Bruno Colbois

Ahmad El Soufi
email: elsoufi@univ-tours.fr.

EXTREMAL EIGENVALUES OF THE LAPLACIAN ON EUCLIDEAN DOMAINS AND CLOSED SURFACES

Keywords: . 1991 Mathematics Subject Classification. 35P15, 58J50, 58E11 Laplacian Eigenvalues, extremal eigenvalues, spectral geometry, closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.

Introduction

A classical topic in spectral geometry is to investigate upper and lower bounds of eigenvalues of the Laplacian subject to various boundary conditions and under the fixed volume constraint. Among the most known results in this topic are the Faber-Krahn inequality for the first Dirichlet eigenvalue, the Szegö-Weinberger inequality for the first positive Neumann eigenvalue on bounded Euclidean domains, and Hersch's inequality for the first positive eigenvalue on closed simply connected surfaces.

Just like most of the results one can find in the literature, these sharp inequalities deal with the lowest order positive eigenvalues. Aside from numerical approaches, mainly in dimension 2, the determination of optimal bounds for eigenvalues of higher order is a problem that remains largely open.

In this article our aim will be to show how it is possible, through quite simple considerations, to establish certain intrinsic relationships between the infima (or the suprema) of eigenvalues of different orders. Let us start by fixing some notations.

Given a regular bounded domain Ω ⊂ R n , n ≥ 2, we designate by {λ k (Ω)} k≥1 (resp. {µ k (Ω)} k≥0 ) the nondecreasing sequence of eigenvalues of the Laplacian on Ω with Dirichlet (resp. Neumann) boundary conditions, each repeated according to its multiplicity. We introduce the following universal sequences of real numbers that are attached to the n-dimensional Euclidean space :

λ * k (n) = inf {λ k (Ω) : Ω ⊂ R n , |Ω| = 1} and µ * k (n) = sup {µ k (Ω) : Ω ⊂ R n , |Ω| = 1}
, where |Ω| stands for the volume of Ω. Notice that thanks to standard continuity results for eigenvalues, the definition of λ * k (n) (resp. µ * k (n)) does not change if the infimum (resp. the supremum) is taken only over connected domains. The famous Faber-Krahn and Szegö-Weinberger isoperimetric inequalities then read respectively as follows:

λ * 1 (n) = λ 1 (B n )|B n | 2 n = j 2 n 2 -1,1 ω 2 n n and µ * 1 (n) = µ 1 (B n )|B n | 2 n = p 2 n 2 ,1 ω 2 n
n , where ω n is the volume of the unit Euclidean ball B n , j n 2 -1,1 is the first positive zero of the Bessel function J n 2 -1 and p n 2 ,1 is the first positive zero of the derivative of the Bessel function J n 2 . It is also well known that (see for instance [13, p. 61

]) λ * 2 (n) = 2 2 n λ * 1 (n).
The same relation is conjectured to hold true between µ * 2 (n) and µ * 1 (n) (see [START_REF] Girouard | Maximization of the second positive Neumann eigenvalue for planar domains[END_REF] for a recent result about this conjecture in the 2-dimensional case). The following inequalities are also expected to be satisfied for every k ≥ 1 (Pólya's conjecture),

µ * k (n) ≤ 4π 2 k ω n 2 n ≤ λ * k (n),
where 4π 2 k ω n 2 n is the first term of the Weyl asymptotic expansion of both Dirichlet and Neumann eigenvalues of domains of volume one. Although this conjecture is still open, it was proved by Berezin [START_REF] Berezin | Covariant and contravariant symbols of operators[END_REF] and Li and Yau [START_REF] Li | On the Schrödinger equation and the eigenvalue problem[END_REF]

that λ * k (n) ≥ n n+2 4π 2 k ω n 2 n
, while Kröger [START_REF] Kröger | Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space[END_REF][START_REF] Kröger | Estimates for sums of eigenvalues of the Laplacian[END_REF]

proved that µ * k (n) ≤ 1 + n 2 2 n 4π 2 k ω n 2 n .
The first observation we make in this paper is that the sequence λ * k (n) n/2 is subadditive while µ * k (n) n/2 is superadditive. Indeed, we prove (Theorem 2.1) that, for every k ≥ 2 and any finite family i 1 , . . . , i p of positive integers such that i

1 + i 2 + • • • + i p = k, λ * k (n) n/2 ≤ λ * i 1 (n) n/2 + λ * i 2 (n) n/2 + • • • + λ * i p (n) n/2 (1) and µ * k (n) n/2 ≥ µ * i 1 (n) n/2 + µ * i 2 (n) n/2 + • • • + µ * i p (n) n/2 . ( 2
)
An immediate consequence of Theorem

k λ * k (n) k 2 n = 4π 2 ω -2 n n (resp. lim k µ * k (n) k 2 n = 4π 2 ω -2 n n , see Corollary 2.2).
Besides their theoretical interest, the inequalities ( 1) and ( 2) provide a "rough test" for the numerical methods used to approximate λ * k (n) and µ * k (n). For example, we observe that the numerical values for λ * k (2) obtained by Oudet [START_REF] Oudet | Numerical minimization of eigenmodes of a membrane with respect to the domain[END_REF] (see also [13, p. 83]) could be improved since the gap between the approximate values given for some successive λ * k (2) exceeds π j 2 0,1 . Improvements of Oudet's calculations leading to approximate values which are consistent with (1) and ( 2) have been obtained recently by Antunes and Freitas [START_REF] Pedro | Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians[END_REF].

Regarding the equality case in [START_REF] Anné | Spectre du laplacien et écrasement d'anses[END_REF] we prove that if it holds, then the infimum λ * k (n) is approximated to any desired accuracy by the λ k of a disjoint union of p domains A j , j = 1, . . . , p, each of which being, up to volume normalization, an "almost" minimizing domain for λ * i j (n) (see Theorem 2.1 for a precise statement). A similar phenomenon occurs for the case of equality in [START_REF] Pedro | Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians[END_REF]. This result complements that by Wolf and Keller [START_REF] Wolf | Range of the first two eigenvalues of the Laplacian[END_REF] where it is proved that if Ω = A ∪ B is a disconnected minimizer of λ k , then there exists a positive integer i < k so that, after volume normalizations, A minimizes λ i and B minimizes λ k-i and, moreover,

λ * k (n) n/2 = λ * i (n) n/2 + λ * k-i (n) n/2
. A Neumann analogue of this result has been recently obtained by Poliquin and Roy-Fortin [START_REF] Poliquin | Wolf-keller theorem for neumann eigenvalues[END_REF] Our next observation is that Wolf-Keller's result extends to "almost minimizing" disconnected domains as follows (Theorem 2.2): If a disconnected domain Ω = A ∪ B minimizes λ k to within some ε ≥ 0, then there exists an integer i so that, after volume normalizations, A minimizes λ n/2 i to within ε and B minimizes λ n/2 k-i to within ε, and, moreover,

0 ≤ λ * i (n) n/2 + λ * k-i (n) n/2 -λ * k (n) n/2 ≤ ε.
A similar property holds for "almost maximizing" disconnected domains of Neumann eigenvalues (Theorem 2.3).

The second part of the paper is devoted to the case of compact surfaces without boundary. If S is an orientable compact surface of the 3dimensional space, we denote by {ν k (S )} k≥0 the spectrum of the Laplace-Beltrami operator acting on S (here ν 0 (S ) = 0). The eigenvalue ν k is not bounded above on the set of compact surfaces of fixed area, as shown in [START_REF] Colbois | Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds[END_REF]Theorem 1.4] (which also justifies why we do not consider higher dimensional hypersurfaces). However, according to Korevaar [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF], for every integer γ ≥ 0, the k-th eigenvalue ν k is bounded above on the set M(γ) of compact surfaces of genus γ and fixed area. As before, we introduce the sequence

ν * k (γ) = sup {ν k (S ) : S ∈ M(γ) and |S | = 1} = sup S ∈M(γ) ν k (S )|S |.
As we will see in Section 3, an equivalent definition of ν * k (γ) consists in taking the supremum of the k-th eigenvalue ν k (Σ γ , g) of the Laplace-Beltrami operator on compact orientable 2-dimensional Riemannian manifolds of genus γ and area one.

For γ = 0, one has, from the results of Hersch [START_REF] Hersch | Quatre propriétés isopérimétriques de membranes sphériques homogènes[END_REF] and Nadirashvili [START_REF] Nadirashvili | Isoperimetric inequality for the second eigenvalue of a sphere[END_REF] ν * 1 (0) = 8π and ν * 2 (0) = 16π. Results concerning extremal eigenvalues on surfaces of genus 1 and 2 can be found in [START_REF] Soufi | A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle[END_REF][START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF][START_REF] Ilias | Riemannian manifolds admitting isometric immersions by their first eigenfunctions[END_REF][START_REF] Ilias | Extremal metrics for the first eigenvalue of the Laplacian in a conformal class[END_REF][START_REF] Jakobson | How large can the first eigenvalue be on a surface of genus two?[END_REF][START_REF] Jakobson | Extremal metric for the first eigenvalue on a Klein bottle[END_REF][START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF][START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF]. On the other hand, we have proved in [START_REF] Colbois | Extremal eigenvalues of the Laplacian in a conformal class of metrics: the 'conformal spectrum[END_REF] that the sequence ν * k (γ) is non decreasing with respect to γ and that it is bounded below by a linear function of k and γ. A. Hassannezhad [START_REF] Hassannezhad | Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem[END_REF] has recently proved that ν * k (γ) is also bounded from below by such a linear function of k and γ.

In Theorem 3.1 we prove that the double sequence ν * k (γ) satisfies the following property (Theorem 3.1): For every γ ≥ 0, k ≥ 1, if γ 1 . . . , γ p ∈ N and i 1 , . . . , i p ∈ N * are such that

γ 1 + • • • + γ p = γ and i 1 + • • • + i p = k, then ν * k (γ) ≥ ν * i 1 (γ 1 ) + • • • + ν * i p (γ p ). (3)
As before, we investigate the equality case in (3) and establish the following Wolf-Keller's type result (Corollary 3.1) : Assume that the disjoint union S 1 ⊔ S 2 of two compact orientable surfaces S 1 and S 2 of genus γ 1 , γ 2 , respectively, satisfies

ν k (S 1 ⊔ S 2 ) = ν * k (γ). ( 4
)
with

|S 1 | + |S 2 | = 1 and γ 1 + γ 2 = γ. Then there exists an integer i ∈ {1, • • • , k -1} such that ν * k (γ) = ν * i (γ 1 ) + ν * k-i (γ 2 ) ν i (S 1 )|S 1 | = ν * i (γ 1 ) and ν k-i (S 2 )|S 2 | = ν * k-i (γ 2 ).
Actually, we give a more general result where S 1 ⊔ S 2 is assumed to maximize ν k to within a positive ε (Theorem 3.2). Similar considerations can be made about nonorienrtable surfaces. This is discussed at the end of the paper.

Dirichlet and Neumann eigenvalue problems on Euclidean domains

To every (sufficiently regular) bounded domain Ω in R n , n ≥ 2, we associate two sequences of real numbers

0 < λ 1 (Ω) ≤ λ 2 (Ω) ≤ • • • ≤ λ k (Ω) ≤ • • • and 0 = µ 0 (Ω) < µ 1 (Ω) ≤ µ 2 (Ω) ≤ • • • ≤ µ k (Ω) ≤ • • • where λ k (Ω) (resp. µ k (Ω))
λ k (t Ω) = t -2 λ k (Ω) , µ k (t Ω) = t -2 µ k (Ω) and |t Ω| = t n |Ω| and, then λ * k (n) = inf {λ k (Ω) : Ω ⊂ R n , |Ω| = 1} = inf {λ k (Ω) : Ω ⊂ R n , |Ω| ≤ 1} (5) = inf {λ k (Ω)|Ω| 2/n : Ω ⊂ R n } and µ * k (n) = sup {µ k (Ω) : Ω ⊂ R n , |Ω| = 1} = sup {µ k (Ω) : Ω ⊂ R n , |Ω| ≥ 1} (6) = sup {µ k (Ω)|Ω| 2/n : Ω ⊂ R n }.
The sequences λ * k (n) and µ * k (n) satisfy the following intrinsic properties. Theorem 2.1. Let n and k be two positive integers and let i

1 ≤ i 2 ≤ • • • ≤ i p be positive integers such that i 1 + i 2 + • • • + i p = k. 1) We have, λ * k (n) n/2 ≤ λ * i 1 (n) n/2 + λ * i 2 (n) n/2 + • • • + λ * i p (n) n/2 (7) and µ * k (n) n/2 ≥ µ * i 1 (n) n/2 + µ * i 2 (n) n/2 + • • • + µ * i p (n) n/2 , (8)
2) If the equality holds in [START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF], then, for every ε > 0, there exist p mutually [START_REF] Soufi | A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle[END_REF], then, for every ε > 0, there exist p mutually

disjoint domains A 1 , A 2 , • • • , A p such that i) λ k (A 1 ∪ • • • ∪ A p ) ≤ (1 + ε)λ * k ; ii) ∀ j ≤ p, λ * i j ≤ λ i j (A j )|A j | 2/n ≤ (1 + ε)λ * i j . iii) |A 1 | + • • • + |A p | = 1 and, ∀ j ≤ p, λ * i j (1+ε)λ * k ≤ |A j | 2/n ≤ (1+ε)λ * i j λ * k ; where λ * k stands for λ * k (n).

3) If the equality holds in

disjoint domains A 1 , A 2 , • • • , A p such that i) µ k (A 1 ∪ • • • ∪ A p ) ≥ (1 -ε)µ * k ; ii) ∀ j ≤ p, (1 -ε)µ * i j ≤ µ i j (A j )|A j | 2/n ≤ µ * i j . iii) |A 1 | + • • • + |A p | = 1 and, ∀ j ≤ p, (1-ε)µ * i j µ * k ≤ |A j | 2/n ≤ µ * i j (1-ε)µ * k ;
where µ * k stands for µ * k (n). Proof. Let ε be any positive real number. For each j ≤ p, let C j be a domain of volume 1 satisfying

λ * i j (n) ≤ λ i j (C j ) ≤ (1 + ε)λ * i j (n) and set B j = λ i j (C j )/λ * k (n) 1 2 C j so that λ i j (B j ) = λ * k (n) and |B j | = λ i j (C j )/λ * k (n) n 2 .
One can assume w.l.o.g. that the domains

B 1 , • • • , B p are mutually dis- joint. Let us introduce the domain Ω = B 1 ∪ • • • ∪ B p . Since for every j ≤ p, λ i j (B j ) = λ * k (n)
and since the spectrum of Ω is the union of the spectra of the B j 's, one has

# l ∈ N * ; λ l (Ω) ≤ λ * k (n) = p j=1 # l ∈ N * ; λ l (B j ) ≤ λ * k (n) ≥ p j=1 i j = k. Thus, λ k (Ω) ≤ λ * k (n). Since λ * k (n) ≤ λ k (Ω)|Ω| 2 n
, the volume of Ω should be greater than or equal to 1. Consequently,

1 ≤ |Ω| = j≤p |B j | = 1 λ * k (n) n 2 j≤p λ i j (C j ) n 2 ≤ (1 + ε) n 2 λ * k (n) n 2 j≤p λ * i j (n) n 2 . ( 9
)
Inequality [START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF] follows immediately from (9) since ε can be arbitrarily small. Assume now that the equality holds in [START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF] and consider for each positive ε, a family B 1 , B 2 , • • • , B p constructed as above. Using [START_REF] Ilias | Riemannian manifolds admitting isometric immersions by their first eigenfunctions[END_REF], one sees that the domain

Ω = B 1 ∪ B 2 ∪ • • • ∪ B p satisfies 1 ≤ |Ω| ≤ (1 + ε) n 2
and it is easy to check that the domains A j := |Ω| -1 n B j , j ≤ p, satisfy the properties (ii) and (iii) of the statement (indeed,

|A j | = |B j | |Ω| with λ * i j (n)/λ * k (n) n 2 ≤ |B j | ≤ (1 + ε)λ * i j (n)/λ * k (n) n 2 )
. As for (i), one has for each j ≤ p,

λ i j (A j ) = |Ω| 2 n λ * k (n). Since k = i 1 +i 2 +• • •+i p , one deduces that λ k (A 1 ∪A 2 ∪• • •∪A p ) = |Ω| 2 n λ * k (n) ≤ (1 + ε)λ * k (n).
The proof in the Neumann case follows the same outline. Indeed, for any positive ε, we consider p mutually disjoint domains

C 1 , C 2 , • • • , C p of volume 1 such that, ∀ j ≤ p, µ * i j (n) ≥ µ i j (C j ) ≥ (1 -ε)µ * i j (n) and set B j = µ i j (C j )/µ * k (n) 1 2 C j and Ω = B 1 ∪ B 2 ∪ • • • ∪ B p . Since for every j ≤ p, µ i j (B j ) = µ * k (n)
, the number of eigenvalues of B j that are strictly less than µ * k (n) is at most i j (recall that µ i j (B j ) denotes the (i j + 1)-th eigenvalue of B j ). As the spectrum of Ω is the union of the spectra of the B j 's, it is clear that the number of eigenvalues of Ω that are strictly less than

µ * k (n) is at most k = i 1 + i 2 + • • • + i p . Thus, µ k (Ω) ≥ µ * k (n) which implies (since µ * k (n) ≥ µ k (Ω)|Ω| 2 n
) that the volume of Ω is less than or equal to 1. To derive Inequality (8) it suffices to observe that 1 ≥ |Ω| = j≤p |B j | and that

|B j | = µ i j (C j )/µ * k (n) n 2 ≥ (1 -ε) n 2 µ * i j (n) n 2 µ * k (n) n 2 .
Assume now that the equality holds in [START_REF] Soufi | A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle[END_REF] and consider for each positive ε, a family

B 1 , B 2 , • • • , B p constructed as above. The domain Ω = B 1 ∪ B 2 ∪ • • • ∪ B p satisfies 1 ≥ |Ω| ≥ (1 -ε) n 2
and it is easy to check that the domains A j := |Ω| -1 n B j , j ≤ p, satisfy the properties (ii) and (iii) of the statement (indeed,

|A j | = |B j | |Ω| with (1 -ε)µ * i j (n)/µ * k (n) n 2 ≤ |B j | ≤ µ * i j (n)/µ * k (n) n 2 ).
Moreover, one has for each j ≤ p,

µ i j (A j ) = |Ω| 2 n µ * k (n). Thus, µ k (A 1 ∪ A 2 ∪ • • • ∪ A p ) = |Ω| 2 n µ * k (n) ≥ (1 -ε)µ k which proves (i).
Corollary 2.1. For every n ≥ 2 and every k ≥ 1, we have

λ * k+1 (n) n/2 -λ * k (n) n/2 ≤ λ * 1 (n) n/2 = j n n 2 -1,1 ω n and µ * k+1 (n) n/2 -µ * k (n) n/2 ≥ µ * 1 (n) n/2 = p n n 2 ,1 ω n . Remark 2.1. (i) The first inequality in Corollary 2.1 is sharp for k = 1 since we know that λ * 2 (n) = 2 2/n λ * 1 (n). (ii) In dimension 2,
λ * k (n) ≤ j 2 n 2 -1,1 ω 2/n n k 2/n and µ * k (n) ≥ p 2 n 2 ,1 ω 2/n n k 2/n
. Combining these inequalities with Pólya conjecture, we expect the following estimates

p 2 n 2 ,1 ω 2/n n k 2/n ≤ µ * k (n) ≤ 4π 2 k ω n 2 n ≤ λ * k (n) ≤ j 2 n 2 -1,1 ω 2/n n k 2/n
which take the following form in dimension 2 :

4 πk ≤ λ * k (2) ≤ 5.784 πk and 3.39 πk ≤ µ * k (2) ≤ 4 πk.
(iv) Let Ω ⊂ R n be the union of k balls of the same radius r = (kω n ) -n so that |Ω| = 1. Then

λ k (Ω) = λ 1 (B n ) = λ 1 (B n )(kω n ) 2/n , and λ k+1 (Ω) = λ 2 (B n ) = λ 2 (B n )(kω n ) 2/n . Thus, λ k+1 (Ω) n/2 -λ k (Ω) n/2 = kω n λ 2 (B n ) n/2 -λ 1 (B n ) n/2 .
This shows that the gap λ k+1 (Ω) n/2λ k (Ω) n/2 cannot be bounded independently of k (see also Proposition 2.1 below). Corollary 2.1 tells us that such a bound exists when we consider the sequence of infima of λ k .

Thanks to Fekete's Lemma, the subadditivity of the sequence λ * k (n) n/2 leads immediately to the following corollary.

Corollary 2.2. For every n ≥ 2, the sequence

λ * k (n) k 2/n converges to a positive limit with lim k λ * k (n) k 2/n = inf k λ * k (n) k 2/n .
In particular, the two following properties are equivalent :

(1) (Pólya's conjecture) For every k ≥ 1 and every domain

Ω ⊂ R n , λ k (Ω) ≥ 4π 2 (|Ω|ω n ) -2/n k 2/n (2) lim k λ * k (n) k 2/n = 4π 2 ω -2/n n .
A similar result holds for the Neumann Laplacian eigenvalues.

The inequality (7) leads to

λ * k (n) n/2 ≤ inf 1≤i≤k-1 λ * i (n) n/2 + λ * k-i (n) n/2 . ( 10
)
Wolf and Keller [START_REF] Wolf | Range of the first two eigenvalues of the Laplacian[END_REF] proved that if λ k is minimized by a non connected domain, that is

λ * k (n) = λ k (A ∪ B)
for a couple of disjoint domains A and B with |A| > 0, |B| > 0 and |A| + |B| = 1, then the equality holds in [START_REF] Ilias | Extremal metrics for the first eigenvalue of the Laplacian in a conformal class[END_REF] and, moreover, A and B are, up to normalizations, minimizers of λ i and λ k-i , respectively. The Neumann's analogue of this result has been established by Poliquin and Roy-Fortin [START_REF] Poliquin | Wolf-keller theorem for neumann eigenvalues[END_REF].

The following theorem shows how Wolf-Keller's result extends to "almost minimizing" disconnected domains.

Theorem 2.2. Let k ≥ 2 and assume that there exists a non connected domain

Ω = A ∪ B in R n with |A| + |B| = 1, |A| > ε/λ * k (n) n/2 , |B| > ε/λ * k (n) n/2 and λ k (A ∪ B) n/2 ≤ λ * k (n) n/2 + ε (11) for some ε ≥ 0. Then there exists an integer i ∈ {1, • • • , k -1} such that 0 ≤ λ * i (n) n/2 + λ * k-i (n) n/2 -λ * k (n) n/2 ≤ ε, 0 ≤ λ i (A) n/2 |A| -λ * i (n) n/2 ≤ ε and 0 ≤ λ k-i (B) n/2 |B| -λ * k-i (n) n/2 ≤ ε.
Proof. Since the spectrum of Ω = A ∪ B is the re-ordered union of the spectra of A and B, the eigenvalue λ k (Ω) belongs to the union of the spectra of A and B and, moreover,

# j ∈ N * ; λ j (A) < λ k (Ω) + # j ∈ N * ; λ j (B) < λ k (Ω) ≤ k -1 (12)
and

# j ∈ N * ; λ j (A) ≤ λ k (Ω) + # j ∈ N * ; λ j (B) ≤ λ k (Ω) ≥ k. ( 13
)
Hence, there exists at least one integer j ∈ {1, . . . , k} such that λ j (A) = λ k (Ω) or λ j (B) = λ k (Ω). Assume that the first alternative occurs and let i be the largest integer between 1 and k such that

λ i (A) = λ k (Ω). Observe first that i ≤ k -1. Indeed, if λ k (A) = λ k (Ω), then λ * k (n) n/2 ≤ λ k (A) n/2 |A| = λ k (Ω) n/2 |A| ≤ λ * k (n) n/2 + ε |A| which implies |A| ≥ λ * k (n) n/2 λ * k (n) n/2 +ε and, then |B| = 1 -|A| ≤ ε λ * k (n) n/2 +ε ≤ ε λ * k (n) n/2 .
This contradicts the volume assumptions of the theorem.

On the other hand, the maximality of i means that

# j ∈ N * ; λ j (A) ≤ λ k (Ω) = i which implies, thanks to (13), λ k-i (B) ≤ λ k (Ω). Thus, λ k (Ω) n/2 = λ k (Ω) n/2 |A| + λ k (Ω) n/2 |B| ≥ λ i (A) n/2 |A| + λ k-i (B) n/2 |B|. (14)
Since

λ i (A) n/2 |A| ≥ λ * i (n) n/2 and λ k-i (B) n/2 |B| ≥ λ * k-i (n) n/2
, we have proved the inequality

λ * k (n) n/2 + ε ≥ λ k (Ω) n/2 ≥ λ * i (n) n/2 + λ * k-i (n) n/2 .

Now, we necessarily have the inequality λ

i (A) n/2 |A| ≤ λ * i (n) n/2 + ε.
Otherwise, we would have, thanks to (14) and Theorem 2.1,

λ k (Ω) n/2 ≥ λ i (A) n/2 |A| + λ k-i (B) n/2 |B| > λ * i (n) n/2 + ε + λ * k-i (n) n/2 ≥ λ * k (n) n/2
+ ε which contradicts the assumption of the theorem. The same argument leads to the inequality

λ k-i (B) n/2 |B| ≤ λ * k-i (n) n/2 + ε. Remark 2.2. (i) Taking ε = 0 in Theorem 2.

2, all the inequalities of the theorem become equalities and we recover the result of Wolf and Keller.

Notice that when ε = 0, it is immediate to see that the integer i is such that

λ * i (n) n/2 + λ * k-i (n) n/2 is minimal. (ii)
The assumption that the volume of each of the components A and B of Ω is bounded below in terms of ε is necessary to guarantee that the integer i is different from 0 and k in Theorem 2.2. Indeed, take for A a domain whose volume is almost equal to one and such that λ k (A) n/2 ≤ λ * k (n) n/2 + ε, and take for B a domain of small volume such that λ

1 (B) n/2 > λ * k (n) n/2 + ε. The domain Ω = A ∪ B would have volume one and λ k (Ω) = λ k (A) < λ 1 (B).
Using similar arguments as in the proof of Theorem 2.2 (see also the proof of Theorem 3.2), we obtain the following Theorem 2.3. Let k ≥ 2 and assume that there exists a non connected domain

Ω = A ∪ B in R n with |A| + |B| = 1 and µ k (A ∪ B) n/2 ≥ µ * k (n) n/2 -ε (15)
for some ε ≥ 0. Then there exists an integer i

∈ {1, • • • , k -1} such that 0 ≤ µ * k (n) n/2 -µ * i (n) n/2 + µ * k-i (n) n/2 ≤ ε, 0 ≤ µ * i (n) n/2 -µ i (A) n/2 |A| ≤ ε and 0 ≤ µ * k-i (n) n/2 -µ k-i (B) n/2 |B| ≤ ε. Remark 2.3. (i) Taking ε = 0 in Theorem 2.

3, all the inequalities of the theorem become equalities and the integer i is necessarily such that µ

* i (n) n/2 + µ * k-i (n) n/2 is maximal. (ii) A consequence of Theorem 2.3 is that if for some ε > 0, there exists a domain Ω in R n with µ k (Ω) n/2 > sup 1≤i≤k-1 µ * i (n) n/2 + µ * k-i (n) n/2 + ε,
then µ * k (n) cannot be approached up to ε by a non connected domain. The following properties are likely well known, we show them here for completeness and comparison with other results in this section. Proposition 2.1. For every n ≥ 2 and k ≥ 1 we have 19) Proof. To see [START_REF] Jakobson | Extremal metric for the first eigenvalue on a Klein bottle[END_REF] it suffices to consider a domain Ω modeled on the disjoint union of k + 1 identical balls of volume 1 k+1 . The k + 1 first Dirichlet eigenvalues of such a domain are almost equal. Now, take any domain D with λ k+1 (D)λ k (D) > 0 and observe that λ k+1 (tD)λ k (tD) → +∞ as t → 0. Then attach to the domain tD a sufficiently long and thin domain in order to obtain a volume 1 domain Ω(t) with λ k (Ω(t)) ≈ λ k (tD) and λ k+1 (Ω(t)) ≈ λ k+1 (tD) (recall that the first eigenvalue of a box of volume 1 goes to infinity as the length of one of its sides becomes very small). Thus, λ k+1 (Ω(t))λ k (Ω(t)) goes to infinity as t → 0 which proves [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF].

inf{λ k (Ω) -λ 1 (Ω) : Ω ⊂ R n , |Ω| = 1} = 0 ; ( 16
) sup{λ k+1 (Ω) -λ k (Ω) : Ω ⊂ R n , |Ω| = 1} = ∞ ; (17) inf{µ k (Ω) -µ 1 (Ω) : Ω ⊂ R n , |Ω| = 1} = 0 ; (18) µ * 1 (n)(k + 1) 2 n ≤ sup{µ k+1 (Ω) -µ k (Ω) : Ω ⊂ R n , |Ω| = 1} ≤ µ * k+1 (n). (
As for the Neumann eigenvalues of a domain Ω modeled on the disjoint union of k + 1 identical balls of volume 1 k+1 , one has µ 0 (Ω) = 0 and µ 1 (Ω), • • • , µ k (Ω) are almost equal to zero, while µ k+1 (Ω) is almost equal to the first positive eigenvalue of one of the balls, that is

µ k+1 (Ω) ≈ µ * 1 (n)(k + 1) 2 n
. This example proves ( 18) and ( 19).

Eigenvalues of closed surfaces

There are two equivalent approaches to introduce the extremal eigenvalues on closed surfaces.

Let us start with the "embedded" point of view. Indeed, if S is a compact connected surface of the 3-dimensional Euclidean space R 3 , we consider on it the Dirichlet's energy functional associated with the tangential gradient, and denote by

0 = ν 0 (S ) < ν 1 (S ) ≤ ν 2 (S ) ≤ • • • ≤ ν k (S ) ≤ • • • .
the spectrum of the corresponding Laplacian. According to [START_REF] Colbois | Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds[END_REF]Theorem 1.4], one has, ∀k ≥ 1, sup

|S |=1 ν k (S ) = +∞.
However, it is known since the work of Korevaar [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF] that for every integer γ ≥ 0, the k-th eigenvalue ν k is bounded above on the set of compact surfaces of genus γ. Thus, for every integer γ ≥ 0 we denote by M(γ) the set of all compact surfaces of genus γ embedded in R 3 and define the sequence

ν * k (γ) = sup {ν k (S ) ; S ∈ M(γ) and |S | = 1} = sup S ∈M(γ) ν k (S )|S |,
where |S | stands for the area of S . Regarding the infimum, it is well known

that inf S ∈M(γ) ν k (S )|S | = 0.
Alternatively, let Σ γ be an abstract closed orientable 2-dimensional smooth manifold of genus γ. To every Riemannian metric g on Σ γ we associate the sequence of eigenvalues of the Laplace-Beltrami operator

∆ g 0 = ν 0 (Σ γ , g) < ν 1 (Σ γ , g) ≤ ν 2 (Σ γ , g) ≤ • • • ≤ ν k (Σ γ , g) ≤ • • • .
Notice that for every positive number t, one has ν k (Σ γ , tg) = t -1 ν k (Σ γ , g) while the Riemannian area satisfies |(Σ γ , tg)| = t|(Σ γ , g)| so that the product ν k (Σ γ , g)|(Σ γ , g)| is invariant under scaling of the metric. Lemma 3.1. Let Σ γ be a closed orientable 2-dimensional smooth manifold of genus γ ≥ 0 and denote by R(Σ γ ) the set of all Riemannian metrics on Σ γ . For every positive integer k one has

ν * k (γ) = sup ν k (Σ γ , g) ; g ∈ R(Σ γ ) and |(Σ γ , g)| = 1 = sup g∈R(Σ γ ) ν k (Σ γ , g)|(Σ γ , g)|.
Proof. Let us first recall the well-known fact (see e.g. Dodziuk's paper [START_REF] Dodziuk | Eigenvalues of the Laplacian on forms[END_REF]) that if two Riemannian metrics g 1 , g 2 on a compact manifold M of dimension m are quasi-isometric with a quasi-isometry ratio close to 1, then the spectra of their Laplacians are close. More precisely, we say that g 1 and g 2 are α-quasi-isometric, with α ≥ 1, if for each v ∈ T M, v 0, we have

α 2 ≤ g 1 (v, v) g 2 (v, v) ≤ α 2 .
The spectra of g 1 and g 2 then satisfy, ∀k ≥ 1,

1 α 2(m+1) ≤ ν k (M, g 1 ) ν k (M, g 2 ) ≤ α 2(m+1) ( 20
)
while the ratio of their volumes is so that

1 α m ≤ |(M, g 1 )| |(M, g 2 )| ≤ α m . ( 21
)
Now, any surface S ∈ M(γ) is of the form S = φ(Σ γ ), where φ : Σ γ → R 3 is a smooth embedding. Denoting by g φ the Riemannian metric on Σ γ defined as the pull back by φ of the Euclidean metric of R 3 , one clearly has

ν k (S ) = ν k (Σ γ , g φ ) and |S | = |(Σ γ , g φ )|.
This immediately shows that

ν * k (γ) ≤ sup g∈R(Σ γ ) ν k (Σ γ , g)|(Σ γ , g)|.
Conversely, given any Riemannian metric g ∈ R(Σ γ ), it is well known that there exists a C 1 -isometric embedding φ from (Σ γ , g) into R 3 (see [START_REF] Nicolaas | On C 1 -isometric imbeddings[END_REF]). Using standard density results, there exists a sequence φ n : Σ γ → R 3 of smooth embeddings that converges to φ with respect to the C 1 -topology. The metrics g n = g φ n induced by φ n are quasi-isometric to g and the corresponding sequence of quasi-isometry ratios converges to 1. Therefore, using ( 20) and ( 21

), lim n ν k (Σ γ , g n ) = ν k (Σ γ , g) and lim n |(Σ γ , g n )| = |(Σ γ , g)|.
Hence, the sequence of surfaces

S n = φ n (Σ γ ) ∈ M(γ) satisfies lim n ν k (S n )|S n | = lim n ν k (Σ γ , g n )|(Σ γ , g n )| = ν k (Σ γ , g)|(Σ γ , g)|.
This completes the proof of the Lemma.

It is known that ν * 1 (0) = ν 1 (S 2 , g s ) = 8π, where g s is the standard metric of the sphere (see [START_REF] Hersch | Quatre propriétés isopérimétriques de membranes sphériques homogènes[END_REF]),

ν * 1 (1) = ν 1 (T 2 , g hex ) = 8π 2 √ 3
, where g hex is the flat metric on the torus associated with the hexagonal lattice (see [START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF]), and ν * 2 (0) = 16π (see [START_REF] Nadirashvili | Isoperimetric inequality for the second eigenvalue of a sphere[END_REF]). Moreover, one has the following inequality (see [START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF][START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF])

ν * 1 (γ) ≤ 8π γ + 3 2 ,
where ⌊•⌋ denotes the floor function. Recently, A. Hassannezhad [START_REF] Hassannezhad | Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem[END_REF] proved that there exist universal constants A > 0 and

B > 0 such that, ∀(k, γ) ∈ N 2 , ν * k (γ) ≤ Aγ + Bk.
On the other hand, ν * k (γ) admits also a lower bound in terms of a linear function of γ and k as shown in our previous work [START_REF] Colbois | Extremal eigenvalues of the Laplacian in a conformal class of metrics: the 'conformal spectrum[END_REF] where we have also proved that ν * k (γ) is nondecreasing with respect to γ. Theorem 3.1. Let γ ≥ 0 and k ≥ 1 be two integers and let γ 1 . . . , γ p ∈ N and i 1 , . . . , i p ∈ N * be such that

γ 1 + • • • + γ p = γ and i 1 + • • • + i p = k. Then ν * k (γ) ≥ ν * i 1 (γ 1 ) + • • • + ν * i p (γ p ). ( 22
)
If the equality holds in [START_REF] Li | On the Schrödinger equation and the eigenvalue problem[END_REF], then, for every ε > 0, there exist p compact orientable surfaces S 1 , • • • , S p of genus γ 1 . . . , γ p , respectively, such that

i) ν k (S 1 ⊔ • • • ⊔ S p ) ≥ (1 -ε)ν * k (γ) ; ii) ∀ j ≤ p, (1 -ε)ν * i j (γ j ) ≤ ν i j (S j )|S j | ≤ ν * i j (γ j ) ; iii) |S 1 | + • • • + |S p | = 1 and, ∀ j ≤ p, ν * i j (γ j ) (1+ε)ν * k (γ) ≤ |S j | ≤ (1+ε)ν * i j (γ j ) ν * k (γ)
.

Before giving the proof of this theorem we recall that if S 1 and S 2 are two closed orientable surfaces in R 3 , then the spectrum {ν k (S 1 ⊔ S 2 )} k≥0 of their disjoint union is given by the re-ordered union of the spectra of S 1 and S 2 (in particular, ν 0 (S 1 ⊔ S 2 ) = ν 1 (S 1 ⊔ S 2 ) = 0). The following lemma shows that this spectrum of S 1 ⊔ S 2 can be approximated, with arbitrary accuracy, by the spectrum of a closed connected orientable surface of genus γ = genus(S 1 ) + genus(S 2 ). Lemma 3.2. Let S 1 and S 2 be two closed surfaces in R 3 of genus γ 1 and γ 2 , respectively. There exists a 1-parameter family S δ ∈ M(γ) of closed surfaces of genus γ = γ 1 + γ 2 such that, for every k ≥ 0,

lim δ→0 ν k (S δ ) = ν k (S 1 ⊔ S 2 )
and lim

δ→0 |S δ | = |S 1 ⊔ S 2 |.
In particular, the definition of ν * k (γ) does not change if we include in M(γ) the disjoint unions of surfaces

S 1 ⊔• • •⊔S p with genus(S 1 )+• • •+genus(S p ) = γ.
Proof of Lemma 3.2. We denote by g 1 and g 2 the Riemannian metrics induced on S 1 and S 2 , respectively. In what follows, we will show how to construct a 1-parameter family g δ of Riemannian metrics on the connected sum S of S 1 and S 2 so that lim δ→0 ν k (S , g δ ) = ν k (S 1 ⊔ S 2 ) and lim δ→0 |(S , g δ )| = |S 1 ⊔ S 2 |. Using arguments as in the proof of Lemma 3.1, we easily see that this family of Riemannian surfaces (S , g δ ) gives rise to a family of embedded surfaces S δ ∈ M(γ 1 + γ 2 ) which satisfies the conditions of the statement. For the sake of clarity we divide the proof into several steps.

Step 1 : Let x 1 ∈ S 1 and x 2 ∈ S 2 be two arbitrary points. For any sufficiently small δ > 0, Lemma 2.3 of [START_REF] Colbois | Extremal eigenvalues of the Laplacian in a conformal class of metrics: the 'conformal spectrum[END_REF] tells us that the metrics g 1 and g 2 of S 1 and S 2 are (1 + δ)-quasi-isometric to other metrics g 1,δ and g 2,δ which are Euclidean around x 1 and x 2 . As in the proof of Lemma 3.1, we use [START_REF] Nicolaas | On C 1 -isometric imbeddings[END_REF] to deduce that lim δ→0 ν k (S i , g i,δ ) = ν k (S i , g i ) and, consequently,

lim δ→0 ν k (S 1 , g 1,δ ) ⊔ (S 2 , g 2,δ ) = ν k ((S 1 , g 1 ) ⊔ (S 2 , g 2 )) . ( 23
)
Step 2 : Let (S , g) be a Riemannian surface which is flat around a point x ∈ S . For every sufficiently small ε > 0, the metric g can be deformed in the complement of the geodesic ball of radius ε into a metric g ε which is (1 + 2ε)-quasi-isometric to g and so that the geodesic annulus A(x, ε, ε + ε 2 ) centered at x with inner and outer radii ε and ε + ε 2 , is isometric to the cylinder S 1 ε × (ε, ε + ε 2 ), where S 1 ε is the circle of radius ε.

Indeed, let us choose ε so that g is flat in the geodesic ball B(x, 2ε) of radius 2ε centered at x that we identify with the Euclidean ball B(O, 2ε) ⊂ R 2 . Using polar coordinates, we may write g = dr 2 + r 2 dθ 2 with r ≤ 2ε and θ ∈ [0, 2π]. We consider the family g ε of metrics on S which coincide with g in the complement of the annulus A(x, ε, 2ε) and whose restriction to this annulus (identified with A(0, ε, 2ε) ⊂ R 2 ) is given by

g ε (r, θ) = dr 2 + ψ 2 ε (r)dθ 2 , with ψ ε (r) = ε if ε ≤ r ≤ ε + ε 2 , ψ(r) = r if ε + 2ε 2 ≤ r ≤ 2ε, and ε ≤ ψ ε (r) ≤ ε + ε 2 if r ∈ (ε + ε 2 , ε + 2ε 2 )
. Notice that we do not need to define ψ ε more explicitly since only ψ ε will be used and not its derivatives.

On the annulus A(0, ε, ε + ε 2 ) the metric g ε coincides with the cylindrical metric

dr 2 + ε 2 dθ 2 , that is A(x, ε, ε + ε 2 ) is isometric to S 1 ε × (ε, ε + ε 2 )
. On the other hand, the metric g ε is clearly quasi-isometric to the Euclidean metric g = dr 2 + r 2 dθ 2 on A(0, ε, 2ε) with min 1, 2 . Since g ε coincides with g in the complement of A(x, ε, 2ε), the metric g ε is in fact globally (1 + 2ε)-quasi-isometric to g.

ψ 2 ε (r) r 2 g ≤ g ε ≤ max 1, ψ 2 ε (r) r 2 g. From the definition of ψ ε one has, ∀r ∈ (ε, 2ε), 1 (1 + 2ε) 2 ≤ ψ 2 ε (r) r 2 ≤ (1 + 2ε)
Step 3 : Construction of the family of metrics g δ . Given a sufficiently small δ > 0, we first apply Step 1 and replace the metrics g 1 and g 2 by g 1,δ and g 2,δ so that, for each i = 1, 2, (S i , g i,δ ) is flat around a point x i ∈ S i . Thanks to Step 2, for every positive ε ε 0 (δ), we define on S i a metric g i,δ,ε which is (1 + 2ε)-quasi-isometric to g and so that the geodesic annulus A(x i , ε, ε + ε 2 ) is isometric to the cylinder S 1 ε × (ε, ε + ε 2 ). Thus, one can smoothly glue (S 1 \ B(x 1 , ε), g 1,δ,ε ) and (S 2 \ B(x 2 , ε), g 2,δ,ε ) along their boundaries and obtain a smooth Riemannian surface (S , g δ,ε ) of genus γ = γ 1 + γ 2 .

Let us denote by λ k (δ, ε) (resp. µ k (δ, ε)) the eigenvalues of the disjoint union of (S 1 \ B(x 1 , ε), g 1,δ ) and (S 2 \ B(x 2 , ε), g 2,δ ) with Dirichlet (resp. Neumann) boundary conditions. Similarily, we denote by λk (δ, ε) (resp. μk (δ, ε)) the eigenvalues of the disjoint union of (S 1 \ B(x 1 , ε), g 1,δ,ε ) and (S 2 \ B(x 2 , ε), g 2,δ,ε ) with Dirichlet (resp. Neumann) boundary conditions. From the min-max principle we have the following inequalities:

μk (δ, ε) ≤ ν k (S , g δ,ε ) ≤ λk (δ, ε).
Moreover, since g i,δ,ε is (1 + 2ε)-quasi-isometric to g i,δ , one has using [START_REF] Nicolaas | On C 1 -isometric imbeddings[END_REF],

(1 + 2ε) -6 µ k (δ, ε) ≤ μk (δ, ε) and λk (δ, ε) ≤ (1 + 2ε) 6 λ k (δ, ε). Therefore, (1 + 2ε) -6 µ k (δ, ε) ≤ ν k (S , g δ,ε ) ≤ (1 + 2ε) 6 λ k (δ, ε).
On the other hand, according to [START_REF] Anné | Spectre du laplacien et écrasement d'anses[END_REF], λ k (δ, ε) (resp. µ k (δ, ε)) converges as ε → ∞, to the k-th eigenvalue of the disjoint union of (S 1 , g 1,δ ) and (S 2 , g 2,δ ). Thus, for every k ≥ 0,

lim ε→0 ν k (S , g δ,ε ) = ν k (S 1 , g 1,δ ) ⊔ (S 2 , g 2,δ ) .
In particular, there exists ε(δ) > 0 such that, for every k ≤

1 δ , |ν k (S , g δ,ε(δ) ) -ν k (S 1 , g 1,δ ) ⊔ (S 2 , g 2,δ ) | < δ.
Thus, if we set g δ = g δ,ε(δ) , then using the last inequality and ( 23), we will have, for every k ≥ 0,

lim δ→0 ν k (S , g δ ) = ν k ((S 1 , g 1 ) (S 2 , g 2 )) .
As for the area, from the construction of g δ , it is clear that |(S , g δ )| tends to

|S 1 | + |S 2 | as δ → 0.
Proof of Theorem 3.1. Let ε be any positive real number and let S 1 , • • • , S p be a family of compact orientable surfaces such that, for each positive j ≤ p, genus(S j ) = γ j and ν i j (S j )|S j | > ν * i j (γ j )ε. After rescaling, we may assume that

ν i j (S j ) = ν * k (γ) and |S j | > ν * i j (γ j ) -ε ν * k (γ)
.

One has, using arguments as in the proof of Theorem 2.1,

# l ∈ N ; ν l (S 1 ⊔ • • • ⊔ S p ) < ν * k (γ) = p j=1 # l ∈ N ; ν l (S j ) < ν * k (γ) ≤ p j=1 i j = k. Consequently, ν k (S 1 ⊔ • • • ⊔ S p ) ≥ ν * k (γ)
. From Lemma 3.2 and the definition of ν * k (γ), one then deduces the following:

|S 1 ⊔ • • • ⊔ S p | = |S 1 | + • • • + |S p | ≤ 1.
This leads to

p j=1 ν * i j (γ j ) -ε ν * k (γ) ≤ 1, that is, p j=1 ν * i j (γ j ) ≤ ν * k (γ) + pε.
This proves the inequality ( 22) since ε can be chosen arbitrarily small. Assume that the equality holds in [START_REF] Li | On the Schrödinger equation and the eigenvalue problem[END_REF] . We can follow the same arguments as in the proof of Theorem 2.1 and conclude. Remark 3.1. A direct consequence of Theorem 3.1 is that, for every γ ≥ 0 and every k ≥ 1, one has

ν * k (γ) ≥ sup i≤k-1 ν * i (γ) + ν * k-i (0) .
In particular, ν * k (γ) ≥ ν * k-1 (γ) + 8π. Therefore, Theorem 3.1 improves our previous results (Theorem C and Corollary 4 of [START_REF] Colbois | Extremal eigenvalues of the Laplacian in a conformal class of metrics: the 'conformal spectrum[END_REF]).

The following theorem deals with the situation where ν * k (γ) is approached by the k-th eigenvalue of a disjoint union of two surfaces. Theorem 3.2. Let γ ≥ 0 and k ≥ 2 be two integers and assume that there exist two compact orientable surfaces S 1 and S 2 of genus γ 1 , γ 2 , respectively, such that

|S 1 | + |S 2 | = 1, γ 1 + γ 2 = γ, and ν k (S 1 ⊔ S 2 ) ≥ ν * k (γ) -ε (24)
for some ε ≥ 0. Then there exists an integer

i ∈ {1, • • • , k -1} such that 0 ≤ ν * k (γ) -ν * i (γ 1 ) + ν * k-i (γ 2 ) ≤ ε, 0 ≤ ν * i (γ 1 ) -ν i (S 1 )|S 1 | ≤ ε and 0 ≤ ν * k-i (γ 2 ) -ν k-i (S 2 )|S 2 | ≤ ε. Proof.
Since the spectrum of S 1 ⊔ S 2 is the re-ordrered union of the spectra of S 1 and S 2 , the eigenvalue ν k (S 1 ⊔S 2 ) belongs to this union and, moreover,

# j ∈ N ; ν j (S 1 ) < ν k (S 1 ⊔ S 2 ) + # j ∈ N ; ν j (S 2 ) < ν k (S 1 ⊔ S 2 ) ≤ k (25) and # j ∈ N ; ν j (S 1 ) ≤ ν k (S 1 ⊔ S 2 ) + # j ∈ N ; ν j (S 2 ) ≤ ν k (S 1 ⊔ S 2 ) ≥ k+1 ( 26
) (recall that the numbering of the eigenvalues start from zero). Hence, there exists at least one integer j ∈ {1, . . . , k} such that ν j (S 1 ) = ν k (S 1 ⊔ S 2 ) or ν j (S 2 ) = ν k (S 1 ⊔ S 2 ). Assume that the first alternative occurs and let i be the least positive integer such that ν i (S 1 ) = ν k (S 1 ⊔ S 2 ). We necessarily have ν k-i (S 2 ) ≥ ν k (S 1 ⊔ S 2 ) since, otherwise, the k + 1 eigenvalues ν 0 (S 1 ), • • • , ν i-1 (S 1 ) and ν 0 (S 2 ), • • • , ν k-i (S 1 ) would be strictly less than ν k (S 1 ⊔ S 2 ) which contradicts [START_REF] Poliquin | Wolf-keller theorem for neumann eigenvalues[END_REF]. Thus, i ≤ k -1 and Extremal eigenvalues of nonorientable surfaces.

ν k (S 1 ⊔ S 2 ) = ν k (S 1 ⊔ S 2 )|S 1 | + ν k (S 1 ⊔ S 2 )|S 2 | ≤ ν i (S 1 )|S 1 | + ν k-i (S 2 )|S 2 |. ( 27
In the non-orientable case, we can similarly define, for every γ ∈ N and every k ∈ N, the number ν * ,k (γ) as the supremum of ν k (S )|S | over compact non-orientable surfaces of genus γ.

We have ν * ,1 (1) = ν 1 (RP 2 , g s ) = 12π where g s is the standard metric of the projective plane (see [START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF]), and ν * ,1 (2) = ν 1 (K 2 , g 0 ) = 12πE(2 √ 2/3) ≃ 13.365 π, where g 0 is a non flat metric of revolution on the Klein bottle and E(2 √ 2/3) is the complete elliptic integral of the second kind evaluated at 2 √ 2 3 (see [START_REF] Soufi | A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle[END_REF]). Moreover, one has the following inequalities (see [START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF][START_REF] Soufi | Le volume conforme et ses applications d'après Li et Yau[END_REF])

ν * ,1 (γ) ≤ 24π γ + 3 2 ,
where ⌊•⌋ denotes the floor function. The same reasoning as in the orientable case leads to the following results : ; iii) ∀ j ≤ p, ν * ,i j (γ j ) ≤ ν i j (S j )|S j | 2/n ≤ (1 + ε)ν * ,i j (γ j ).

Corollary 3 . 1 .

 31 ) Since ν i (S 1 )|S 1 | ≤ ν * i (γ 1 ) and ν k-i (S 2 )|S 2 | ≤ ν * k-i (γ 2 ), we get ν * k (γ)ε ≤ ν k (S 1 ⊔ S 2 ) ≤ ν * i (γ 1 ) + ν * k-i (γ 2 ). Now, ν i (S 1 )|S 1 | ≥ ν * i (γ 1 )ε.Otherwise, we would have, thanks to (27) and Theorem 3.1,ν k (S 1 ⊔ S 2 ) ≤ ν i (S 1 )|S 1 | + ν k-i (S 2 )|S 2 | < ν * i (γ 1 )ε + ν * k-i (γ 2 ) ≤ ν * k (γ)ε which contradicts the assumption of the theorem. The same argument leads to the inequalityν k-i (S 2 )|S 2 | ≥ ν k-i (n) + ε.As a consequence of Theorem 3.2, we obtain the following Wolf-Keller type result. Let γ ≥ 0 and k ≥ 2 be two integers and assume that there exist two compact orientable surfaces S 1 and S 2 of genus γ 1 , γ 2 , respectively, such that|S 1 | + |S 2 | = 1, γ 1 + γ 2 = γ, and ν k (S 1 ⊔ S 2 ) = ν * k (γ). (28)Then there exists an integer i∈ {1, • • • , k -1} such that ν * k (γ) = ν * i (γ 1 ) + ν * k-i (γ 2 ) = sup j=1,••• ,k-1 ν * j (γ 1 ) + ν * k-j (γ 2 ) ν i (S 1 )|S 1 | = ν * i (γ 1 ) and ν k-i (S 2 )|S 2 | = ν * k-i (γ 2 ).

Theorem 3 . 3 .

 33 Let γ ≥ 0 and k ≥ 1 be two integers and let γ 1 . . . , γ p and i 1 , . . . , i p be such thatγ 1 + • • • + γ p = γ and i 1 + • • • + i p = k. Then ν * ,k (γ) ≥ ν * ,i 1 (γ 1 ) + • • • + ν * ,i p (γ p ). (29)If the equality holds in[START_REF] Li | On the Schrödinger equation and the eigenvalue problem[END_REF], then, for every ε > 0, there exist p compact orientable surfaces S 1 , • • • , S p of genus γ 1 . . . , γ p , respectively, such thati) ν k (S 1 ⊔ • • • ⊔ S p ) ≤ (1 + ε)ν * ,k (γ) ; ii) |S 1 | + • • • + |S p | = 1 and, ∀ j ≤ p, ν * ,i j (γ j ) (1+ε)ν * ,k (γ) ≤ |S j | ≤ (1+ε)ν * ,i j (γ j ) ν * ,k (γ)

  2.1 and Fekete's SubadditiveLemma is that the sequences λ *

	k (n)/k	2 n and µ * k (n)/k	2 n are convergent and that
	Pólya's conjecture for Dirichlet (resp. Neumann) eigenvalues is equivalent
	to the following
	lim

  denotes the k-th eigenvalue of the Laplacian in Ω with Dirichlet (resp. Neumann) boundary conditions on ∂Ω. If t is a positive number, the notation t Ω will designate the image of the domain Ω under the Euclidean dilation of ratio t. One has

  the inequalities of Corollary 2.1 lead to

	λ * k+1 (2) -λ * k (2) ≤ π j 2 0,1 ≈ 18.168
	and
	µ * k+1 (2) -µ * k (2) ≥ πp 2 1,1 ≈ 10.65, which provides a simple tool to test the accuracy of numerical approxima-
	tions.
	(iii) Iterating the inequalities of Corollary 2.1 we get

The second author has benefited from the support of the ANR (Agence Nationale de la Recherche) through FOG project ANR-07-BLAN-0251-01.


Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Eighth order Peregrine breather solution of the NLS equation and their deformations with fourteen parameters

We construct new families of quasi-rational solutions of the NLS equation of order 8 with 14 real parameters. We obtain new patterns of different types of rogue waves. We recover the triangular configurations as well as isolated rings as found for the lower orders. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.

Introduction

From the pioneer work of Zakharov and Shabat in 1972 [START_REF] Zakharov | Exact theory of two dimensional self focusing and one dimensinal self modulation of waves in nonlinear media[END_REF], and the first expressions of the quasi-rational solutions given by Peregrine in 1983 [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF], a considerable number of studies were carried out. Akhmediev, Eleonski and Kulagin constructed the first higher order analogue of the Peregrine breather [START_REF] Akhmediev | Generation of periodic trains of picosecond pulses in an optical fiber : exact solutions[END_REF][START_REF] Akhmediev | Exact first order solutions of the nonlinear Schrödinger equation[END_REF] in 1986. Akhmediev et al. [START_REF] Akhmediev | Rogue waves and rational solutions of nonlinear Schrödinger equation[END_REF][START_REF] Akhmediev | Rogue waves, rational solutions, the patterns of their zeros and integral relations[END_REF], constructed other families of order 3 and 4, using Darboux transformations. In 2010, rational solutions of the NLS equation have been written as a quotient of two Wronskians in [START_REF] Dubard | On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation[END_REF]. In 2011, an other representation of the solutions of the NLS equation has been constructed in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF], also in terms of a ratio of two Wronskians determinants of order 2N .

In 2012, the solutions of the focusing NLS equation has been written as a ratio of two determinants in [START_REF] Guo | Nonlinear Schrodinger equation: Generalized Darboux transformation and rogue wave solutions[END_REF] using generalized Darboux transform. Ohta and Yang [START_REF] Ohta | General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation[END_REF] have given an other representation of solutions of the focusing NLS equation by means of determinants, obtained from Hirota bilinear method. A the beginning of the year 2012, the author obtained a representation in terms of determinants which does not involve limits [START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF]. These first two formulations given in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF][START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF] did depend in fact only on two parameters; this remark was first pointed out by Matveev. In 2012 Matveev and Dubard obtained for the first time, with the approach defined in [START_REF] Dubard | On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation[END_REF], the solutions of NLS explicitly, for the order 3 depending on 4 parameters and those for the order 4 depending on 6 parameters [START_REF] Dubard | preprint RIMS-1777[END_REF]. Then the author found for the order N (for determinants of order 2N), solutions depending on 2N -2 real parameters. With this new method, we construct here news deformations at order 8 with 14 real parameters.

Determinant representation of solutions of NLS equation

2.1 Quasi-rational limit solutions of the NLS equation

We recall the results obtained in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF]. We consider the focusing NLS equation

iv t + v xx + 2|v| 2 v = 0. (1)
In the following, we consider 2N parameters λ ν , ν = 1, . . . , 2N satisfying the relations

0 < λ j < 1, λ N +j = -λ j , 1 ≤ j ≤ N. (2)
We define the terms κ ν , δ ν , γ ν by the following equations,

κ ν = 2 1 -λ 2 ν , δ ν = κ ν λ ν , γ ν = 1 -λ ν 1 + λ ν , (3)
and

κ N +j = κ j , δ N +j = -δ j , γ N +j = 1/γ j , j = 1 . . . N. (4)
The terms x r,ν (r = 3, 1) are defined by

x r,ν = (r -1) ln γ ν -i γ ν + i , 1 ≤ j ≤ 2N. (5)
The parameters e ν are defined by

e j = ia j -b j , e N +j = ia j + b j , 1 ≤ j ≤ N, (6)
where a j and b j , for 1 ≤ j ≤ N are arbitrary real numbers. The terms ǫ ν are defined by :

ǫ ν = 0, 1 ≤ ν ≤ N ǫ ν = 1, N + 1 ≤ ν ≤ 2N.
We use the following notations :

Θ r,ν = κ ν x/2 + iδ ν t -ix r,ν /2 + γ ν y -ie ν , 1 ≤ ν ≤ 2N .
We consider the functions

φ r,ν (y) = sin Θ r,ν , 1 ≤ ν ≤ N, φ r,ν (y) = cos Θ r,ν , N + 1 ≤ ν ≤ 2N. ( 7
) W r (y) = W (φ r,1 , . . . , φ r,2N ) is the Wronskian W r (y) = det[(∂ µ-1 y φ r,ν ) ν, µ∈[1,...,2N ] ]. (8)
Then we get the following statement [START_REF] Gaillard | Wronskian representation of solutions of the NLS equation and higher Peregrine breathers[END_REF] Theorem 2.1 The function v defined by

v(x, t) = W 3 (0) W 1 (0) exp(2it -iϕ). ( 9
)
is solution of the NLS equation ( 1)

iv t + v xx + 2|v| 2 v = 0.
To obtain quasi-rational solutions of the NLS equation, we take the limit when the parameters

λ j → 1 for 1 ≤ j ≤ N and λ j → -1 for N +1 ≤ j ≤ 2N .
Fot that, we consider the parameter λ j written in the form

λ j = 1 -2j 2 ǫ 2 , 1 ≤ j ≤ N. (10)
When ǫ goes to 0, we obtain quasi-rational solutions of the NLS equation given by :

Theorem 2.2 The function v defined by v(x, t) = exp(2it -iϕ) lim ǫ→0 W 3 (0) W 1 (0) , (11)
is a quasi-rational solution of the NLS equation ( 1)

iv t + v xx + 2|v| 2 v = 0.

Expression of solutions of NLS equation in terms of a ratio of two determinants

We construct the solutions of the NLS equation expressed as a quotient of two determinants which does not involve a passage to the limit. We use the following notations :

A ν = κ ν x/2 + iδ ν t -ix 3,ν /2 -ie ν /2, B ν = κ ν x/2 + iδ ν t -ix 1,ν /2 -ie ν /2, (12)
for 1 ≤ ν ≤ 2N , with κ ν , δ ν , x r,ν defined in (3), ( 4) and ( 5).

The parameters e ν are defined by [START_REF] Dubard | On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation[END_REF].

Here, the parameters a j and b j , for 1 ≤ N are chosen in the form

a j = N -1 k=1 ãk ǫ 2k+1 j 2k+1 , b j = N -1 k=1 bk ǫ 2k+1 j 2k+1 , 1 ≤ j ≤ N. (13)
We consider the following functions :

f 4j+1,k = γ 4j-1 k sin A k , f 4j+2,k = γ 4j k cos A k , f 4j+3,k = -γ 4j+1 k sin A k , f 4j+4,k = -γ 4j+2 k cos A k , (14)
for 1 ≤ k ≤ N , and

f 4j+1,N +k = γ 2N -4j-2 k cos A N +k , f 4j+2,N +k = -γ 2N -4j-3 k sin A N +k , f 4j+3,N +k = -γ 2N -4j-4 k cos A N +k , f 4j+4,N +k = γ 2N -4j-5 k sin A N +k , (15)
for 1 ≤ k ≤ N . We define the functions g j,k for 1 ≤ j ≤ 2N , 1 ≤ k ≤ 2N in the same way, we replace only the term A k by B k .

g 4j+1,k = γ 4j-1 k sin B k , g 4j+2,k = γ 4j k cos B k , g 4j+3,k = -γ 4j+1 k sin B k , g 4j+4,k = -γ 4j+2 k cos B k , (16)
for 1 ≤ k ≤ N , and

g 4j+1,N +k = γ 2N -4j-2 k cos B N +k , g 4j+2,N +k = -γ 2N -4j-3 k sin B N +k , g 4j+3,N +k = -γ 2N -4j-4 k cos B N +k , g 4j+4,N +k = γ 2N -4j-5 k sin B N +k , (17)
for 1 ≤ k ≤ N . Then we get the following result :

Theorem 2.3 The function v defined by v(x, t) = det((n jk) j,k∈[1,2N ] ) det((d jk) j,k∈[1,2N ] ) e 2it-iϕ (18)
is a quasi-rational solution of the NLS equation ( 1)

iv t + v xx + 2|v| 2 v = 0,
depending on 2N -2 parameters ãj , bj , 1 ≤ j ≤ N -1,
where

n j1 = f j,1 (x, t, 0), n jk = ∂ 2k-2 f j,1 ∂ǫ 2k-2 (x, t, 0), n jN +1 = f j,N +1 (x, t, 0), n jN +k = ∂ 2k-2 f j,N +1 ∂ǫ 2k-2 (x, t, 0), d j1 = g j,1 (x, t, 0), d jk = ∂ 2k-2 g j,1 ∂ǫ 2k-2 (x, t, 0), d jN +1 = g j,N +1 (x, t, 0), d jN +k = ∂ 2k-2 g j,N +1 ∂ǫ 2k-2 (x, t, 0), 2 ≤ k ≤ N, 1 ≤ j ≤ 2N (19)
The functions f and g are defined in ( 14),( 15), ( 16), (17).

We don't have the space to give the proof in this publication. We will give it in an other forthcoming paper. The solutions of the NLS equation can also be written in the form :

v(x, t) = exp(2it -iϕ) × Q(x, t)
where Q(x, t) is defined by :

Q(x, t) := f 1,1 [0] . . . f 1,1 [N -1] f 1,N +1 [0] . . . f 1,N +1 [N -1] f 2,1 [0] . . . f 2,1 [N -1] f 2,N +1 [0] . . . f 2,N +1 [N -1] . . . . . . . . . . . . . . . . . . f 2N,1 [0] . . . f 2N,1 [N -1] f 2N,N +1 [0] . . . f 2N,N +1 [N -1] g 1,1 [0] . . . g 1,1 [N -1] g 1,N +1 [0] . . . g 1,N +1 [N -1] g 2,1 [0] . . . g 2,1 [N -1] g 2,N +1 [0] . . . g 2,N +1 [N -1] . . . . . . . . . . . . . . . . . . g 2N,1 [0] . . . g 2N,1 [N -1] g 2N,N +1 [0] . . . g 2N,N +1 [N -1] ( 20
)
3 Quasi-rational solutions of order 8 with twelve parameters

Wa have constructed in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF] solutions for the cases from N = 1 until N = 7, and in [START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF] with two parameters. We don't give the analytic expression of the solution of NLS equation with fourteen parameters because of the length of the expression. When all the parameters a j and b j , 1 ≤ j ≤ 7 are equal to 0, we get the analogue of the Peregrine breather of order 8. For parameters non equal to 0, we construct figures to show deformations of the eighth Peregrine breather.

We get different types of symmetries in the plots in the (x, t) plane. We give some examples of this fact in the following.

If we choose ãi = bi = 0 for 1 ≤ i ≤ , we obtain the classical Peregrine breather : With other choices of parameters, we obtain all types of configurations : triangles and multiple concentric rings configurations with a maximum of 36 peaks.

In the case of the variation of one parameter, we obtain different types of configurations with a maximum of 36 peaks.

In the cases a 1 = 0 or b 1 = 0 we obtain triangles with a maximum of 36 peaks; for a 2 = 0 or b 2 = 0, we have 5 rings with respectively 7, 7, 8, 8, 5 peaks with in the center one peak. For a 3 = 0 or b 3 = 0, we obtain 5 rings with 7 peaks on each of them with a central peak. For a 4 = 0 or b 4 = 0, we have 4 rings with 9 peaks on each of them without central peak. For a 5 = 0 or b 5 = 0, we have 3 rings of 11 peaks with in the center the Peregrine of order 2. For a 6 = 0 or b 6 = 0, we have 2 rings with 13 peaks and in the center the Peregrine breather of order 4. For a 7 = 0 or b 7 = 0, we have one ring with 15 peaks and in the center the Peregrine breather of order 6.

Conclusion

In the present paper we have constructed solutions of the NLS equation of order N = 8 with 2N -2 = 14 real parameters. The explicit representation in terms of polynomials in x and t are too monstrous to be published. We can't give in this text. By different choices of these parameters, we obtained new patterns in the (x; t) plane; we recognized ring shape as already observed in the case of deformations depending on two parameters [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF][START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF]. We get news triangular shapes and multiple concentric rings. From the studies already carried out for the order 1 until the order 8, it would be important to obtain a classification for the order N in general as already done for orders 1 -6 by Akhmediev et al. in [START_REF] Kedziora | Classifying the hierarchy of the nonlinear Schrödinger equation rogue waves solutions[END_REF]. Research on this subject should be done in the next years.

Figure 1 :

 1 Figure 1: Solution of NLS, N=8; left : all parameters equal to 0, the Peregrine P 8 ; right : a regular triangle with 36 peaks.

Figure 2 :

 2 Figure 2: Solution of NLS, N=8, ã2 = 10 6 : 5 rings with respectively 7, 7, 8, 8, 5 peaks with in the center one peak; on the right, sight of top.

Figure 3 :

 3 Figure 3: Solution of NLS, N=8, b2 = 10 6 : 5 rings with respectively 7, 7, 8, 8, 5 peaks with in the center one peak; on the right, sight of top.

Figure 4 :

 4 Figure 4: Solution of NLS, N=8, ã3 = 10 5 : 5 rings with 7 peaks on each of them with a central peak; on the right, sight of top.

Figure 5 :

 5 Figure 5: Solution of NLS, N=8, b3 = 10 5 : 5 rings with 7 peaks on each of them with a central peak; on the right, sight of top.

Figure 6 :

 6 Figure 6: Solution of NLS, N=8, ã4 = 10 10 : 4 rings with 9 peaks on each of them without central peak; on the right, sight of top.

Figure 7 :

 7 Figure 7: Solution of NLS, N=8, b4 = 10 10 : 4 rings with 9 peaks on each of them without central peak; on the right, sight of top.

Figure 8 :

 8 Figure 8: Solution of NLS, N=8, ã5 = 10 15 : 3 rings of 11 peaks with in the center the Peregrine of order 2; on the right, sight of top.

Figure 9 :

 9 Figure 9: Solution of NLS, N=8, b5 = 10 15 : 3 rings of 11 peaks with in the center the Peregrine of order 2; on the right, sight of top.

Figure 10 :

 10 Figure 10: Solution of NLS, N=8, ã6 = 10 20 : 2 rings with 13 peaks and in the center the Peregrine breather of order 4; on the right, sight of top.

Figure 11 :

 11 Figure 11: Solution of NLS, N=8, b6 = 10 20 : 2 rings with 13 peaks and in the center the Peregrine breather of order 4; on the right, sight of top.

Figure 12 :

 12 Figure 12: Solution of NLS, N=8, ã7 = 10 20 : one ring with 15 peaks and in the center the Peregrine breather of order 6; on the right, sight of top.

Figure 13 :

 13 Figure 13: Solution of NLS, N=8, b7 = 10 20 : one ring with 15 peaks and in the center the Peregrine breather of order 6; on the right, sight of top.