Corentin Léna

Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions

Keywords: Neumann eigenvalues, Robin eigenvalues, Nodal domains, Courant's theorem, Pleijel's theorem. MSC classification. Primary: 35P05; Secondary: 35P15, 35P20, 58J50

est destinée au dépôt

1 Introduction

Problem and results

The main objective of this paper is to extend Pleijel's nodal domain theorem to eigenfunctions of the Laplacian which satisfy a Neumann boundary condition. Let Ω ⊂ R n , with n ≥ 2, be a connected open set, which we assume to be bounded, with a sufficiently regular boundary. For technical reasons (appearing in Section 3), we ask for ∂Ω to be C 1,1 . In the rest of the paper, we denote by -∆ N Ω the self-adjoint realization of the (non-negative) Laplacian in Ω, with the Neumann boundary condition, and by (µ k (Ω)) k≥1 its eigenvalues, arranged in non-decreasing order and counted with multiplicities. Similarly, in the case of the Dirichlet boundary condition, we denote the self-adjoint realization of the Laplacian by -∆ D Ω and its eigenvalues by (λ k (Ω)) k≥1 . For any function f continuous in Ω, we call nodal set of f the closed set

N (f ) := {x ∈ Ω ; f (x) = 0}
and nodal domains the connected components of Ω \ N (f ). We denote by ν(f ) the cardinal of the set of nodal domains. We are interested in estimating ν(u) from above when u is an eigenfunction of -∆ N Ω . A fundamental result of this type was first obtained by R. [START_REF] Courant | Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke[END_REF] (see [START_REF] Courant | Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke[END_REF] or [START_REF] Courant | Methods of mathematical physics[END_REF]VI.6]).

Theorem 1.1. If k is a positive integer and u an eigenfunction associated with λ k (Ω) or µ k (Ω), ν(u) ≤ k.

Å. Pleijel showed in 1956 that, in the Dirichlet case, equality in the previous theorem can only occur for a finite number of eigenvalues. He originally proved it for domains in R 2 [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF]. The result was extended by J. Peetre in 1957 to some domains on two-dimensional Riemannian manifolds [START_REF] Peetre | A generalization of Courant's nodal domain theorem[END_REF], and a general version, valid for n-dimensional Riemannian manifolds with or without boundary, was obtained by P. Bérard and D. Meyer in 1982 [2, II.7]. In those three works, the authors actually proved a stronger result, in the form of an asymptotic upper bound. To state it, let us denote by ν D k (Ω) the largest possible value of ν(u), when u is an eigenvalue of -∆ D Ω associated with λ k (Ω). Let us also define

γ(n) := 2 n-2 n 2 Γ n 2 2 j n n 2 -1,1
, where j n 2 -1,1 is the smallest positive zero of the Bessel function of the first kind J n 2 -1 . We recall the inequality γ(n) < 1, proved in [2, II.9] (see also [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF]Section 5] for more precise results).

Theorem 1.2. If Ω ⊂ R n is
an open, bounded, and connected set which is Jordan measurable,

lim sup k→+∞ ν D k (Ω) k ≤ γ(n).
Theorem 1.2 is actually proved in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] for closed Riemannian manifolds, or Riemannian manifolds with smooth boundary in the Dirichlet case. However, the results in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] do not include the Neumann case. Let us note that the Jordan measurability is imposed in Theorem 1.2 so that the Weyl's asymptotics holds for the sequence (λ k (Ω)) k≥1 (see [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF]Section XIII.15]). The hypothesis that ∂Ω is C 1,1 would be more than sufficient.

The constant γ(n) has the following interpretation:

γ(n) = (2π) n λ 1 (B n ) n 2 ω n ,
where ω n is the volume of the unit ball in R n , and B n is a ball of volume 1 in R n . It was proved in [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF]Section 5] that the sequence (γ(n)) n≥2 is strictly decreasing and converges to 0 exponentially fast.

Since γ(n) < 1, Theorem 1.2 implies that there exists a finite smallest rank k D (Ω) ≥ 1 such that, for all k > k D (Ω), an eigenfunction of -∆ D Ω associated with λ k (Ω) as strictly less than k nodal domains. Two recent papers [START_REF] Bérard | The weak Pleijel theorem with geometric control[END_REF][START_REF] Van Den Berg | On the number of Courant-sharp Dirichlet eigenvalues[END_REF] give upper bounds of k D (Ω) in term of the geometry of Ω.

As in the Dirichlet case, let us denote by ν N k (Ω) the largest possible value for ν(u) when u is an eigenfunction of -∆ N Ω associated with µ k (Ω). The question of finding an asymptotic upper bound of ν N k (Ω) was already raised by Å. Pleijel, who showed that, for a square, the same upper bound as in the Dirichlet case holds true [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF]Section 7]. To our knowledge, the most general result known so far involving ν N k (Ω) has been obtained by I. Polterovich in 2009 [START_REF] Polterovich | Pleijel's nodal domain theorem for free membranes[END_REF]. His proof uses estimates by J. A. Toth and S. Zelditch [START_REF] Toth | Counting nodal lines which touch the boundary of an analytic domain[END_REF] of the number of nodal lines touching the boundary, and is therefore restricted to two-dimensional domains with quite regular boundaries.

Theorem 1.3. If Ω ⊂ R 2 is
an open, bounded, and connected set with a piecewise-analytic boundary,

lim sup k→+∞ ν N k (Ω) k ≤ γ(2) = 4 j 2 0,1 .
In this paper, we prove the following result, valid in any dimension.

Theorem 1.4. If Ω ⊂ R n , with n ≥ 2,
is an open, bounded, and connected set with a C 1,1 boundary,

lim sup k→+∞ ν N k (Ω) k ≤ γ(n).
Furthermore, our proof can be quite easily extended to some Robin-type boundary conditions. Let us be more specific: we assume that Ω satisfies the same hypotheses as above and that h is a Lipschitz function in Ω such that h ≥ 0 on ∂Ω. By analogy with the Dirichlet and Neumann cases, we denote by -∆ R,h Ω the self-adjoint realization of the Laplacian in Ω with the Robin boundary condition

∂u ∂ν + hu = 0 on ∂Ω.
Here ∂u ∂ν is the exterior normal derivative. We denote by (µ k (Ω, h)) k≥1 the eigenvalues of -∆ R,h Ω and by ν R k (Ω, h) the maximal number of nodal domains for an eigenfunction of -∆ R,h Ω associated with µ k (Ω, h). Let us note that in dimension 2, this eigenvalue problem gives the natural frequencies of a membrane elastically held at its boundary [24, 9.5]. The condition h ≥ 0 implies that each point x on the boundary is subject either to no force (if h(x) = 0) or to a binding elastic force pulling it back to its equilibrium position (if h(x) > 0). We prove the following result.

ν R k (Ω, h) k ≤ γ(n).
Theorem 1.4 is of course a special case of Theorem 1.5, corresponding to h = 0. However, in order to make the argument more readable, we prefer to treat first the Neumann case, and then outline the few changes to be made in order to prove Theorem 1.5.

Let us note that the same constant γ(n) appears in the Dirichlet, Neumann, and Robin cases. It is known from the work of J. Bourgain [START_REF] Bourgain | On Pleijel's nodal domain theorem[END_REF] and S. Steinerberger [START_REF] Steinerberger | A geometric uncertainty principle with an application to Pleijel's estimate[END_REF] that this constant is not optimal when n = 2 in the Dirichlet case. See [START_REF] Helffer | A review on large k minimal spectral k-partitions and Pleijel's theorem[END_REF] for an extensive discussion, in connection with minimal partition problems. I. Polterovich [START_REF] Polterovich | Pleijel's nodal domain theorem for free membranes[END_REF] conjectures that for a sufficiently regular open set Ω ⊂ R 2 , lim sup

k→+∞ ν D k (Ω) k ≤ 2 π and lim sup k→+∞ ν N k (Ω) k ≤ 2 π .
The constant 2 π is the smallest possible, as can be seen by considering rectangles [START_REF] Polterovich | Pleijel's nodal domain theorem for free membranes[END_REF][START_REF] Helffer | A review on large k minimal spectral k-partitions and Pleijel's theorem[END_REF][START_REF] Bonnaillie-Noël | Nodal and spectral minimal partitions -The state of the art in 2015[END_REF]. Let us finally point out that the analogue to Theorems 1.2, 1.4, and 1.5, with the same constant γ(n), holds for the Schrödinger operator -∆ + V in R n , for a class of radially symmetric potentials V . This was shown recently by P. Charron [START_REF] Charron | A Pleijel-type theorem for the quantum harmonic oscillator[END_REF] for the harmonic oscillator and P. Charron, B. Helffer, and T. Hoffmann-Ostenhof in a more general situation [START_REF] Charron | Pleijel's theorem for Schrödinger operators with radial potentials[END_REF].

Overview of the paper

Let us now introduce the main ideas of the paper. The proof of Theorem 1.4 is given in Section 2, and follows quite closely Pleijel's original argument [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF]. This consisted in obtaining a control of ν D k (Ω) in terms of λ k (Ω) and |Ω|, the volume of Ω, by applying the Faber-Krahn inequality to each nodal domain of an eigenfunction u, associated with λ k (Ω). The key fact at this point is the equality λ k (Ω) = λ 1 (D) where D is a nodal domain. The upper bound in Theorem 1.2 then follows from Weyl's law. In the Neumann case, the same method cannot be applied to the nodal domains touching the boundary of Ω, since the eigenfunctions do not satisfy a Dirichlet boundary condition there. The proof in [START_REF] Polterovich | Pleijel's nodal domain theorem for free membranes[END_REF] relied on the fact that the number of nodal domains touching the boundary is controlled by µ k (Ω), under the hypotheses of Theorem 1.3. As far as we know, a similar estimate is not available in dimension higher than 2, nor for a less regular boundary. To overcome this obstacle, we classify the nodal domains of the eigenfunction u into two types: those for which the L 2 -norm of u is mostly concentrated inside Ω (bulk domains), and those for which a significant proportion of the L 2 -norm is concentrated near ∂Ω (boundary domains). To control the number of boundary domains, we reflect them through ∂Ω before applying the Faber-Krahn inequality. To make this approach precise, we use some rather standard partition-of-unity arguments. Let us point out that P. Bérard and B. Helffer propose a closely related strategy for proving a version of Pleijel's theorem, on a manifold with boundary M , in the Neumann case. They suggest to consider the double manifold M , obtained by gluing two copies of M along the boundary ∂M . They obtain in this way a manifold which is symmetric with respect to ∂M . They can then identify the Neumann eigenfunctions on M with the symmetric eigenfunctions of the Laplacian on M . The results in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] would then give

lim k→+∞ ν N k (M ) k ≤ 2γ(n),
with 2γ(n) < 1 for n ≥ 3 (see [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF]Section 5]). However, for this geometric approach to work, one has to require that the metric on M is sufficiently regular, so that the asymptotic isoperimetric inequality of [2, II.15] holds, and therefore also the asymptotic Faber-Krahn inequality of [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF]II.16]. This would impose strong constraints on ∂M , for instance that ∂M is totally geodesic, which are unnecessary in the approach of the present paper.

For the sake of completeness, we give in Section 3 the proof of two technical results, which are used in Section 2. The first is a regularity result up to the boundary for Neumann eigenfunctions. Proposition 1.6. Let Ω ⊂ R n be an open, bounded, and connected set with a C 1,1 boundary. An eigenfunction u of -∆ N Ω belongs to C 1,1 -Ω := α∈(0,1) C 1,α Ω . In particular, u ∈ C 1 Ω . The proof follows a remark from [15, 1.2.4] and uses the regularity results for elliptic boundary value problems contained in the classical monograph [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF].

The second result, used repeatedly in Section 2, is the following Green's formula for Neumann eigenfunctions.

Proposition 1.7. Let Ω ⊂ R n be an open, bounded, and connected set with a C 1,1 boundary. If u is an eigenfunction of -∆ N Ω associated with the eigenvalue µ, and if D is a nodal domain of u, then

D |∇u| 2 dx = µ D u 2 dx. (1)
In dimension 2, the nodal set of u is the union of a finite number of C 1 curves, possibly crossing or hitting the boundary of Ω at a finite number of points, where they form equal angles (see for instance [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF]Section 2]). In particular, this implies that D is a Lipschitz domain (even piecewise-C 1 ). We can therefore apply Green's formula for bounded Lipschitz domains (see for instance [10, Theorem 1.5.3.1]), and obtain Equation (1) directly. In higher dimension, there exists as far as we know no proof that the nodal domains are Lipschitz (see however [START_REF] Hardt | Critical sets of solutions to elliptic equations[END_REF]). A way around this difficulty is indicated in [2, Appendix D]. The authors approximate a nodal domain by super-level sets of the eigenfunction for regular values. The boundary of these sets is regular enough to apply Green's formula, and Sard's theorem provides a sequence of regular values converging to 0. We give a proof a Proposition 1.7 along similar lines. This method is also used in Section 2.5 to carry out the reflection argument.

In Section 4, we indicate the changes to be made in order to treat a Robin boundary condition. We give a precise formulation of the eigenvalue problem and prove Theorem 1.5.

Proof of the main theorem 2.1 Preliminaries

For any δ > 0, we write ∂Ω δ := {x ∈ R n ; dist(x, ∂Ω) < δ}.

and ∂Ω + δ := {x ∈ Ω ; dist(x, ∂Ω) < δ}. Let us first note that, since the boundary of Ω is C 1,1 , we can locally straighten it. More explicitly, there exists a finite covering (U i ) 1≤i≤N of ∂Ω by coordinate charts. We mean by coordinate charts that, for i ∈ {1, . . . , N }, U i is an open set in R n , and there exists an open ball B(0, r i ) in R n and a C 1,1 diffeomorphism ψ i : B(0, r i ) → U i , such that ψ i and ψ -1 i are bounded, with first order derivatives bounded and Lipschitz, and such that

U i ∩ Ω = ψ i B + (0, r i ) where B + (0, r i ) = {y = (y ′ , y n ) ∈ R n-1 × R, ; y ∈ B(0, r i ) and y n > 0}.
There exists an associated family (χ i ) 1≤i≤N of C 1,1 non-negative functions and a positive constant δ 1 such that i. supp(χ i ) is compactly included in U i for i ∈ {1, . . . , N };

ii.

N i=1 χ 2 i ≤ 1 in R n and N i=1 χ 2 i ≡ 1 in ∂Ω δ1 .
Let us note that N , δ 1 , and the family (χ i ) 1≤i≤N depend only on Ω, and are fixed in the rest of this section. As a consequence of this local straightening of the boundary, we have the existence of partitions of unity adapted to our problem. Lemma 2.1. There exist two positive constants 0 < a < A such that, for all 0 < δ < δ 1 /A, there exists two non-negative functions ϕ δ 0 and

ϕ δ 1 , C 1,1 in Ω, satisfying i. (ϕ δ 0 ) 2 + (ϕ δ 1 ) 2 ≡ 1 on Ω;
ii. supp(ϕ δ 0 ) ⊂ Ω \ ∂Ω + aδ and supp(ϕ δ 1 ) ⊂ ∂Ω + Aδ ; iii. ∇ϕ δ i (x) ≤ C δ for x ∈ Ω and i ∈ {0, 1}, with C a constant independent of δ. Proof. This construction is rather standard, and we merely give an outline of the demonstration. It is enough to find a C 1,1 function f δ such that a. 0 ≤ f δ ≤ 1 on Ω; b. f δ ≡ 1 on ∂Ω + aδ and supp(f δ ) ⊂ ∂Ω + Aδ ; c. ∇f δ (x) ≤ K δ for x ∈ Ω and i ∈ {0, 1}, with K a constant independent of δ. Indeed, we can then set

ϕ δ 1 (x) = f δ (x) f δ (x) 2 + (1 -f δ (x)) 2 and ϕ δ 0 (x) = 1 -f δ (x) f δ (x) 2 + (1 -f δ (x)) 2 ,
and we obtain functions satisfying Properties iiii of Lemma 2.1. In order to construct f δ , we fix 0 < b < B such that for all i ∈ {1, . . . , N } and all y = (y ′ , y n ) ∈ B + (0, r i ),

by n ≤ dist(ψ i (y), ∂Ω) ≤ By n .

We also fix a non-increasing smooth function g : R → R such that g(t) = 1 for all t ∈ (-∞, 1/4] and g(t) = 0 for all t ∈ [3/4, +∞). We now define the function

f δ i for each i ∈ {1, . . . , N } by f δ i (x) = 0 if x / ∈ U i and f δ i (x) = g y n δ χ i (x) for x ∈ U i , with y = (y ′ , y n ) = ψ -1 i (x). Then the function f δ = N i=1 f δ i 2 satisfies Properties a-c with a = b/4 and A = B.
Let us now write µ = µ k (Ω), with k ≥ 2, and let u be an associated eigenfunction with ν N k (Ω) nodal domains. We set δ := µ -θ , with θ a positive constant to be determined later. We write u 0 := ϕ δ 0 u and u 1 := ϕ δ 1 u. According to property i of Lemma 2.1, we have, for every nodal domain D,

D u 2 dx = D u 2 0 dx + D u 2 1 dx.

First main step: two types of nodal domains

We fix ε > 0, and we distinguish between the bulk domains, i.e. the domains D satisfying

D u 2 0 dx ≥ (1 -ε) D u 2 dx,
and the boundary domains, i.e. the domains D satisfying

D u 2 1 dx > ε D u 2 dx,
We denote by ν 0 (ε, µ) the number of bulk domains, and the bulk domains themselves by D 0 1 , . . . , D 0 ν0(ε,µ) . Similarly, the number of boundary domains and the boundary domains themselves are denoted by ν 1 (ε, µ) and D 1 1 , . . . , D 1 ν1(ε,µ) respectively.

Bulk domains

We begin by giving an upper bound of the number of bulk domains ν 0 (ε, µ). Let us fix j ∈ {1, . . . , ν 0 (ε, µ)}.

According to the Faber-Krahn inequality (see for instance [15, 3.2]) and the variational characterization of λ 1 (D 0 j ), we have

λ 1 (B n ) D 0 j -2 n ≤ D 0 j |∇u 0 | 2 dx D 0 j u 2 0 dx ≤ 1 1 -ε D 0 j |∇u 0 | 2 dx D 0 j u 2 dx . ( 2
)
We have, in Ω, ∇u 0 = ϕ δ 0 ∇u + u∇ϕ δ 0 , and therefore, according to Young's inequality,

|∇u 0 | 2 ≤ (1 + ε)(ϕ δ 0 ) 2 |∇u| 2 + 1 + 1 ε ∇ϕ δ 0 2 u 2 .
Integrating over D 0 j , and using property i and iii of Lemma 2.1, we find

D 0 j |∇u 0 | 2 dx ≤ (1 + ε) D 0 j |∇u| 2 dx + 1 + 1 ε C 2 δ 2 D 0 j u 2 dx. (3)
Injecting Inequality (3) into Inequality (2), we find

λ 1 (B n ) D 0 j -2 n ≤ 1 + ε 1 -ε D 0 j |∇u| 2 dx D 0 j u 2 dx + 1 + 1 ε C 2 (1 -ε)δ 2
According to Proposition 1.7, we have

D 0 j |∇u| 2 dx = µ D 0 j u 2 dx,
and therefore

λ 1 (B n ) D 0 j -2 n ≤ 1 + ε 1 -ε µ + 1 + 1 ε 1 -ε C 2 µ 2θ .
We therefore obtain

1 ≤ D 0 j λ 1 (B n ) n 2 1 + ε 1 -ε µ + 1 + 1 ε 1 -ε C 2 µ 2θ n 2
and, summing over j ∈ {1, . . . , ν 0 (ε, µ)}, we get

ν 0 (ε, µ) ≤ 1 λ 1 (B n ) n 2 1 + ε 1 -ε µ + 1 + 1 ε 1 -ε C 2 µ 2θ n 2 ν0(ε,µ) j=1 D 0 j ≤ 1 λ 1 (B n ) n 2 1 + ε 1 -ε µ + 1 + 1 ε 1 -ε C 2 µ 2θ n 2 |Ω| . (4)

Boundary domains

Let us now give an upper bound of the number of boundary domains ν 1 (ε, µ). We further decompose the function u 1 in the following way. Using the family (χ i ) 1≤i≤N introduced at the beginning of this section, we set u i 1 := χ i u 1 . Let us now fix j ∈ {1, . . . , ν 1 (ε, µ)}. According to Property ii for the family (χ i ) 1≤i≤N (here we use the assumption δ < δ 1 /A), we have

D 1 j u 2 1 dx = N i=1 D 1 j (u i 1 ) 2 dx.
If α is a regular value for the function v, the set Σ α := ∂V α \ {y n = 0} is a C 1 submanifold with boundary in R n , the normal being given by ∇v(y) for any y ∈ Σ α . The boundary of Σ α is γ α := y = (y ′ , y n ) ∈ V ; y n = 0 and w(y) = α .

Let us impose the additional condition that α is a regular value for the function

w |Γ : Γ → R y ′ → w(y ′ , 0), where Γ is the open set in R n-1 defined by Γ = {y ′ ∈ R n-1 ; (y ′ , 0) ∈ V and w(y ′ , 0) > 0}.
Then, for any y ∈ γ α , the component of ∇w(y) tangential to {y n = 0} is non-zero. This implies that that Σ α touches the hyperplane {y n = 0} transversally. Let us now denote by V R α and v R α the reflection of V α and v α through {y n = 0}, defined in the same way as V R and v R in Equations ( 9) and [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. By

our choice of α, V R α = 2 |V α | and the function v R α belongs to H 1 0 V R α .
The Faber-Krahn inequality, applied to the open set V R α , gives us

λ 1 (B n )2 -2 n |V α | -2 n = λ 1 (B n ) V R α -2 n ≤ V R α ∇v R α 2 dy V R α (v R α ) 2 dy = Vα |∇v α | 2 dy Vα v 2 α dy . (11)
According to Sard's theorem, applied to the functions v : D → R and w |Γ : Γ → R, we can find a sequence (α m ) m≥1 of positive regular values for both functions satisfying α m → 0. Using Inequality [START_REF] Hardt | Critical sets of solutions to elliptic equations[END_REF] for α = α m and passing to the limit, we find

λ 1 (B n )2 -2 n |V | -2 n ≤ V |∇v| 2 dy V v 2 dy .
Using Inequalities ( 7), [START_REF] Courant | Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke[END_REF], and (5), we get

λ 1 (B n ) D 1 j ∩ ∂Ω + Aδ -2 n ≤ λ 1 (B n ) |U | -2 n ≤ (2C ′′ ) 2 n C ′′′ U ∇u ij 1 2 dx U (u ij 1 ) 2 dx ≤ N ε (2C ′′ ) 2 n C ′′′ U ∇u ij 1 2 dx D 1 j u 2 dx . (12)
A computation similar to the one done for u 0 , combined with Inequality (6), gives us

U ∇u ij 1 2 dx ≤ 2 D 1 j |∇u| 2 dx + 2 (C ′ ) 2 δ 2 D 1 j u 2 dx. (13)
Combining Inequalities ( 12) and ( 13), as in the case of u 0 , we obtain

1 ≤ C|D 1 j ∩ ∂Ω + Aδ |ε -n 2 µ + (C ′ ) 2 µ 2θ n 2 ,
with C a constant depending only on Ω. Summing over j ∈ {1, . . . , ν 1 (ε, µ)}, we get

ν 1 (ε, µ) ≤ C|∂Ω + Aδ |ε -n 2 µ + (C ′ ) 2 µ 2θ n 2 .
Since |∂Ω + δ | ∼ H n-1 (∂Ω) Aδ as δ → 0, we obtain finally

ν 1 (ε, µ) ≤ C ′ µ -θ ε -n 2 µ + (C ′ ) 2 µ 2θ n 2 , (14)
with C ′ a constant depending only on Ω.

End of the proof

We now fix θ ∈ (0, 1/2), for instance θ = 1/4, and consider the limits when k → +∞, keeping ε fixed (we recall that µ = µ k (Ω)). We have lim sup

k→+∞ ν N k (Ω) k ≤ lim sup k→+∞ ν 0 (ε, µ k (Ω)) k + lim sup k→+∞ ν 1 (ε, µ k (Ω)) k .
Using Inequality (4), we get lim sup

k→+∞ ν 0 (ε, µ k (Ω)) k ≤ lim sup k→+∞ 1 kλ 1 (B n ) n 2 1 + ε 1 -ε µ k (Ω) + 1 + 1 ε 1 -ε C 2 µ k (Ω) 2θ n 2
|Ω| .

We recall that µ k (Ω) ≤ λ k (Ω) for all positive integer k [20, XIII.

n 2 |Ω| k = (2π) n ω n .
We therefore have the asymptotic upper bound lim sup

k→+∞ µ k (Ω) n 2 |Ω| k ≤ (2π) n ω n . ( 15
)
This gives us lim sup

k→+∞ ν 0 (ε, µ k (Ω)) k ≤ 1 + ε 1 -ε n 2 (2π) n λ 1 (B n ) n 2 ω n = 1 + ε 1 -ε n 2 γ(n).
On the other hand, Inequality [START_REF] Helffer | On nodal domains in Euclidean balls[END_REF] implies

lim sup k→+∞ ν 1 (ε, µ k (Ω)) k ≤ lim sup k→+∞ C ′ kµ k (Ω) θ 2 ε µ k (Ω) + 2(C ′ ) 2 ε µ k (Ω) 2θ n 2
, and therefore, according to Inequality [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF],

lim sup k→+∞ ν 1 (ε, µ k (Ω)) k = 0.
We obtain lim sup

k→+∞ ν N k (Ω) k ≤ 1 + ε 1 -ε n 2 γ(n).
Letting ε tend to 0, we get finally lim sup

k→+∞ ν N k (Ω) k ≤ γ(n).
3 Proof of the auxiliary results with f = (µ + 1)u. We know by Lemma 3.1 that this system has a solution w ∈ H 2 (Ω). By uniqueness of the weak solution, w = u. We therefore obtain u ∈ H 2 (Ω).

If n ≤ 4, the Sobolev embedding theorem tells us that for any p ∈ [1, ∞[, u ∈ L p (Ω), and another application of Lemma 3.1 gives us u ∈ W 2,p (Ω). If n > 4, we still obtain u ∈ L p (Ω), and therefore u ∈ W 2,p (Ω), for all p ∈ [1, ∞[, after a standard bootstrap argument, using repeatedly Lemma 3.1 and the Sobolev embedding theorem. Since W 2,p (Ω) ⊂ C 1,1-n p Ω for all p > n, we have proved Proposition 1.6.

Proof of Proposition 1.7

We use the method of [2, Appendix D]. We consider an eigenfunction u of -∆ N Ω associated with µ, and a nodal domain D of u. Up to replacing u by -u, we assume that u is positive in D. Since u ∈ C 1 Ω , there exists an open neighborhood O of Ω in R n and a C 1 function g : O → R such that g = u in Ω. We denote by E the nodal domain of g containing D. For α > 0 small enough, we write

D α := {x ∈ D ; u(x) > α} and E α := {x ∈ E ; g(x) > α}.
Let us note that

∂E α ∩ O = {x ∈ E ; g(x) = α}.
We have the following decomposition of ∂D α ⊂ Ω into disjoint subsets:

∂D α = Σ α ∪ Γ α ∪ γ α , (16)
where

Σ α := ∂E α ∩ Ω is a closed set in Ω, Γ α := E α ∩ ∂Ω
is an open set in ∂Ω, and

γ α := ∂E α ∩ ∂Ω is a closed set in ∂Ω.
We now assume that α is a regular value for the function g. Then ∂E α is a C 1 -regular surface, and for each x ∈ ∂E α , ∇g(x) is orthogonal to ∂E α at x. Since u satisfies a Neumann boundary condition on ∂Ω, we have ν(x) • ∇g(x) = 0 for any x ∈ ∂Ω, where ν(x) is the exterior normal unit vector to ∂Ω at x. This implies that the two C 1 -submanifolds ∂E α and ∂Ω intersect transversally, and therefore that γ α is a C 1 -submanifold of ∂Ω, with dimension n -2. From this and the decomposition ( 16), we conclude that ∂D α is Lipschitz. We can therefore apply Green's formula to the function u α := uα in D α (see [

The Robin boundary condition

Let us begin this Section with a definition of the operator -∆ R,h Ω . We follow the method of [21, 3.1], although this reference uses slightly stronger regularity assumption on the domain. We define the real bilinear form q h on the domain H 1 (Ω) by

q h (u, v) = Ω ∇u • ∇v dx + ∂Ω uv dσ
for all u and v in H 1 (Ω). The form q h is closed, symmetric, and non-negative. We define the selfadjoint operator -∆ R,h Ω as the Friedrichs extension of q h [START_REF] Reed | Methods of Modern Mathematical physics. II. Fourier Analysis, Self-Adjointness[END_REF]Theorem X.23]. The compact embedding H 1 (Ω) ⊂ L 2 (Ω) ensures that -∆ R,h Ω has compact resolvent. The spectrum of -∆ R,h Ω therefore consists in a sequence of isolated non-negative eigenvalues of finite multiplicity tending to +∞, which we denote by (µ k (Ω, h)) k≥1 (with repetition according to the multiplicities). Let us point out that for all positive integer k, µ k (Ω, h) ≤ λ k (Ω). Indeed,

3. 1 6 Lemma 3 . 1 .

 1631 Proof of Proposition 1.We use the following regularity result, which can be found for instance in[10, 2.1]. If Ω is a bounded open set with a C 1,1 boundary and f ∈ L p (Ω) with p ∈ (1, ∞), there exists a unique w ∈ W 2,p (Ω) which solves-∆w + w = f in Ω; ∂w ∂ν = 0 on ∂Ω.We now follow the method indicated in[START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF] Remark 1.2.11]. Let us consider u, an eigenfunction of -∆ N Ω associated with µ. The function u is in H 1 (Ω), and is the unique weak solution of the boundary value problem -∆u + u = f in Ω;∂u ∂ν = 0 on ∂Ω;

u 2 dx -α m µ Dα m u dx = Dα m |∇u| 2 Du 2

 222 10, Theorem 1.5.3.1]), and we obtain Dα (-∆u α ) u α dx = -We have u α = 0 on Σ α and ∂uα ∂ν = 0 on Γ α , and therefore µ Dα u 2 dxαµ Dα u dx = Dα |∇u| 2 dx. According to Sard's theorem, there exists a sequence (α m ) m≥1 of positive regular values for the function g, converging to 0. For any m large enough, we have µ Dα m dx. Letting m → +∞, we get µ dx = D |∇u| 2 dx, which concludes the proof of Proposition 1.7.

  q h (u, v) = Ω ∇u • ∇v dxfor all u and v in H 1 0 (Ω), so that the inequality follows from the minmax characterization of µ k (Ω, h) and λ k (Ω)[START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF] Theorem XIII.2]. Weyl's law for the sequence (λ k (Ω)) k≥1 then implies lim sup k→+∞ µ k (Ω, h) be generalized in the following way. Proposition 4.1. Let Ω ⊂ R n be an open, bounded, and connected set with a C 1,1 boundary, and h be a Lipschitz function in Ω with h ≥ 0 on ∂Ω. An eigenfunction u of -∆ R,h Ω belongs to C 1,1 -Ω := α∈(0,1) C 1,α Ω . In particular, u ∈ C 1 Ω . To prove Proposition 4.1, we use the following regularity result, which is a special case of [10, Theorem 2.4.2.7]. Lemma 4.2. Let Ω ⊂ R n be an open, bounded, and connected set with a C 1,1 boundary, and let h be a Lipschitz function in Ω with h ≥ 0 on ∂Ω, and f ∈ L p (Ω) with p ∈ (1, ∞). There exists a unique w ∈ W 2,p (Ω) which solves -∆w + w = f in Ω;∂w ∂ν + hw = 0 on ∂Ω. We then repeat the steps in the proof of Proposition 4.1. For the type of Robin boundary condition studied here, the Green identity given in Proposition 1.7 can be replaced by the following inequality.

Proposition 4 . 3 .

 43 Let Ω ⊂ R n be an open, bounded, and connected set with a C 1,1 boundary, and let h be a Lipschitz function in Ω with h ≥ 0 on ∂Ω. If u is an eigenfunction of -∆ R,h Ω associated with the eigenvalue µ, and if D is a nodal domain of u, then D |∇u| 2 dx ≤ µ D u 2 dx.

  Theorem 1.5. If Ω ⊂ R n , with n ≥ 2, is an open, bounded, and connected set with a C 1,1 boundary, and if h is a Lipschitz function in Ω with h ≥ 0 on ∂Ω,

	lim sup
	k→+∞

Acknowledgements The author thanks Bernard Helffer for introducing him to this problem, for his suggestions and corrections, as well as advice and encouragement, and Pierre Bérard for his careful reading of this work, for suggesting numerous improvements, and correcting the reflection argument of Section 2.5. The author also thanks Susanna Terracini for discussions of related results and possible extensions, Alessandro Iacopetti for his help with the regularity questions, and James Kennedy for discussions of the Robin case. This work was partially supported by the ANR (Agence Nationale de la Recherche), project OPTIFORM n • ANR-12-BS01-0007-02, and by the ERC, project COMPAT n • ERC-2013-ADG.

As a consequence, there exists i j ∈ {1, . . . , N } such that

Let us note that u ij 1 = χu, with χ = χ ij ϕ δ j . The function χ is C 1,1 , 0 ≤ χ ≤ 1, supp( χ) ⊂ ∂Ω Aδ , and

with C ′ a constant depending only on Ω. Up to replacing u by -u, we assume that

We define the open set

We now straighten the boundary locally, that is to say we set v

The set V is open and contained in B + (0, r ij ). There exists positive constants C ′′ and C ′′′ , depending only on Ω, such that

and

Second main step: reflection of the boundary domains

The basic idea consists in extending V and v by reflection through the hyperplane {y n = 0}. We denote by σ the reflection

Intuitively, we define

We expect V R = 2 |V |, and v R ∈ H 1 0 V R . Indeed, if ∂V is regular enough, v satisfies a Dirichlet boundary condition on ∂V \ {y n = 0}, and therefore v R satisfies a Dirichlet boundary condition on ∂V R . We would then apply the Faber-Krahn inequality to the domain V R to obtain a lower bound of the Rayleigh quotient

The above reasoning is of course not valid in general, since we do not know if ∂V is regular enough. To overcome this difficulty, we follow a method used in [2, Appendix D] (we use the same method to prove Proposition 1.7). Let us first note that Proposition 1.6 implies that v can be extended to a function w ∈ C 1 D . For α > 0 small enough, we consider the super-level sets Proof. Using Proposition 4.1 instead of Proposition 1.6, we essentially repeat the steps in the proof of Proposition 1.7. However, two points have to be modified. First, since we do not in general have ∂u ∂ν = 0 on ∂Ω, the argument in Section 3.2 showing that ∂E α and ∂Ω intersect transversally does not apply. However, if α > 0 is a regular value for the function g : ∂Ω → R, the tangential part of ∇g(x) is non-zero when x ∈ γ α . If α is also a regular value for g, we can proceed as in Section 3.2. Applying Green's formula to u α , and using the Robin boundary condition, we obtain

Using Sard's theorem for g and its restriction g : ∂Ω → R, we find a sequence (α m ) m≥1 of positive regular values for both functions, such that α m → 0. We conclude by applying Inequality [START_REF] Reed | Methods of Modern Mathematical physics. II. Fourier Analysis, Self-Adjointness[END_REF] with α = α m and passing to the limit.

The proof of Theorem 1.5 then follows the steps of Section 2, using Inequality (17) instead of Inequality (15), and Inequality (18) instead of Equation ( 1).