Caroline Kalla

Christian Klein

NEW CONSTRUCTION OF ALGEBRO-GEOMETRIC SOLUTIONS TO THE CAMASSA-HOLM EQUATION AND THEIR NUMERICAL EVALUATION

New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation

Introduction

The Camassa-Holm (CH) equation (1.1)

u t + 3 uu x = u xxt + 2 u x u xx + uu xxx -2k u x ,
was first found by Fokas and Fuchssteiner [START_REF] Fokas | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF] with the method of recursion operators and shown to be a bi-hamiltonian equation with an infinite number of conserved functionals. Camassa and Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] showed that it appeared as a model for unidirectional propagation of waves in shallow water, u(x, t) representing the height of the free surface about a flat bottom, k being a constant related to the critical shallow water speed. In this context, only real-valued solutions are physically meaningful.

To be able to formulate a Cauchy problem for the CH equation which implies the solution of (1.1) for a given function u(x, 0), it is convenient to write (1.1) in the nonlocal evolutionary form, (1.2)

u t = D -1 (-3uu x + 2 u x u xx + uu xxx -2k u x ),
where the operator D is given by D = 1-∂ xx , and where its inverse is defined by giving certain boundary conditions. This non-locality has mathematically interesting consequences: the CH equation has traveling wave solutions of the form u(x, t) = c exp{-|x -vt|} (v being the speed) called peakons that have a discontinuous first derivative at the wave peak.

In particular, Camassa and Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] described the dynamics of the peakons in terms of a finite-dimensional completely integrable Hamiltonian system, namely, each peakon solution is associated with a mechanical system of moving particles. The class of mechanical systems of this type was further extended by Calogero and Françoise in [START_REF] Calogero | An integrable Hamiltonian system[END_REF][START_REF] Calogero | Solvable quantum version of an integrable Hamiltonian system[END_REF]. Multipeakon solutions were studied using different approaches in a series of papers [4, [START_REF] Beals | Multi-peakons and a theorem of Stietjes[END_REF][START_REF] Beals | Multipeakons and the classical moment[END_REF][START_REF] Camassa | Characteristic variables for a completely integrable shallow water equation[END_REF]. Periodic solutions of the shallow water equation were discussed in [START_REF] Mckean | A shallow water equation on the circle[END_REF].

A further consequence of this non-locality is that almost periodic solutions of the CH equation in terms of multi-dimensional theta functions do not depend explicitly on the physical coordinates. Such solutions were first given by Alber and Fedorov in [START_REF] Alber | Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians[END_REF] by solving a generalized Jacobi inversion problem. In contrast to the well known cases of Kortewegde Vries (KdV), nonlinear Schrödinger and sine-Gordon equations, see for instance [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] and references therein, complex solutions of the CH equation are not meromorphic functions of (x, t) but have several branches. This is due to the presence of an implicit function y(x, t) of the variables x and t in the argument of the theta function appearing in the solutions. This monodromy effect is present in the profile of real-valued solutions such as cusps and peakons. This means that even bounded solutions to the CH equation can have discontinuous or infinite derivatives in contrast to KdV solutions. Algebro-geometric solutions of the Camassa-Holm equation and their properties are studied in [START_REF] Alber | The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type[END_REF][START_REF] Alber | Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians[END_REF][START_REF] Alber | Algebraic geometrical solutions certain evolution equations and Hamiltonian flows Beals[END_REF].

Our goal in this paper is to give an independent derivation of such solutions based on identities between multi-dimensional theta functions, which naturally arise from Fay's identity [START_REF] Fay | Theta functions on Riemann surfaces[END_REF]. While the authors in [START_REF] Alber | The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type[END_REF][START_REF] Alber | Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians[END_REF][START_REF] Alber | Algebraic geometrical solutions certain evolution equations and Hamiltonian flows Beals[END_REF] employ the trace formula for the function u in terms of (projections of) auxiliary divisors, and used generalized theta functions and generalized Jacobians (going back to investigations of Clebsch and Gordan [START_REF] Clebsch | Theorie der Abelschen Funktionen[END_REF]), we derive our solutions from a purely transcendental identity satisfied by theta functions. This fits into the program formulated by Mumford [27] that all algebro-geometric solutions to integrable equations should be obtained from Fay's identity and suitable degenerations thereof. This approach was recently successfully in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF][START_REF] Kalla | On the numerical evaluation of algebro-geometric solutions to integrable equations[END_REF] to the multi-component nonlinear Schrodinger equations and the Davey-Stewartson equations. We provide here the first example for an integrable equation with nonlocal terms (1.2). The spectral data for these solutions consist of a hyperelliptic curve of the form µ 2 = 2g+2 j=1 (λλ j ) with three marked points: two of them are interchanged under the involution σ(λ, µ) = (λ, -µ), and the third is a ramification point (λ j 0 , 0).

Our construction of real valued solutions is based on the description of the real and imaginary part of the Jacobian associated to a real hyperelliptic curve (i.e., the branch points λ j are real or pairwise conjugate non-real). In this way, one gets purely transcendental conditions on the parameters (i.e., without reference to a divisor defined by the solution of a Jacobi inversion problem), such that the solutions are real-valued and smooth. It turns out that continuous real valued solutions are either smooth or have an infinite number of cusp-type singularities.

Concrete examples for the resulting solutions are studied numerically by using the code for real hyperelliptic surfaces [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF]. This code uses so-called spectral methods to compute periods on the surfaces. It allows also to study numerically almost degenerate surfaces where the branch points collapse pairwise. In this limit, the theta functions break down to elementary functions, and the solutions describe solitons or cusps. The quality of the numerics is ensured by testing the identities between theta functions which are used to construct the CH solutions in this paper. In addition, the solutions are computed on a grid and are numerically differentiated. These independent tests ensure that the shown solutions are correct to much better than plotting accuracy.

The paper is organized as follows: in section 2 we summarize important facts on Riemann surfaces, especially Fay's identities for theta functions and results on real surfaces. In section 3 we use Fay's identities to rederive theta-functional solutions to the CH equation, and give reality and smoothness conditions. In section 4 we study numerically concrete examples, also in almost degenerate situations. We add some concluding remarks in section 5.

Theta functions and real Riemann surfaces

In this section we recall basic facts on Riemann surfaces, in particular real surfaces and multi-dimensional theta functions defined on them.

2.1. Theta functions. Let R g be a compact Riemann surface of genus g > 0. Denote by (A, B) := (A 1 , . . . , A g , B 1 , . . . , B g ) a canonical homology basis, and by (ω 1 , . . . , ω g ) the basis of holomorphic differentials normalized via (2.1)

A k ω j = 2iπδ kj , k, j = 1, . . . , g.

The matrix B =

B k ω j of B-periods of the normalized holomorphic differentials ω j , j = 1, . . . , g, is symmetric and has a negative definite real part. The theta function with

(half integer) characteristic δ = [δ 1 , δ 2 ] is defined by (2.2) Θ B [δ](z) = m∈Z g exp 1 2 B(m + δ 1 ), m + δ 1 + m + δ 1 , z + 2iπδ 2 ,
for any z ∈ C g ; here δ 1 , δ 2 ∈ 0,

(2.3) Θ[δ](z) = Θ(z + 2iπδ 2 + Bδ 1 ) exp 1 2 Bδ 1 , δ 1 + z + 2iπδ 2 , δ 1 .
Denote by Λ the lattice Λ = {2iπN + BM, N, M ∈ Z g } generated by the A and Bperiods of the normalized holomorphic differentials ω j , j = 1, . . . , g. The complex torus J := J(R g ) = C g /Λ is called the Jacobian of the Riemann surface R g . The theta function (2.2) has the following quasi-periodicity property with respect to the lattice Λ:

Θ[δ](z + 2iπN + BM) (2.4) = Θ[δ](z) exp -1 2 BM, M -z, M + 2iπ( δ 1 , N -δ 2 , M ) .
Denote by Π the Abel map Π : R g -→ J defined by

(2.5) Π(p) = p p 0 ω,
for any p ∈ R g , where p 0 ∈ R g is the base point of the application, and where ω = (ω 1 , . . . , ω g ) t is the vector of the normalized holomorphic differentials. In the whole paper we use the notation b a = Π(b) -Π(a). Now let k a denote a local parameter near a ∈ R g and consider the following expansion of the normalized holomorphic differentials ω j , j = 1, . . . , g, (2.6) ω j (p) = (V a,j + W a,j k a (p) + . . .) dk a (p), for any point p ∈ R g in a neighborhood of a, where V a,j , W a,j ∈ C. Denote by D a (respectively D a ) the operator of the directional derivative along the vector V a = (V a,1 , . . . , V a,g ) t (respectively W a ). According to [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF], the theta function satisfies the following identities derived from Fay's identity [START_REF] Fay | Theta functions on Riemann surfaces[END_REF]:

D b ln Θ(z + a c ) Θ(z) = p 1 + p 2 Θ(z + a b ) Θ(z + b c ) Θ(z + a c ) Θ(z) , (2.7) D a D b ln Θ(z) = q 1 + q 2 Θ(z + b a ) Θ(z - b a ) Θ(z) 2 , (2.8)
for any z ∈ C g and any distinct points a, b ∈ R g ; here the scalars p i , q i , i = 1, 2 depend on the points a, b and are given by

p 1 (a, b, c) = -D b ln Θ[δ]( b a ) Θ[δ]( b c ) , (2.9) p 2 (a, b, c) = Θ[δ]( a c ) Θ[δ]( a b )Θ[δ]( c b ) D b Θ[δ](0) , (2.10) q 1 (a, b) = D a D b ln Θ[δ]( b a ), (2.11) q 2 (a, b) = D a Θ[δ](0) D b Θ[δ](0) Θ[δ]( b a ) 2 , (2.12)
where δ is a non-singular odd characteristic.

Real Riemann surfaces.

A Riemann surface R g is called real if it admits an antiholomorphic involution τ : R g → R g , τ 2 = id. The connected components of the set of fixed points of the anti-involution τ are called real ovals of τ . We denote by R g (R) the set of fixed points. According to Harnack's inequality [START_REF] Harnack | Ueber die Vieltheiligkeit der ebenen algebraischen Curven[END_REF], the number χ of real ovals of a real Riemann surface of genus g cannot exceed g + 1: 0 ≤ χ ≤ g + 1. Curves with the maximal number χ = g + 1 of real ovals are called M-curves.

The complement R g \R g (R) has either one or two connected components. The curve R g is called a dividing curve if R g \ R g (R) has two components, and R g is called non-dividing if R g \ R g (R) is connected (notice that an M-curve is always a dividing curve).

Example 2.1. Consider the hyperelliptic curve of genus g defined by the equation

(2.13) µ 2 = 2g+2 i=1 (λ -λ i ),
where the branch points λ i ∈ R are ordered such that λ 1 < . . . < λ 2g+2 . On such a curve, we can define two anti-holomorphic involutions τ 1 and τ 2 , given respectively by τ 1 (λ, µ) = (λ, µ) and τ 2 (λ, µ) = (λ, -µ). Projections of real ovals of τ 1 on the λ-plane coincide with the intervals [λ 2g+2 , λ 1 ], . . . , [λ 2g , λ 2g+1 ], whereas projections of real ovals of τ 2 on the λ-plane coincide with the intervals [λ 1 , λ 2 ], . . . , [λ 2g+1 , λ 2g+2 ]. Hence the curve (2.13) is an M-curve with respect to both anti-involutions τ 1 and τ 2 .

Let (A, B) be a basis of the homology group H 1 (R g ). According to Proposition 2.2 in Vinnikov's paper [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] (see also [START_REF] Gross | Real algebraic curves[END_REF]), there exists a canonical homology basis such that

(2.14) τ A τ B = I g 0 H -I g A B ,
where I g is the g × g unit matrix, and H is a block diagonal g × g matrix defined as follows:

1) if R g (R) = ∅, H =              0 1 1 0 . . . 0 1 1 0 0 . . . 0              if R g is dividing, H =          1 . . . 1 0 . . . 0          if R g is non-dividing; rank(H) = g + 1 -χ in both cases.
2) if R g (R) = ∅, (i.e. the curve does not have real ovals), then

H =        0 1 1 0 . . . 0 1 1 0        or H =          0 1 1 0 . . . 0 1 1 0 0          ; rank(H) = g if g is even, rank(H) = g -1 if g is odd.
In what follows we choose the canonical homology basis in H 1 (R g ) satisfying (2.14) and take a, b ∈ R g such that τ a = b. Denote by a contour connecting the points a and b which does not intersect the canonical homology basis. Then the action of τ on the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) is given by

(2.15)   τ A τ B τ   =   I g 0 0 H -I g 0 N t 0 -1     A B   ,
for some N ∈ Z g . In the case where τ a = a and τ b = b, the action of τ on the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) is given by

(2.16)   τ A τ B τ   =   I g 0 0 H -I g 0 N t M t 1     A B   ,
where the vectors N, M ∈ Z g are related by (see [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF] for more details)

(2.17)

2 N + HM = 0.
Now let us study the action of τ on Abelian differentials and the action of the complex conjugation on the theta function with zero characteristic. Denote by τ * the action of τ lifted to the space of differentials: τ * ω(p) = ω(τ p) for any p ∈ R g . By (2.14) the A-cycles of the homology basis are invariant under τ . Due to the normalization conditions (2.1), this leads to the following action of τ on the normalized holomorphic differentials:

(2.18) τ * ω j = -ω j .
Let a, b ∈ R g satisfy τ a = b. Using the action (2.15) of τ on the A-cycles in the homology group H 1 (R g \ {a, b}), by the uniqueness of the normalized differential of the third kind Ω b-a which has residue 1 at b and residue -1 at a, we get

(2.19) τ * Ω b-a = -Ω b-a .
Analogously, in the case where τ a = a and τ b = b, by (2.16) we deduce that

(2.20) τ * Ω b-a = Ω b-a + M t ω,
where ω denotes the vector of normalized holomorphic differentials. From (2.14) and (2.18) we obtain the following reality property for the Riemann matrix B:

(2.21) B = B -2iπ H. Moreover, according to Proposition 2.3 in [29], for any z ∈ C g , relation (2.21) implies (2.22) Θ(z) = κ Θ(z -iπ diag(H)),
where diag(H) denotes the vector of diagonal elements of the matrix H, and κ is a root of unity which depends on the matrix H (knowledge of the exact value of κ is not needed for our purpose).

2.3.

Action of τ on the Jacobian and the theta divisor of real Riemann surfaces.

In this part, we review known results about the theta divisor of real Riemann surfaces (see [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF][START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF]). Let us choose the canonical homology basis satisfying (2.14) and consider the Jacobian J := J(R g ) of a real Riemann surface R g . The anti-holomorphic involution τ on R g gives rise to an anti-holomorphic involution on the Jacobian: if D is a positive divisor of degree n on R g , then τ D is the class of the point (

τ D n τ p 0 ω) = ( D n p 0 τ * ω)
in the Jacobian. Therefore, by (2.18) τ lifts to the anti-holomorphic involution on J, denoted also by τ , given by (2.23)

τ ζ = -ζ + n ζ Π(τ p 0 ), for any ζ ∈ J, where n ζ ∈ Z, n ζ ≤ g, is the degree of the divisor D such that Π(D) = ζ.
Now consider the following two subsets of the Jacobian

S 1 = {ζ ∈ J; ζ + τ ζ = iπ diag(H)}, (2.24) S 2 = {ζ ∈ J; ζ -τ ζ = iπ diag(H)}, (2.25)
where the matrix H was introduced in (2.14). Below we study their intersections S 1 ∩ (Θ) and S 2 ∩ (Θ) with the theta divisor (Θ), the set of zeros of the theta function. Let us introduce the notation: (e i ) k = δ ik , B i = B e i . The following proposition was proved in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF].

Proposition 2.1. The set S 1 is the disjoint union of the tori T v defined by

(2.26) T v = {ζ ∈ J; ζ = iπ (diag(H)/2 + v 1 e r+1 + . . . + v g-r e g ) + β 1 Re(B 1 ) + . . . + β g Re(B g ) , β 1 , . . . , β r ∈ R/2Z , β r+1 , . . . , β g ∈ R/Z},
where v = (v 1 , . . . , v g-r ) ∈ (Z/2Z) g-r and r is the rank of the matrix H. Moreover, if R g (R) = ∅, then T v ∩ (Θ) = ∅ if and only if the curve is dividing and v = 0.

The last statement means that among all curves which admit real ovals, the only torus T v which does not intersect the theta divisor is the torus T 0 corresponding to dividing curves. This torus is given by (2.27)

T 0 = {ζ ∈ J; ζ = β 1 Re(B 1 ) + . . . + β g Re(B g ), β 1 , . . . , β r ∈ R/2Z , β r+1 , . . . , β g ∈ R/Z}.
It is straightforward to prove that the set S 1 is the disjoint union of the tori (2.26). Moreover, in the case where R g (R) = ∅ and R g is non-dividing, for all v the torus T v contains an odd half-period which yields T v ∩ (Θ) = ∅. The same holds for all v = 0 in the case where the curve is dividing or does not have real ovals. Corollary 4.3 in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] proves that if

R g is dividing then T 0 ∩ (Θ) = ∅.
The following proposition was proved in [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF].

Proposition 2.2. The set S 2 is the disjoint union of the tori Tv defined by

(2.28) Tv = {ζ ∈ J ; ζ = 2iπ (α 1 e 1 + . . . + α g e g ) + (v 1 /2) B r+1 + . . . + (v g-r /2) B g , α 1 , . . . , α g ∈ R/Z}, where v = (v 1 , . . . , v g-r ) ∈ (Z/2Z) g-r and r is the rank of the matrix H. Moreover, if R g (R) = ∅, then Tv ∩ (Θ) = ∅ if
and only if the curve is an M-curve and v = 0.

Algebro-geometric solutions of the Camassa-Holm equation

In this section we will use Fay's identities to construct solutions to the CH equation on hyperelliptic surfaces. For the resulting formulae we establish conditions to obtain real and smooth solutions. In what follows R g denotes a hyperelliptic curve of genus g > 0, written as

(3.1) µ 2 = 2g+2 i=1 (λ -λ i ),
where the branch points λ i ∈ C satisfy the relations λ i = λ j for i = j. We denote by σ the hyperelliptic involution defined by σ(λ, µ) = (λ, -µ). Note that the CH equation can be expressed in the following simple form,

(3.2) m t + u m x + 2 m u x = 0,
where we put m := uu xx + k.

3.1.

Identities between theta functions. In our approach to construct algebro-geometric solutions of the CH equation we use the corollaries (2.7) and (2.8) of Fay's identity.

Proposition 3.1. Let a, b ∈ R g such that σ(a) = b and let e ∈ R g be a ramification point, namely, e = (λ j , 0) for some j ∈ {1, . . . , 2g + 2}. Denote by g 1 and g 2 the following functions of the variable z ∈ C g :

(3.3) g 1 (z) = Θ z + r 2 Θ(z) , g 2 (z) = Θ z -r 2 Θ(z) ,
where r = b a ω and ω is the vector of normalized holomorphic differentials. Then the two following identities hold:

(3.4) D b D e ln g 1 g 2 = - p 2 g 1 g 2 D b ln g 1 g 2 , (3.5) D b D e ln(g 1 g 2 ) = q2 p2 1 g 1 g 2 D b ln g 1 g 2 -2 p1 -2 q2 g 1 g 2 .
Here we used the notation:

(3.6) p 2 = p 2 (b, e, a), pi = p i (e, b, a), q2 = q 2 (b, e),
where the scalars q 2 (., .) and p i (., ., .), i = 1, 2, are defined in (2.12) and (2.9), (2.10); D b (respectively D e ) denotes the directional derivative along the vector V b (respectively V e ) defined in (2.6).

Proof. Under the changes of variables (a, b, c) → (b, e, a) and z → zr/2, identity (2.7) becomes

(3.7) D e ln g 1 g 2 = p 1 + p 2 g 1 g 2 .
Here we used the fact that σ * ω j =ω j , j = 1, . . . , g.

Applying the differential operator D b to the equation (3.7) one gets:

D b D e ln g 1 g 2 = -p 2 D b (g 1 g 2 ) (g 1 g 2 ) 2 = - p 2 g 1 g 2 D b ln g 1 g 2 ,
which proves (3.4). To prove (3.5), consider the change of variables (a, b, c) → (e, b, a) in (2.7), which leads to

D b ln g 1 = p1 + p2 g 2 Θ(z + r) Θ z + r 2 .
Changing z to -z in the last equality, one gets

D b ln g 2 = -p1 -p2 g 1 Θ(z -r) Θ z -r 2 .
From these two identities, it can be deduced that

Θ(z + r) Θ z + r 2 = (p 2 g 2 ) -1 (D b ln g 1 -p1 ), (3.9) Θ(z -r) Θ z -r 2 = -(p 2 g 1 ) -1 (D b ln g 2 + p1 ). (3.10) Moreover, since D b D e ln(g 1 g 2 ) = D b D e ln Θ z + r 2 + D b D e ln Θ z - r 2 -2 D b D e ln Θ(z), using (2.8) one gets (3.11) D b D e ln(g 1 g 2 ) = q2 g 1 Θ(z + r) Θ z + r 2 + q2 g 2 Θ(z -r) Θ z -r 2 -2 q2 g 1 g 2 ,
which by (3.9) and (3.10) leads to (3.5).

With identities (3.4) and (3.5) we are now able to construct theta-functional solutions of the CH equation:

(3.12) x + α 1 y + α 2 t + ζ = ln Θ Z -d + r 2 Θ Z -d -r 2 ,
where r = ω. Here the vector Z is given by

(3.13) Z(x, t) = V e y(x, t) + V b t,
where the vectors V e and V b are defined in (2.6). The scalars α 1 and α 2 satisfy

(3.14) α 1 = p 1 (b, e, a), α 2 = 2 p 1 (e, b, a) + k,
where the function p 1 is defined in (2.9). Then the following function of the variables x and t is solution of the CH equation:

(3.15) u(x, t) = D b ln Θ Z -d + r 2 Θ Z -d -r 2 -α 2 .
Here D b denotes the directional derivative along the vector V b .

Proof. Let β, δ ∈ C and α 1 , α 2 ∈ C be arbitrary constants. Let us look for solutions u of CH having the form

(3.16) u(x, t) = β D b ln Θ Z -d + r 2 Θ Z -d -r 2 + δ = β D b ln g 1 g 2 + δ,
where Z(x, t) is defined in (3.13), and the functions g 1 , g 2 were introduced in (3.3) with z = Z(x, t)d. By (3.12), the derivative with respect to the variable x of the implicit function y(x, t) is given by

y x = D e ln g 1 g 2 -α 1 -1
.

Putting α 1 = p 1 (b, e, a), relation (3.7) implies (3.17)

y x = g 1 g 2 p 2 .
Analogously it can be checked that (3.18)

y t = -y x u β - δ β -α 2 .

Now let us express the function

m(x, t) = u -u xx + k introduced in (3.
2) in terms of the functions g 1 and g 2 of (3.3). By (3.4) and (3.17), the first derivative of the function u (3.16) with respect to the variable x is given by

(3.19) u x = -β D b ln(g 1 g 2 ).
By (3.5) and (3.17) we obtain for the second derivative of u with respect to x:

(3.20) u xx = β D b ln g 1 g 2 -2 p1 -2β p2 (g 1 g 2 ) 2 ;
here we used the identity q2 = -p2 p 2 relating the scalars q2 , p2 and p 2 defined in (3.6). Therefore, with (3.16) and (3.20), the function m reads

(3.21) m(x, t) = δ + k + 2β p1 + 2β p2 (g 1 g 2 ) 2 .
Taking the derivative of m with respect to x, and the derivative of m with respect to t, one gets respectively:

m x (x, t) = 4β p2 (g 1 g 2 ) 2 y x D e ln(g 1 g 2 ), (3.22) m t (x, t) = 4β p2 (g 1 g 2 ) 2 (y t D e ln(g 1 g 2 ) + D b ln(g 1 g 2 )) . ( 3
2 p2 D e ln(g 1 g 2 ) y x (g 1 g 2 ) 2 u 1 - 1 β + δ β + α 2 -D b ln(g 1 g 2 ) δ + k + 2β p1 + 2 p2 (g 1 g 2 ) 2 (β -1) = 0.
In particular, the previous equality holds for β = 1, δ = -2 p1k and α =δ, which completes the proof.

3.2.

Real-valued solutions and smoothness conditions. In this part, we identify realvalued and smooth solutions of the CH equation. Assume that R g is a real hyperelliptic curve which admits real ovals with respect to an anti-holomorphic involution τ . Let us choose the homology basis satisfying (2.14). Recall that R g (R) denotes the set of fixed points of the anti-holomorphic involution τ .

The following propositions provide reality and smoothness conditions for the solutions u(x, t) (3.15) in the case where the points a and b are stable under τ , and in the case where τ a = b. In particular, it is proved that, for fixed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x when a and b are stable under τ . In the case where τ a = b, the function u(x, t 0 ) is either smooth, or it has cusp-like singularities. Using the representation in terms of multi-dimensional theta functions of the differential Ω c-a (see, for instance, [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]), one gets:

(3.26) Ω c-a (b) = p 1 (a, b, c) dk b .
We deduce that the scalars p 1 (b, e, a) and p 1 (e, b, a) appearing respectively in α 1 and α 2 (see (3. Therefore, from (2.20) it can be deduced that p 1 (b, e, a) and p 1 (e, b, a) are real, which involves (3.28)

α 1 = α 1 , α 2 = α 2 .
Now let us fix y, t ∈ R and denote by h the function

(3.29) h(y, t) = Θ Z -d + r Θ Z -d -r 2
.

By
h(y, t) = Θ Z -d + r 2 Θ Z -d -r 2 exp{-r, L },
where we used the quasi-periodicity property (2.4) of the theta function. Therefore, the function h is real if L = 0, that is d ∈ iR g . Now let us check that h has a constant sign with respect to y, t ∈ R. Since Zd ± r 2 ∈ iR g , by Proposition 2.2 the functions Θ(Zd ± r 2 ) of the real variables y and t do not vanish if the hyperelliptic curve is an M-curve, i.e., if all branch points in (3.1) are real. Hence h is a real continuous non vanishing function with respect to the real variables y and t, which means it has a constant sign. Therefore, x is a real-valued function of y and t if the constant ζ in (3.12) is chosen such that ln(h(y, t))ζ is real. It is straightforward to see that the solution u (3.15) is a real-valued function of the real variables y and t, and then is real-valued with respect to the real variables x and t. Now fix t 0 ∈ R and let us study smoothness conditions for the function u(x, t 0 ) with respect to the variable x. First let us check that the solution u (3.15) is a smooth function of the real variable y. The function u is smooth with respect to y if and only if it does not have singularities. Since the theta function is entire, singularities of the solution u are located at the zeros of its denominator. By the same argument used previously to prove the real-valuedness of u, if the curve is an M-curve and d ∈ iR g , the functions Θ(Zd ± r

2 ) and Θ(Zd) do not vanish. In this case, the function u is smooth with respect to the real variable y. Now let us prove that u is smooth with respect to the real variable x. By (3.12), the function x(y) is smooth with respect to the variable y. Moreover, it can be seen from (3.12) and (3.7) that (3.32) x y (y

) = p 2 g 1 g 2 = p 2 Θ(Z -d) 2 Θ Z -d + r 2 Θ Z -d -r 2 .
Since the functions Θ(Zd ± r 2 ) and Θ(Zd) do not vanish, we deduce that x(y) is a strictly monotonic real function, and thus the inverse function y(x) has the same property. Therefore, the function u(x, t 0 ) = u(y(x)) is a smooth real-valued function with respect to the real variable x. for some n ∈ N \ {0} and x 0 ∈ R, which correspond to cusps.

Proof. Analogously to the case where a and b are stable under τ , let us prove that solutions u (3.15) are real-valued. First let us check that the vector Z (3.13) satisfies:

(3.33) Z = Z.
From (2.6) and (2.18) one gets V a = -V b and V e = V e . Moreover, the vectors V a and

V b satisfy V a + V b = 0 because of (3.8); thus we have V b = V b and V e = V
(3.36) h(y, t) = Θ Z -d + r 2 + p Θ Z -d -r 2 + p
, where p = -iπNiπ diag(H). Let us choose the vector d ∈ C g such that

d ≡ d + p (mod 2iπZ g + BZ g ),
which is, since dd and p are purely imaginary, equivalent to d = d + p + 2iπT, for some T ∈ Z g . Here we used the action (2.21) of the complex conjugation on the Riemann matrix B, and the fact that B has a negative definite real part. Hence, the vector d can be written as

(3.37) d = d R + iπ 2 (N + diag(H) -2 T),
for some d R ∈ R g and T ∈ Z g . For this choice of the vector d, by (3.36) the function h is a real-valued function of the real variables y and t. Now let us study in which cases the function h has a constant sign. The sign of the function h is constant with respect to y and t if the functions Θ(Zd ± r 2 ) do not vanish. By (3.33), (3.35) and (3.37), the vectors Z-d± r 2 belong to the set S 1 introduced in (2.24). Hence by Proposition 2.1, the functions Θ(Zd ± r

2 ) do not vanish if the hyperelliptic curve is dividing (in this case diag(H) = 0), and if the arguments Zd ± r 2 in the theta function are real (modulo 2iπZ g ). The vector

Z -d + r 2 is real if T = 0 in (3.37).
With this choice of the vector T, the imaginary part of the vector Z-d-r 2 equals -iπN. Therefore, the vector Z-d-r 2 is real modulo 2iπZ g if all components of the vector N are even. To summarize, the function h(y, t) defined in (3.29) is a real-valued function with constant sign if the hyperelliptic curve is dividing (i.e., all ramification points are stable under τ , since the ramification point e is stable under τ ), if T = 0 and N = 2L for some L ∈ Z g , where vector N ∈ Z g is defined in (2.15). Analogously to the proof of Proposition 3.3, we conclude that x is a real-valued continuous function of the real variables y and t, and thus solutions u(x, t) (3.15) are real-valued functions of the real variables x and t. Now let us study smoothness conditions for fixed t 0 ∈ R. Notice that the function u(y) (3.15) is a smooth function of the real variable y since the denominator does not vanish, as we have seen before. Put z = Zd. Let us consider the function x y (y) given in (3.32) in both cases: N = 0 and N = 0.

-If N = 0, the function x y (y) does not vanish, since in this case z ∈ R g which implies that the function Θ(z) does not vanish. Hence, analogously to the case where a and b are stable under τ , for fixed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x.

-If N = 0, the function Θ(z) vanishes when z belongs to the theta divisor. Fix x 0 , t 0 ∈ R and denote by z 0 and y 0 the corresponding values of z and y. Assume that z 0 is a zero of the theta function of order n ≥ 1. Then by (3.32), the function x y (y) has a zero at y 0 of order 2n. It follows that function x(y)x(y 0 ) has a zero of order 2n + 1 at y 0 , and then

(3.38) y(x) -y 0 = O (x -x 0 ) 1 2n+1
.

On the other hand, it can be seen from (3.4) that (3.39) u y (y

) = p 2 Θ(z) Θ z + r 2 Θ z -r 2 [Θ(z) ψ(z) -2 D b Θ(z)] ,
where ψ(z) = D b ln Θ(z + r 2 ) Θ(z -r 2 ) . Identity (3.39) implies that function u y (y) has a zero at y 0 of order 2n -1, namely,

µ 2 = 2g+2 i=1 (λ -λ i ),
where g is the genus of the Riemann surface, and where we have for the branch points λ i ∈ R the relations λ i = λ j for i = j.

For the numerical evaluation of the CH solutions (3.15) we use the code presented in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF] for real hyperelliptic Riemann surfaces. The reader is referred to these publications for details. The basic idea is to introduce a convenient homology basis on the related surfaces, see Fig. 1. It is related to the basis used in the previous sections by the simple

A 1 B 1 < > λ 1 λ 2 λ 3 λ 4 λ 2g+1 λ 2g+2 B g A g
. . . relation A → -B, B → A. This choice of the homology basis has the advantage that the limit in which branch points encircled by the same A-cycle collapse can be treated essentially numerically. Below we will consider examples where the distance of such a pair of branch points is of the order of machine precision (10 -14 ) and thus numerically zero. This limit is interesting since the B-periods diverge, which implies that the corresponding theta functions reduce to elementary functions. The CH solutions (3.15) reduce in this case to solitons or cuspons. Since we want to study also this limit numerically, we use the homology basis of Fig. 1, and not the one of the previous sections.

The sheets are identified at the point a by the sign of the root picked by Matlab. We denote a point in the first sheet with projection λ into the complex plane by λ (1) , and a point in the second sheet with the same projection by λ (2) . The theta functions are in general approximated numerically by a truncated series as explained in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF] and [START_REF]Computational Approach to Riemann Surfaces[END_REF]. A vector of holomorphic differentials for these surfaces is given by (1, λ, . . . , λ g-1 ) t dλ/µ. The periods of the surface are computed as integrals between branch points over these differentials as detailed in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF]. The Abel map to the point a (and analogously for b) is computed in a similar way, see [START_REF] Kalla | On the numerical evaluation of algebro-geometric solutions to integrable equations[END_REF], as the integral between a and the branch point with minimal distance to a. It is well known (see for instance [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]) that the Abel map between two branch points is a half period.

To control the accuracy of the numerical solutions, we use essentially two approaches. First we check the theta identity (2.7), which is the underlying reason for the studied functions being solutions to CH. Since this identity is not implemented in the code, it provides a strong test. This computation for various combinations of the points a, b and e ensures that the theta functions are computed with sufficient precision, and that the quantities α 1 and α 2 in (3.12) are known with the wanted precision (we always use machine precision here). In addition, the smooth solutions are computed on Chebyshev collocation points (see, for instance, [START_REF] Trefethen | Spectral Methods in Matlab[END_REF]) for x and t. This can be used to approximate the computed solution via Chebyshev polynomials, a so-called spectral method having in practice exponential convergence for smooth functions. Since the derivatives of the Chebyshev polynomials can be expressed linearly in terms of Chebyshev polynomials, a derivative acts on the space of polynomials via a so called differentiation matrix. With these standard Chebyshev differentiation matrices (see [START_REF] Trefethen | Spectral Methods in Matlab[END_REF]), the solution can be numerically differentiated. The computed derivatives allow to check with which numerical precision the partial differential equation (PDE) is satisfied by a numerical solution. With these two independent tests, we ensure that the shown solutions are correct to much better than plotting accuracy (the code reports a warning if the above tests are not satisfied to better than 10 -6 ). We do not use the expansion in terms of Chebyshev polynomials for the cusped solutions since the convergence is slow for functions with cusps.

We first consider smooth solutions in the case τ a = a, τ b = b. To obtain non-trivial solutions in the solitonic limit, we use a vector d corresponding to the characteristic 1 2 [ 1 ... 1 0 ... 0 ] t in all examples. To plot a solution u in dependence of x and t, we compute it on a numerical grid for y and t to obtain x(y, t) and u(y, t). These are then used to obtain a plot of u(x, t) without having to solve the implicit relation (3.12). In all examples we have k = 1.

The solutions for τ a = b, i.e., for points a and b on the cuts encircled by the A-periods in Fig. 1 look very similar and are therefore not shown here. Solutions on elliptic surfaces describe travelling waves and will not be discussed here. In genus 2 we obtain CH solutions of the form shown in Fig. 2. The typical soliton collision known from the KdV equation is also present here, the unchanged shape of the solitons after the collision, but an asymptotic change of phase.

In genus 6 the CH solutions have the form shown in Fig. 3. In the solitonic limit one can recognize a 6 soliton event.

The reality properties of the quantities entering the solution (3.15) depend on the choice of the homology basis. For instance, in the homology basis of Fig. 1 and for a and b stable under τ , the Abel map r up to a vector proportional to iπ and the vectors V b , V e are real, whereas these quantities are purely imaginary in the homology basis used in the previous sections. Thus the easiest way to obtain cusped solutions is in this case to put e = λ 2 and to choose d corresponding to the characteristic 1 2 [ 1 ... 1 1 ... 1 ] t . It can be easily checked that the theta functions Θ(Zd ± r/2) cannot vanish since the argument is real, whereas the Θ(Zd) will have zeroes since the argument is complex. This implies that the derivative u x in (3.19) diverges which corresponds to cups for the solution u. Peakons do not appear in such a limit of theta-functional solutions to CH and are thus not discussed here. To obtain them one would have to glue solutions in the solitonic limit on finite intervals to obtain a continuous solution that is piecewise C 1 . 4) (1) , b = (-4) (2) and e = (-3, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right. 8) (1) , b = (-8) (2) and e = (-7, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.

We show the cusped solutions always in a comoving frame x = x + vt to allow a better visualization of the solutions. In genus 2 we obtain cusped CH solutions of the form shown in Fig. 4, where also cusped solitons can be seen in the degenerate situation. Obviously the collision between cuspons is analogous to soliton collisions.

In genus 6 the CH solutions have the form shown in Fig. 5. In the solitonic limit one can recognize a 6-cuspon.

Conclusion

In this paper we have shown at the example of the CH equation that Mumford's program to construct algebro-geometric solutions to integrable PDEs can be also applied to nonlocal (here in x) equations. For the studied case the solutions in terms of multi-dimensional theta functions depend not directly on the physical coordinates x and t, but via an implicit -4) (1) , b = (-4) (2) and e = (-2, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right. 8) (1) , b = (-8) (2) and e = (-6, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.

function. One consequence of this non-locality is the existence of non-smooth solitons. A numerical study of smooth and non-smooth solutions was presented.

A further example in this context would be the equation from the Dym-hierarchy for which theta-functional solutions were studied in [START_REF] Alber | Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians[END_REF], which will be treated elsewhere with Mumford's approach. It is an interesting question whether the 2+1 dimensional generalization of the CH equation [START_REF] Falqui | On a camassa-holm type equation with two dependent variables[END_REF][START_REF] Chen | A two-component generalization of the camassa-holm equation and its solutions[END_REF], for which algebro-geometric solutions are so far unknown, can be also treated with these methods.

Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex, France E-mail address: Caroline.Kalla@u-bourgogne.fr Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex, France E-mail address: Christian.Klein@u-bourgogne.fr

e a ω = b e ω = r/ 2 ,

 2 according to the action of σ * (the action of σ lifted to the space of one-forms) on the normalized holomorphic differentials ω j :(3.8)

Proposition 3 . 2 .

 32 Let a, b ∈ R g such that σ(a) = b, and let e ∈ R g be a ramification point. Denote by an oriented contour between a and b which contains point e. Assume that does not cross cycles of the canonical homology basis. Choose arbitrary constants d ∈ C g and k, ζ ∈ C. Let y(x, t) be an implicit function of the variables x, t ∈ R defined by

Proposition 3 . 3 .

 33 Assume that all ramification points of R g are stable under τ and denote by e ∈ R g (R) one of them. Let a, b ∈ R g (R) such that σ(a) = b. For any c ∈ {a, b, e}, choose the local parameter k c such that k c (τ p) = k c (p) for any point p in a neighborhood of c. Denote by an oriented contour between a and b containing point e, which does not intersect cycles of the canonical homology basis. Choose such that the closed path τ -is homologous to zero in H 1 (R g ). Take d ∈ iR g and k ∈ R. Choose ζ ∈ C in (3.12) such that ln Θ(d+r/2) Θ(d-r/2)ζ is real.Then solutions u(x, t) of the CH equation given in (3.15) are real-valued, and for fixed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x.Proof. Let us check that under the conditions of the proposition, the function u(x, t) (3.15) is real-valued. First of all, invariance with respect to the anti-involution τ of the points e and b implies(3.24) Z = -Z,where the vector Z is defined in (3.13). In fact, using the expansion (2.6) of the normalized holomorphic differentials ω j near e (respectively b) one gets τ * ω j (e)(p) = V e,j + W e,j k e (p) + o k e (p) 2 dk e (p), for any point p in a neighborhood of e (respectively b). Then by (2.18), the vectors V e and V b appearing in the vector Z are purely imaginary, which leads to(3.24). Moreover, since the closed contour τis homologous to zero in H 1 (R g ), from (2.18) one gets (3.25) r =r.

  14)) satisfy (3.27) Ω a-b (e) = p 1 (b, e, a) dk e , Ω a-e (b) = p 1 (e, b, a) dk b .

  Now let us study real-valuedness and smoothness of the solutions in the case where τ a = b. Proposition 3.4. Assume that all ramification points of R g are stable under τ and denote by e ∈ R g (R) one of them. Let a, b ∈ R g such that σ(a) = b and assume that τ a = b. Choose the local parameters such that k b (τ p) = k a (p) for any point p in a neighborhood of a, and k e (τ p) =k e (p) for any p lying in a neighborhood of e. Denote by an oriented contour between a and b containing point e, which does not intersect cycles of the canonical homology basis. Assume that N = 2L for some L ∈ Z g , where N ∈ Z g is defined in (2.15). Take k ∈ R and define d = d R + iπ 2 N for some d R ∈ R g . Choose ζ ∈ C in (3.12) such that ln Θ(d+r/2) Θ(d-r/2)ζ is real. Then solutions u (3.15) of the CH equation are real-valued. Moreover, for fixed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x in the case where N = 0, otherwise it has an infinite number of singularities of the type O (xx 0 ) 2n 2n+1

( 3 .

 3 40) u(y)u(y 0 ) = O (yy 0 ) 2n . Finally with (3.38) and (3.40), the function u(x, t 0 ) has an infinite number of singularities of the type O (xx 0 ) 2n 2n+1 which correspond to cusps. 4. Numerical study of algebro-geometric solutions to the Camassa-Holm equation In this section we will numerically study concrete examples for the CH solutions (3.15). As shown in the previous sections, real and bounded solutions are obtained on hyperelliptic M-curves, i.e., curves of the form

Figure 1 .

 1 Figure 1. Homology basis on real hyperelliptic M-curves, contours on sheet 1 are solid, contours on sheet 2 are dashed.

Figure 2 .

 2 Figure 2. Solution(3.15) to the CH equation on a hyperelliptic curve of genus 2 with branch points -3, -2, 0, , 2, 2 + and a = (-4)(1) , b = (-4)(2) and e = (-3, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.

Figure 3 .

 3 Figure 3. Solution(3.15) to the CH equation on a hyperelliptic curve of genus 2 with branch points -7, -6, -5, -5 + , -3, -3 + , -1, -1 + , 1, 1 + , 3, 3 + , 5, 5 + and a = (-8)(1) , b = (-8)(2) and e = (-7, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.

Figure 4 .

 4 Figure 4. Cusped solution(3.15) to the CH equation on a hyperelliptic curve of genus 2 with branch points -3, -2, 0, , 2, 2 + and a = (-4)(1) , b = (-4)(2) and e = (-2, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.

Figure 5 .

 5 Figure 5. Solution(3.15) to the CH equation on a hyperelliptic curve of genus 6 with branch points -7, -6, -5, -5 + , -3, -3 + , -1, -1 + , 1, 1 + , 3, 3 + , 5, 5 + and a = (-8)(1) , b = (-8)(2) and e = (-6, 0) for = 1 on the left and = 10 -14 , the almost solitonic limit, on the right.


Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Maria J Esteban
email: esteban@ceremade.dauphine.fr

Michael Loss
email: loss@math.gatech.edu

SYMMETRY OF EXTREMALS OF FUNCTIONAL INEQUALITIES VIA SPECTRAL ESTIMATES FOR LINEAR OPERATORS

Keywords: September 28, 2011 Hardy-Sobolev inequality, Caffarelli-Kohn-Nirenberg inequality, extremal functions, Kelvin transformation, Emden-Fowler transformation, radial symmetry, symmetry breaking, rigidity, Lieb-Thirring inequalities, generalized Poincaré inequalities, estimates of the best constants, cylinder, Riemannian manifold, Ricci curvature AMS classification

We prove new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in any dimension larger or equal than 2 , in a range of parameters for which no explicit results of symmetry were previously known.

Dedicated to Elliott Lieb on the occasion of his 80th birthday

Introduction

The Caffarelli-Kohn-Nirenberg inequalities [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] in space dimension N ≥ 2 can be written as (1) and in this paper C N a,b denotes the optimal constant. Typically, Inequality (1) is stated with a < a c (see [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]) so that the space D a,b is obtained as the completion of C ∞ c (R N ) , the space of smooth functions in R N with compact support, with respect to the norm w 2 = |x| -b w 2 p + |x| -a ∇w 2 2 . Actually (1) holds also for a > a c , but in this case D a,b is obtained as the completion with respect to • of the space 1) is sometimes called the Hardy-Sobolev inequality, as for N > 2 it interpolates between the usual Sobolev inequality (a = 0 , b = 0) and the weighted Hardy inequalities corresponding to b = a + 1 (see [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], a key paper by F. Catrina and Z.-Q. Wang, on this topic).

R N |w| p |x| b p dx 2/p ≤ C N a,b R N |∇w| 2 |x| 2 a dx ∀ w ∈ D a,b with a ≤ b ≤ a + 1 if N ≥ 3 , a < b ≤ a + 1 if N = 2 ,
C ∞ c (R N \ {0}) := w ∈ C ∞ c (R N ) : supp(w) ⊂ R N \ {0} . Inequality (
For b = a < 0 , N ≥ 3 , equality in (1) is never achieved in D a,b . For b = a + 1 and N ≥ 2 , the best constant in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] is given by C N a,a+1 = (a c -a) 2 and it is never achieved (see [5, Theorem 1.1, (ii)]). In contrast, for a < b < a + 1 and N ≥ 2 , the best constant in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] is always achieved at some extremal function w a,b ∈ D a,b . However w a,b is not explicitly known unless we have the additional information that it is radially symmetric with respect to the origin. In the class of radially symmetric functions, the extremals of (1) are all given (see [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF][START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]) up to scaling and multiplication by a constant, by

w * a,b (x) = 1 + |x| 2 (N-2-2 a)(1+a-b) N-2 (1+a-b) - N-2 (1+a-b) 2 (1+a-b)
.

See [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF] for more details and in particular for a modified inversion symmetry property of the extremal functions, based on a generalized Kelvin transformation, which relates the parameter regions a < a c and a > a c .

In the parameter region 0 ≤ a < a c , a ≤ b ≤ a + 1 , if N ≥ 3 , the extremals are radially symmetric (see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF][START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF][START_REF] Glaser | A family of optimal conditions for the absence of bound states in a potential[END_REF] and more specifically [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF]; also see [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] for a proof based on symmetrization and [START_REF] Dolbeault | Radial symmetry and symmetry breaking for some interpolation inequalities[END_REF] for an extension to the larger class of inequalities considered in Section 5). On the other hand, extremals are known to be non-radially symmetric for a certain range of parameters (a, b) identified first in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] and subsequently improved by V. Felli and M. Schneider in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF], where it was shown that in the region a < 0 , By contrast, few symmetry results were available in the literature for a < 0 , and they are all of a perturbative nature. We refer the reader to [START_REF]About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] for a detailed review of existing results. When N ≥ 3 and for a fixed b ∈ (a, a + 1) , radial symmetry of the extremals has been proved for a close to 0 (see [START_REF]Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Lin | Erratum to: "Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF]; also see [START_REF] Smets | Partial symmetry and asymptotic behavior for some elliptic variational problems[END_REF]Theorem 4.8] for an earlier but slightly less general result). In the particular case N = 2 , a symmetry result was proved in [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF] for a in a neigbourhood of 0 -, which asymptotically complements the symmetry breaking region described in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF][START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF] as a → 0 -. Later, in [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF], it was proved that for every a < 0 and b sufficiently close to a + 1 , the global minimizers are also symmetric. Due to the perturbative nature of these results, there were currently no explicit values a < 0 for which one knew the radial symmetry of the minimizer. For instance, up to now, it was not known whether the extremals of the inequality (2)

R 3 |w| 3 dx 2/3 ≤ C 3 -1/2,0 R 3 |x| |∇w| 2 dx
were radial or not, and as a consequence, the value of C 3 -1/2,0 was also unknown.

For any N ≥ 2 , it has been proved in [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] that, in the two-dimensional region of the parameters a and b , the symmetry and symmetry breaking regions are both simply connected, and separated by a continuous curve starting at the point (a = 0, b = 0) , contained in the region a < 0 , b FS (a) < b < a + 1. The curve b FS can be parametrized by a as a function of b -a , such that a → -∞ as b -a → 1 -. It is also known from [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] that the region of symmetry contains a neighborhood of the set a < 0 , b = a + 1 and a = 0 -, 0 < b < 1 in the set a < 0 , b FS (a) ≤ b < a + 1 , but no explicit estimate of this neighborhood has been given yet. These results are all based on compactness arguments. Since radial symmetry is broken in certain parameter ranges, it seems unlikely that a universal tool, like symmetrization, can be applied in the case a < 0 .

In this paper we determine a large region for a < 0 where the extremals of the Caffarelli-Kohn-Nirenberg inequalities (1) are radial. The result will be expressed in terms of the following function

b ⋆ (a) := N (N -1) + 4 N (a -a c ) 2 6 (N -1) + 8 (a -a c ) 2 + a -a c . Theorem 1. Let N ≥ 2 . When a < 0 and b ⋆ (a) ≤ b < a + 1 , the extremals of the Caffarelli-Kohn-Nirenberg inequality (1) are radial and (3)
C N a,b = |S N -1 | p-2 p (a-ac) 2 (p-2) 2 p+2 p-2 2 p p+2 2 p (a-ac) 2 4 p+2 6-p 2 p Γ 2 p-2 + 1 2 √ π Γ 2 p-2 p-2 p
.

An elementary computation shows that

a → b ⋆ (a) -b FS (a) = N 2 1 - ac-a √ (ac-a) 2 +N -1 - 2 (N -1) 4 (ac-a) 2 +3 (N -1)
is an increasing function of a ≤ 0 , so that, for any a ≤ 0 ,

0 ≤ b ⋆ (a) -b FS (a) ≤ b ⋆ (0) -b FS (0) = 1 1 + N (N -1)
.

Hence a → b ⋆ (a) -b FS (a) is controlled by its value at a = 0 . The plots are qualitatively similar in any dimension N ≥ 2 , and a → b ⋆ (a) -b FS (a) uniformly converges for a ≤ 0 to 0 as N increases. See Fig. 1 for an illustration of Theorem 1. As an example, for all N ≥ 2 , for a = -1 2 and b = 0 , we find that b ⋆ (-

1 2 ) = - 1 2 N -2 N + 2 ≤ 0 ,
so that all extremals of the corresponding Caffarelli-Kohn-Nirenberg inequality, (2), are radial and, as a consequence,

1 C N -1/2,0 = inf w∈D a,b (R N )\{0} R N |x| |∇w| 2 dx R N |w| 2 N N-1 dx N-1 N = 4 N (N -1) π N-1 2 Γ N + 1 2 Γ N 2 Γ(N ) 1 N
.

It has been already observed in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] that Caffarelli-Kohn-Nirenberg inequalities on R N are equivalent to interpolation inequalities of Gagliardo-Nirenberg-Sobolev type on the cylinder C := R × S N -1 . On the other hand, a classical observation in the Euclidean space is that Gagliardo-Nirenberg-Sobolev interpolation inequalities are equivalent to estimates on eigenvalues of a Schrödinger type operator, in terms of a Lebesgue norm of the potential: see [START_REF] Lieb | Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities[END_REF][START_REF] Dolbeault | Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems[END_REF]. Such estimates are known as Lieb-Thirring estimates. In the present context we shall consider only the one-bound state version of the Lieb-Thirring inequality, in which only the lowest eigenvalue is taken into account. The strategy of the proof of Theorem 1 is to exploit a similar equivalence of the two inequalities on the cylinder, and rely on the one-dimensional version of the Lieb-Thirring inequality (i.e. which depends only on the s coordinate corresponding to the axis of the cylinder) and on a generalized Poincaré inequality on S N -1 . This paper is organized as follows. In Section 2, Inequality (1) is rewritten on the cylinder C = R × S N -1 and the one-dimensional, one-bound state version of the Lieb-Thirring inequality is stated. Section 3 is devoted to the proof of Theorem 1, which, as mentioned above, relies on the one-bound state Lieb-Thirring inequality and on a generalized Poincaré inequality on the sphere. Theorem 1 is also reformulated as a rigidity result for the eigenfunction associated to the lowest eigenvalue of a Schrödinger operator on the cylinder. Rigidity means that the optimizing potential for the Lieb-Thirring inequality depends only on the variable s and not the angular variable. Details will be given in Section 4. In Section 5, a larger class of Caffarelli-Kohn-Nirenberg inequalities than (1) is considered: see [START_REF] Smets | Partial symmetry and asymptotic behavior for some elliptic variational problems[END_REF]. Neither rigidity nor symmetry results can be achieved in such a general case, but the method used for (1) still provides estimates on the optimal constants. Finally, in Section 6 we explain how the proofs of our results for the case C = R × S N -1 can be adapted to general cylinders C = R × M , where M is a compact manifold without boundary and with positive Ricci curvature. Critical for this result is an extension of the sharp generalized Poincaré inequality to

a b 0 -1 -1 2 1 b = a + 1 b = a b = b FS (a)

Symmetry region

Symmetry breaking region such manifolds M due to M.-F. Bidaut-Véron and L. Véron that can be found in [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF].

N -2 2 = b = b ⋆ (a) a c

Lieb-Thirring type inequalities on the cylinder

Before we state the one-dimensional, one-bound state version of the Lieb-Thirring inequality, let us introduce a transformation which removes the weights in the Caffarelli-Kohn-Nirenberg inequalities.

Emden-Fowler transformation. We reformulate the Caffarelli-Kohn-Nirenberg inequalities in cylindrical variables (see [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF]) using the Emden-Fowler transformation

s = log |x| , ω = x |x| ∈ S N -1 , u(s, ω) = |x| ac-a w(x) .
Inequality (1) for w is equivalent to a Gagliardo-Nirenberg-Sobolev inequality on the cylinder

C := R × S N -1 , that is (4) u 2 L p (C) ≤ C N a,b ∇u 2 L 2 (C) + Λ u 2 L 2 (C) , for any u ∈ H 1 (C) ,
Λ FS (p) := 4 N -1 p 2 -4 and Λ ⋆ (p) := (N -1) (6 -p) 4 (p -2) . It is straightforward to check that Λ ⋆ < Λ FS and Λ⋆ Λ FS = 1 16 (6 -p)(p + 2) is a strictly decreasing function of p ∈ (2, 2 N N -2 ] for any N ≥ 3 , such that 1 = lim p→2 Λ⋆ Λ FS ≥ Λ⋆ Λ FS ≥ lim p→ 2 N N-2 Λ⋆ Λ FS = (N -1)(N -3) (N -2) 2 . If N = 2 , b ≥ b ⋆ (a) means p ≤ 6 -16 a 2 1 + 4 a 2 < 6 . Radial symmetry of w = w(x) means that u = u(s, ω) is independent of ω .
Up to multiplication by a constant, the extremal functions in the class of functions depending only on s ∈ R solve the equation ( 5)

-u ′′ * + Λ u * = u p-1 * in R .
Up to translations in s and multiplication by a constant, non-negative solutions of this equation are all equal to the function

(6) u * (s) := A cosh(B s) 2 p-2 ∀ s ∈ R , with (7) A = 1 2 Λ and B = 1 2 √ Λ (p -2)
(see Section 5 for details).

We can restate Theorem 1 in terms of the variables p and Λ as follows.

Theorem 2. Let N ≥ 2 . For all p > 2 and a 2 c < Λ ≤ Λ ⋆ (p) , the extremals of (4) depend only on the variable s . That is, all extremals of (4) are equal to u * , up to a translation and a multiplication by a constant.

Note that Λ ∈ (a 2 c , Λ ⋆ (p)] implies p < 6 if N = 2 and p < 2 N N -2 if N ≥ 3 .
Interpolation and Lieb-Thirring inequalities. Before we prove Theorem 2, we state a well-known result, the Lieb-Thirring inequality for one-bound state in dimension 1 . We will later use it for potentials depending on the variable s ∈ R of the cylinder.

Lemma 3. [17] Let V = V (s) be a non-negative real valued potential in L γ+ 1 2 (R) for some γ > 1/2 and let -λ 1 (V ) be the lowest eigenvalue of the Schrödinger operator -d 2 ds 2 -V . Define c LT (γ) = π -1/2 γ -1/2 Γ(γ + 1) Γ(γ + 1/2) γ -1/2 γ + 1/2 γ+1/2 . Then (8) λ 1 (V ) γ ≤ c LT (γ) R V γ+1/2 (s) ds ,
with equality if and only if, up to scalings, translations and a multiplication by a positive constant,

V (s) = γ 2 -1/4 cosh 2 (s) =: V 0 (s) , in which case λ 1 (V 0 ) = (γ -1/2) 2 .
Furthermore, the corresponding eigenspace is generated by by

ψ γ (s) = π -1/4 Γ(γ) Γ(γ -1/2) 1/2 cosh(s) -γ+1/2 .

Proof of main result

We will consider functions u = u(s, ω) where the variable s and ω are respectively in R and S N -1 . By dω we denote the uniform probability measure on S N -1 and we will denote by L 2 the Laplace-Beltrami operator on S N -1 , that can be written as

L 2 = α L 2 α
where the sum is over all ordered pairs α = (i, j) , 1 ≤ i < j ≤ N , and

L α u = ω i ∂ j u -ω j ∂ i u ,
where ∂ i denotes the derivative along the coordinate ω i and the sphere S N -1 is defined as the set ω = (ω i ) N i=1 ∈ R N : Σ N i=1 ω 2 i = 1 . We shall use the abbreviation

S N-1 |Lu| 2 dω = α S N-1 |L α u| 2 dω .
Naturally, we have

S N-1 |Lu| 2 dω = - S N-1 u (L 2 u) dω .
The main result of our paper goes as follows.

Theorem 4. Let N ≥ 2 and let u be a non-negative function on

C = R × S N -1 that satisfies (9) -∂ 2 s u -L 2 u + Λ u = u p-1
on C , and consider the solution u * given by [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF]. Assume that

(10) C |u(s, ω)| p ds dω ≤ R |u * (s)| p ds , for some p > 2 . If a 2 c < Λ ≤ Λ ⋆ (p)
, then for a.e. ω ∈ S N -1 and s ∈ R , we have u(s, ω) = u * (s -C) for some constant C . Remark 5. Up to the multiplication by a constant, an optimal function for (4) solves [START_REF]About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF]

so that C N a,b = u 2-p L p (C) .
Requiring [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] is then natural if we look for extremal functions. Furthermore, notice that C N a,b is bounded from below by the optimal constant for the inequality restricted to symmetric functions.

The symmetry result of Theorem 4 is in fact a uniqueness statement for the Euler-Lagrange equations associated with the variational problem (4), under an energy condition. It has been proved in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF] that optimal functions exist for the sharp Caffarelli-Kohn-Nirenberg inequalities. Hence Theorem 1 (or equivalently Theorem 2) is a consequence of the stronger Theorem 4.

A few comments about the strategy of the proof are in order. Equation ( 5) can be viewed as a Schrödinger equation with V * := u p-2 * as a potential and -Λ as the smallest eigenvalue. It can be readily solved and yields, up to translations, the function u * given by ( 6) and the potential

V * (s) := A p-2 | cosh(B s)| 2 ∀ s ∈ R ,
with A , B given by [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF]. This function can be viewed as the solution for the single bound state Lieb-Thirring inequality in Lemma 3 and Corollary 6. Up to a translation and a multiplication by a positive constant, V * is the unique optimizing potential for Inequality [START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF] for

γ = 1 2 p + 2 p -2 ,
if the normalization is chosen such that the optimal eigenvalue is given by -Λ . Moreover, one can check that

(11) c LT (γ) R V γ+ 1 2 * ds = Λ γ .
Proof of Theorem 4. We will apply a number of inequalities, for which equality cases will be achieved. Let

F[u] := S N-1 R |∂ s u| 2 -u p ds dω + C |Lu| 2 ds dω .
If u is a solution to ( 9), then we have

F[u] = -Λ C |u(s, ω)| 2 ds dω .
First step. It relies on Lemma 3. With V = u p-2 , we find that a.e. in ω ,

R |∂ s u(s, ω)| 2 -|u(s, ω)| p ds ≥ -c LT (γ) 1/γ R |u(s, ω)| p ds 1/γ |v(ω)| 2 , with v(ω) := R |u(s, ω)| 2 ds . (12)
Hence we get

F[u] ≥ -c LT (γ) 1/γ S N-1 R |u(s, ω)| p ds 1/γ |v(ω)| 2 dω + C |Lu(s, ω)| 2 ds dω .
Second step. Since

L α v(ω) = R u(s, ω) L α u(s, ω) ds R u(s, ω) 2 ds
, it follows from Schwarz's inequality that

(13) C |Lu(s, ω)| 2 ds dω ≥ S N-1 |Lv(ω)| 2 dω .
Third step. Since γ > 1 , we apply Hölder's inequality,

S N-1 R |u(s, ω)| p ds 1 γ |v(ω)| 2 dω ≤ C |u(s, ω)| p ds dω 1 γ S N-1 |v(ω)| 2 γ γ-1 dω γ-1 γ (14)
and thus, with

D := c LT (γ) 1/γ C u p ds dω 1 γ
, we obtain

F[u] ≥ S N-1 (Lv) 2 dω -D S N-1 v 2 γ γ-1 dω γ-1 γ =: E[v] .
Fourth step. The generalized Poincaré inequality [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF][START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF] states that for all q ∈ 1,

N +1 N -3 , (15)
q-1 N -1 S N-1 (Lv) 2 dω ≥ S N-1 v q+1 dω 2 q+1 - S N-1 v 2 dω . Choosing q + 1 = 2 γ γ -1 = 2 p + 2 6 -p ,
i.e. q = 3 p-2 6-p , we arrive at

E[v] ≥ N -1 q-1 -D S N-1 v q+1 dω 2 q+1 -N -1 q-1 S N-1 v 2 dω .
Note that q is in the appropriate range, since we may indeed notice that

q ≤ N +1 N -3 is equivalent to p ≤ 2 N N -2 .
Fifth step. Using the fact that dω is a probability measure, by Hölder's inequality, we get ( 16)

S N-1 v q+1 dω 2 q+1 ≥ S N-1 v 2 dω . Thus, if D ≤ N -1 q -1 , we get N -1 q-1 -D S N-1 v q+1 (ω) dω 2 q+1 ≥ N -1 q-1 -D S N-1 v 2 dω ,
that is,

E[v] ≥ -D S N-1 v 2 dω .
By [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF] and [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF], we know that

D = c LT (γ) 1 γ C u p ds dω 1 γ ≤ c LT (γ) 1 γ C u p * ds dω 1 γ = Λ . Thus, if Λ ≤ N -1 q -1 = (N -1) (6 -p) 4 (p -2) = Λ ⋆ (p) , then D ≤ N -1
q-1 and the chain of inequalities

-Λ C u 2 ds dω = F[u] ≥ E[v] ≥ -D C u 2 ds dω ≥ -Λ C u 2 ds dω
shows that D = Λ and equality holds at each step. Now, let us investigate the consequences of such equalities:

(1) By Lemma 3, equality in (12) yields the existence of three functions, A , B and C on S N -1 such that, for a.e. (s, ω) ∈ C ,

u p-2 (s, ω) = A(ω) p-2 cosh B(ω) (s -C(ω)) 2 .
Note that, since cosh(x) is an even function, we may choose B(ω) to be a non-negative function.

(2) Equality in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF] means that v is proportional to R |u(s, ω)| p ds 1/p on the sphere, so that B does not depend on ω .

(3) Equality in [START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF] means that v is constant on the sphere and, as a consequence, A 2 /B does not depend on ω .

If Λ < Λ ⋆ (p) there must be equality in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF] and in ( 16) and hence A and B do not depend on ω . Thus, u satisfes the equation [START_REF]About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] we have necessarily that L 2 u = 0 , which means that C and hence u are independent of ω .

-∂ 2 s u + Λu = u p-1 on C . Since u satisfies
The problem is a bit trickier if Λ = Λ ⋆ (p) and p ≤ 2 N N -2 . In this case equality in ( 16) is not required. But, recall that equality in [START_REF] Dolbeault | Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems[END_REF] means that for some B(ω) , C(ω) ,

∂ s log(u p-2 (s, ω)) = -2 B(ω) tanh B(ω) (s -C(ω)) := f (s)
does not depend on the variable ω . As s → ∞ , f (s) converges to -2 B(ω) and hence B(ω) must be constant. Since x → tanh(x) is a strictly monotone function, C(ω) is also a constant. Let X(s) := cosh(B (s -C)) -2/(p-2) . Since u solves (9), we find

0 = -∂ 2 s u -L 2 u + Λ u -u p-1 = A 2 p B 2 (p-2) 2 -A p-2 X 2 (p-1) + A Λ -4 B 2 (p-2) 2 -L 2 A X 2 ,
so that A p-2 = 2 p B 2 (p-2) 2 must be constant too. This is again enough to conclude that A, B and C do not depend on ω .

Next we state a rigidity result which is a consequence of the proof of Theorem 2. The connection with Theorem 4 will be made clear in Section 4.

Corollary 6. Let N ≥ 2 . Fix γ > 1 such that γ ≥ N -1 2 if N ≥ 4 and let q = γ+1 γ-1 . Further fix D ≤ N -1 q-1 . Among all potentials V = V (s, ω) with c LT (γ) 1 γ C V γ+ 1 2 ds dω 1 γ = D ,
the potential V that minimizes the first eigenvalue of -∂ 2 s -L 2 -V on L 2 (C, ds dω) does not depend on ω . Moreover, u = V (2 γ-1)/4 is optimal for (4).

Interpolation and one-bound state Lieb-Thirring inequalities in higher dimensions

As a straightforward consequence of their definitions, both C(Λ, p, N ) and C * (Λ, p, N ) are monotone non-increasing functions of Λ and we have [START_REF] Lieb | Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities[END_REF] C(Λ, p, N ) ≥ C * (Λ, p, N ) , where

C * (Λ, p, N ) = |S N -1 | -p-2 p Λ (p-2) 2 p+2 p-2 2 p p+2 2 p Λ 4 p+2 6-p 2 p Γ 2 p-2 + 1 2 √ π Γ 2 p-2 p-2 p
according, e.g., to [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF][START_REF] Horiuchi | Best constant in weighted Sobolev inequality with weights being powers of distance from the origin[END_REF][START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Del Pino | A logarithmic Hardy inequality[END_REF]. We observe that

C * (Λ, p, N ) = C * (1, p, N ) Λ -p+2 2 p , so that lim Λ→0 + C * (Λ, p, N ) = ∞ . From [5
N = 2 , lim Λ→∞ Λ ac-N p C(Λ, p, N ) = inf w∈H 1 (R d )\{0} R d |∇u| 2 + |u| 2 dx R d |u| p dx 2/p , so that lim Λ→∞ C(Λ, p, N ) = 0 .
With these observations in hand and γ = 1 2 p+2 p-2 , we can define

Λ N γ (µ) := inf Λ > 0 : µ 2 γ 2 γ+1 = 1/C(Λ, p, N ) . If N = 1 , we observe that C(Λ, p, 1) = C * (Λ, p, 1) , so that Λ 1 γ (µ) = Λ 1 γ (1) µ and Λ 1 γ (1) = C * (1, p, N ) 2 p p+2 . With γ = 1 2 p+2 p-2 , notice that the condition p ∈ (2, 6) means γ ∈ (1, ∞) while the condition p ≤ 2 N N -2 means γ ≥ N -1 2 .
Next, consider on C = R×S N -1 the Schrödinger operator -∂ 2 s -L 2 -V and denote by -λ 1 (V ) its lowest eigenvalue. We assume that V is non-negative, so that λ 1 (V ) , if it exists, is non-negative. The main point of this section is that λ 1 (V ) can be estimated using Λ N γ (µ) provided V is controlled in terms of µ . The Gagliardo-Nirenberg-Sobolev inequality (4) on C is equivalent to the following one-bound state version of the Lieb-Thirring inequality.

Lemma 7. For any γ ∈ (2, ∞) if N = 1 , or for any γ ∈ (1, ∞) such that γ ≥ N -1 2 if N ≥ 2 , if V is a non-negative potential in L γ+ 1 2 (C)
, then the operator -∂ 2 -L 2 -V has at least one negative eigenvalue, and its lowest eigenvalue, -λ 1 (V ) , satisfies

(18) λ 1 (V ) ≤ Λ N γ (µ) with µ = µ(V ) := C V γ+ 1 2 ds dω 1 γ
.

Moreover, equality is achieved if and only if the eigenfunction u corresponding to λ 1 (V ) satisfies u = V (2 γ-1)/4 and u is optimal for (4).

Proof of Lemma 7. Let

Q[u, V ] := C |∇u| 2 ds dω -C V |u| 2 ds dω C |u| 2 ds dω so that -λ 1 (V ) = inf u∈H 1 (C)\{0} Q[u, V ]
is achieved by some function u ∈ H 1 (C) such that u L 2 (C) = 1 . Using Hölder's inequality, we find that

C V |u| 2 ds dω ≤ µ 2 γ 2 γ+1 u 2 L p (C) ,
with p = 2 2 γ+1 2 γ-1 and µ = µ(V ) , and equality in the above inequality holds if and only if [START_REF] Lin | Erratum to: "Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] V γ+ 1 2 = κ |u| p for some κ > 0 , which is actually such that κ u p L p (C) = µ γ . Then we have found that

Q[u, V ] ≥ ∇u 2 L 2 (C) -µ 2 γ 2 γ+1 u 2 L p (C)
. By definition of Λ N γ (µ) and ( 4), we get that, for all u , Q[u, V ] ≥ -Λ N γ (µ) , thus proving ( 18) and ( 20)

-Λ N γ (µ) = inf V ∈ L γ+ 1 2 (C) C V γ+ 1 2 ds dω = µ γ inf u∈H 1 (C)\{0} Q[u, V ]
where equality holds if and only if [START_REF] Lin | Erratum to: "Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] holds and u is optimal for (4). This concludes the proof.

A symmetry result for the one-bound state Lieb-Thirring inequality. It is remarkable that optimality in ( 18) is equivalent to optimality in (4). As a non trivial consequence of the above considerations and of Theorem 2, symmetry results for interpolation inequalities are also equivalent to symmetry results for the one-bound state Lieb-Thirring inequality in the cylinder.

Corollary 8. Let N ≥ 2 . For all γ > 1 such that γ ≥ N -1 2 if N ≥ 4 , if V is a non-negative potential in L γ+ 1 2 (C) such that µ(V ) := C V γ+ 1 2 ds µ(dω) 1 γ ≤ Λ ⋆ (p) C * (1, p, N ) 2 p p+2 with p = 2 2 γ + 1 2 γ -1 , then, the lowest (non-positive) eigenvalue of -∂ 2 s -L 2 -V , -λ 1 (V ) , satisfies λ 1 (V ) ≤ Λ 1 γ (1) µ(V )
. Moreover, equality in the above inequality is achieved by a potential V which depends only on s and u = V (2γ-1)/4 is optimal for (4).

Proof. Based on the definition of Λ N γ (µ) and ( 17), we know that

µ 2 γ 2 γ+1 = 1 C(Λ N γ (µ), p, N ) ≤ 1 C * (Λ N γ (µ), p, N ) = 1 C * (1, p, N ) Λ N γ (µ) p+2 2 p .
We observe that γ = 1 2 p+2 p-2 means 2 γ 2 γ+1 = p+2 2 p and hence

Λ N γ (µ) ≥ C * (1, p, N ) 2 p p+2 µ .
However, there is equality in the above inequality as long as Λ N γ (µ) ≤ Λ ⋆ (p) (see [START_REF]Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] and Theorem 4). Since

µ → Λ N γ (µ) is monotone increasing, requir- ing Λ N γ (µ) ≤ Λ ⋆ (p) is equivalent to asking µ ≤ Λ ⋆ (p) C * (1, p, N ) -2 p/(p+2) .
Then the optimality in ( 4) is achieved among symmetric functions, by Theorem 2. This completes the proof. Details are left to the reader.

5.

Beyond symmetry and symmetry breaking: getting estimates for the non-radial optimal constants

Caffarelli-Kohn-Nirenberg inequalities are actually much more general than the ones considered in Section 1 and in view of previous results (see for instance [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF][START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF][START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF]) it is very natural to consider another family of interpolation inequalities, which can be introduced as follows.

* := 2 N N -2 if N ≥ 3 and 2 * := ∞ if N = 1 or 2 .
For any a < a c , we consider the following Caffarelli-Kohn-Nirenberg inequalities, which were introduced in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] (also see [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF]):

Let b ∈ (a + 1/2, a + 1] and θ ∈ (1/2, 1] if N = 1 , b ∈ (a, a + 1] if N = 2 and b ∈ [a, a + 1] if N ≥ 3 . Assume that p = p(a, b) , and θ ∈ [ϑ(p, N ), 1] if N ≥ 2 . Then, there exists a finite positive constant K CKN (θ, Λ, p) with Λ = Λ(a) such that, for any u ∈ C ∞ c (R N \ {0}) , (21)
|x| -b u 2 L p (R N ) ≤ K CKN (θ, Λ, p) |S N -1 | p-2 p |x| -a ∇u 2 θ L 2 (R N ) |x| -(a+1) u 2 (1-θ) L 2 (R N ) .
We denote by K * CKN (θ, Λ, p) the best constant among all radial functions. We recall that this constant is explicit and equal to

K * CKN (θ, Λ, p) = K * θ,p Λ -(2 θ-1) p+2 2 p
where

K * θ,p := (p-2) 2 (2 θ-1) p+2 p-2 2 p (2 θ-1) p+2 2 p θ θ 4 p+2 6-p 2 p Γ 2 p-2 + 1 2 √ π Γ 2 p-2 p-2 p
according to [7, Lemma 3]. In the special case θ = 1 , we have

K * CKN (1, Λ, p) = S N -1 p-2 p C N a,b . Define the function C by C(p, θ) := (p+2) p+2 (2 θ-1) p+2 (2 θ-1) p+2 2-p (1-θ) 2 2 2-p (1-θ) (2 θ-1) p+2 • Γ( p p-2 ) Γ( θ p p-2 ) 4 (p-2) (2 θ-1) p+2 Γ( 2 θ p p-2 ) Γ( 2 p p-2 ) 2 (p-2) (2 θ-1) p+2
.

Notice that C(p, θ) ≥ 1 and C(p, θ) = 1 if and only if θ = 1 . Theorem 9. With the above notations, for any N ≥ 3 , any p ∈ (2, 2 * ) and any θ ∈ [ϑ(p, N ), 1) , we have the estimate

K * CKN (θ, Λ, p) ≤ K CKN (θ, Λ, p) ≤ K * CKN (θ, Λ, p) C(p, θ) (2 θ-1) p+2 2 p
under the condition

(22) a 2 c < Λ ≤ (N -1) C(p, θ) (2 θ -3) p + 6 4 (p -2) .
Although we do not establish here a symmetry result, it is interesting to notice that a symmetry result would amount to prove that K CKN (θ, Λ, p) = K * CKN (θ, Λ, p) , except maybe on the threshold curve in the set of parameters. Theorem 9 does not establish such a symmetry result for θ < 1 , but we recover the already known fact that lim θ→1 K CKN (θ, Λ, p) = K * CKN (1, Λ, p) , with an explicit estimate, in the appropriate region of the parameters. This is essentially the result of Theorem 1.

For the convenience of the reader, we split our computations in several steps and provide some details which have been skipped in the previous sections. For instance, we give the expression of c LT (γ) in Lemma 3, which is also needed to establish the expression of C(p, θ) .

1. Preliminary computations. Consider the equation ( 23)

-(p -2) 2 w ′′ + 4 w -2 p |w| p-2 w = 0 in R .
The function

w(s) = (cosh s) -2 p-2
∀ s ∈ R is, up to translations, the unique positive solution of (23). As a consequence, the unique positive solution of -θ w ′′ + η w = |w| p-2 w in R which reaches its maximum at s = 0 can be written as

w(s) = A w(B s) ∀ s ∈ R with A = p η 2 1 p-2 and B = p-2 2 η θ .
As in [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF], define

I q := R |w(s)| q ds and J 2 := R |w ′ (s)| 2 ds . Using the formula R ds (cosh s) q = √ π Γ q 2 Γ q+1 2 =: f (q) ,
we can compute

I 2 = f 4 p -2 , I p = f 2 p p -2 = f 4 p -2 + 2 ,
and get the relations

I 2 = √ π Γ 2 p-2 Γ p+2 2 (p-2) , I p = 4 I 2 p+2 and J 2 := 4 (p-2) 2 (I 2 -I p ) = 4 I 2 (p+2)(p-2) .
2. Further preliminary computations in the case θ < 1. Consider now the solution of the Euler-lagrange equation satisfied by the extremals for [START_REF] Smets | Partial symmetry and asymptotic behavior for some elliptic variational problems[END_REF] written in the cylinder C , that is,

(24) -θ ∂ 2 s u + L 2 u + [(1 -θ) t[u] + Λ] u = u p-1 on C , where t[u] := C (∂ s u) 2 + (Lu) 2 ds dω C u 2 ds dω .
Such an extremal function always exists for all θ > ϑ(p, d): see [START_REF] Dolbeault | Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF]. The case θ = ϑ(p, d) in the theorem will be achieved by passing to the limit.

Multiplying (24) by u and integrating on C , we find that

C (∂ s u) 2 + (Lu) 2 + Λ u 2 ds dω = C u p ds dω .
To relate K CKN (θ, Λ, p) and K * CKN (θ, Λ, p) we have to compare

Q[u] := C (∂ s u) 2 + (Lu) 2 + Λ u 2 ds dω θ C u 2 ds dω 1-θ C u p ds dω 2 p = C u p ds dω θ-2 p C u 2 ds dω 1-θ
where equality holds because u is a solution of (24), with the same quantity written for u * , an extremal for [START_REF] Smets | Partial symmetry and asymptotic behavior for some elliptic variational problems[END_REF] in the cylinder C , in the class of functions depending on s . Either K CKN (θ, Λ, p) = K * CKN (θ, Λ, p) and then Theorem 9 is proved, or the inequality

(25) 1 K CKN (θ, Λ, p) = Q[u] = C u p ds dω θ-2 p C u 2 ds dω 1-θ ≤ 1 K * CKN (θ, Λ, p) = Q[u * ] = R u p * ds θ-2 p R u 2 * ds 1-θ
is strict.

3. The symmetric optimal function for θ < 1. The solution u * can be explicitly computed. On the one hand, it solves

-θ (u * ) ′′ + η u * = u p-1 * in R , with η = (1 -θ) t[u * ] + Λ .
After multiplying by u * , integrating with respect to s ∈ R and dividing by R u 2 * ds , we find

t[u * ] + Λ = R u p * ds R u 2
* ds where u * (s) = A w(B s) , for all s ∈ R , has been computed in the first step of this section. From this expression, we deduce that 4. Collecting estimates: proof of Theorem 9. As in the case θ = 1 , we start by estimating the functional

t[u * ] = B 2 J 2 I 2 = p -2 p + 2
F[u] := C (∂ s u) 2 -u p + (Lu) 2 ds dω
from below. From Lemma 3 applied with γ replaced by some well-chosen γ θ , we get the lower bound

F[u] ≥ -c LT (γ θ ) 1 γ θ S N-1 R u θ p ds 1 γ θ R u 2 ds dω + C (Lu) 2 ds dω ,
where γ θ is now chosen such that (γ θ + 1 2 ) (p -2) = θ p , that is

γ θ = (2 θ -1) p + 2 2 (p -2) .
Exactly as in the proof of Theorem 4, except that γ and p are now replaced respectively by γ θ and by θ p , we find the lower bound

F[u] ≥ S N-1 (Lv) 2 (ω) dω -c LT (γ θ ) 1 γ θ C |u(s, ω)| θ p ds dω 1 γ θ S N-1 v q+1 (ω) dω 2 q+1 with q + 1 = 2 γ θ γ θ -1 , i.e. q =
(2 θ + 1) p -2 (2 θ -3) p + 6 .

By Hölder's inequality, we find

C u θ p ds dω ≤ C u 2 ds dω (1-θ) p p-2 C u p ds dω θ p-2 p-2
.

Altogether, we have shown that

F[u] ≥ S N-1 (Lv) 2 dω -D S N-1 v q+1 dω 2 q+1
where we abbreviated

D := c LT (γ θ ) 1 γ θ   C u 2 ds dω (1-θ) p p-2 C u p ds dω θ p-2 p-2   1 γ θ . With γ θ = (2 θ-1) p+2 2 (p-2)
and using Assumption (25), we know that

D = c LT (γ θ ) 1 γ θ Q[u] p γ θ (p-2) ≤ c LT (γ θ ) 1 γ θ Q[u * ] p γ θ (p-2) = C(p, θ) Λ ,
where the last equality is a definition of C(p, θ) (see the computation of its precise value below).

As in the case θ = 1 , using again the generalized Poincaré inequality, if D ≤ N -1 q-1 and q ≤ N +1 N -3 , we find that

F[u] ≥ -D S N-1 v 2 dω . A sufficient condition for D ≤ N -1 q-1 is C(p, θ) Λ ≤ N -1 q -1 = (N -1) (2 θ -3) p + 6 4 (p -2)
which is equivalent to [START_REF] Talenti | Best constant in Sobolev inequality[END_REF], while the condition

(2 θ+1) p-2 (2 θ-3) p+6 = q ≤ N +1 N -3 amounts to p ≤ 2 N N -2 θ , that is θ ≥ ϑ(p, N
) . Because of the first equality in (25), u is a minimizer of Q[u] and therefore solves (24). A multiplication of the equation by u and an integration on C shows that

-Λ C u 2 ds µ(dω) = F[u] .
Hence, if θ ≥ ϑ(p, N ) and Condition ( 22) holds, we have proved that

-Λ C u 2 ds µ(dω) = F[u] ≥ -D S N-1 v 2 dω = -D C u 2 ds µ(dω) ,
that is, Λ ≤ D , and then we have the chain of inequalities

(26) Λ ≤ D = c LT (γ θ ) 1 γ θ Q[u] p γ θ (p-2) ≤ c LT (γ θ ) 1 γ θ Q[u * ] p γ θ (p-2) = C(p, θ) Λ where Q[u] = 1/K CKN (θ, Λ, p) and Q[u * ] = 1/K * CKN (θ, Λ, p) by (25). Recall- ing that γ θ = (2 θ-1) p+2
2 (p-2) , this allows to express (26) as

c LT (γ θ ) p-2 p C(p, θ) (2 θ-1) p+2 2 p Λ -(2 θ-1) p+2 2 p = K * CKN (θ, Λ, p) ≤ K CKN (θ, Λ, p) ≤ c LT (γ θ ) p-2 p Λ -(2 θ-1) p+2 2 p
, which concludes the proof of Theorem 9.

5. Computation of c LT (γ) in Lemma 3. When θ = 1 and d = 1 , equality is achieved in (26) and we actually have u = u * , up to multiplication by constants, translations and scalings. As a consequence, we can compute

c LT (γ) = Λ γ Q[u * ] p p-2 = Λ γ R u p * ds where γ = p+2 2 (p-2) , i.e. p = 2 2 γ+1 2 γ-1 . With θ = 1 and η = Λ , we know that u * (s) = A w(B s) with A = p Λ 2 1 p-2 and B = p-2 2 √ Λ , so that R u p * ds = A p B I p = 8 √ π p 2 -4 Γ 2 p-2 Γ p+2 2 (p-2) p Λ 2 p p-2 1 √ Λ and hence c LT (γ) = p 2 -4 8 √ π Γ p+2 2 (p-2) Γ 2 p-2 2 p p p-2 where γ = 1 2 p+2 p-2 , that is p = 2 2 γ+1 2 γ-1 . All computations done, we get c LT (γ) = 2 γ-1 2 γ+1 γ-1 2 2 γ 2 γ+1 Γ(γ) √ π Γ(γ + 1 2 )
.

6. Computation of C(p, θ). From (26), we know that

C(p, θ) = c LT (γ θ ) 1 γ θ 1 Λ Q[u * ] 2 p
(2 θ-1) p+2 .

Using the results of Step 5 allows to compute C(p, θ) . Notice that the term

Q[u * ] 2 p
(2 θ-1) p+2 is proportional to Λ , so that Λ does not enter in the expression of C(p, θ) .

Interpolation and one-bound state Lieb-Thirring inequalities on general cylinders

In this section we extend the results of the previous sections to the more general case of the cylinders R × M , where M is a Riemannian manifold, using the results of [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. For this purpose we need the following assumptions: (M, g) is a compact Riemannian manifold of dimension N -1 ≥ 2 , without boundary, ∆ g is the Laplace-Beltrami operator on M , the Ricci tensor R and the metric tensor g satisfy R ≥ N -2 N -1 (q -1) λ g in the sense of quadratic forms, with q > 1 , λ > 0 and q ≤ N +1 N -3 . Moreover, one of these two inequalities is strict if (M, g) is S N -1 with the standard metric.

For brevity, we shall say that (H) holds if these assumptions are satisfied. Theorem 10. [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] Assume that (H) holds. If u is a positive solution of ∆ g u -λ u + u q = 0 , then u is constant with value λ 1/(q-1) .

As a consequence (see [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]Corollary 6.2]), with D(M, q) := max λ > 0 : R ≥ N -2 N -1 (q -1) λ g , we get the following generalized Poincaré inequality, an extension of [START_REF] Glaser | A family of optimal conditions for the absence of bound states in a potential[END_REF]. In the particular case θ = 1 , C(p, 1) = 1 , so that K(θ, Λ, p) = K CKN (θ, Λ, p) and the extremals of (27) are equal to u * , up to translation and multiplication by a constant.

Sketch of the proof. As in step 4 of the proof of Theorem 9, we can apply the generalized Poincaré inequality of Corollary 11 if D is such that D ≤ D(M, q) with q = (2 θ+1) p-2 (2 θ-3) p+6 . The equality case can be handled directly using Theorem 10.

As in Theorem 4, in the particular case θ = 1 , we actually have a slightly stronger result. Assume that N ≥ 3 , M = S N -1 , and let u be a nonnegative function of -∆ g u + Λ u = u p-1 on C . If Λ ≤ D(M, 3p-2 6-p ) and C |u(s, ω)| p ds dω ≤ R |u * (s)| p ds , where u * is the solution given by ( 6), then for a.e. ω ∈ M and s ∈ R , we have u(s, ω) = u * (s -C) for some constant C . Also in the case of a general cylinder C , for θ = 1 , we also have results similar to the one-bound state version of the Lieb-Thirring inequality of Lemma 7 and to Corollary 8, that can be summarized as follows.

Corollary 13. Assume that (H) holds, M = S N -1 , θ = 1 and N ≥ 3 . For any γ ∈ (1, ∞) such that γ ≥ N -1 2 , if V is a non-negative potential in L γ+ 1 2 (C) , then the operator -∆ g -V has at least one negative eigenvalue, and its lowest eigenvalue, -λ 1 (V ) satisfies for some positive constant Λ N γ (µ) . Moreover, equality is achieved if and only if the eigenfunction u corresponding to λ 1 (V ) , satisfies u = V (2 γ-1)/4 , where u is optimal for (27).

-2 2 .

 2 and a = a c defined by a c = a c (N ) := N The exponent p = p(a, b) := 2 N N -2 + 2 (b -a) is determined by scaling considerations. Inequality (1) holds in the space D a,b := w ∈ L p (R N , |x| -b dx) : |x| -a |∇w| ∈ L 2 (R N , dx)

  a < b < b FS (a) with b FS (a) := 2 N (a c -a) (a c -a) 2 + (N -1) + a -a c , extremals are non-radially symmetric. The proof is based on an analysis of the second variation of the functional associated to (1) around the radial extremal w * a,b . Above the curve b = b FS (a) , all corresponding eigenvalues are positive and w * a,b is a strict local minimum, while there is at least one negative eigenvalue if b < b FS (a) and w * a,b is then a saddle point. As a → -∞ , b = b FS (a) is asymptotically tangent to b = a + 1 .

Figure 1 .

 1 Figure 1. In dimension N = 3 , the symmetry region is shown in light grey: the best constant in (1) is achieved among radial functions if either a < 0 and b ⋆ (a) ≤ b < a + 1 , or 0 ≤ a < a c and a ≤ b ≤ 1 . The known symmetry breaking region appears in dark grey and contains at least the region a < 0 , a ≤ b < b FS (a) . Symmetry in the tiny (white) zone in between those regions is still an open question.

  with Λ and p given in terms of a , a c = a c (N ) and N by Λ = (a c -a) 2 and p = 2 N N -2 + 2 (b -a) , and the same optimal constant C N a,b as in (1). It turns out to be convenient to reparametrize the problem, originally written in terms of a and b , in terms of the parameters Λ and p . Hence we shall use the notation C(Λ, p, N ) := C N a,b and define C * (Λ, p, N ) as the best constant in (4) among functions depending on s only. For a < 0 , the conditions b < b FS (a) and b ≥ b ⋆ (a) respectively become Λ > Λ FS (p) and Λ ≤ Λ ⋆ (p) , where

  and recall that a c := N -2 2 , Λ(a) := (a -a c ) 2 and p(a, b) := 2 N N -2+2 (b-a) . We shall also set 2

Proposition 11 . [ 3 ]∀.and 9 . 12 .

 113912 Under Assumption (H), if |M | = 1 and D(M, q) > 0 v ∈ W 1,1 (M) .With these tools in hand, we can now consider the general cylinderC := R × M .Using the notations of Section 5, let b ∈ [a, a + 1] and assume that N ≥ 3 , p = p(a, b) , and θ ∈ [ϑ(p, N ), 1] . Under the assumptions of Proposition 11, there exists a finite positive constant K(θ, Λ, p) such that, for any u∈ C ∞ c (R N \ {0})(C) ,Such an interpolation inequality is easy to establish using Hölder and Sobolev inequalities on C . We are now in a position to state a result which generalizes Theorems 2Theorem Assume that (H) holds and M = S N -1 . With the above notations and C(p, θ) defined as in Section 5, for any p ∈ (2, 2 * ] and any θ ∈ [ϑ(p, N ), 1] , we have the estimateK * CKN (θ, Λ, p) ≤ K(θ, Λ, p) ≤ K * CKN (θ, Λ, p) C(p, θ) (2 θ-1) p+2 2 punder the conditions p < 6 if N = 3 , anda 2 c < Λ ≤ 1 C(p, θ)D M, (2 θ + 1) p -2 (2 θ -3) p + 6 .

λ 1 (C

 1 V ) ≤ Λ N γ (µ) with µ = µ(V ) :=

1 ,

 1 then λ 1 (V ) ≤ Λ 1 γ (1) µ(V ) = Λ N γ (µ) with Λ 1 γ (1) = K * (1, p, N ) 2 pp+2 , and equality in the above inequality is achieved by a potential V which depends only on s .The constant Λ N γ (µ) can be related to K(θ, Λ, p) as in the case M = S N -1 .

Acknowlegments. J.D. and M.J.E. have been supported by the projects CBDif and EVOL of the French National Research Agency (ANR). M.J.E. has also been partially supported by the ANR project NONAP. M.L. has been supported in part by NSF grant DMS-0901304. c 2011 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.