is non-zero. It follows that on an arbitrarily small neighborhood of that point, the degree for ∇ Γ is non-zero for all sufficiently small ω. Hence there exists a zero ζ ω ∈ (R × {0}) × (R 2 ) N -1 of ∇ Γ as close to ζ as we wish. From the rotation invariance, it follows that ζ ω is also a critical point of Γ . The proof of Theorem 3 is concluded.

Appendix A. Facts on Relative Equilibria

In this Appendix we have collected some results on the N -body problem introduced in Section 1 which are of interest for the proofs of Theorems 1-3, with a list of relevant references.

Non-degeneracy of relative equilibria in a standard form. Relative equilibria are by definition critical points of the function V m : R 2N → R defined by

Here we assume that N ≥ 2, and m i > 0, i = 1, . . . N are given parameters.

Following Smale in [START_REF] Smale | Topology and mechanics II[END_REF], we can rewrite this problem as follows. Let us consider the (2N -3)-dimensional manifold

2 |q i | 2 ) = (0, 1) , q i = q j if i = j .

The problem of finding critical points of the functional U m (q 1 , . . . q N ) = 1 8π