B Helffer

T Hoffmann-Ostenhof

F Jauberteau

C Léna

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains -with some numerics-

The motivating problem is to analyze the multiplicity of the k-th eigenvalue of the Dirichlet problem in a domain Ω in R 2 . It is for example an old result of Cheng [START_REF] Cheng | Eigenfunctions and nodal sets[END_REF], that the multiplicity of the second eigenvalue is at most 3. In [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] an example with multiplicity 3 is given as a side product of the production of a counter example to the nodal line conjecture (see also [START_REF] Hoffmann-Ostenhof | The nodal line of the second eigenfunction of the Laplacian in R 2 can be closed[END_REF], and the papers by Fournais [START_REF] Fournais | The nodal surface of the second eigenfunction of the Laplacian in R d can be closed[END_REF] and Kennedy [START_REF] Kennedy | Closed nodal surfaces for simply connected domains in higher dimensions[END_REF] who extend to higher dimensions these counter examples, introducing new methods). This example is based on the spectral analysis of the Laplacian in domains consisting of a disc in which we have introduced on an interior concentric circle suitable cracks.

We discuss the initial proof and complete it by one missing argument. For completion, we will also extend the validity of a theorem of Cheng to less regular domains. Although not needed for the positive results, we complete the paper with numerical results illustrating why some argument has to be modified and propose a fine theoretical analysis of the spectral problem when the cracks are closed.

Main statement

The starting point for the construction of counterexamples to the nodal line conjecture [START_REF] Hoffmann-Ostenhof | The nodal line of the second eigenfunction of the Laplacian in R 2 can be closed[END_REF][START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] is the introduction of two concentric open discs B R1 and B R2 with 0 < R 1 < R 2 and the corresponding annulus M R1,R2 = B R2 \ BR1 . The authors choose R 1 and R 2 such that

λ 1 (B R1 ) < λ 1 (M R1,R2 ) < λ 2 (B R1 ) , (2.1)
where, for ω ⊂ R 2 bounded, λ j (ω) denotes the j-th eigenvalue of the Dirichlet Laplacian H in ω.

We observe indeed that for fixed R 1 , λ 1 (M R1,R2 ) tends to +∞ as R 2 → R 1 (from above) and tends to 0 as R 2 → +∞ . Moreover R 2 → λ 1 (M R1,R2 ) is decreasing. Hence there is some interval (a(R 1 ), b(R 1 )) with a(R 1 ) > R 1 such that (2.1) is satisfied if and only if R 2 ∈ (a(R 1 ), b(R 1 )).

Then we introduce

D R1,R2 = B R1 ∪ M R1,R2
and observe that λ 1 (D R1,R2 ) = λ 1 (B R1 ) λ 2 (D R1,R2 ) = λ 1 (M R1,R2 ) λ 3 (D R1,R2 ) = min(λ 2 (B R1 ), λ 2 (M R1,R2 )) .

(2.2)

If Condition (2.1) was important in the construction of the counter-example to the nodal line conjecture, the weaker assumption max(λ 1 (B R1 ), λ 1 (M R1,R2 )) < min(λ 2 (B R1 ), λ 2 (M R1,R2 )) .

(2.3) suffices for the multiplicity question. Under this condition, we have:

λ 1 (D R1,R2 ) = min(λ 1 (B R1 ), λ 1 (M R1,R2 )) λ 2 (D R1,R2 ) = max(λ 1 (B R1 ), λ 1 (M R1,R2 )) λ 3 (D R1,R2 ) = min(λ 2 (B R1 ), λ 2 (M R1,R2 )) , (2.4)
and it is not excluded (we are in the non connected situation) to consider the case λ 1 (D R1,R2 ) = λ 2 (D R1,R2 ). We now carve holes in ∂B R1 such that D R1,R2 becomes a domain. For N ∈ N * := N \ {0} and ∈ [0, π N ], we introduce (see Figure 1 for N = 2, 3)

D(N, ) = D R1,R2 ∪ N -1 j=0 {x ∈ R 2 , r = R 1 , θ ∈ ( 2πj N -, 2πj N + )} . (2.5)
The theorem stated in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] is the following:

Theorem 2.1. Let N ≥ 3, then there exists ∈ (0, π N ) such that λ 2 (D(N, )) has multiplicity 3.

We prove below that the theorem is correct. But the proof given in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] works only for even integers N ≥ 4 and in this case there is a need for additional arguments. So we improve in this paper the result in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] by giving an example Ω := D(3, ) where the number of components of ∂Ω equals 4, hence N = 3.

Remark 2.2. Theorem 2.1 leads to the following question: Is there a bounded domain Ω ⊂ R 2 whose boundary ∂Ω has strictly less than 4 components so that λ 2 (Ω) has multiplicity 3? This is also a motivation for analyzing the cases N = 1, 2. The natural conjecture (see Remark 4.3 for further discussion) would be that for simply connected domains Ω, λ 2 (Ω) has at most multiplicity 2.

Remark 2.3. For a specific choice of the pair (R 1 , R 2 ) which will be introduced in Subsection 8.1, the numerics (see Figure 2) illustrates the statement of Theorem 2.1 when N = 3 and N = 4. Although the precision is not very good for close to 0 and π 3 (see Section 8), we can predict as N = 3 a second eigenvalue of multiplicity 3 for ∼ 0.29. A second crossing appears for ∼ 0.96 but corresponds to a third eigenvalue of multiplicity 3. The eigenvalues correspond (with the notation of Section 3) to = 0 and to = 1, the eigenvalues for = 1 having multiplicity 2. When N = 4, we also see a first crossing for ∼ 0.54 where the multiplicity becomes 3, as the theory will show. The eigenvalues correspond (with the notation of Section 3) to = 0, = 1 and = 1, the eigenvalues for = 1 having multiplicity 2.The eigenvalues for = 0 and 2 are simple for ∈ (0, π 2 ).

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 epsilon (between 0 and pi/3) lambda lambda 1 (l=0) lambda 2 (l=0) lambda 1 (l=1) lambda 2 (l=1) ), with R 1 = 0.4356, R 2 = 1.

Figure 2: N=3. Six lowest eigenvalues of the Laplacian in D(N, )) in function of ∈ (0, π ), with R 1 = 0.4356, R 2 = 1. 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 epsilon (between 0 and pi/2) lambda lambda 1 (l=0) lambda 2 (l=0) lambda 1 (l=1) lambda 2 (l=1) lambda 1 (l=2) lambda 2 (l=2)
We now explain what were the difficulties arising in the sketch of the proof given in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF].

The authors introduce a notion of symmetry or antisymmetry with respect to the inversion x → -x but this does not work for odd N since D(N, ) has no center of inversion. So the proof can only work for N even.

Considering N even (N ≥ 4), the idea behind the proof in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] is that there is a crossing for increasing between an eigenvalue associated with an antisymmetric eigenspace of multiplicity 2 and an eigenvalue associated with a symmetric eigenspace. With the considered antisymmetry proposed by the authors, it seems wrong that the multiplicity 2 results simply from the information that the eigenvalue corresponds to a non trivial antisymmetric eigenspace. We will give a theoretical analysis in Section 7 completed by a numerical study in Section 8 giving evidence that this guess is at least wrong in the simpler case N = 2 which is not considered in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF]. Hence one has also to change the argument for even N .

Symmetry spaces

Before proving Theorem 2.1, we recall some basic representation theory. We consider a Hamiltonian which is the Dirichlet realization of the Laplacian in an open set Ω which is invariant by the action of the group G N generated by the rotation g by 2π N . The Hilbert space is H := L 2 (Ω, R) but it is also convenient to work in H C := L 2 (Ω, C). In this case, it is natural to analyze the eigenspaces attached to the irreducible representations of the group G N . This is standard, see for example [START_REF] Helffer | Spectral theory for the dihedral group[END_REF] and references therein, but note that these authors work with a larger group of symmetry, i.e. the dihedral group D 2N . Here we prefer to start with the smaller group G N and it is important to note that we do not assume in our work that Ω is homeomorphic to a disk or to an annulus. The theory of this section will in particular apply for the family of open sets Ω = D(N, ) (which satisfy the D 2N -symmetry). Hence in this case, Theorems 1.2 and 1.3 of [START_REF] Helffer | Spectral theory for the dihedral group[END_REF] do not fully apply. The theory is simpler for complex Hilbert spaces i.e. H C := L 2 (Ω, C), but the multiplicity property appears when considering operators on real Hilbert spaces, i.e H := L 2 (Ω, R). If we work in H C , we introduce for = 0,

• • • , N -1, B = {w ∈ H C | gw = e 2πi /N w} . (3.1)
For = 0, this corresponds to the invariant situation. Hence in the model above (where Ω = B R2 ) u 0 and u 6 belong to B 0 . We also observe that the complex conjugation sends B onto B N -. Hence, except in the cases = 0 and = N 2 the corresponding eigenspace are of even dimension.

The second case appears only if N is even. For 2 = N , one can alternately come back to real spaces by introducing for 0

< < N 2 ( ∈ N) C = B ⊕ B N - (3.2)
and observing that C can be recognized as the complexification of the real space

A A = {u ∈ H | u -2 cos(2 π/N )gu + g 2 u = 0} (3.3) such that C = A ⊗ C (3.4)
where (3.3) follows from an easy computation based on (3.1). For = 0 and = N 2 (if N is even), we define A by

B = A ⊗ C . (3.5)
Under the invariance condition on the domain, the Dirichlet Laplacian commutes with the natural action of g in L 2 . Hence we get for 0 ≤ ≤ N/2 a family of well defined selfadjoint operators H ( ) obtained by restriction of H to A (with domain D(H) ∩ A ). Note that except for = 0 and = N 2 all the eigenspaces of H ( ) have even multiplicity.

The other point is that Stollmann's theory [START_REF] Stollmann | A convergence theorem for Dirichlet forms with applications to boundary problems with varying domains[END_REF] works for the spectrum of H ( ) ( , N ) associated with the Dirichlet realization H( , N ) of the Laplacian in D N, . Hence we have continuity and monotonicity with respect to of the eigenvalues. Note also that

σ(H( , N )) = ∪ 0≤ ≤ N 2 σ(H ( ) ( , N )) . Remark 3.1.
When N is even, a particular role is played by g N 2 which corresponds to the inversion considered in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF]. One can indeed decompose the Hilbert space H (or H C ) using the symmetry with respect to g N 2 and get the decomposition

H = H even ⊕ H odd , (3.6)
and

H( , N ) = H even ( , N ) ⊕ H odd ( , N ) . (3.7)
One can compare this decomposition with the previous one. We observe that A belongs to H even if is even and to H odd if is odd.

4 Upper bound: the regularity assumptions in Cheng's statement revisited

In [START_REF] Cheng | Eigenfunctions and nodal sets[END_REF], S.Y. Cheng proved that the multiplicity of the second eigenvalue is at most 3. Cheng's proof is actually using a regularity assumption which is not satisfied by D(N, ). This domain has indeed cracks and we need a description of the nodal line structure near corners or cracks. But we will explain how to complete the proof in this case. We recall that for an eigenfunction u the nodal set N (u) of u is defined by

N (u) := {x ∈ Ω , | u(x) = 0} .
For other reasons (this was used in the context of spectral minimal partitions) this analysis was needed and treated in the paper of Helffer, Hoffmann-Ostenhof, and Terracini [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] (Theorem 2.6). With this complementary analysis near the cracks, we can follow the main steps of the proof given in the first part of [START_REF] Hoffmann-0stenhof | Bounds on the multiplicity of eigenvalues for fixed membranes[END_REF] (Theorem B). This proof includes an extended version of Euler's Polyhedral formula (Proposition 2.8 in [START_REF] Hoffmann-0stenhof | Bounds on the multiplicity of eigenvalues for fixed membranes[END_REF] with a stronger regularity assumption).

Proposition 4.1. Let Ω be a C1,+ -domain 1 with possibly corners of opening2 απ for 0 < α ≤ 2. If u is an eigenfunction of the Dirichlet Laplacian in Ω, N denotes the nodal set of u and µ(N ) denotes the cardinality of the components of Ω \ N , i.e. the number of nodal domains, then

µ(N ) ≥ x∈N ∩Ω (ν(x) -1) + 2 , (4.1)
where ν(x) is the multiplicity of the critical point x ∈ N (i.e. the number of lines crossing at x).

For a second eigenfunction µ(N ) = 2, and the upper bound of the multiplicity by 3 comes by contradiction. Assuming that the multiplicity of the second eigenvalue is ≥ 4, one can, for any x ∈ Ω, construct some u in the second eigenspace such that ν(x) ≥ 2. This gives the contradiction with (4.1). Hence we have Proposition 4.2. Let Ω be a C 1,+ -domain with possibly corners of opening απ for 0 < α ≤ 2. Then the multiplicity of the second eigenvalue of the Dirichlet Laplacian in Ω is not larger than 3.

Remark 4.3. An upper bound of the multiplicity by 2 is obtained by C.S. Lin when Ω is convex (see [START_REF] Lin | On the second eigenfunction of the Laplacian in R 2[END_REF]). As observed at the end of Section 2 in [START_REF] Zhang Liqun | On the multiplicity of the second eigenvalue of Laplacian in R 2[END_REF], Lin's theorem can be extended to the case of a simply connected domain for which the nodal line conjecture holds. If the multiplicity of the second eigenvalue is larger than 2, one can indeed find in the associated spectral space an eigenfunction whose nodal set contains a point in the boundary where two half lines hit the boundary. This will contradict either the nodal line conjecture or Courant's theorem. See also [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] for some sufficient conditions on domains for the nodal line conjecture to hold.

Proof of Theorem 2.1

We first observe that for the disk of radius R we have

λ 1 (B R ) < λ 2 (B R ) = λ 3 (B R ) < λ 4 (B R ) = λ 5 (B R ) < λ 6 (B R ) .
(5.1)

The eigenfunctions u 1 and u 6 are radial. We will use this property with R = R 2 .

Proposition 5.1.

For N ≥ 3, there exists ∈ (0, π N ) such that λ 2 (H( , N )) belongs to σ(H ( ) ( , N )) for some 0 < < N 2 AND to σ(H ( ) ( , N )) for = 0 or (in the case N even) N 2 .
In particular, the multiplicity of λ 2 for this value of is exactly 3.

Proof.

Note that the condition N ≥ 3 implies the existence of at least one ∈ (0, N 2 ). We now proceed by contradiction. Suppose the contrary. By continuity of the second eigenvalue, we should have

• either λ 2 (H( , N )) belongs to ∪ 0< < N 2 σ(H ( ) ( , N ))) and not to σ(H (0) ( , N )) ∪ σ(H (N/2) ( , N )) for any , • or λ 2 (H( , N )) belongs to σ(H (0) ( , N )) ∪ σ(H (N/2) ( , N )) and not to ∪ 0< < N 2 σ(H ( ) ( , N ))) for any .
But, as we shall see below, the analysis for > 0 small enough shows that we should be in the first case and the analysis for close to π N that we should be in the second case. Hence a contradiction.

The analysis for > 0 very small is by perturbation a consequence of the analysis of = 0. Here we see from (2.2) that λ 2 (D R1,R2 ) is simple and belongs to σ(H (0) (0, N )).

Remark 5.2. If we only have (2.3), we observe that the two first eigenvalues belong to σ(H (0) (0, N )) and the argument is unchanged.

The analysis for close to π

N is by perturbation a consequence of the analysis of = π N . More details (which are not necessary for the argument) will be given in Section 7. Here we see from (5.1) that λ 2 (B R2 ) has multiplicity two corresponding to σ(H (1)

( π N , N )).
So we have proven that for this value of the multiplicity is at least three, hence equals three by the extension of Cheng's statement [START_REF] Cheng | Eigenfunctions and nodal sets[END_REF] proven in the previous section.

Comparison with the former proof proposed in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] When N/2 is even, we deduce from Remark 3.1 that

σ(H (odd) ) ⊂ ∪ 0< < N 2 σ(H ( ) ( , N ))) and σ(H (1) ( , N )) ⊂ σ(H (odd) ) ,
with equality for N = 4. From these two properties which imply that the eigenvalues in σ(H (odd) ) have even multiplicity we can rewrite the previous proof in the way presented in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF]: Proposition 5.3. For N ≥ 4 and N/2 even, there exists ∈ (0, π N ) such that λ 2 (H( , N )) belongs to σ(H even ( , N )) AND to σ(H odd ( , N )). In addition, the multiplicity of λ 2 for this value of is exactly 3.

For N/2 odd integer some extra argument is necessary to exclude that an eigenvalue in σ(H (N/2) ( , N )) (which belongs to σ(H (odd) )) becomes a second eigenvalue. More precisely we should prove that λ 1 (H (N/2) ( , N )) > λ 1 (H (1) ( , N )) for any > 0. Here we have to use the additional dihedral invariance and use the arguments in [START_REF] Helffer | Spectral theory for the dihedral group[END_REF]. The inequality follows from comparing the nodal sets of corresponding eigenfunctions (see (3.3) and (3.4) in [START_REF] Helffer | Spectral theory for the dihedral group[END_REF] after having verified that the proof does not use the assumption that Ω is homeomorphic to a disk or an annulus). Hence, we have completed the proof sketched in [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] but the new proof looks more natural.

6 Further discussion for the case N = 2

In the previous sections, we have excluded the case N = 2 because we were unable to prove that the eigenspaces of H (N/2) have even dimension and there were no more spaces A with 0 < < N 2 to play with. We now assume N = 2 and consider D(2, ). Note that this time it will be quite important to have not only the dihedral symmetry but also the property that the cracks are on a circle.

As in [START_REF] Helffer | Spectral theory for the dihedral group[END_REF] (see (1.16) and (1.17) there), we will use the decomposition of L 2 :

H := L 2 (D(2, )) = A 0 ⊕ A a 1 ⊕ A s 1 .
Here

A s 1 = {u ∈ H , gu = -u , T u = u} , A a 1 = {u ∈ H , gu = -u , T u = -u} , where T u(x 1 , x 2 ) = u(x 1 , -x 2 ).
We also observe that g is for N = 2 the inversion. We similarly define the operators H (1,a) ( , 2) and H (1,s) ( , 2). The question is then to compare the spectra of these two operators and more specifically the first eigenvalue.

If we observe what is imposed by the symmetry or the antisymmetry with respect to {x 1 = 0} or {x 2 = 0} we can replace D(2, ) by

D(2, ) := D(2, ) ∩ {x 1 > 0, x 2 > 0} .
The problem corresponding to H (1,s) 2 the two operators are isospectral. Hence the question is: Are the ground state energies of the two problems the same or are they different for = 0, π 2 ? We will show in Section 7 that for a given pair (R 1 , , R 2 ) with R 1 < R 2 this can only be true, in any closed subinterval of (0, π 2 ], for a finite number of different values of 's. Moreover, for a specific natural pair (R 1 , R 2 ) we can give in Section 8 the following numerically assisted answer:

The ground state energies of H (1,a) ( , 2) and H (1,s) ( , 2) are equal if and only if = 0 or π 2 .

Theoretical asymptotics in domains with cracks

In this section, we analyze theoretically the behavior of the eigenvalue as tends to π 2 . This improves the general results based on [START_REF] Stollmann | A convergence theorem for Dirichlet forms with applications to boundary problems with varying domains[END_REF] and explains why we have to modify the sketch of [START_REF] Hoffmann-Ostenhof | On the nodal line conjecture[END_REF] for the proof of Theorem 2.1.

Preliminaries

We now fix N = 2 and consider 0 < R 1 < R 2 . Motivated by the previous question, we analyze the different spectral problems according to the symmetries. This leads us to consider on the quarter of a disk (0 < θ < π

2 ) four different models. On the exterior circle and on the cracks, we always assume the Dirichlet condition and then, according to the boundary conditions retained for θ = 0 and θ = π/2, we consider four test cases :

• Case NND (homogeneous Neumann boundary conditions for θ = 0 and θ = π/2).

• Case DDD (homogeneous Dirichlet boundary conditions for θ = 0 and θ = π/2).

• Case NDD (homogeneous Neumann boundary conditions for θ = 0 and homogeneous Dirichlet boundary conditions for θ = π/2).

• Case DND (homogeneous Dirichlet boundary conditions for θ = 0 and homogeneous Neumann boundary conditions for θ = π/2).

This is immediately related to the problem on the cracked disk by using the symmetries with respect to the two axes. The symmetry properties lead either to Dirichlet or Neumann.

The cases NND and DND

We use the notation

B + R2 := B R2 ∩ {x 2 > 0} ; x ± := (0, ±R 1 ) ; δ := π 2 -; K δ := {x ∈ R 2 ; r = R 1 , θ ∈ [-π/2 -δ, -π/2 + δ] ∩ [π/2 -δ, π/2 + δ]} ; K + δ := K δ ∩ {x 2 > 0} ; K - δ := K δ ∩ {x 2 < 0} .
By the symmetry arguments of Section 6,

λ N N D 1 ( D(2, )) = λ 1 (B R2 \ K δ ); λ DN D 1 ( D(2, )) = λ 1 (B + R2 \ K + δ ).
The family of compact sets (K δ ) δ>0 concentrates to the set {x + , x -}, in the sense that K δ is contained in any open neighborhood of {x + , x -} for δ small enough. Reference [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] provides two-term asymptotic expansions in this situation.

A direct application of Theorem 1.7 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] gives

λ 1 (B + R2 \ K + δ ) = λ 1 (B + R2 ) + u(x + ) 2 2π log(diam(K + δ ) + o 1 log(diam(K + δ )
, where diam(K + δ ) is the diameter of K + δ and u an eigenfunction associated with λ 1 (B + R2 ), normalized in L 2 (B + R2 ). Using diam(K + δ ) = 2R 1 sin(δ) and the nor-malized eigenfunction given by Proposition 1.2.14 in [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF] we find, after simplification

λ DN D 1 ( D(2, )) = j 2 1,1 + + 8 R 2 2 J 1 (j 1,1 R 1 /R 2 ) J 1 (j 1,1 ) 2 1 |log(π/2 -)| + o 1 |log(π/2 -)| , (7.1)
where j ,k is the k-th zero of the Bessel function J corresponding to the integer ∈ N (see Subsection 8.2 for more details and numerical values).

We obtain a similar expansion for the other eigenvalue, starting from Theorem 1.4 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF], which gives us

λ 1 (B R2 \ K δ ) = λ 1 (B R2 ) + Cap B R 2 (K δ , u) + o Cap B R 2 (K δ , u) .
In this formula, u is an eigenfunction associated with λ 1 (B R2 ) and normalized in L 2 (B R2 ), and Cap B R 2 (K δ , u) is defined by Equation ( 6) in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF]. Since u is radially symmetric, u(x + ) = u(x -). We then observe that the proof of Proposition 1.5 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] can be adapted to give

Cap B R 2 (K δ , u) = u(x ± ) 2 Cap B R 2 (K δ ) + o Cap B R 2 (K δ ) , where Cap B R 2 (K δ ) is the classical (condenser) capacity of K δ relative to B R2 . Since K δ = K + δ ∪ K - δ
, and since K + δ and K - δ concentrate to x + and x -respectively, we have

Cap B R 2 (K δ ) ∼ Cap B R 2 (K + δ ) + Cap B R 2 (K - δ )
as δ → 0. This last fact seems to be well known (see [START_REF] Flucher | Approximation of Dirichlet eigenvalues on domains with small holes[END_REF], page 178), but we give a proof in Appendix A for completeness. Finally, Proposition 1.6 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] gives an asymptotic expansion for Cap B R 2 (K ± δ ). Gathering these estimates, we find

λ N N D 1 ( D(2, )) = j 2 0,1 + + 4 R 2 2 J 0 (j 0,1 R 1 /R 2 ) J 0 (j 0,1 ) 2 1 |log(π/2 -)| + o 1 |log(π/2 -)| . (7.2)

Analysis of the cases NDD and DDD

In these cases, the results in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] give an estimate of the eigenvalue variation but no explicit first term for the expansion. However, they strongly suggest the form of this term, which we present as a conjecture in each case. By the symmetry arguments of Section 6, we have, for close to π/2,

λ DDD 1 ( D(2, )) = λ 2 (B + R2 \ K + δ ).
We further note that λ 2 (B + R2 ) (= j 2 2,1 ) is simple and that an associated eigenfunction u, normalized in L 2 (B + R2 ), is given by

u(r cos θ, r sin θ) = 2 √ π 1 R 2 |J 2 (j 2,1 )| J 2 j 2,1 r R 2 sin(2θ).
In particular, it follows that

∂ x1 u(x + ) = 1 R 1 4 √ π 1 R 2 |J 2 (j 2,1 )| J 2 j 2,1 R 1 R 2 ,
so that,

u(x + + ρ(cos t, sin t)) = 4 √ π 1 R 1 R 2 |J 2 (j 2,1 )| J 2 j 2,1 R 1 R 2 ρ cos t + O ρ 2 .
Theorem 1.4 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] gives us

λ 1 (B + R2 \ K + δ ) = λ 1 (B + R2 ) + Cap B + R 2 (K + δ , u) + o Cap B + R 2 (K + δ , u) . (7.
3) Proposition 7.1. There exists C > 0 such that

j 2 2,1 ≤ λ DDD 1 ( D(2, )) ≤ j 2 2,1 + C π 2 - 2 . (7.4)
Proof. By monotonicity of the Dirichlet eigenvalues with respect to the domain, we immediately have

j 2 2,1 = λ DDD 1 ( D(2, π/2)) ≤ λ DDD 1 ( D(2, )).
On the other hand, since x + is a (regular) point in the nodal set of u, we have [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF]. The upper bound follows from Equation (7.3). Conjecture 7.2. As δ → 0 + ,

Cap B + R 2 (K + δ , u) = O δ 2 as δ → 0, according to Lemma 2.2 in
Cap B + R 2 (K + δ , u) ∼ Cap B + R 2 (s δ , u),
where s δ is the segment

[-δR 1 , δR 1 ] × {R 1 }.
The first term of the asymptotics of Cap B + R 2

(s δ , u) is given by Theorem 1.13 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF], so that Conjecture 7.2 would imply that as → π/2 -,

λ DDD 1 ( D(2, )) = j 2 2,1 + + 16 R 2 2 J 2 (j 2,1 R 1 /R 2 ) J 2 (j 2,1 ) 2 π 2 - 2 + o π 2 - 2 . (7.5)
Using again Section 6, we have

λ N DD 1 ( D(2, )) = λ 2 (B R2 \ K δ ) = µ 2 (B + R2 \ K + δ ),
where (µ j ) j≥1 denotes the eigenvalues of the Laplacian in B + R2 , with a Dirichlet condition on the semi-circle {|x| = R 2 } ∩ x 2 > 0 and a Neumann condition on the diameter [-R 2 , R 2 ] × {0}. We note that µ 2 (B + R2 ) is simple. The eigenvalue λ 2 (B R2 ) is double when the Laplacian is understood as acting on L 2 (B R2 ), but simple if we restrict the Laplacian to A s 1 (as defined in Section 6). We denote by (λ s j ) j≥1 the spectrum of H (1,s) ( , 2) (by a slight abuse of notation we do not specify the value of ). We remark that

µ 2 (B + R2 ) = λ 2 (B R2 ) = λ s 1 (B R2
).

An eigenfunction associated with λ s 1 (B R2 ) and normalized in A s 1 is given by

u(r cos θ, r sin θ) = 2 π 1 R 2 |J 1 (j 1,1 )| J 1 j 1,1 r R 2 cos θ .
An eigenfunction associated with µ 2 (B + R2 ) and normalized in

L 2 (B + R2 ) is given by √ 2u |B + R 2 .
Proposition 7.3. There exists C > 0 such that

j 2 1,1 ≤ λ N DD 1 ( D(2, )) ≤ j 2 1,1 + C π 2 - 2 . (7.6)
Proof. We use λ N DD

1 ( D(2, )) = λ 2 (B R2 \ K δ ).
To find an upper bound, we would like to apply Theorem 1.4 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] to get

λ 2 (B R2 \ K δ ) = λ 2 (B R2 ) + Cap B R 2 (K δ , u) + o Cap B R 2 (K δ , u) .
Unfortunately, this cannot be done directly, since λ 2 (B R2 ) is a double eigenvalue. But the result is easily obtained by repeating the proof in Appendix A of [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] for the Laplacian acting on the symmetric space A s 1 . A straightforward adaptation of the proof of Lemma 2.2 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF], taking into account the fact that K δ concentrates to two points, gives Cap B R 2 (K δ , u) = O δ 2 as δ → 0.

Let us assume again that Conjecture 7.2 holds. Let us also assume that Theorem 1.4 in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] holds for the eigenvalue problems with mixed boundary conditions which define (µ j ) j≥1 . We obtain, as δ → 0 + ,

µ 2 (B + R2 \ K + δ ) = µ 2 (B + R2 ) + 2π|∂ x1 u(x + )| 2 R 2 1 δ 2 + o δ 2 .
Alternatively, we could work in the symmetric space A s 1 . Repeating the proof in Appendix A of [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] in this space and assuming that the u-capacity defined in [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF] is asymptotically additive for small distant sets, we obtain

λ s 1 (B R2 \ K δ ) = λ s 1 (B R2 ) + π|∂ x1 u(x + )| 2 R 2 1 δ 2 + π|∂ x1 u(x -)| 2 R 2 1 δ 2 + o δ 2 .
Both methods would give

λ N DD 1 ( D(2, )) = j 2 1,1 + + 4 R 2 2 J 1 (j 1,1 R 1 /R 2 ) J 1 (j 1,1 ) 2 π 2 - 2 + o π 2 - 2 . (7.7)

Comparison

As a consequence of Propositions 7.1 and 7.3 and using also the analyticity with respect to , we obtain Proposition 7.4. There exists 0 ∈ (0, π 2 ) such that for ∈ [ 0 , π 2 ) we have

λ N DD 1 ( D(2, )) < λ DN D 1 ( D(2, )) . Moreover δ( ) := λ N DD 1 ( D(2, )) -λ DN D 1 ( D(2,
)) can at most vanish in (0, π 2 ) on a sequence of 's with no accumulation point except possibly at 0.

A more accurate analysis as → 0 would be useful for excluding the possibility of a sequence of zeros of δ tending to 0. We will see in the next section that numerics strongly suggests that δ( ) is negative in (0, π 2 ). The argument used in Proposition 7.4 is general and not related to N = 2.

8 Some illustrating numerics

Preliminaries

In this section, we complete the theoretical study of the previous section by using numerics. With the discussion around (2.4) in mind, a particular choice for the pair (R 1 , R 2 ) is to start from B R2 with R 2 = 1, and then to take as 0 < R 1 < 1 the radius of the circle on which the second radial eigenfunction (which is associated with the sixth eigenvalue) vanishes. In this case, we have

λ 1 (B R1 ) = λ 1 (M R1,R2 ) and λ 1 (B R1 ) = λ 6 (B R2 ) is an eigenvalue of the Dirichlet Laplacian in D(N, ) for any ∈ [0, π N ].
Its labelling as eigenvalue is 2 for 0 < ≤ 0 and becomes 6 for sufficienly close π N . In addition, there is a unique * 0 such that the labelling is 2 for 0 < ≤ * 0 and becomes > 2 for > * 0 . This follows from the piecewise analyticity of the eigenvalues (Kato's theory) and a more detailed analysis as → 0 or → π N (see [START_REF] Dauge | Eigenvalues variation II. Multidimensional problems[END_REF] for the technical details, [START_REF] Colorado | Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions[END_REF][START_REF] Hillairet | The eigenvalues of the Laplacian on domains with small cracks[END_REF] for related questions and our previous section). The two next subsections recall what will be used for a theoretical verification of our numerical approach in the limits → 0 and → + π 2 . We have indeed in these cases enough theoretical information for controlling the method.

8.2 Reminder: the case of the disk (Dirichlet) This leads to the following ordering of the zeros : j 0,1 < j 1,1 < j 2,1 < j 0,2 < j 3,1 < j 1,2 < j 4,1 < j 2,2 < j 0,3 < . . .

• • • < j 5,1 < j 3,2 < j 6,1 < j 1,3 < j 7,1 < j 2,3 < j 0,4 < j For = 0, we should recover the eigenvalue corresponding to the fourth Dirichlet eigenvalue in B R1 i.e. 139.05 (which is also an eigenvalue of the N N D problem) with a labelling 5. This suggests that the four first eigenvalues correspond to eigenvalues of the annulus with the right symmetry as confirmed by our computations in Subsection 8.4. They correspond to the pairs for the annulus (1, 2), [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Colorado | Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions[END_REF], [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Flucher | Approximation of Dirichlet eigenvalues on domains with small holes[END_REF] and [START_REF] Abramowitz | Handbook of mathematical functions[END_REF][START_REF] Abramowitz | Handbook of mathematical functions[END_REF]. The sixth one corresponding to the pair [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF]. 2 and we are mainly interested in the ground state energy. For = 0, the first eigenvalue is the second eigenvalue ( = 1) either in B R1 or in the annulus B 1 \ B R1 . In the first case, this would be (j 0,2 /j 0,1 ) 2 λ DN D 2 ( D(2, 0)) = (j 0,2 /j 0,1 ) 2 λ D 2 (B 1 ) ∼ 77.21, which appears with labelling 3. Hence, we have to look at the first DND-eigenvalue of the annulus corresponding to (k, ) = (1, 1), which is approximately 32.53. Note that the second eigenvalue is obtained for (k, ) = (1, 3), and is approximately 48.78. For = π 2 , we get as ground state

λ DN D 1 ( D(2, π 2 )) = λ D 2 (B 1 ) ∼ 14.67 λ DN D 2 ( D(2, π 2 )) = λ D 7 (B 1 ) ∼ 40.70 λ DN D 3 ( D(2, π 2
)) = λ D 9 (B 1 ) ∼ 49 . We also recover the behavior announced in (7.1). For = 0, we recover the pairs (1, 1), [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Cheng | Eigenfunctions and nodal sets[END_REF] and (1, 5) of the annulus.

The case of the quarter of a disk (NDD) with D-cracks

Here we keep the same pair R 1 , R 2 and assume Neumann for θ = 0 and Dirichlet for θ = π 2 and we are mainly interested in the ground state energy.

Comparison between (NDD) and (DND) with D-cracks

For = 0 and π 2 the theory says that the two spectra coincide. We recall from Section 6, that the union of these two spectra corresponds to the odd eigenfunctions on D(2, ) which are antisymmetric by inversion. For the ground state energies, the two curves do not cross and have different curvature properties. This strongly suggests that they are only equal for = 0 and π 2 . Some crossing (two points) is observed for the curves corresponding to the second eigenvalues. No crossing is observed for the curves corresponding to the third eigenvalues. Putting the whole spectrum together, we see clearly in Figure 8, as mentioned in the introduction of Section 6, why there was no hope to get the multiplicity 3 in the case N = 2 by the successful approach presented for N > 2. We have indeed a first crossing but it only leads to an eigenvalue of multiplicity 2. Look at Figure 2 corrresponding to N = 3 for an interesting comparison.

On the numerical approach

Here we detail the numerical method used to obtain the different figures. We look for the numerical computation of the eigenvalue problem :

-∆u(x, y) = λu(x, y) (8.7)
in the case of a domain Ω = {(x, y) ∈ R 2 , x = r cos(θ), y = r sin(θ), r ∈ ]0, 1[ and θ ∈]0, π/2[} (a quarter of a disk).

In polar coordinates, Problem (8.7) becomes :

-∆u(r, θ) = λu(r, θ) (8.8) where ∆u(r, θ) = ∂ For the numerical discretization of the Laplacian in polar coordinates we use a second order centered finite difference scheme : u i+1,j -2u i,j + u i-1,j δr 2 + 1 r i u i+1,j -u i-1,j 2δr + 1 r 2 i u i,j+1 -2u i,j + u i,j-1 δθ 2 (8.9) where u i,j is a numerical approximation of u(r i , θ j ) on the grid r i = i δr and θ j = j δθ for i, j = 1, . . . , M -1, where δr and δθ are the steps in each direction δr = R2 M and δθ = π 2M (i = 0, M and j = 0, M correspond to the boundary conditions). After discretization we obtain a non symmetric tridiagonal M × M matrix A. For the simulations we have retained M = 180.

To compute the eigenvalues of the previous matrix A we use the function DGEEV of the Lapack library. To validate the code we have considered the case of the unit disk with Dirichlet conditions, computing the six first eigenvalues to compare with (8.3). This allows us in particular the treatment of the singularity of the coefficients of the operator appearing at r = 0. We can also control some limits as → 0 and → π 2 where again we have theoretical values or numerical values obtained by different methods.

For the numerical tests we have considered R 1 = 0.4356, corresponding to an approximation of the radius of the nodal line of the second radial eigenfunction in B 1 .

Let us now fix R > 0 such that the closed balls B i := B(x i , R) are contained in Ω and mutually disjoint. By the above claim, δ ε → 0 when ε → 0, where δ ε := max i∈{1,...,N } max ∂Bi V ε .

For i ∈ {1, . . . , N }, we define

v ε i := 1 1 -δ ε (V ε -δ ε ) + 1 Bi .
We have v ε i ∈ H 1 0 (Ω), and furthermore v ε i ∈ Γ K ε i . Indeed, let us pick a sequence (ϕ n ) converging in H 1 0 (Ω) to V ε and such that, for all n, ϕ n ≥ 1 almost everywhere in a neighborhood of K ε i . Setting

ψ n := 1 1 -δ ε (ϕ n -δ ε ) + 1 Bi ,
we get ψ n ∈ H 1 0 (Ω) for all n and (ψ n ) converges to v ε i in H 1 0 (Ω). Furthermore, for all n, ϕ n (x) ≥ 1 implies ψ n (x) ≥ 1, so that ψ n ≥ 1 almost everywhere in a neighborhood of K ε i . It follows that

Cap Ω (K ε i ) ≤ Ω |∇v ε i | 2 .
Summing for i ranging from 1 to N , we find

N i=1 Cap Ω (K ε i ) ≤ N i=1 Ω |∇v ε i | 2 ≤ 1 (1-δε) 2 Ω |∇V ε | 2 = 1
(1-δε) 2 Cap Ω (K ε ).

Figure 1 :

 1 Figure 1: The domains with cracks for N = 2, N = 3 and N = 4.

Figure 3 :

 3 Figure 3: N=4. Eight lowest eigenvalues of the Laplacian in D(N, )) in function of ∈ (0, π), with R 1 = 0.4356, R 2 = 1.

  ( , 2) is the problem where we assume on ∂ D(2, ) ∩ {x 2 = 0} the Neumann condition and on ∂ D(2, ) ∩ {x 1 = 0} the Dirichlet condition, keeping the Dirichlet condition on the other parts of the boundary. The problem corresponding to H (1,a) ( , 2) is the problem where we assume on ∂ D(2, ) ∩ {x 2 = 0} the Dirichlet condition and on ∂ D(2, ) ∩ {x 1 = 0} the Neumann condition, keeping the Dirichlet condition on the other parts of the boundary. For = 0 and = π

Figure 5 :

 5 Figure 5: Case Dirichlet-Dirichlet: six first eigenvalues (R 1 = 0.4356)

8. 6 Figure 6 :

 66 Figure 6: Case Dirichlet-Neumann: three first eigenvalues (R 1 = 0.4356)

Figure 7 :

 7 Figure 7: Case Neumann-Dirichlet: three first eigenvalues (R 1 = 0.4356)

Figure 8 :

 8 Figure 8: Case Neumann-Dirichlet and case Dirichlet-Neumann: three first eigenvalues (R 1 = 0.4356)

2 (

 2 singularity for r = 0). For the boundary conditions we impose Dirichlet boundary conditions on r = 1. Moreover, we impose Dirichlet boundary conditions on a line (D-crack) corresponding to

Figure 9 :

 9 Figure 9: Six first eigenvalues (R 1 = 0.4356)

  Let j ,k be the k-th zero of the Bessel function J corresponding to the integer ∈ N. Here is a list of approximate values after the celebrated handbook of[START_REF] Abramowitz | Handbook of mathematical functions[END_REF], p. 409, we keep only the first values including at least all the eigenvalues which are less than approximately 13.

	=	0	1	2	3	4	5	6	7	8
	k = 1	2.404 3.831 5.135 6.380 7.588 8.771 9.936 11.086 12.225
	2	5.520 7.015 8.417 9.761 11.064 12.338 13.589 14.821 16.037
	3	8, 653 10.173 11.619 13.015 14.372 15.700 17.003 18.287 19.554
	4	11.791 13.323 14.796 16.223 17.616 18.980 20.320 21.641 22.942
									(8.1)

C 1,+ means C 1, for some > 0.

α = 2 corresponds to the crack case.

Acknowledgements. B.H and T.H-O would like to thank S. Fournais for many discussions on the subject along the years. B.H and C.L would like to thank the Mittag-Leffler institute where part of this work was achieved. C.L was partially supported by the Swedish Research Council (Grant D0497301).

The corresponding eigenvalues for the disk B 1 of radius 1, with Dirichlet condition are given by j 2 ,k . The multiplicity is 1 if = 0 (radial case) and 2 if = 0. Hence we get for the six first eigenvalues (ordered in increasing order): 8. [START_REF] Colorado | Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions[END_REF] The case of the quarter of a disk (NND) with D-cracks When = 0, the two first eigenvalues of this problem with Neumann on the two radii (θ = 0, π 2 ) are equal (due to our choice of R 1 ) and correspond to λ D 6 (B 1 ) ∼ 30.47.

The third eigenvalue is either the fourth Dirichlet eigenvalue in B R1 or the second N N D eigenvalue in the annulus. By dilation, this would be in the first case (j 0,2 /j 0,1 ) 2 λ N N D 2 (B 1 ) = (j 0,2 /j 0,1 ) 2 λ D 4 (B 1 ) ∼ 139.05 . We are actually in the second case, the next eigenvalues for the (NND) problem corresponding for = 0 to the pairs (k, ) for the annulus: (1, 2), [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Colorado | Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions[END_REF], [START_REF] Abatangelo | Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators[END_REF][START_REF] Flucher | Approximation of Dirichlet eigenvalues on domains with small holes[END_REF] and (2, 0).

When = π

2 , we should recover the eigenvalues of the Dirichlet problem in B 1 which have the right symmetry. We get:

)) ∼ 30.47 .

We recover the result predicted by our two terms asymptotics in (7.2).

A Asymptotic additivity of the capacity

We recall the definition of the condenser capacity of a compact set K ⊂ Ω, relative to Ω:

Here Γ K is the closed convex subset of H 1 0 (Ω) consisting of the functions v satisfying v ≥ 1 in the following sense: there exists a sequence (v n ) of functions in H 1 0 (Ω) such that v n ≥ 1 almost everywhere in an open neighborhood of K and v n → v in H 1 0 (Ω) (see for instance Definition 3.3.19 in [START_REF] Henrot | Variation et optimisation de formes: une analyse géométrique[END_REF]). By the Projection Theorem in the Hilbert space H 1 0 (Ω), there exists a unique V K ∈ Γ K realizing the infimum, called the capacitary potential. From the minimization property, it follows immediately that V K is harmonic in Ω \ K. Furthermore, V K is non-negative in Ω (see for instance Item 3 of Theorem 3.3.21 in [START_REF] Henrot | Variation et optimisation de formes: une analyse géométrique[END_REF]).

Let us now fix an integer N ≥ 1, N distinct points x 1 , . . . , x N in Ω, and N families of compact subsets of Ω, (K ε i ) ε>0 for i ∈ {1, . . . , N }. We assume that, for all i ∈ {1, . . . , N }, (K ε i ) ε>0 concentrates to x i as ε → 0, that is to say, for any open neighborhood U of x i , there exists ε U > 0 such that

, with lim ε→0 r ε i = 0. By monotonicity of the capacity,

Furthermore, by subadditivity of the capacity,

Let us now show that in this situation, the capacity is asymptotically additive.

Proposition A.1. If Cap Ω (K ε ) > 0 for all ε > 0, we have, as ε → 0,

Proof. Taking into account Inequality (A.2), we only have to prove

Let us set V ε := V K ε . We claim that for any (fixed) K compact subset of Ω \ {x 1 , . . . , x N }, V ε converges to 0 as ε tends to 0, uniformly in K. Indeed, for ε small enough, K ⊂ Ω \ K ε , so that V ε is harmonic in an open neighborhood U of K. Let us fix r > 0 such that B(x, r) ⊂ U for all x ∈ K. From the Mean Value Formula, for all x ∈ K,

The claim follows from the fact that V ε tends to 0 in L 2 (Ω).