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# Upper bounds on eigenvalue multiplicities for surfaces of genus 0 revisited (Work in progress)

Pierre Bérard, Bernard Helffer

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Pierre Bérard, Bernard Helffer. Upper bounds on eigenvalue multiplicities for surfaces of genus 0 revisited (Work in progress). 2022. hal-03575434

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Distributed under a Creative Commons Attribution 4.0 International License
# UPPER BOUNDS ON EIGENVALUE MULTIPLICITIES FOR SURFACES OF GENUS 0 REVISITED
## WORK IN PROGRESS

PIERRE BÉRARD AND BERNARD HELFFER

ABSTRACT. We revisit two 1999 papers: [1] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili. On the multiplicity of eigenvalues of the Laplacian on surfaces. Ann. Global Anal. Geom. 17 (1999) 43–48 and [2] T. Hoffmann-Ostenhof, P. Michor, and N. Nadirashvili. Bounds on the multiplicity of eigenvalues for fixed membranes. Geom. Funct. Anal. 9 (1999) 1169–1188.

The main result of these papers is that the multiplicity mult($\lambda_k(M)$) of the kth eigenvalue of the surface M is bounded from above by $(2k - 3)$ provided that $k \ge 3$. Here, M is [1] a closed surface with genus 0, or [2] a planar domain with Dirichlet boundary condition (in both cases, the starting label of eigenvalues is 1). The proofs given in [1,2] are not very detailed, and often rely on figures or special configurations of nodal sets. In the case of closed surfaces with genus 0, we provide full details of the proof that mult($\lambda_k$) $\le (2k - 3)$ by introducing and carefully studying the combinatorial type (defined in Subsection 5.2) of the nodal sets involved. In the case of planar domains, we consider the three boundary conditions, Dirichlet, Neumann, Robin, and we also carefully study the combinatorial types of the nodal sets involved. We provide full details of the proof of the inequality mult($\lambda_k$) $\le (2k - 2)$ ($k \ge 3$). Unfortunately, we are so far unable to complete the proof of the inequality mult($\lambda_k$) $\le (2k - 3)$, even when the domain is simply connected. In the last section, we explain the difficulties we met. Our work on the subject is still ongoing.

## CONTENTS

<table><tr><td>1. Introduction</td><td>3</td></tr><tr><td>2. Historical sketch</td><td>4</td></tr><tr><td>3. Eigenvalue problems</td><td>6</td></tr><tr><td>3.1. Definitions, notation and preliminary results</td><td>6</td></tr><tr><td>3.2. Local structure of eigenfunctions near a zero</td><td>8</td></tr><tr><td>3.3. Eigenfunctions with prescribed singular points</td><td>10</td></tr><tr><td>3.4. A global property of nodal sets</td><td>14</td></tr><tr><td>4. Partitions, graphs and Euler type formulas</td><td>15</td></tr><tr><td>4.1. Partitions</td><td>15</td></tr><tr><td>4.2. Graphs associated with a partition</td><td>17</td></tr><tr><td>4.3. Euler type formulas for partitions</td><td>19</td></tr><tr><td>4.3.1. The case of the sphere</td><td>20</td></tr><tr><td>4.3.2. The general case</td><td>20</td></tr></table>

Date: February 15, 2022 (berard-helffer-2022-HoMiNa-0215.tex).

Key words and phrases. Spectral theory, Möbius strip, Laplacian, Partitions.

We consulted several colleagues during the preparation of this paper: A. Berdnikov, L. Friedlander, T. Hoffmann-Ostenhof, P. Jammes, M. Karpukhin, J. Kennedy, Z. Liqun, N. Nadirashvili, I. Polterovich, S. Zelditch. We thank them all for their comments.
4.3.3. The case of the Möbius strip 21

4.4. Nodal partitions 21

5. Revisiting the multiplicity bounds for closed surfaces of genus 0 22

5.1. Introduction 23

5.2. Surfaces with genus zero, proof that $\dim U(\lambda_k) \le (2k-2)$ for $k \ge 3$ 23

5.2.1. Structure and combinatorial type of $\mathcal{Z}(u)$, for $0 \ne u \in U_x$ 24

5.2.2. The rotating function argument 25

5.3. Surfaces with genus zero, proof that $\dim U(\lambda_k) \le (2k-3)$ for $k \ge 3$ 27

6. Plane domains, bounding $\mult(\lambda_k)$ from above, preliminaries 28

6.1. Introduction 28

6.2. Notation 28

6.3. The initial inequalities 29

6.4. Multiplicity of the second eigenvalue and the nodal line conjecture 30

6.4.1. Nodal sets of second eigenfunctions 30

6.4.2. The nodal line conjecture 32

6.4.3. Dirichlet boundary condition 33

6.4.4. Neumann boundary condition 34

6.4.5. Robin boundary condition 34

7. Plane domains, the estimate $\mult(\lambda_k) \le (2k-2)$ for $k \ge 3$ 34

7.1. An abstract setting 35

7.2. Describing nodal sets near a boundary singular point 35

7.3. Analysis of eigenfunctions with two prescribed boundary singular points 36

7.3.1. $\Omega$ simply connected. Properties of $U_{x,y}$ 36

7.3.2. $\Omega$ simply connected. Structure and combinatorial type of nodal sets in $U_{x,y}$ 37

7.3.3. $\Omega$ with one hole. Properties of $U_{x,y}$ 38

7.3.4. $\Omega$ with one hole. Structure and combinatorial type of nodal sets in $U_{x,y}$ 39

7.3.5. $\Omega$ with $k$ holes. Properties of $U_{x,y}$ 40

7.3.6. $\Omega$ with $k$ holes. Structure and combinatorial type of nodal sets in $U_{x,y}$ 41

7.4. Analysis of eigenfunctions with one prescribed boundary singular point 43

7.4.1. Properties of $U_x^1$ and $U_x^2$ 43

7.4.2. Structure and combinatorial type of nodal sets in $U_x^1$ and $U_x^2$ 45

7.5. Application of the previous analysis 45

7.6. Proof that $\mult(\lambda_k) \le (2k-2)$ for all $k \ge 3$, assuming $\Omega$ to be simply connected 46

7.6.1. Preparation 46

7.6.2. Structure and combinatorial types for $v_1$ and $v_2$ 47

7.6.3. Labeling and comparing nodal domains 49

7.6.4. The rotating function argument 50

7.6.5. Proof that $\mult(\lambda_k) \le (2k-2)$ for $k \ge 3$, general case 52

8. Plane domains, properties of $\lambda_k$-eigenfunctions assuming that $\mult(\lambda_k) = (2k-2)$ 53

8.1. Eigenfunctions with one prescribed interior singular point 53

8.1.1. Properties of $U_x$ 53

8.1.2. Structure and combinatorial type of nodal sets in $U_x$ 55

8.2. Eigenfunctions with one prescribed boundary singular point 56
<table>
  <tbody>
    <tr>
      <td>8.2.1. Properties of W<sub>y</sub></td>
      <td>56</td>
    </tr>
    <tr>
      <td>8.2.2. Structure and combinatorial type of nodal sets in W<sub>y</sub></td>
      <td>57</td>
    </tr>
    <tr>
      <td>8.2.3. Properties of Γ<sub>1,(2k-3)</sub> and Γ<sub>1,(2k-2)</sub></td>
      <td>59</td>
    </tr>
    <tr>
      <td>9. Simply connected planar domains, properties of λ<sub>k</sub>-eigenfunctions assuming that mult(λ<sub>k</sub>) = (2k - 2)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>9.1. Boundary behaviour of the map x ↦→ u<sub>x</sub></td>
      <td>62</td>
    </tr>
    <tr>
      <td>9.2. Eigenfunctions with two prescribed boundary singular points</td>
      <td>67</td>
    </tr>
    <tr>
      <td>9.2.1. First properties of V<sub>y,s</sub></td>
      <td>68</td>
    </tr>
    <tr>
      <td>9.2.2. Structure and combinatorial type of nodal sets in V<sub>y,s</sub></td>
      <td>75</td>
    </tr>
    <tr>
      <td>9.2.3. Precise description of V<sub>y,s</sub></td>
      <td>77</td>
    </tr>
    <tr>
      <td>10. Possible steps towards a proof that mult(λ<sub>k</sub>) ≤ (2k - 3) for k ≥ 3</td>
      <td>84</td>
    </tr>
    <tr>
      <td>10.1. Preamble</td>
      <td>85</td>
    </tr>
    <tr>
      <td>10.2. Steps for the proof that mult(λ<sub>k</sub>) ≤ (2k - 3) when k ≥ 3</td>
      <td>85</td>
    </tr>
    <tr>
      <td>Appendix A. A remark on upper bounds for multiplicities vs Courant-sharp eigenvalues</td>
      <td>92</td>
    </tr>
    <tr>
      <td>Appendix B. The local structure of the nodal set in the neighborhood of a critical zero</td>
      <td>93</td>
    </tr>
    <tr>
      <td>Appendix C. Local structure of the nodal set near a critical zero, another approach</td>
      <td>96</td>
    </tr>
    <tr>
      <td>Appendix D. Labeling loops and nodal domains</td>
      <td>99</td>
    </tr>
    <tr>
      <td>D.1. Labeling of loops and bouquets of loops</td>
      <td>99</td>
    </tr>
    <tr>
      <td>D.2. Labeling of nodal domains</td>
      <td>100</td>
    </tr>
    <tr>
      <td>D.2.1. Working out an example</td>
      <td>100</td>
    </tr>
    <tr>
      <td>D.2.2. Proof of Property D.3</td>
      <td>102</td>
    </tr>
    <tr>
      <td>References</td>
      <td>105</td>
    </tr>
  </tbody>
</table>

# 1. INTRODUCTION

In these notes, we are concerned with upper bounds for the multiplicities of the eigenvalues {$\lambda_k, k \ge 1$} of an operator $-\Delta + V$ on a compact, smooth (ie. $C^\infty$), connected Riemannian surface (when the boundary $\partial M$ is not empty, we consider the Dirichlet, Neumann or Robin boundary conditions). We do not consider the Steklov eigenvalue problem for which we refer to the papers Karpukhin, Kokarev and Polterovich [KaKP2014], Fraser and Schoen [FrSc2016], Jammes [Jam2016], and their reference lists.

Our main purpose is to revisit the papers [HoHN1999] (closed surfaces with genus zero) and [HoMN1999] (planar domains with smooth boundary) whose proofs are not very detailed and often rely on figures and special configurations of nodal sets. We introduce and carefully study the *combinatorial type* (defined in Subsection 5.2) of the nodal sets involved. We provide a unified treatment for the three boundary conditions (Dirichlet, Neumann, Robin). We also illustrate our proofs with many figures.
In the sequel $\Delta$ is the Laplace-Beltrami operator for some smooth metric $g$, and $V$ is a smooth real valued function. We list the eigenvalues in nondecreasing order, multiplicities accounted for. Our convention is that, in all cases, we label the eigenvalues starting from the label 1,

$$ \lambda_1 < \lambda_2 \le \lambda_3 \le \dots, $$

and we denote the multiplicity of $\lambda_k$ by $\mathrm{mult}(\lambda_k)$. We refer to Section 3 for more precise definitions.

In Section 2 we provide a panorama of the main results on the multiplicity problem, and the ideas behind their proofs. Section 3 is devoted to definitions, notations and preliminary results. Section 4 deals with Euler type formulas in the framework of partitions. In Section 5, we revisit [HoHN1999]. Introducing the combinatorial type of some particular nodal sets, see Definition 5.4, we provide a complete proof of the inequality $\mathrm{mult}(\lambda_k) \le (2k-3)$ in the case of closed surfaces with genus 0, see Subsection 5.3. This section constitutes a preamble to the remaining sections. Sections 6 to 10 are devoted to the upper bounds on multiplicities for smooth bounded domains in $\mathbb{R}^2$. They form the core of this paper.

Section 6 contains some preliminary results, in particular a proof of Nadirashvili's estimate $\mathrm{mult}(\lambda_k) \le (2k-1)$ (in our restricted framework). In Section 7, we give a detailed proof of the upper bound $\mathrm{mult}(\lambda_k) \le (2k-2)$ for a smooth bounded domain $\Omega \subset \mathbb{R}^2$ (not assumed to be simply connected), see Proposition 7.17. The strategy of the proof is as follows: assume that $\mathrm{mult}(\lambda_k) = (2k-1)$, construct $\lambda_k$-eigenfunctions with prescribed singular points, analyze their nodal sets, and derive a contradiction.

In [HoMN1999, Theorem B, p. 1172], the authors state that the previous bound can be improved to $\mathrm{mult}(\lambda_k) \le (2k-3)$ for all $k \ge 3$. The strategy of the proof is similar, except that we now start from the assumption $\mathrm{mult}(\lambda_k) = (2k-2)$. Unfortunately, we have so far not succeeded in writing down complete details for the arguments given in [HoMN1999], even when $\Omega$ is simply connected (for comments, see Section 2 or [Berd2018, Section 4]). In Sections 8 and 9, we provide some properties of $\lambda_k$-eigenfunctions under the assumption that $\mathrm{mult}(\lambda_k) = (2k-2)$. In Section 10, we indicate the main steps of the proof sketched in [HoMN1999, Section 3], and we point out where we are stuck.

In Appendix A we relate the problem of bounding multiplicities from above to the question of Courant-sharp eigenvalues. The other appendices are devoted to technical results.

## 2. HISTORICAL SKETCH

In the case of closed surfaces, the first upper bounds on multiplicities were obtained by Cheng [Chen1976], Besson [Bess1980], and Nadirashvili [Nadi1987]. We denote their respective upper bounds on $\mathrm{mult}(\lambda_k)$ by $m_k^*$, with $* \in \{b, c, n\}$, where $b$ stands for “Besson”, $c$ for “Cheng”, and $n$ for “Nadirashvili”, and provide a summary of their results in Table 2.1.

The upper bounds for the multiplicity of the second eigenvalue (ie, the least positive eigenvalue of a closed surface) given in the fifth column are sharp. For the sphere the bound is achieved for the canonical (round) metric, [Chen1976]; for the projective space the bound is achieved for the metric induced by the canonical metric of the
sphere, [Bess1980]; for the torus the bound is achieved for the equilateral torus $T_e$ with metric induced from $\mathbb{R}^2$, [Bess1980]; for the Klein bottle, the bound is achieved for a nontrivial pair $(g, V)$ constructed in [Nadi1987, § 2], and for smooth metrics constructed in [ColV1987, Théorème 4.2]. An interesting feature of $\mathbb{S}^2, \mathbb{R}P^2$ and $T^2$ is that the bounds for $\mathrm{mult}(\lambda_2)$ are also achieved for metrics different from the ones mentioned above, see [Bess1980].

In [ColV1987, Théorème 1.5], Colin de Verdière shows that for a closed surface $M$, $\sup\{\mathrm{mult}(\lambda_2(M, -\Delta_g + V)) | (g, V)\} \ge C(M) - 1$,

where the supremum is taken over the Riemannian metrics and potentials on $M$, and where $C(M)$ is the chromatic number of $M$ (the maximal $N$ such that the complete graph on $N$ vertices $K_N$ can be embedded into $M$). Table 2.1 shows that equality holds for $\mathbb{S}^2, \mathbb{R}P^2, T^2$ and $K^2$; it also holds for surfaces with $\chi(M) \ge -3$, [Seve2002]. It is conjectured that equality holds for all closed surfaces.

<table><caption>TABLE 2.1. Closed surfaces: multiplicity upper bounds obtained by Cheng, Besson, and Nadirashvili</caption><thead><tr><th>M</th><th>χ(M)</th><th>γ(M)</th><th>γ̃(M)</th><th>mult(λ<sub>2</sub>) ≤</th><th>mult(λ<sub>k</sub>) ≤</th></tr></thead><tbody><tr><td>S<sup>2</sup></td><td>2</td><td>0</td><td>—</td><td>3</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = <span style="display: block; border-bottom: 1px solid black; padding: 0 5px;">1/2k(k + 1)</span></span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = 2k − 1</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k − 1</span></div></td></tr><tr><td>RP<sup>2</sup></td><td>1</td><td>—</td><td>0</td><td>5</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = not considered</span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = 4k − 1</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k + 1</span></div></td></tr><tr><td>T<sup>2</sup></td><td>0</td><td>1</td><td>—</td><td>6</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = <span style="display: block; border-bottom: 1px solid black; padding: 0 5px;">1/2(k + 2)(k + 3)</span></span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = 2k + 3</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k + 2</span></div></td></tr><tr><td>K<sup>2</sup></td><td>0</td><td>—</td><td>1</td><td>5</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = not considered</span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = not considered</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k + 1</span></div></td></tr><tr><td>M<sup>2</sup></td><td>χ(M) < 0</td><td>γ</td><td>—</td><td>—</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = <span style="display: block; border-bottom: 1px solid black; padding: 0 5px;">1/2(k + 2γ)(k + 2γ + 1)</span></span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = 2k + 4γ − 1</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k + 4γ − 3</span></div></td></tr><tr><td>M<sup>2</sup></td><td>χ(M) < 0</td><td>—</td><td>γ̃</td><td>—</td><td><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>c</sup> = not considered</span><span style="display: inline-block; padding: 0 5px;">m<sub>k</sub><sup>b</sup> = 4k + 4γ̃ − 1</span></div><div style="text-align: center;"><span style="display: inline-block; border-bottom: 1px solid black; padding: 0 5px;">m<sub>k</sub><sup>n</sup> = 2k + 2γ̃ − 1</span></div></td></tr></tbody></table>

Concerning Table 2.1, recall that for an orientable surface, $\chi(M) = 2$ (the sphere) or $\chi(M) = 2 - 2\gamma(M)$, where the genus $\gamma(M)$ is the number of handles; in this case, $M$ is the connected sum of $\gamma(M)$ tori. For a non-orientable surface, $\chi(M) = 1$ (the real projective space) or $\chi(M) = 2 - \tilde{\gamma}(M)$ if the surface is a connected sum of $\tilde{\gamma}(M)$ projective spaces.

Better upper bounds were later obtained by M. and T. Hoffmann-Ostenhof and Nadirashvili [HoHN1999] (surfaces with genus 0), Sévennec [Seve2002] (improved
bounds on the multiplicity of $\lambda_2$ when $\chi(M) < 0$), Berdnikov, Nadirashvili and Penskoi [BeNP2016] (improved bounds for the multiplicities on the projective plane), Fortier Bourque and Petri [FoBP2021] (Klein quartic).

In [Nadi1987, Theorem 2], Nadirashvili considers smooth bounded domains $\Omega \subset \mathbb{R}^2$ and proves that the multiplicity of the $k$th eigenvalue $\lambda_k$ of an operator $-\Delta + V$ with Dirichlet or Neumann boundary condition is at most $(2k-1)$.

In the paper [HoMN1999], Hoffmann-Ostenhof, Michor and Nadirashvili, improve Nadirashvili's bound for bounded planar domains with $C^\infty$ boundary with Dirichlet boundary condition. More precisely, they state that the multiplicity of $\lambda_k$ is at most $(2k-3)$. Berdnikov [Berd2018] considers the case of compact surfaces with boundary, under the assumption that $\chi(M) + b_0(\partial M)$ is negative. He points out some problem in the proof in [HoMN1999] when the domain is not simply connected.

The general idea to prove upper bounds for the eigenvalue multiplicities is a combination of the following ingredients:

(i) Courant's nodal domain theorem.

(ii) Local structure theorems for eigenfunctions near a singular point.

(iii) Existence of eigenfunctions with prescribed singular points, provided the dimension of the eigenspace is large enough.

(iv) Euler's formula for the graph associated with the nodal set of an eigenfunction.

(v) An argument which we call the *rotating function argument* (see § 5.2.2).

(vi) Energy arguments and eigenvalue monotonicity.

In one form or another, these arguments go back to Cheng [Chen1976], Besson [Bess1980], and Nadirashvili [Nadi1987].

We refer to the papers of Burger, Colbois and/or Colin de Verdière [BuCo1985, Colb1985, ColV1986, ColV1987, CoCo1988] for results of a different flavour.

Two other papers, respectively [HeHO1999] by Helffer, M. and T. Hoffmann-Ostenhof and Owen, and [HeHN2002] by Helffer, M. and T. Hoffmann-Ostenhof and Nadirashvili, have used the same techniques for related purposes (for example the Aharonov-Bohm operators).

Finally similar techniques are used in the analysis of the properties of minimal partitions [HeHT2009, BoHe2017].

### 3. EIGENVALUE PROBLEMS

**3.1. Definitions, notation and preliminary results.** In this section, $M$ denotes a closed surface (compact, no boundary), or a compact surface with boundary. The boundary is denoted by $\partial M$, and the interior $M\\partial M$ is denoted by $\text{int}(M)$. Unless otherwise stated, the surface is assumed to be smooth and connected. We equip $M$ with a smooth Riemannian metric $g$, and we consider a (non-magnetic) Schrödinger operator of the form $-\Delta_g + V$, where $\Delta_g$ is the Laplace-Beltrami operator for the metric $g$ and $V$ is a smooth real valued function on $M$.

The notation are as follows. The Riemannian measure is denoted by $v_g$; $h$ is a nonnegative given constant. When $\partial M \neq \emptyset$, $\sigma_g$ is the Riemannian measure of $\partial M$ for the metric induced by $g$, and $\nu$ is the unit normal to $\partial M$ pointing outward.
We consider the closed<fnref n="1" /> eigenvalue problem

$$ (3.1) \qquad -\Delta u + Vu = \lambda u \quad \text{in } M \quad (\text{here, } \partial M = \emptyset), $$

associated with the quadratic form

$$ (3.2) \qquad \int_M (|du|_g^2 + Vu^2) \, dv_g, \quad \text{with domain } H^1(M). $$

When $\partial M \neq \emptyset$, we consider the boundary eigenvalue problem

$$ (3.3) \qquad \left\{ \begin{array}{ll} -\Delta u + Vu = \lambda u & \text{in } \text{int}(M), \\ B(u) = 0 & \text{on } \partial M, \end{array} \right. $$

where $B(u)$ is one the following boundary conditions:

$$ (3.4) \qquad B(u) = \begin{cases} u & (\text{Dirichlet}), \\ \frac{\partial u}{\partial \nu} & (\text{Neumann}), \\ \frac{\partial u}{\partial \nu} + h u & (\text{h-Robin}). \end{cases} $$

The associated quadratic forms are

$$ (3.5) \qquad \int_M (|du|_g^2 + Vu^2) \, dv_g, \quad \text{with domain } H_0^1(M). $$

for the Dirichlet problem, and

$$ (3.6) \qquad \int_M (|du|_g^2 + Vu^2) \, dv_g + h \int_{\partial M} (u_{\partial M})^2 d\sigma_g, \quad \text{with domain } H^1(M), $$

for the Neumann problem (in this case $h = 0$) and for the h-Robin problem.

For these eigenvalue problems, the spectrum is discrete, and consists of a sequence of non-negative eigenvalues with finite multiplicities,

$$ (3.7) \qquad \lambda_1 < \lambda_2 \le \lambda_3 \le \cdots \nearrow \infty, $$

which we list in nondecreasing order, multiplicities accounted for, starting from the label 1.

**Remark 3.1.** Whenever necessary, we shall indicate the dependence on $M, g, V, h$ and the boundary condition, for example, $\lambda_k(M, g, V, \delta)$ for the Dirichlet eigenvalues of $-\Delta + V$ on $(M, g)$ with the Dirichlet boundary condition on $\partial M$.

In the sequel, we only consider *real valued* eigenfunctions. The *eigenspace* associated with the eigenvalue $\lambda_k$ will be denoted by $U(\lambda_k)$. Its dimension, the *multiplicity* of $\lambda_k$, will be denoted by $\text{mult}(\lambda_k)$ or $\text{dim } U(\lambda_k)$.

**Definitions 3.2 (Terminology).**

(i) The *nodal set* of a nontrivial eigenfunction $u$ is denoted by $\mathcal{Z}(u)$, and defined by

$$ (3.8) \qquad \mathcal{Z}(u) = \overline{\{x \in \text{int}(M) \mid u(x) = 0\}}. $$

When $\partial M \neq \emptyset$, $\mathcal{Z}(u)$ is the closure in $M$ of the set of interior zeros of $u$.

(ii) In dimension 2, the nodal set $\mathcal{Z}(u)$ of an eigenfunction $u$ is also called the *nodal line* of $u$.

<footnote n="1">¹By closed eigenvalue problem, we mean an eigenvalue problem on a closed manifold (no boundary condition).</footnote>
(iii) The nodal domains of the eigenfunction $u$ are the connected components<fnref n="2" /> of $\text{int}(M) \setminus \mathcal{Z}(u)$. We denote by $\kappa(u)$ the number of nodal domains of $u$.

According to Courant's nodal domain theorem, the number of nodal domains of an eigenfunction associated with $\lambda_k$ is at most $k$. Eigenfunctions associated with $\lambda_1$ are characterized by the fact that they have precisely one nodal domain. An eigenfunction associated with $\lambda_k, k \ge 2$, has at least two nodal domains. An eigenfunction associated with $\lambda_2$ has precisely two nodal domains. It turns out that, for $k$ large enough (depending on $M, g, V$), $\sup\{\kappa(u) | 0 \ne u \in U(\lambda_k)\} < k$ (see [Plej1956], [Polt2009], [Lena2019]), and that the bound $\kappa(u) \ge 2$, for $k \ge 2$, can generally speaking not be improved (see [Ster1925, Lewy1977], [BeHe2015s, BeHe2015r]).

**3.2. Local structure of eigenfunctions near a zero.** We say that a function $u$ vanishes at order $n \ge 1$ at a point $x$, and we write $\text{ord}(u, x) = n$, if (in a local coordinate system) the function and all its derivatives of order less than or equal to $(n-1)$ vanish at $x$, and at least one derivative of order $n$ does not vanish at $x$. A critical zero of $u$ is a point at which $u$ vanishes at order at least 2 (i.e. $u(x) = 0$ and $\nabla_x u = 0$). A critical zero $x$ of $u$ is called an interior critical zero if $x \in \text{int}(M)$, and a boundary critical zero if $x \in \partial M$.

**Theorem 3.3.** Let $u$ be a nontrivial eigenfunction of the Schrödinger operator $-\Delta + V$ on a smooth compact Riemannian surface $M$ (with or without boundary), where $V$ is a smooth real valued potential. Then, $u \in C^\infty(M)$, and does not vanish at infinite order at any point of $M$. Furthermore, depending on the boundary condition on $\partial M$, $u$ has the following properties.

(i) If $x_0 \in M$ is an interior point, and if $u$ has a zero of order $\ell$ at $x_0$, then there exist local polar coordinates $(r, \omega)$ centered at $x_0$ such that the Taylor expansion of $u$ is

$$ (3.9) \qquad u(x) = r^{\ell} (a \sin(\ell\omega) + b \cos(\ell\omega)) + \mathcal{O}(r^{\ell+1}), $$
where $a, b \in \mathbb{R}$, $a^2 + b^2 \ne 0$.

(ii) If $x_0 \in \partial M$, and if a Dirichlet eigenfunction $u$ has a zero of order $\ell$ at $x_0$, then there exist local polar coordinates $(r, \omega)$ centered at $x_0$, such that the Taylor expansion of $u$ is

$$ (3.10) \qquad u(x) = a r^{\ell} \sin(\ell\omega) + \mathcal{O}(r^{\ell+1}) $$
for some $a \in \mathbb{R}$, $a \ne 0$. The angle $\omega$ is chosen so that the tangent to the boundary at $x_0$ is given by the equation $\omega = 0$.

(iii) If $x_0 \in \partial M$, and if a Robin eigenfunction $u$ has a zero of order $\ell$ at $x_0$, then there exist local polar coordinates $(r, \omega)$ centered at $x_0$, such that the Taylor expansion of $u$ is

$$ (3.11) \qquad u(x) = b r^{\ell} \cos(\ell\omega) + \mathcal{O}(r^{\ell+1}) $$
for some $b \in \mathbb{R}$, $b \ne 0$. The angle $\omega$ is chosen so that the tangent to the boundary at $x_0$ is given by the equation $\omega = 0$.

<footnote n="2">²In the sequel, unless otherwise stated, we shall use the word *component* for the expression *connected component*.</footnote>
*Proof.* For a proof of this theorem, and for references to the literature, we refer to [GiHe2019, Appendix A] (which deals with the more delicate case in which the boundary of the domain is not $C^\infty$). The starting point is to use the unique continuation theorem, see [Aron1957] when $x_0$ is an interior point, and [DoFe1990a] when $x_0$ is a boundary point (see also [YaZh2021]). □

From a local point of view, we have the following properties.

**Corollary 3.4.**

(i) Let $x_0 \in M$. If $u$ has a zero of order $\ell$ at $x_0$, then exactly $\ell$ nodal curves pass through $x_0$. More precisely, in a neighborhood of $x_0 \in \text{int}(M)$, the nodal set $\mathcal{Z}(u)$ consists of $2\ell$ semi-arcs emanating from $x_0$ tangentially to the rays $\{\omega = \omega_j\}$ where $\omega_j := j\frac{\pi}{\ell}, 0 \le j < 2\ell$. The semi-tangents to these semi-arcs dissect the full unit circle in the tangent plane at $x_0$ into $2\ell$ equal parts.

(ii) Let $x_0 \in \partial M$. Let $u$ be a Dirichlet eigenfunction. If $u$ has a zero of order $\ell \ge 2$ at $x_0$, then exactly $(\ell - 1)$ semi-arcs hit $\partial M$ at $x_0$, their semi-tangents at $x_0$ dissect the half unit circle in the tangent plane at $x_0$ into $\ell$ equal angles given by the equation $\sin(\ell\omega) = 0$.

(iii) Let $x_0 \in \partial M$. Let $u$ be a Robin eigenfunction. If $u$ has a zero of order $\ell \ge 1$ at $x_0$, then exactly $\ell$ semi-arcs hit $\partial M$ at $x_0$, their semi-tangents at $x_0$ dissect the half unit circle in the tangent plane at $x_0$ into $\ell$ equal angles given by the equation $\cos(\ell\omega) = 0$.

*Proof.* Assertion (i) is proved in Appendix B. For Assertions (ii) and (iii), see the references in [GiHe2019, Appendix]. □

Points at which nodal arcs meet in the interior $\text{int}(M)$, and points at which the nodal set hits the boundary $\partial M$ play an important role in the global understanding of nodal sets. The terminology in the following definition comes from the framework of partitions, see Definition 4.2.

**Definition 3.5.** Define the singular points of an eigenfunction $u$ as follows.

(i) A point $x_0 \in \text{int}(M)$ is an interior singular point of $u$ if and only if it is an interior critical zero; the set of interior singular points of $u$ is denoted by $S_i(u)$. The index $\nu(u, x_0)$ of the interior singular point $x_0$ is defined as the number of nodal semi-arcs emanating from $x_0$, $\nu(u, x_0) = 2 \text{ord}(u, x_0)$.

(ii) A point $x_0 \in \partial M$ is a boundary singular point of $u$ if and only if the nodal set $\mathcal{Z}(u)$ hits the boundary $\partial M$ at $x_0$; the set of boundary singular points of $u$ is denoted by $S_b(u)$. The index $\rho(u, x_0)$ of the boundary singular point $x_0$ is defined as the number of nodal arcs hitting $\partial M$ at $x_0$. If $u$ is a Dirichlet eigenfunction, $\rho(u, x_0) = \text{ord}(u, x_0) - 1$; if $u$ is a Robin eigenfunction, $\rho(u, x_0) = \text{ord}(u, x_0)$.

The set $S(u)$ of singular points of $u$ is the set $S(u) = S_i(u) \cup S_b(u)$.

**Remark 3.6.** The order of vanishing is semi-continuous in the following sense. Let $\{v_n\}$ be a sequence of functions which converges to some $v$ uniformly in $C^{k+2}$ and let $\{x_n\}$ be a sequence of points which converges to some $x$ in $M$. Assume that $\text{ord}(v_n, x_n) \ge k$ for all $n$ and some fixed $k$, then $\text{ord}(v, x) \ge k$. Since they are defined in terms of order of vanishing, the indices $\nu$ and $\rho$ inherit this property.
From a more global point of view, the connected components of $\mathcal{Z}(u) \setminus \mathcal{S}(u)$ are smooth 1-dimensional submanifolds homeomorphic to either circles or open intervals whose boundaries consist of singular points.

**Definitions 3.7 (Terminology).**

(i) We call a circle-like component, a nodal circle; we call an interval-like component, a nodal arc.

(ii) Let $I_{x,y}$ be a nodal arc with boundary $\{x, y\}$. In this case, we call $I_{x,y} \cup \{x, y\}$ a nodal semi-arc emanating from $x$ and arriving at $y$, with some semi-tangents at $x$ and $y$ given by the local structure theorem. The point $x$ (resp. $y$) might be an interior singular point, or a boundary singular point (if the nodal set $\mathcal{Z}(u)$ hits the boundary $\partial M$ at $x$, resp. $y$). If $x = y$, we say that the component $I_{x,x}$ is a nodal loop at $x$. In this case, the semi-tangents at $x$ might not be co-linear, so that the (closed) loop is not a smooth circle, but a continuous, piecewise $C^1$ circle.

(iii) The collection {singular points, nodal circles and nodal arcs} constitutes a multigraph called the nodal graph of $u$, see Subsection 4.2.

**Corollary 3.8.**

(1) The nodal set of $u$ is the union of the finitely many singular points, nodal circles in the interior of $M$, and nodal arcs some of which may hit $\partial M$.

(2) Each component of $\partial M$ is hit by an even number of nodal arcs. If $\Gamma$ is a component of $\partial M$, then

$$ \sum_{z \in S_b(u) \cap \Gamma} \rho(u, z) \in 2\mathbb{N}. $$

*Proof.* The first assertion is well-known. We give the proof of the second assertion for completeness.

◇ Dirichlet case. The component $\Gamma$ is topologically a circle which meets $S_b(u)$ at finitely many points $z_j$, $1 \le j \le k$, which are precisely the zeros of the normal derivative $\partial_\nu u(z)$. Choosing a parametrization $z$ of $\Gamma$ and taking the local structure of $u$ at each $z_j$ into account, we see that each time $z$ passes some $z_j$, the sign of $\partial_\nu u$ is multiplied by $(-1)^{\rho(z_j)}$.

◇ Robin case. The proof is similar, actually simpler. $\square$

<figure><img src="image_4.png" /></figure>

**3.3. Eigenfunctions with prescribed singular points.** In order to bound multiplicities, we will use eigenfunctions with prescribed singular points of sufficiently high index. Their existence is given by the following lemmas. These lemmas appear in one form or another in [Chen1976, Theorem 3.4], [Bess1980, Theorem 2.1], [Nadi1987, Lemma 4], [HoHN1999, Proposition 2], [HoMN1999, Lemma 2.9].

The first lemma prescribes an interior singular point.

**Lemma 3.9.** Let $M$ be a compact surface (with or without boundary), and $x$ an interior point. Let $U$ be a linear subspace of an eigenspace of the eigenvalue problem (3.1) or (3.3), with $\dim U = m \ge 2$.

(i) There exists a function $0 \ne u \in U$ such that $x$ is a singular point of $u$ with index $\nu(u, x) \ge 2$ $\lfloor \frac{m}{2} \rfloor$ (the integer part of $\frac{m}{2}$), equivalently $\text{ord}(u, x) \ge \lfloor \frac{m}{2} \rfloor$.

(ii) Furthermore, if $m$ is odd, there exist at least two linearly independent such functions.
*Proof.* We use induction on $m$. Recall that $\nu(u, x) = 2 \operatorname{ord}(u, x)$. The assertion is clear when $m = 2$. Assume $m = 3$, and let $\{u_1, u_2, u_3\}$ be a basis of $U$. Then, we can find $0 \neq v_1 \in \operatorname{span}\{u_1, u_2\}$ such that $\operatorname{ord}(v_1, x) \geq 1$. The subspace $V_1$ of $U$ orthogonal to $v_1$ has dimension 2, and hence there exists $0 \neq v_2 \in V_1$ such that $\operatorname{ord}(v_2, x) \geq 1$. Then $v_1$ and $v_2$ are two linearly independent functions in $U$ vanishing at order at least 1 at $x$.

Assume that the lemma holds for $2p$ and $2p + 1$ for some $p \geq 1$.

Let $U$ be linear subspace of an eigenspace with dimension $(2p + 2)$, and basis

$$ \{u_1, \dots, u_{2p+2}\}. $$

By the induction hypothesis, in the subspace $V_1 := \operatorname{span}\{u_1, \dots, u_{2p+1}\}$, we can find two linearly independent functions $v_1, v_2$ such that $\operatorname{ord}(v_i, x) \geq p$. If one of them vanishes at order at least $(p+1)$, the assertion for $U$ is satisfied. If not, according to Theorem 3.3 (i), there exist $(a_i, b_i)$, $i = 1, 2$, with $a_i^2 + b_i^2 \neq 0$, such that

$$ v_i = r^p (a_i \sin(p\omega) + b_i \cos(p\omega)) + \mathcal{O}(r^{p+1}). $$

The subspace $V_2$ of $U$ orthogonal to $v_1$ and $v_2$ has dimension $2p$ and hence, there exists $0 \neq v_3 \in V_2$ such that $\operatorname{ord}(v_3, x) \geq p$. If $v_3$ vanishes at order at least $(p+1)$ at $x$, we are done. Otherwise, there exist $(a_3, b_3)$, with $a_3^2 + b_3^2 \neq 0$ such that

$$ v_3 = r^p (a_3 \sin(p\omega) + b_3 \cos(p\omega)) + \mathcal{O}(r^{p+1}). $$

The functions $r^p(a \sin(p\omega) + b \cos(p\omega))$ are the homogeneous harmonic polynomials of degree $p$ in $\mathbb{R}^2$, a vector space of dimension 2. The three polynomials $r^p(a_i \sin(p\omega) + b_i \cos(p\omega))$, $i \in \{1, 2, 3\}$ must be linearly dependent, and hence there exists a nontrivial linear combination of $v_1, v_2, v_3$ which vanishes at order at least $(p+1)$ at $x$.

Let $U$ be an eigenspace with dimension $2p + 3$, with basis $\{u_1, \dots, u_{2p+3}\}$. By the previous proof, in the subspace $V_1 := \operatorname{span}\{u_1, \dots, u_{2p+2}\}$, there exists $0 \neq v_1$ such that $\operatorname{ord}(v_1, x) \geq (p+1)$. For the same reason, in the subspace $V_2$ orthogonal to $v_1$, there exists $0 \neq v_2$ such that $\operatorname{ord}(v_2, x) \geq (p+1)$. The functions $v_1, v_2$ are two linearly independent functions in $U$ vanishing at order at least $(p+1)$ at $x$.

The proof of Lemma 3.9 is complete. $\square$

The next lemmas prescribe respectively one or two boundary singular points.

**Lemma 3.10.** Let $M$ be a compact surface with boundary, and $x \in \partial M$. Let $U$ be a linear subspace of an eigenspace of the eigenvalue problem (3.3), with $\dim U = m \geq 2$. Then, there exists a function $0 \neq u \in U$ such that $x$ is a boundary singular point of $u$ with index $\rho(u, x) \geq (m-1)$.

*Proof.* We use induction on $m$. Recall that $\rho(u, x) = (\operatorname{ord}(u, x) - 1)$ for Dirichlet eigenfunctions, resp. $\rho(u, x) = \operatorname{ord}(u, x)$ for Robin eigenfunctions.

◇ Dirichlet boundary condition. When $m = 2$, the assertion is clear. Assume it is true for some $m \geq 2$. Let $U$ be a linear subspace of an eigenspace with dimension $(m+1)$, and basis $\{u_1, \dots, u_{m+1}\}$. Consider the subspace $V_1 = \operatorname{span}\{u_1, \dots, u_m\}$. By the induction hypothesis, there exists $0 \neq v_1 \in V_1$ such that $\operatorname{ord}(v_1, x) \geq m$.
If $v_1$ vanishes at order at least $(m+1)$, we are done. Otherwise, by Theorem 3.3, Equation (3.10), there exists $a_1 \neq 0$ such that, in local polar coordinates at $x$,

$$v_1(z) = a_1 r^m \sin(m \omega) + \mathcal{O}(r^{m+1}).$$

The subspace $V_2 = \{u \in U \mid u \perp v_1\}$ orthogonal to $v_1$ has dimension $m$, and hence there exists $0 \neq v_2 \in V_2$ such that $\mathrm{ord}(v_2, x) \geq m$. If $v_2$ vanishes at order at least $(m+1)$, we are done. Otherwise, as above we can write

$$v_2(z) = a_2 r^m \sin(m \omega) + \mathcal{O}(r^{m+1})$$

for some $a_2 \neq 0$, and hence the linear combination $v = a_2v_1 - a_1v_2$ vanishes at order at least $(m+1)$. ✓

◇ Robin boundary condition. When $m = 2$, the assertion is clear. Assume it is true for some $m \geq 2$. Let $U$ be a linear subspace of an eigenspace with dimension $(m+1)$, with basis $\{u_1, \dots, u_{m+1}\}$. Consider the subspace $U_1 = \mathrm{span}\{u_1, \dots, u_m\}$. By the induction hypothesis, there exists $0 \neq v_1 \in U_1$ such that $\mathrm{ord}(v_1, x) \geq (m-1)$. If $v_1$ vanishes at order at least $m$, we are done. Otherwise, by Theorem 3.3, Equation (3.11), there exists $b_1 \neq 0$ such that, in local polar coordinates at $x$,

$$v_1(z) = b_1 r^m \cos((m-1)\omega) + \mathcal{O}(r^{m+1}).$$

We can then consider the subspace $U_2$ orthogonal to $v_1$ in $U$, and conclude by arguing as above. ✓

The proof of Lemma 3.10 is complete.

<figure><img src="image_5.png" /></figure>

**Lemma 3.11.** Let $M$ be a compact surface with boundary, and $x, y \in \partial M$, with $x \neq y$. Let $U$ be a linear subspace of an eigenspace of the eigenvalue problem (3.3), with $\dim U = m \geq 3$. Then, there exists a function $0 \neq u \in U$ such that $x$ and $y$ are boundary singular points of $u$ with indices $\rho(u, x) \geq (m-2)$ and $\rho(u, y) \geq 1$.

*Proof.*

◇ Dirichlet boundary condition. Choose $\{u_1, \dots, u_m\}$ a basis of $U$. Looking at a general element $u = \sum \alpha_j u_j$ in $U$, the condition at $y$ reads

$$\sum_{j=1}^{m} \alpha_j (\partial_{\nu} u_j)(y) = 0.$$

There are two cases.

* If $\partial_{\nu} u_j(y) = 0$ for all $j$, the condition at $y$ is satisfied for any $u \in U$;

* If $\partial_{\nu} u_j(y) \neq 0$ for some $j$, then there exists a subspace $U' \subset U$ of dimension $(m-1) \geq 2$ such that the condition at $y$ is satisfied for any $u \in U'$.

We can then apply Lemma 3.10 with $U$ in the first case and with $U'$ in the second case. ✓

◇ Robin boundary condition. The condition $\rho(u, y) \geq 1$ holds if and only if $u$ vanishes at $y$. Since $m \geq 3$ there exists a linear subspace $U' \subset U$, with $\dim U' \geq (m-1) \geq 2$ such that any $u \in U'$ satisfies $u(y) = 0$. Then, Lemma 3.10 implies that there exists $0 \neq u \in U'$ such that $\rho(u, x) \geq (m-2)$.

The proof of Lemma 3.11 is complete.

<figure><img src="image_6.png" /></figure>

**Lemma 3.12.** Let $M$ be a compact surface. Let $U$ be a linear subspace of an eigenspace of the eigenvalue problem (3.1) or (3.3).
(i) Let $x \in \text{int}(M)$, and let $u_1, u_2, u_3$ be three linearly independent functions in $U$, such that $\nu(u_1, x) = \nu(u_2, x) = \nu(u_3, x) \ge 2$. Then, there exists $0 \ne u \in \text{span}\{u_1, u_2, u_3\}$ such that $\nu(u, x) \ge \nu(u_1, x) + 2$.

(ii) Let $x \in \partial M$, and let $u_1, u_2$ be two linearly independent functions in $U$, such that $\rho(u_1, x) = \rho(u_2, x) \ge 1$. Then, there exists $0 \ne u \in \text{span}\{u_1, u_2\}$ such that $\rho(u, x) \ge \rho(u_1, x) + 1$.

*Proof.* Since the index of a singular point can be expressed in terms of the vanishing order, the lemma follows from Theorem 3.3. Indeed, under the assumption of Assertion (i), we can write

$$u_i(z) = p_i(z - x) + \mathcal{O}(|z - x|^{k+1}),$$

in local coordinates centered at $x$, where $p_i$ is a nonzero harmonic homogeneous polynomial of degree $k = \frac{\nu(u_1, x)}{2}$ in two variables. Since the vector space of such polynomials has dimension 2, there exist real numbers $\alpha_1, \alpha_2$ and $\alpha_3$, not all of them equal to zero, such that $\alpha_1 p_1 + \alpha_2 p_2 + \alpha_3 p_3 = 0$. It follows that $\alpha_1 u_1 + \alpha_2 u_2 + \alpha_3 u_3$ vanishes at order at least $(k+1)$ at $x$. This proves Assertion (i).

The proof of Assertion (ii) is similar, using the local forms (3.10) or (3.11) depending on the boundary condition, Dirichlet or Robin. $\square$

For later purposes, we introduce the following notation.

**Notation 3.13.** Let $u$ be an eigenfunction of the eigenvalue problem (3.3) in the compact surface $M$. Define the function $\check{u}$ on $\partial M$ by

$$(3.12) \qquad \check{u} = \begin{cases} u|_{\partial M} & \text{in the Robin case,} \\ \partial_{\nu} u & \text{in the Dirichlet case.} \end{cases}$$

Then, for any $y \in \partial M$, $\rho(u, y) \ge 1$ if and only if $\check{u}(y) = 0$.

The following lemma will be useful later on.

**Lemma 3.14.** Let $u$ be an eigenfunction of the eigenvalue problem (3.3).

(i) If $u$ is a Dirichlet eigenfunction, and $y \in \partial M$, then $u$ vanishes at order $k$ at $y$ if and only if the function $\partial_{\nu} u$ vanishes at order $(k-1)$ at $y$ along $\partial M$.

(ii) If $u$ is a Robin eigenfunction, and $y \in \partial M$, then $u$ vanishes at order $k$ at $y$ if and only if the function $u|_{\partial M}$ vanishes at order $k$ at $y$ along $\partial M$.

Therefore, the order of vanishing of the function $\check{u}$ at some boundary point $y$ is precisely the number $\rho(u, y)$ of nodal arcs hitting $\partial M$ at $y$.

*Proof.* The proof is by induction on $k$. The equation $\Delta u = (V - \lambda)u$ implies relations between the derivatives of $u$ of degree $k$, evaluated at $y$, assuming that the derivatives of order less than or equal to $(k-1)$ vanish at $y$.

More precisely, according to [YaZh2021, Section 2], fixing some $y \in \partial M$, we can choose local boundary isothermal coordinates at $y$ such that the equation $(-\Delta + V)u = \lambda u$ in a neighborhood of $y$ is transformed into the equation

$$(e) \qquad \Delta v = Av$$

in some half-ball $\{(\xi_1, \xi_2) \in \mathbb{R}^2 \mid \xi_1^2 + \xi_2^2 < a^2, \xi_2 > 0\}$, where $0$ is the image of $y$. Here, $\Delta$ is the ordinary Laplacian in the variables $(\xi_1, \xi_2)$, $a$ is some given positive number, $A$ and $v$ are $C^\infty$ up to the boundary, and correspond to $(V - \lambda)$ and $u$ respectively.

In the proof, we use the following conventions.
◇ The symbol $\stackrel{t}{\equiv}$ indicates a trivial identity.

◇ The symbol $\stackrel{e}{\equiv}$ indicates an identity which follows from the above identity (e).

◇ The symbol $\stackrel{(m)}{\equiv}$ indicates an identity which holds up to a linear combination of derivatives of $v$ of order less than or equal to $m$.

◇ The symbol $\partial^{(p,q)}$ stands for $\frac{\partial^{p+q}}{\partial\xi_1^p \partial\xi_2^q}$.

For $k \ge 2$, we have

$$ \begin{aligned} \partial^{(k-2q,2q)} v &\stackrel{t}{\equiv} \partial^{(k-2q,2q-2)} \partial^{(0,2)} v \\ &\stackrel{e}{\equiv} \partial^{(k-2q,2q-2)} (-\partial^{(2,0)} v + Av) \\ &\stackrel{(k-2)}{\equiv} -\partial^{(k-2q+2,2q-2)} v. \end{aligned} $$

Assuming that $u$ vanishes at order larger than or equal to $(k-1)$ at $(0,0)$, we obtain that

$$ (a) \quad \partial^{(k-2q,2q)} u(0,0) = (-1)^q \partial^{(k,0)} u(0,0), \text{ for } q \in \left\{0, 1, \dots, \left\lfloor \frac{k}{2} \right\rfloor\right\}. $$

Similarly,

$$ \begin{aligned} \partial^{(k-2q-1,2q+1)} v &\stackrel{t}{\equiv} \partial^{(k-2q-1,2q-1)} \partial^{(0,2)} v \\ &\stackrel{e}{\equiv} \partial^{(k-2q-1,2q-1)} (-\partial^{(2,0)} v + Av) \\ &\stackrel{(k-2)}{\equiv} -\partial^{(k-2q+1,2q-1)} v. \end{aligned} $$

Assuming that $u$ vanishes at order larger than or equal to $(k-1)$ at $(0,0)$, we obtain that

$$ (b) \quad \partial^{(k-2q-1,2q+1)} u(0,0) = (-1)^q \partial^{(k-1,1)} u(0,0), \text{ for } q \in \left\{0, 1, \dots, \left\lfloor \frac{k-1}{2} \right\rfloor\right\}. $$

◇ Dirichlet case. In this case, $\check{v}(\xi_1) := \partial^{(0,1)}v(\xi_1, 0)$. Since $v(\xi_1, 0) \equiv 0$, Equation (a) implies that $\partial^{(k-2q,2q)}v(0,0) = 0$ for all $q \in \{0, \dots, \left\lfloor \frac{k}{2} \right\rfloor\}$. If $v$ vanishes at order greater than or equal to $(k-1)$, Equation (b) implies that $\partial^{(k-2q-1,2q+1)}v(0,0) = (-1)^q \partial^{(k-1)}\check{v}(0)$ for all $q \in \{0, \dots, \left\lfloor \frac{k-1}{2} \right\rfloor\}$. It follows that if $v$ vanishes at order greater than or equal to $(k-1)$, then $v$ vanishes at order greater than or equal to $k$ if and only if $\partial^{(k-1)}\check{v}(0) = 0$.

◇ Robin case. In this case, $\check{v}(\xi_1) := v(\xi_1, 0)$. Assuming that $v$ vanishes at order at least $(k-1)$, Equation (a) implies that $\partial^{(k-2q,2q)}v(0,0) = (-1)^q \partial^k \check{v}(0)$. Since $\partial^{(0,1)}v(\xi_1, 0) \equiv B(\xi_1)u(\xi_1, 0)$ (Robin condition), Equation (b) implies that $\partial^{(k-2q-1,2q+1)}u(0,0) = 0$. Therefore, if $v$ vanishes at order at least $(k-1)$ at $(0,0)$, then $v$ vanishes at order at least $k$ if and only if $\partial^k \check{v}(0) = 0$.

We have proved that $v$ vanishes at order at least $k$ (resp. equal to $k$) at $(0,0)$ if and only if $\check{v}$ vanishes at order $\rho(v, (0,0))$ at $(0,0)$. $\square$

### 3.4. A global property of nodal sets.

**Lemma 3.15.** Let $(M, g)$ be a compact Riemannian surface. Let $w_n, w: M \to \mathbb{R}$ be continuous functions with zero sets $K_n := w_n^{-1}(0)$ and $K := w^{-1}(0)$. Assume that $w_n \to w$ uniformly.
(i) The limit points of the sequence {$K_n$} with respect to the Hausdorff distance associated with the Riemannian distance of $(M, g)$ are compact and contained in $K$. They are connected if the sets $K_n$ are connected.

(ii) If $K_n, K$ are nodal sets of eigenfunctions, then the sequence {$K_n$} converges to $K$ in the Hausdorff distance.

*Proof.* The properties that the sequence {$K_n$} has limit points, and that they are compact (and connected if the sets $K_n$ are connected) are general. If a subsequence {$K_{n,1}$} tends to $K'$ in the Hausdorff distance, for any $z' \in K'$ there exists a sequence $z_n$ with $z_n \in K_n$, such that $z_n \to z'$. Write $w(z') = w(z') - w(z_n) + w(z_n) - w_n(z_n)$, and use the fact that $w_n \to w$ uniformly to conclude that $K' \subset K$. The second assertion uses the nodal character: assume that there exists $z \in K \setminus K'$. Then $d(z, K') =: 2\eta > 0$, and for $n$ large enough, $d(z, K_n) > \eta$. Since $K$ is a nodal set, there exists a small arc through $z$, from $z_-$ to $z_+$ such that $w(z_-) < 0$ and $w(z_+) > 0$. It follows that for $n$ large enough we also have $w_n(z_-) < 0$ and $w_n(z_+) > 0$. This shows that $w_n$ must vanish on the small arc, and hence there is some $z'_n \in K_n$ close to $z$ contradicting the fact that $d(z, K_n) > \eta$. Since the only possible limit point of {$K_n$} is $K$, the assertion follows. □

## 4. PARTITIONS, GRAPHS AND EULER TYPE FORMULAS

In this section, $M$ denotes a closed surface (compact, no boundary), or a compact surface $M$ with nonempty boundary $\partial M$, and interior $\text{int}(M)$. Unless otherwise stated, the surfaces are assumed to be smooth and connected. We are interested in Euler type formulas for partitions of $M$, in particular for partitions associated with the eigenfunctions of the eigenvalue problems (3.1) or (3.3).

**4.1. Partitions.** We first recall (or modify) some of the definitions given in the papers [BeHe2014], [BoHe2017].

A $k$-partition of $M$ is a collection, $\mathcal{D} = \{D_j\}_{j=1}^k$, of $k$ pairwise disjoint, connected, open subsets of $M$. We furthermore assume that the $D_j$'s are piecewise $C^1$, and that

$$ (4.1) \qquad \int(\cup_j \overline{D_j}) = \int(M). $$

The set of such partitions is denoted by $\mathfrak{D}_k(M)$.

The boundary set $\partial \mathcal{D}$ of a partition $\mathcal{D} = \{D_j\}_{j=1}^k \in \mathfrak{D}_k(M)$ is the closed set,

$$ (4.2) \qquad \partial \mathcal{D} = \overline{\cup_j (\partial D_j \cap M)}. $$

**Definition 4.1.** A partition $\mathcal{D} = \{D_j\}_{j=1}^k$ is called *essential* if, for all $j$, $1 \le j \le k$,

$$ (4.3) \qquad \int(\overline{D_j}) = D_j. $$

**Definition 4.2.** A regular $k$-partition is a $k$-partition whose boundary set $\partial \mathcal{D}$ satisfies the following properties:

(i) The boundary set $\partial \mathcal{D}$ is locally a piecewise $C^1$ immersed curve in $M$, and it is embedded except possibly at finitely many points {$y_i \in \partial \mathcal{D} \cap M$} in a neighborhood of which $\partial \mathcal{D}$ is the union of $\nu(y_i) C^1$ semi-arcs meeting at $y_i$, $\nu(y_i) \ge 3$. These points are called *interior singular points*. The integer $\nu(y_i)$ is called the *index of y$_i$*.
(ii) The set $\partial\mathcal{D} \cap \partial M$ consists of finitely many points $\{z_j\}$. Near the point $z_j$, the set $\partial\mathcal{D}$ is the union of $\rho(z_j) \ge 1$ $C^1$ semi-arcs hitting $\partial M$ at $z_j$. These points are called *boundary singular points*. The integer $\rho(z_j)$ is called the *index of $z_j$*.

(iii) The boundary set $\partial\mathcal{D}$ has the following *transversality property*: at any interior singular point $y_i$, the semi-arcs meet transversally; at any boundary singular point $z_j$, the semi-arcs meet transversally, and they meet the boundary $\partial M$ transversally.

The subset of regular $k$-partitions is denoted by $\mathcal{R}_k(M) \subset \mathcal{D}_k(M)$. When $\mathcal{D}$ is a regular partition, we denote by $S(\mathcal{D}) = S_i(\mathcal{D}) \cup S_b(\mathcal{D})$ the set of singular points of $\partial\mathcal{D}$, where $S_i(\mathcal{D})$ denotes the set of interior singular points, and $S_b(\mathcal{D})$ the set of boundary singular points.

**Definition 4.3.** A regular $k$-partition $\mathcal{D} = \{D_j\}_{j=1}^k$ is called *normal*, if it satisfies the additional condition, for all $j$, $1 \le j \le k$,

$$ (4.4) \qquad \forall x \in \partial D_j, \exists r > 0 \text{ s.t. } B(x, r) \cap D_j \text{ is connected.} $$

**Remark 4.4.** The definition of a normal partition implies that each domain in the partition is a topological surface with boundary (actually a piecewise $C^1$ surface with boundary, possibly with corners). A normal partition is essential.

**Example 4.5.** Nodal partitions are special examples of essential, regular partitions. They are not necessarily normal, see Subsection 4.4.

**Notation 4.6.** For a partition $\mathcal{D} \in \mathcal{D}(M)$, we introduce the following numbers.

(a) $\beta(\mathcal{D})$ is defined as $\beta(\mathcal{D}) = b_0(\partial\mathcal{D} \cup \partial M) - b_0(\partial M)$, the difference between the number of components of $\partial\mathcal{D} \cup \partial M$, and the number of components of $\partial M$;

(b) $\kappa(\mathcal{D})$ denotes the number of domains of the partition;

(c) $\sigma(\mathcal{D}) = \sigma_i(\mathcal{D}) + \sigma_b(\mathcal{D})$ weighs the singular points of a regular partition $\mathcal{D}$,

$$ \left\{ \begin{array}{l} \sigma_i(\mathcal{D}) = \frac{1}{2} \sum_{x \in S_i(\mathcal{D})} (\nu(x) - 2), \\ \sigma_b(\mathcal{D}) = \frac{1}{2} \sum_{x \in S_b(\mathcal{D})} \rho(x); \end{array} \right. $$

(d) $\omega(\mathcal{D})$ denotes the *orientability character* of the partition,

$$ \left\{ \begin{array}{l} \omega(\mathcal{D}) = 0, \text{ if all the domains of the partition are orientable,} \\ \omega(\mathcal{D}) = 1, \text{ if at least one domain of the partition is non-orientable.} \end{array} \right. $$

Obviously, $\omega(\mathcal{D}, M) = 0$ whenever the surface $M$ is orientable.

**Remarks 4.7.**

(i) We use the definition of orientability given in [BeGo1988, Chap. 5.3] (via differential forms of degree 2), or the similar form given in [GaXu2013, Chap. 4.5] in the setting of topological manifolds (via the degree). A topological surface is orientable if one can choose an atlas whose changes of chart are homeomorphisms with degree 1.

(ii) A compact surface (with boundary) is non-orientable if and only if it contains the homeomorphic image of a Möbius strip. One direction is clear since the Möbius strip is not orientable. For the other direction, one can
use the classification of compact surfaces (with boundary), see for example [GaXu2013].

**Lemma 4.8 (Normalization).** Let $\mathcal{D}$ be an essential, regular partition of $M$. Then, one can construct a normal partition $\tilde{\mathcal{D}}$ of $M$ such that

$$ \left\{ \begin{array}{l} \beta(\tilde{\mathcal{D}}) = \beta(\mathcal{D}), \\ \kappa(\tilde{\mathcal{D}}) - \sigma(\tilde{\mathcal{D}}) = \kappa(\mathcal{D}) - \sigma(\mathcal{D}), \\ \omega(\tilde{\mathcal{D}}) = \omega(\mathcal{D}). \end{array} \right. $$

*Proof.* Using condition (4.3), we see that an essential, regular partition $\mathcal{D} = \{D_j\}_{j=1}^k$ is normal except possibly at points $x$ in $S(\mathcal{D})$, with index bigger than 1, and for which there exists some domain $D_j$ such that $B(x, \varepsilon) \cap M_j$ has at least two components for all $\varepsilon > 0$. Here, $B(x, \varepsilon)$ denotes the disk with center $x$ and radius $\varepsilon$ in $M$ (we can for example fix a Riemannian metric on $M$ to have a distance function). Let $x$ be such a point. For $\varepsilon$ small enough, introduce the partition $\mathcal{D}_x$ whose elements are the $D_j \setminus \overline{B(x, \varepsilon)}$, and the extra domain $B(x, \varepsilon) \cap M$. In this procedure, we have $\kappa(\mathcal{D}_x) = \kappa(\mathcal{D}) + 1$; an interior singular point $x$, with $\nu(x) \ge 4$ is replaced by $\nu(x)$ singular points of index 3 for which condition (4.4) is satisfied. Hence, $\sigma(\mathcal{D}_x) = \sigma(\mathcal{D}) + 1$. It follows that $\kappa(\mathcal{D}_x) - \sigma(\mathcal{D}_x) = \kappa(\mathcal{D}) - \sigma(\mathcal{D})$. A similar procedure is applied at a boundary singular point. This is illustrated by Figure 4.1. By recursion, we can in this way eliminate all the singular points at which condition (4.4) is not satisfied. This procedure does not change $\beta$; choosing $\varepsilon$ small enough, it does not change $\omega$ either, since the added disks are orientable. $\square$

<figure><img src="image_7.png" /><figcaption>FIGURE 4.1. Normalization at a singular point</figcaption></figure>

**Remarks 4.9.** (i) By the same process, one could also remove all singular points with index $\nu > 3$ or $\rho > 1$. We do not need to do that for our purposes. (ii) We will use the previous lemma to prove an Euler-type formula for partitions of the Möbius strip, see Theorem 4.21 and [BeHe2020m]

**Notation 4.10.** We occasionally use the notation $\kappa(\mathcal{D}, M)$ for $\kappa(\mathcal{D})$, $S(\mathcal{D}, M)$ for $S(\mathcal{D})$, etc. if we need stress the fact that $\mathcal{D}$ is a partition of $M$.

## 4.2. Graphs associated with a partition.

**Notation 4.11.** We use the definition of a graph given in [Gibl2010] (i.e. “graph = simple graph”). Given a graph $G$, we denote by $\alpha_0(G)$ the number of vertices, by $\alpha_1(G)$ the number of edges, and by $c(G)$ the number of components of $G$. For a graph $G$ embedded in a surface $M$, we denote by $r(G, M)$ the number of components of $M \setminus G$.
Let $M$ be a compact surface, with boundary $\partial M$, possibly empty. Let $\mathcal{D}$ be a regular partition of $M$, with boundary set $\partial \mathcal{D}$, and singular set $S(\mathcal{D})$. The general idea is to associate a graph to the set $\partial \mathcal{D} \cup \partial M$. The vertices of such a graph should comprise the singular points of $\mathcal{D}$, and the edges should comprise both sub-arcs of $\partial \mathcal{D}$ and sub-arcs of $\partial M$. One snag is that the components of $(\partial \mathcal{D} \cup \partial M) \setminus S(\mathcal{D})$ might not all be intervals. Indeed, some of them might be circles (components of $\partial \mathcal{D} \cup \partial M$ which do not intersect $S(\mathcal{D})$). For this reason, we first define a 'graph-to-be' $G_0 := G_0(\mathcal{D}, M)$, as follows.

* Let $e := e(\mathcal{D}, M)$ be the number of components of $(\partial \mathcal{D} \cup \partial M) \setminus S(\mathcal{D})$ which are not an interval. We first choose one vertex for each such component. Call $\{v_1, \dots, v_e\}$ these vertices, if any. Define the set $V_0$ of vertices of $G_0$ as $S(\mathcal{D}) \cup \{v_1, \dots, v_e\}$.

* Define the set $E_0$ of edges of $G_0$ as the set of components of $(\partial \mathcal{D} \cup \partial M) \setminus V_0$ (they are all intervals).

**Lemma 4.12.** The pair $G_0 = (V_0, E_0)$ is a multigraph. The number of vertices $\alpha_0(G_0)$, and the number of edges $\alpha_1(G_0)$ of $G_0$ are given by

$$ (4.5) \qquad \left\{ \begin{array}{l} \alpha_0(G_0) = e + |S_i(\mathcal{D})| + |S_b(\mathcal{D})|, \\ \alpha_1(G_0) = e + \frac{1}{2} \left( \sum_{y \in S_i(\mathcal{D})} \nu(y) + \sum_{z \in S_b(\mathcal{D})} \rho(z) \right) + |S_b(\mathcal{D})|, \end{array} \right. $$

where $e$ is defined above, where $|S_i(\mathcal{D})|$ (resp $|S_b(\mathcal{D})|$) denotes the number of interior (resp. boundary) singular points of $\mathcal{D}$, and where the numbers $\nu$ and $\rho$ are as in Definition 4.2. In particular,

$$ (4.6) \qquad \alpha_1(G_0) - \alpha_0(G_0) = \frac{1}{2} \left( \sum_{y \in S_i(\mathcal{D})} (\nu(y) - 2) + \sum_{z \in S_b(\mathcal{D})} \rho(z) \right). $$

*Proof.* Clearly $G_0$ is a multigraph in the sense of [Dies2017, Section 1.10]. The formula for $\alpha_0$ is clear. The first term in the right-hand side for $\alpha_1$ counts both the number of points $v_j$ and the number of edges contained in the components of $(\partial \mathcal{D} \cup \partial M) \setminus S(\mathcal{D})$ which are circles. The second term counts the number of edges between two singular points, each one being a simple curve contained in $\partial \mathcal{D} \setminus S(\mathcal{D})$. The third term counts the number of edges determined by the boundary singular points on the components of $\partial M$ which intersect $S(\mathcal{D})$. $\square$

**Remark 4.13.** The second relation in (4.5) also follows from the relation

$$ (4.7) \qquad 2\alpha_1(\Gamma) = \sum_{x \in V(\Gamma)} \deg_{\Gamma}(x) $$

which holds for any multi-graph $\Gamma$ (here, $\deg_{\Gamma}(x)$, the degree of the vertex $x$, is the number of edges of $\Gamma$ one of whose ends is $x$).

Note that the multigraph $G_0(\mathcal{D}, M)$,

(a) might contain loops at some singular point or vertex $v_j$;

(b) might contain pairs of distinct vertices in $V_0$ linked by more than one edge.

In order to apply the results of [Gibl2010], we transform the multigraph $G_0(\mathcal{D}, M)$ into a graph $G(\mathcal{D}, M)$ (in the sense of [Dies2017, Section 1.1] or [Gibl2010, p. 10]), keeping track of the number of vertices and edges. More precisely, if necessary, we introduce additional vertices and edges by performing one of the following vertex-edge additions.
**Definition 4.14.** We call *vertex-edge additions* the following modifications of the graph $G_0(\mathcal{D}, M)$.

(i) If a component $\Gamma$ of $(\partial\mathcal{D} \cup \partial M) \setminus V_0$ is bounded by only one vertex $v$ (i.e., there is a loop at $v$), we add two extra vertices $v_1, v_2$ on $\Gamma$, and replace the edge $\Gamma$ by three edges, the components of $\Gamma \setminus \{v_1, v_2\}$.

(ii) If two distinct vertices of $V_0$ are the endpoints of more than one edge in $E_0$, i.e., of more than one components $\Gamma_j$ of $(\partial\mathcal{D} \cup \partial M) \setminus V_0$, we add one extra vertex $w_j$ to each $\Gamma_j$, and replace $\Gamma_j$ by two edges, the components of $\Gamma_j \setminus \{w_j\}$.

Figure 4.2 illustrates the transformation of $\partial\mathcal{D} \cup \partial M$ into a graph. Lines contained in the boundary appear in black and lines contained in $\partial\mathcal{D}$ appear in red. Blue dots represent vertices $v_i$ initially attached to each circle component. Green dots represent vertices added in the vertex-edge additions. Sub-figure (A) illustrated the transformation of circle components of $\partial M$ or $\partial\mathcal{D}$, and loop in $\partial\mathcal{D}$ into graphs. Sub-figure (B) illustrates the transformation of multiple edges into graphs.

<figure><img src="image_8.png" /><figcaption>FIGURE 4.2. Vertex-edge additions</figcaption></figure>

**Lemma 4.15.** *Performing finitely many vertex-edge additions transforms the multigraph $G_0(\mathcal{D}, M)$ into a graph $G(\mathcal{D}, M)$.*

The following lemma follows from the fact that the number $\alpha_0 - \alpha_1$ remains unchanged if we perform a vertex-edge addition.

**Lemma 4.16.** For the graph $G(\mathcal{D}, M)$ obtained from the multigraph $G_0(\mathcal{D}, M)$ by performing vertex-edge additions, we have,

$$ egin{cases} 
\alpha_1(G) - \alpha_0(G) = \alpha_1(G_0) - \alpha_0(G_0) = \sigma(\mathcal{D}), \\ 
c(G) = c(G_0) = b_0(\partial\mathcal{D} \cup \partial M), \\ 
r(G, M) = r(G_0, M) = \kappa(\mathcal{D}). 
\end{cases} $$

4.3. Euler type formulas for partitions. Let $\mathcal{D}$ be a regular partition of a compact surface $M$, with boundary $\partial M$. In this subsection, we look at three examples.
### 4.3.1. The case of the sphere.

**Proposition 4.17.** Let $M_0$ be the sphere $\mathbb{S}^2$. For $q \ge 1$, let $M_q = M_0 \cup \cup_{j=1}^q U_j$, where the $U_j$, $1 \le j \le q$, are simply connected, pairwise disjoint, open sets, with piecewise $C^1$ boundary. Let $\mathcal{D}_q$ be a regular partition of $M_q$, $q \ge 0$. Then,

$$ (4.8) \qquad \begin{aligned} \kappa(\mathcal{D}_q) &= 1 + \beta(\mathcal{D}_q) + \sigma(\mathcal{D}_q) \\ &= 1 + b_0(\partial\mathcal{D}_q \cup \partial M_q) - b_0(\partial M_q) + \sigma(\mathcal{D}_q) \\ &\ge 2 - q + \sigma(\mathcal{D}_q). \end{aligned} $$

*Proof.* Let $G_0 := G(\mathcal{D}_0, M_0)$ be a graph associated with $\mathcal{D}_0$ as in Subsection 4.2. In this case, $\beta(\mathcal{D}_0) = b_0(\partial\mathcal{D}_0) = c(G_0)$. By Lemma 4.16, we can write

$$ \kappa(\mathcal{D}_0) - \beta(\mathcal{D}_0) - \sigma(\mathcal{D}_0) = r(G_0, M_0) - c(G_0) - \alpha_1(G_0) + \alpha_0(G_0) = 1, $$

where the equality on the right is given by Euler's formula for graphs in $\mathbb{S}^2$ (planar graphs).

Let $M_q = M_0 \cup \cup_{j=1}^q U_j$. Let $\mathcal{D}_q$ be a partition of $M_q$, and let $G_q = G(\mathcal{D}_q, M_q)$ be a graph as constructed in Subsection 4.2. Then, $\mathcal{D}'_q = \mathcal{D}_q \cup \{U_1, \dots, U_q\}$ is a partition of the sphere $M_0$. The graph $G_q$ viewed as a graph in $M_0$ is also a graph of the form $G(\mathcal{D}'_q, M_0)$. We have $r(G_q, M_0) = r(G_q, M_q) + q$. We also have

$$ \beta(\mathcal{D}_q) := b_0(\partial\mathcal{D}_q \cup \partial M_q) - b_0(\partial M_q) = c(G_q) - q. $$

Applying Lemma 4.16 to the pair $(\mathcal{D}_q, M_q)$, we obtain

$$ \begin{aligned} \kappa(\mathcal{D}_q) - \beta(\mathcal{D}_q) - \sigma(\mathcal{D}_q) &= r(G_q, M_q) + q - c(G_q) + \alpha_0(G_q) - \alpha_1(G_q) \\ &= r(G_q, M_0) - c(G_q) + \alpha_0(G_q) - \alpha_1(G_q) = 1, \end{aligned} $$

where the last equality follows from Euler's formula for the graph $G_q$ viewed as a graph in the sphere. $\square$

**Remark 4.18.** Formula (4.8) applies in particular to a bounded domain $\Omega \subset \mathbb{R}^2$, with piecewise $C^1$ boundary, and $q$ boundary components.

### 4.3.2. The general case.
More generally, we consider, for $c \in \mathbb{Z}$, a closed surface $\Sigma_c$ with Euler characteristic $\chi(\Sigma_c) = c$.

**Proposition 4.19.** Let $\Sigma_c$ be a closed surface with Euler-characteristic $\chi(\Sigma_c) = c$. For $q \ge 1$, let $\Sigma_{c,q} = \Sigma_c \cup \cup_{j=1}^q U_j$, where the $U_j$, $1 \le j \le q$, are simply connected, pairwise disjoint, open sets, with piecewise $C^1$ boundary. Let $\mathcal{D}_q$ be a regular partition of $\Sigma_{c,q}$, $q \ge 0$. Then,

$$ (4.9) \qquad \kappa(\mathcal{D}_q, \Sigma_{c,q}) \ge \chi(\Sigma_c) - q + \sigma(\mathcal{D}_q, \Sigma_{c,q}) = \chi(\Sigma_{c,q}) + \sigma(\mathcal{D}_q, \Sigma_{c,q}). $$

*Proof.* The proof mimics the proof of Proposition 4.17, and make use of Euler's inequality for graphs in a surface $\Sigma_c$, [Gibl2010, Corollary 9.27]. To the partition $\mathcal{D}_q$, we associate a graph $G_q = G(\mathcal{D}_q, \Sigma_{c,q})$ as in Subsection 4.2. We consider $\mathcal{D}'_q = \mathcal{D}_q \cup \{U_1, \dots, U_q\}$. This is a partition of $\Sigma_c$. The graph $G_q$, viewed as a graph in $\Sigma_c$, is also of the form $G(\mathcal{D}'_q, \Sigma_c)$, and $r(G_q, \Sigma_c) = r(G_q, \Sigma_{c,q}) + q$. We have
$\beta(D_q) = b_0(\partial D_q \cup \partial \Sigma_{c,q}) - b_0(\partial \Sigma_{c,q}) = c(G_q) - q$. Then, using Lemma 4.16 and applying Euler's inequality [Gib12010, Corollary 9.27], we can write

$$ 
\begin{aligned}
\kappa(D_q, \Sigma_{c,q}) &= r(G_q, \Sigma_{c,q}) = r(G_q, \Sigma_c) - q \\
&\geq \chi(\Sigma_c) + \alpha_1(G_q) - \alpha_0(G_q) - q = \chi(\Sigma_c) - q + \sigma(D_q, \Sigma_{c,q}) \\
&\geq \chi(\Sigma_{c,q}) + \sigma(D_q, \Sigma_{c,q}).
\end{aligned}
$$

□

**Remark 4.20.** Obviously, formula (4.9) is less precise than formula (4.8). In the case of the Möbius strip, we can give a better statement.

### 4.3.3. The case of the Möbius strip.

**Theorem 4.21.** Let $\mathcal{D}$ be an essential, regular partition of the Möbius strip $M$. Then, with the Notation 4.6,

$$ (4.10) \qquad \kappa(\mathcal{D}) = \omega(\mathcal{D}) + \beta(\mathcal{D}) + \sigma(\mathcal{D}). $$

The proof of this theorem is given in [BeHe2020m]. The Möbius strip is homeomorphic to a projective plane with a disk removed. We can rewrite Formula 4.10 as follows.

$$ \kappa(\mathcal{D}) = \omega(\mathcal{D}) + b_0(\partial \mathcal{D} \cup \partial M) - 1 + \sigma(\mathcal{D}). $$

Since $\omega(\mathcal{D}) \geq 0$ and $b_0(\partial \mathcal{D} \cup \partial M) \geq 1$, we have

$$ \kappa(\mathcal{D}) \geq \chi(\mathbb{P}^2) - 1 + \sigma(\mathcal{D}) = \chi(\mathbb{M}) + \sigma(\mathcal{D}). $$

Formula 4.10 is more precise than Formula 4.9 applied to the projective plane with a disk removed.

### 4.4. Nodal partitions.
The partition $D_u$ of a compact surface $M$ (with or without boundary) associated with an eigenfunction $u$ of the operator $-\Delta + V$ (with some prescribed boundary condition if $\partial M \neq \emptyset$) is called a *nodal partition*. The domains of the partition $D_u$ are the nodal domains of $u$, the boundary set $\partial D_u$ is the nodal set $\mathcal{Z}(u)$, the singular set $S(D_u)$ is the set $S(u)$ of singular points of $u$ (Definition 3.5), and we write $S(u) = S_i(u) \cup S_b(u)$. This is an example of an essential, regular partition.

Nodal partitions are not necessarily normal as can be seen in Figure 4.3 which displays the nodal partitions of two eigenfunctions of the flat Möbius strip with fundamental domain $(0, \pi) \times (0, \pi)$; the horizontal sides are identified, $(x, 0) \sim (\pi - x, \pi)$, and the Dirichlet boundary condition holds on the sides $\{x = 0\}$ and $\{x = \pi\}$, see [BeHK2021m]. Other examples can be found in [BeHK2021k].
<figure><img src="image_9.png" /><figcaption>FIGURE 4.3. Partition associated with eigenfunctions of the Möbius strip</figcaption></figure>

For nodal partitions, we write $\kappa(u), \beta(u), \dots$ for the corresponding $\kappa(\mathcal{D}_u), \beta(\mathcal{D}_u), \dots$, and we have $\nu(\mathcal{D}_u, z) = \nu(u, z)$ and $\rho(\mathcal{D}_u, z) = \rho(u, z)$ for the corresponding indices.

For nodal partitions we have the following additional properties.

**Proposition 4.22.** Let $u$ be an eigenfunction of $M$. For any component $\Gamma$ of $\partial M$, we have

$$ (4.11) \qquad \sum_{z \in S_b(u) \cap \Gamma} \rho(z) \in 2\mathbb{N}. $$

Furthermore,

$$ (4.12) \qquad b_0(\mathcal{Z}(u) \cup \partial M) - b_0(\partial M) + \frac{1}{2} \sum_{y \in S_b(u)} \rho(y) \geq 1, $$

*Proof.* The first assertion is general and contained in Corollary 3.8. To prove the second assertion, we divide the components of $\partial M$ into two sets: the components $\Gamma'_i, 1 \le i \le p$, which meet $\mathcal{Z}(u)$, and the components $\Gamma''_j, 1 \le j \le q$, which do not meet $\mathcal{Z}(u)$. Let $\Gamma(u) = \cup_{i=1}^p \Gamma'_i$. Clearly, we have the relation

$$ b_0(\mathcal{Z}(u) \cup \partial M) - b_0(\partial M) = b_0(\mathcal{Z}(u) \cup \Gamma(u)) - b_0(\Gamma(u)). $$

On the other-hand, according to (4.11), for each $1 \le i \le p$, we have

$$ \sum_{z \in S_b(u) \cap \Gamma'_i} \rho(z) \ge 2. $$

Relation (4.12) follows from the fact that $b_0(\mathcal{Z}(u) \cup \Gamma(u)) \ge 1$. $\square$

**Corollary 4.23.** If $u$ is an eigenfunction of the Möbius strip $M$, with $\mathcal{Z}(u) \ne \emptyset$,

$$ (4.13) \qquad \kappa(u) \ge \omega(\mathcal{D}_u) + 1 + \frac{1}{2} \sum_{x \in S_i(u)} (\nu(x) - 2). $$

*Proof.* Relation (4.13) is a consequence of (4.10) applied to the nodal partition associated with $u$, and (4.12). $\square$

# 5. REVISITING THE MULTIPLICITY BOUNDS FOR CLOSED SURFACES OF GENUS 0

In this section, we revisit the paper [HoHN1999] by M. and T. Hoffmann-Ostenhof and N. Nadirashvili.
5.1. **Introduction.** Let $(M, g)$ be a closed connected smooth Riemannian surface with genus 0, i.e. $M$ is topologically a sphere (in particular it is simply connected with Euler characteristic 2), and $g$ is a smooth metric. We are also given a smooth, real valued function $V$ on $M$, and we consider the eigenvalue problem (3.1) for the Schrödinger operator $-\Delta + V$ on $M$, with eigenvalues ${\lambda_k}^{\infty}_{k=1}$ (with first label 1).

According to Cheng, [Chen1976], $\mathrm{mult}(\lambda_2) \le 3$ and this bound is sharp, achieved for the round metric on $\mathbb{S}^2$ and $V = 0$. Nadirashvili, [Nadi1987], proved that, for $k \ge 1$, $\mathrm{mult}(\lambda_k) \le (2k-1)$. In [HoHN1999], M. and T. Hoffmann-Ostenhof and Nadirashvili, prove the following result.

**Theorem 5.1.** For $(M, g, V)$ as above, and for $k \ge 3$, $\mathrm{mult}(\lambda_k) \le (2k-3)$.

Here is the underlying idea of the proof of Theorem 5.1. Fix some $k \ge 3$, and some $x \in M$. Consider the eigenspace $U(\lambda_k)$, and recall that $M$ is topologically a sphere. Assume that $\mathrm{mult}(\lambda_k) = \dim U(\lambda_k) = m$ for some integer $m \ge 2$. Then, according to Lemma 3.9, there exists a function $0 \ne u \in U(\lambda_k)$ such that $\nu(u, x) \ge 2 \lfloor \frac{m}{2} \rfloor$ (where $\lfloor \frac{m}{2} \rfloor$ denotes the integer part of $\frac{m}{2}$), and actually two linearly independent such functions when $m$ is odd. By Courant's theorem, the number of nodal domains of $u$ satisfies $\kappa(u) \le k$. Since $M$ is topologically a sphere, we can apply Euler's formula (4.8) to the nodal partition $\mathcal{D}_u$ of the function $u$,

$$ (5.1) \qquad \kappa(u) = 1 + b_0(\mathcal{Z}(u)) + \frac{1}{2} \sum_{z \in S(u)} (\nu(u, z) - 2). $$

Summing up the above information, we have

$$ (5.2) \qquad \begin{aligned} 0 \ge \kappa(u) - k &= \{b_0(\mathcal{Z}(u)) - 1\} + \sum_{\substack{z \in S(u) \\ z \ne x}} \frac{\nu(u, z) - 2}{2} + \{\frac{\nu(u, x)}{2} - k + 1\} \\ &\ge \{b_0(\mathcal{Z}(u)) - 1\} + \sum_{\substack{z \in S(u) \\ z \ne x}} \frac{\nu(u, z) - 2}{2} + \{\lfloor \frac{m}{2} \rfloor - k + 1\}. \end{aligned} $$

Notice that the first two terms in the right-hand side of both lines of (5.2) are nonnegative. It follows that

$$ (5.3) \qquad \lfloor \frac{m}{2} \rfloor \le k-1 \quad \text{i.e. } \dim U(\lambda_k) \le (2k-1). $$

The above inequality is sharp for $k=1$ and $2$, see Table 2.1.

In view of (5.3), to prove Theorem 5.1, it suffices to show that the cases $\dim U(\lambda_k) = (2k-1)$, and $\dim U(\lambda_k) = (2k-2)$ cannot occur. This is the purpose of the following two subsections, revisiting the arguments of [HoHN1999].

5.2. **Surfaces with genus zero, proof that $\dim U(\lambda_k) \le (2k-2)$ for $k \ge 3$.**
The proof is by contradiction. Taking (5.3) into account, we assume that

$$ (5.4) \qquad k \ge 3 \quad \text{and} \quad \dim U(\lambda_k) = 2k-1. $$

We fix some $x \in M$, and choose an orientation in $T_x M$.

Introduce the linear subspace

$$ (5.5) \qquad U_x := \{u \in U(\lambda_k) \mid \nu(u, x) \ge 2(k-1)\}. $$
5.2.1. Structure and combinatorial type of $\mathcal{Z}(u)$, for $0 \neq u \in U_x$.

**Properties 5.2.** Assume that (5.4) holds. Then, for any $0 \neq u \in U_x$,

(i) $\nu(u, x) = 2(k-1)$, and $x$ is the only singular point of the function $u$;

(ii) $\mathcal{Z}(u)$ is connected;

(iii) $\kappa(u) = k$.

Furthermore, $\dim U_x = 2$, and there exists a basis $\{v_1, v_2\}$ of $U_x$ such that, in some local polar coordinates $(r, \omega)$ centered at $x$,

$$ (5.6) \qquad \left\{ \begin{array}{l} v_1 = r^{k-1} \sin((k-1)\omega) + \mathcal{O}(r^k), \\ v_2 = r^{k-1} \cos((k-1)\omega) + \mathcal{O}(r^k). \end{array} \right. $$

*Proof.* According to Lemma 3.9 (ii), there exist two linearly independent functions $u_1, u_2 \in U(\lambda_k)$, with $\nu(u_i, x) \ge 2 \lfloor \frac{m}{2} \rfloor = 2(k-1)$, for $i=1,2$. Given any $0 \neq u \in U_x$, we can apply Inequality (5.2) to $u$, with $m = (2k-1)$, so that

$$ (5.7) \qquad \begin{aligned} 0 \ge \kappa(u) - k &= \{b_0(\mathcal{Z}(u)) - 1\} + \sum_{\substack{z \in S(u) \\ z \ne x}} \frac{\nu(u, z) - 2}{2} + \{\frac{\nu(u, x)}{2} - k + 1\} \ge \\ & \ge \{b_0(\mathcal{Z}(u)) - 1\} + \sum_{\substack{z \in S(u) \\ z \ne x}} \frac{\nu(u, z) - 2}{2} \ge 0. \end{aligned} $$

The terms in the right-hand sides of both lines of (5.7) are nonnegative, and their sum is zero. This proves Assertions (i)–(iii).

From the previous paragraph, we know that $U_x$ has dimension at least 2. Assume that it has dimension at least 3, and let $v_1, v_2, v_3$ be three linearly independent functions in $U_x$. Since $\nu(v_i, x) = 2(k-1)$, Lemma 3.12 implies the existence of a nontrivial linear combination $v$ of these functions with $\nu(v, x) \ge 2k$, contradicting Assertion (i). Hence, $\dim U_x = 2$.

In view of (3.9), choosing the local polar coordinates $(r, \omega)$ appropriately, and scaling the functions if necessary, we can start from a basis $\{v_1, w_2\}$ of $U_x$ such that

$$ \left\{ \begin{array}{l} v_1 = r^{k-1} \sin((k-1)\omega) + \mathcal{O}(r^k), \\ w_2 = r^{k-1} \sin((k-1)\omega + \beta) + \mathcal{O}(r^k), \end{array} \right. $$

for some angle $\beta \in (0, \pi)$. For $\theta \in (0, \pi)$, consider the functions

$$ \cos \theta v_1 + \sin \theta w_2 = r^{k-1} (\cos \theta + \sin \theta \cos \beta) \sin((k-1)\omega) + r^{k-1} \sin \theta \sin \beta \cos((k-1)\omega) + \mathcal{O}(r^k). $$

Choose $\theta \in (0, \pi)$ such that $\cos \theta + \sin \theta \cos \beta = 0$, then $\sin \theta \sin \beta \neq 0$, and the function

$$ v_2 = (\cos \theta v_1 + \sin \theta w_2) / (\sin \theta \sin \beta) $$

satisfies the second condition in (5.6). This proves the last assertion, and the proof of Properties 5.2 is now complete. ✓

Let $0 \neq u \in U_x$. The local structure theorem – Corollary 3.4 (i) – implies that, in a neighborhood of $x$, the nodal set $\mathcal{Z}(u)$ consists of $2(k-1)$ nodal semi-arcs which emanate from $x$, tangentially to $2(k-1)$ rays dividing the unit circle in $T_x M$ into equal parts. Since $\mathcal{S}(u) = \{x\}$, if we follow a semi-nodal arc emanating from
$x$, we will eventually come back to $x$. Using the fact that $S(u) = \{x\}$ and the connectedness of $Z(u)$, we conclude that $Z(u)$ consists of $(k-1)$ simple loops at $x$, and that these loops only intersect each other at $x$.

**Definition 5.3.** Define a p-bouquet of loops at $x$ as a collection of p piecewise $C^1$ loops at $x$, whose semi-tangents at $x$ are pairwise transverse in $T_x M$, and which do not intersect away from $x$.

Therefore, for any $0 \neq u \in U_x$, the nodal set $Z(u)$ is a $(k-1)$-bouquet of loops at the point $x$.

Choosing a direct frame $\{\vec{e}_1, \vec{e}_2\}$ in $T_x M$, we label the rays tangent to $Z(u)$ at $x$ counter-clockwise, according to their angle with respect to $\vec{e}_1$. We obtain an ordered list $\{\vartheta_1, \dots, \vartheta_{2(k-1)}\}$ in which two consecutive rays make an angle $\frac{\pi}{k-1}$. The loops in the $(k-1)$-bouquet of loops $Z(u)$ can now be described by a map

$$ (5.8) \qquad \tau_{x,u} : \{1, \dots, 2(k-1)\} \to \{1, \dots, 2(k-1)\}, $$

which is defined in the following way. For $j \in \{1, \dots, 2(k-1)\}$, we consider the loop which emanates from $x$ tangentially to the ray $\vartheta_j$, and define $\tau_{x,u}(j)$ as the labelling of the ray tangent to the loop when it arrives back at $x$. We then denote this loop by $\gamma_{j, \tau_{x,u}(j)}^{x,u}$. We shall write $\tau$ instead of $\tau_{x,u}$ when there is no ambiguity.

The properties of nodal sets imply that

$$ (5.9) \qquad \left\{ \begin{array}{l} \tau_{x,u}(j) \neq j \quad \text{for all } j \in \{1, \dots, 2(k-1)\}, \\ \tau_{x,u}^2 = \mathrm{Id}. \end{array} \right. $$

Note that changing the frame $\{\vec{e}_1, \vec{e}_2\}$ at $x$ amounts to conjugating the map $\tau_{x,u}$ by a circular permutation of the set $\{1, \dots, 2(k-1)\}$.

**Definition 5.4.** We call the map $\tau_{x,u}$ the combinatorial type of the nodal set $Z(u)$ at the point $x$.

5.2.2. *The rotating function argument.* Fix the basis $\{v_1, v_2\}$ of $U_x$ provided by Properties 5.2. Fix the direct frame $\{\vec{e}_1, \vec{e}_2\}$ in $T_x M$ corresponding to the local polar coordinates $(r, \omega)$ at $x$ such that (5.6) holds.

We now analyze the nodal sets of the one-parameter family,

$$ (5.10) \qquad w_{\theta} = \cos((k-1)\theta) v_1 - \sin((k-1)\theta) v_2, $$
for $\theta \in [0, \frac{\pi}{k-1}]$. In particular, we have

$$ (5.11) \qquad v_1 = w_0 = -w_{\frac{\pi}{k-1}} \quad \text{and} \quad v_2 = -w_{\frac{\pi}{2(k-1)}}, $$

and

$$ (5.12) \qquad w_{\theta} = r^{k-1} \sin((k-1)(\omega - \theta)) + \mathcal{O}(r^k). $$

Properties 5.2 state that $x$ is the sole singular point of the eigenfunction $w_{\theta}$, and that the nodal set $Z(w_{\theta})$ is connected. With respect to the frame $\{\vec{e}_1, \vec{e}_2\}$ in $T_x M$, there are $2(k-1)$ nodal semi-arcs which emanate from $x$, tangentially to the $2(k-1)$ rays $\{\omega = \omega_j(\theta) := \omega_j + \theta\}$, where $\omega_j := j \frac{\pi}{k-1}, j \in \{0, \dots, 2k-3\}$. We call these rays $\omega_j(\theta)$ for short, and we view $j$ as defined modulo $2(k-1)$.

The nodal set $Z(w_{\theta})$ is a $(k-1)$-bouquet of loops described by the map $\tau_{x,w_{\theta}}$ associated with the rays $\omega_j(\theta)$. Call this map $\tau_{\theta}$ for short, and call $\gamma_{j, \tau_{\theta}(j)}^{\theta}$ the corresponding loops at $x$.
**Property 5.5.** Assume that (5.4) holds. Considering $\tau_\theta(j)$ instead of $j$ if necessary, we can assume that $0 \le j < \tau_\theta(j) \le 2k-3$. The loop $\gamma_{j,\tau_\theta(j)}^\theta$ separates (the topological sphere) $M$ into two components. The rays $\omega_k(\theta)$ such that $j < k < \tau_\theta(j)$ point inside one of the two components; the rays $\omega_k(\theta)$ with $k < j$ or $k > \tau_\theta(j)$ point inside the other component. In particular, $\tau_\theta(j) - j$ is an odd integer.

The proof of this property is clear.

<figure><img src="image_10.png" /></figure>

**Property 5.6.** Assume that (5.4) holds. The map $\tau_\theta$ (i.e. the combinatorial type of $w_\theta$) does actually not depend on $\theta$. More precisely, for any $j \in \{0, \dots, 2k-3\}$, and for any $\theta \in [0, \frac{\pi}{k-1}]$, the loop which emanates from $x$ tangentially to the ray $\omega_j(\theta)$ arrives at $x$ tangentially to the ray $\omega_{\tau_0(j)}(\theta)$. We shall henceforth denote this map by $\tau$.

*Proof.* Since all the functions $w_\theta$ share the same properties, it suffices to show that $\tau_\theta = \tau_0$ for $\theta$ small enough. Assume the contrary. Then, there exists a sequence $\theta_n$ tending to zero, and a sequence $\{j_n\} \subset \{0, \dots, 2k-3\}$ such that $\tau_{\theta_n}(j_n) \ne \tau_0(j_n)$. Since the sequence $\{j_n\}$ takes finitely many values, we can find a constant subsequence $\{j_{n,1}\} \subset \{j_n\}$. Similarly, we can take a subsequence $\{j_{n,2}\} \subset \{j_{n,1}\}$ such that $\tau_{\theta_{n,2}}(j_{n,2})$ is constant. Hence, there exists some $\ell \in \{0, \dots, 2k-3\}$, and a sequence $\theta_n$ such that $\tau_{\theta_n}(\ell) \ne \tau_0(\ell)$. Without loss of generality, we may assume that $\ell = 0$, so that there exists a sequence $\theta_n$ tending to zero such that $\tau_{\theta_n}(0) \equiv \ell_0 \ne \tau(0)$.

We now use a more precise version of the local structure theorem, see Appendix B. For any $\alpha > 0$ small enough, there exists $r_0 > 0$ such that, for all $\theta$, $\mathcal{Z}(w_\theta) \cap B(x, 2r_0)$ consists of $2(k-1)$ nodal semi-arcs

$$ (5.13) \qquad A_j(r, \theta) : (0, 2r_0) \ni r \mapsto \exp_x (r \tilde{\omega}_j(r, \theta)) \in B(x, 2r_0), $$

for $j \in \{0, \dots, 2k-3\}$. Here, we assume that an orthonormal frame $\{e_1, e_2\}$ has been chosen in $T_x M$, is such a way that the vector $e_1$ directs the ray $\omega_0$. In the polar coordinates $(r, \omega)$ associated with this frame, we identify the angle $\omega$ with a point on the unit circle. Furthermore, the functions $\tilde{\omega}_j$ are smooth in $(r, \theta) \in (0, 2r_0) \times [0, 2\pi]$, and they satisfy,

$$ (5.14) \qquad \left\{ \begin{array}{l} \tilde{\omega}_j(r, \theta) \in (\omega_j + \theta - \alpha, \omega_j + \theta + \alpha), \\ \lim_{r \to 0} \tilde{\omega}_j(r, \theta) = \omega_j + \theta, \end{array} \right. $$

for all $j, 0 \le j \le 2k-3$. The semi-arc $A_j(r, \theta)$ is semi-tangent to the ray $\omega_j + \theta$ at the point $x$.

We now reason as in the proof of Lemma 3.15. In the closed ball $\bar{B}(x, r_0)$, the nodal set $\mathcal{Z}(\omega_{\theta_n})$ consists of $2(k-1)$ nodal semi-arcs $A_j(\cdot, \theta_n)$, with end points $x$ and $\exp_x(r_0 \tilde{\omega}_j(r_0, \theta_n))$ which converge to the corresponding semi-arcs $A_j(\cdot, 0)$ with end points $x$ and $\exp_x(r_0 \tilde{\omega}_j(r_0, 0))$.

In the compact set $M \setminus B(x, r_0)$, the nodal set $\mathcal{Z}(\omega_{\theta_n})$ consists of $(k-1)$ dis-joint connected nodal arcs $C_j(r_0, \theta_n)$ with two end points $\exp_x(r_0 \tilde{\omega}_j(r_0, \theta_n))$ and $\exp_x(r_0 \tilde{\omega}_{\tau(j)}(r_0, \theta_n))$, which correspond to the intersections of the loops in $\mathcal{Z}(w_{\theta_n})$ with $M \setminus B(x, r_0)$. We look more precisely at the arcs $C_0(r_0, \theta_n)$. From this sequence of compact connected subsets of $M \setminus B(x, r_0)$, we can extract a subsequence which converges in the Hausdorff distance to some compact connected set $C_0$. Since any $z \in C_0$ is the limit of a sequence $z_n \in C_0(r_0, \theta_n)$, and since $w_{\theta_n}$ tends to $w_0$ uniformly on $M$, we conclude that $w_0(z) = 0$, i.e., that $C_0 \subset \mathcal{Z}(w_0)$. The set $C_0$ contains the
points $\exp_x(r_0 \tilde{\omega}_0(r_0, 0))$ and $\exp_x(r_0 \tilde{\omega}_{\ell_0}(r_0, 0))$. Since $C_0$ is connected and contained in $\mathcal{Z}(w_0)$, and in view of the structure of $\mathcal{Z}(w_0)$, we must have $\ell_0 = \tau(0)$, and we reach a contradiction. The proof of Property 5.6 is complete. ✓

**Conclusion of the rotating function argument.** Assuming that (5.4) holds, since $w_{\frac{\pi}{k-1}} = -v_1$, we infer from Property 5.6 that $\gamma_{0,\tau(0)}^{\frac{\pi}{k-1}} = \gamma_{1,\tau(0)+1}$. Since there is only one nodal semi-arc tangent to a given ray at $x$, we conclude that $\tau(0) \neq 1$ and $\tau(0) \neq 2k-3$. It follows that $0 < 1 < \tau(0)$, and that $\tau(0) < \tau(0)+1 = \tau(1) \leq 2k-3$ (here we have used the assumption $k \geq 3$). This contradicts Property 5.5, and proves that $\dim U(\lambda_k) = (2k-1)$ cannot occur. ✓

We have proved the inequality

$$ (5.15) \qquad \mathrm{mult}(\lambda_k) \leq (2k-2), \quad \text{for any } k \geq 3. \qquad \square $$

**Remark 5.7.** As far as we know, the idea to consider the family of functions $w_\theta$ was introduced by Besson [Bess1980], proof of Theorem 3.C.1 in which he improves the upper bound for the multiplicity of the second eigenvalue of a torus from 7 to 6. A similar idea was used by Nadirashvili [Nadi1987], p. 231 lines 1–8, for higher eigenvalues as well. It is used in [HoHN1999, HoMN1999] also, and will appear several times, in one form or another, in the present paper.

**Remark 5.8.** In the next section, we will introduce the combinatorial type for different kinds of nodal sets on a compact surface $M$ with boundary, and we will repeatedly use the “rotating function argument”.

**5.3. Surfaces with genus zero, proof that** $\dim U(\lambda_k) \leq (2k-3)$ for $k \geq 3$. The proof is by contradiction. Taking Subsection 5.2 into account, we assume that $\dim U(\lambda_k) = (2k-2)$.

**Properties 5.9.** Assume that $\dim U(\lambda_k) = (2k-2)$. Then, the linear subspace $U_x = \{u \in U(\lambda_k) | \nu(u, x) \geq 2(k-1)\}$ has the following properties.

(i) $\dim U_x = 1$, and for any $0 \neq u \in U_x$, we have

(ii) $\nu(u, x) = 2(k-1)$, and $x$ is the only singular point of the eigenfunction $u$;

(iii) $\mathcal{Z}(u)$ is connected;

(iv) $\kappa(u) = k$.

*Proof.* By Lemma 3.9, $\dim U_x \geq 1$. Given any $0 \neq u \in U_x$, Euler's formula gives

$$ (5.16) \qquad 0 \geq \kappa(u) - k = \{b_0(\mathcal{Z}(u)) - 1\} + \sum_{\substack{z \in S(u) \\ z \neq x}} \frac{\nu(u, z) - 2}{2} + \{\frac{\nu(u, x)}{2} - k + 1\}, $$

and we conclude that (ii)–(iv) hold. To prove (i), assuming that $\dim U_x \geq 2$, we can repeat the arguments of Subsection 5.2, and reach a contradiction. □

Given any $x \in M$, there exists a function $0 \neq u_x \in U$, uniquely defined up to multiplication by a nonzero scalar, such that $\nu(u_x, x) = 2(k-1)$. Choosing a basis $\{u_1, \dots, u_{2k-2}\}$ for the eigenspace $U(\lambda_k)$, we can write $u_x = \sum_{j=1}^{2k-2} \alpha_j(x)u_j$, and the condition $\mathrm{ord}(u_x, x) = (k-1)$ is equivalent to a linear system in the $(2k-2)$ unknowns $\alpha_j(x)$. This linear system as constant rank 1. It follows that, in a neighborhood of any point $x_0$, we can find a $C^\infty$ function $x \to (\alpha_1(x), \dots, \alpha_{2k-2}(x))$, with values
in $\mathbb{R}^{2k-2} \setminus \{0\}$, and defined up to multiplication by a nonzero scalar $C^\infty$ function. These local solutions yield a unique global $C^\infty$ function with values in $\mathbb{P}(U(\lambda_k))$, the projective space of the eigenspace $U(\lambda_k)$. Since $M$ is simply connected, this map can be lifted to a smooth map $x \mapsto u_x$, from $M$ to $\mathbb{S}(U(\lambda_k))$, the unit sphere of $U(\lambda_k)$ (for example with respect to the $L^2$ norm).

To each $x \in M$, we associate the homogeneous polynomial $p_x$ on $T_x M$, of degree $(k-1)$, defined by $p_x: T_x M \ni v \mapsto \frac{d^{k-1}}{dt^{k-1}} u_x(\exp_x(tv))$. It is harmonic with respect to the Riemannian metric $g_x$ in $T_x M$. The map $x \mapsto p_x$ is smooth. The restriction of this polynomial to the unit circle $S_x M$ in $(T_x M, g_x)$ has simple zeros. Choose some $x_0 \in M$, and some root $e_{x_0}$ of $p_{x_0}$ in $S_{x_0} M$. Given any $x_1 \in M$, and any curve $c$ from $x_0$ to $x_1$, we can follow this root by continuity along the curve $c$ so that $e_{c(t)}$ is a root of $p_{c(t)}$. Since $M$ is simply connected, the root $e_{c(1)}$ at $x_1$ does not depend on the choice of the curve $c$. It follows that we have defined a continuous unit vector-field $x \mapsto e_x$ on $M$, contradicting the Poincaré-Hopf theorem [Miln1997, p. 35] which asserts that the sum of the indices of the zeros of a vector-field on $M$ equals the Euler characteristic of $M$ (here, we have a vector-field without zero, and $\chi(M) = 2$). We have proved that the assumption $\dim U(\lambda_k) = (2k-2)$ leads to a contradiction, and hence that $\dim U(\lambda_k) \le (2k-3)$. $\square$

## 6. PLANE DOMAINS, BOUNDING mult($\lambda_k$) FROM ABOVE, PRELIMINARIES

**6.1. Introduction.** Let $\Omega$ be a regular bounded domain<fnref n="3" /> in $\mathbb{R}^2$. We are interested in the eigenvalue problem (3.3) for the Laplacian or for a Schrödinger operator of the form $-\Delta + V$ in $\Omega$. In [HoMN1999], T. Hoffmann-Ostenhof, P. Michor and N. Nadirashvili state the following theorem, and mention that the result carries over to the Neumann boundary condition (see Theorem A and comments, p. 1170).

**Theorem 6.1.** For $k \ge 3$, the Dirichlet eigenvalues of the operator $-\Delta + V$ in a bounded domain $\Omega \subset \mathbb{R}^2$ satisfy $\mathrm{mult}(\lambda_k) \le (2k-3)$.

More precisely, they give a 3-step proof. They first give an easy proof of the upper bound $(2k-1)$ (which is due to Nadirashvili [Nadi1987] in the case of surfaces with genus 0). This implies in particular that the second eigenvalue has multiplicity at most 3. In [HoMN1999, Section 2], they improve this bound to $(2k-2)$ when $k \ge 3$. In [HoMN1999, Section 3], they prove the bound $(2k-3)$ (same bound as in the case of closed surfaces with genus 0, [HoHN1999]). However, in the non simply connected case, the validity of the proof of the bound $(2k-3)$ is questioned in [Berd2018], (Introduction, line 5; Section 2, page 544, line 10; Section 4). Therefore, the non simply connected case seems to remain open.

In this section, we introduce some notation and basic inequalities to be used in Sections 6–9, and we prove that $\mathrm{mult}(\lambda_k) \le (2k-1)$ for any $k \ge 1$. In Section 7, we will prove that $\mathrm{mult}(\lambda_k) \le (2k-2)$ for any $k \ge 3$.

**6.2. Notation.** Let us fix some notation for Sections 6–9.

Let $U$ denote a linear subspace of an eigenspace of the eigenvalue problem (3.3) in $\Omega$. We assume that $U$ satisfies the inequality

$$ (6.1) \qquad \{\kappa(u) \mid 0 \ne u \in U\} \le \ell, \quad \text{for some integer } \ell \ge 2. $$

<footnote n="3">³By “domain”, we mean a connected open subset.</footnote>
We write $\partial\Omega$ as the union of its components, $\partial\Omega = \cup_{j=1}^q \Gamma_j$, with $q \ge 1$.
Given $0 \neq u \in U$, define the sets

$$ (6.2) \qquad \left\{ \begin{array}{l} J(u) := \{j \mid \Gamma_j \cap \mathcal{Z}(u) \neq \emptyset\}, \\ \Gamma(u) := \cup_{j \in J(u)} \Gamma_j. \end{array} \right. $$

Given a function $0 \neq u \in U$, we denote by

$$ (6.3) \qquad [u] := \{a \mid u \mid a \in \mathbb{R} \setminus \{0\}\} $$

the corresponding line in the projective space $\mathbb{P}(U)$. We will say that $u$ is a generator of the line $[u]$. If a function $u$ is uniquely determined by some condition, up to multiplication by a nonzero scalar, we will say that $u$ is uniquely determined up to scaling or, equivalently, that $[u]$ is uniquely determined.

**6.3. The initial inequalities.** We shall make an extensive use of Euler's formula for the nodal partition $D_u$ associated with an eigenfunction $u$, see Paragraph 4.3.1, taking into account the assumption (6.1) on $U$,

$$ (6.4) \qquad \ell \geq \kappa(u) = 1 + \beta(u) + \sigma_i(u) + \sigma_b(u), $$

where,

$$ (6.5) \qquad \left\{ \begin{array}{l} \beta(u) := b_0(\mathcal{Z}(u) \cup \partial\Omega) - b_0(\partial\Omega) \\ \qquad = b_0(\mathcal{Z}(u) \cup \Gamma(u)) - b_0(\Gamma(u)), \end{array} \right. $$

$$ (6.6) \qquad \left\{ \begin{array}{l} \sigma_i(u) = \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2), \\ \sigma_b(u) = \frac{1}{2} \sum_{z \in S_b(u)} \rho(u, z) = \sum_{j \in J(u)} \frac{1}{2} \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z). \end{array} \right. $$

From the nodal character of the partition $D_u$, we also have, using Proposition 4.22 and the definition of $J(u)$,

$$ (6.7) \qquad \forall j \in J(u), \quad \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) \text{ is even and } \geq 2. $$

We now rewrite the Euler inequality (6.4) in the form,

$$ (6.8) \qquad \left\{ \begin{array}{l} 0 \geq \kappa(u) - \ell = (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \\ \qquad + \sum_{j \in J(u)} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right) - (\ell - 2). \end{array} \right. $$

We first observe that the first three terms in the right-hand side of the equality in (6.8) are nonnegative. We apply this inequality to eigenfunctions with prescribed singular points.

Fix some $x \in \Gamma_1$, and let $m := \dim U$. By Lemma 3.10, there exists $0 \neq u \in U$ such that $\rho(u, x) \geq (m - 1)$. Rewrite (6.8) as,

$$ (6.9) \qquad \left\{ \begin{array}{l} 0 \geq \kappa(u) - \ell = (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(z) - 2) \\ \qquad + \sum_{j \in J(u), j \neq 1} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right) \\ \qquad + \frac{1}{2} \sum_{z \in S_b(u) \cap \Gamma_1} \rho(u, z) - \ell + 1. \end{array} \right. $$

The first three terms in the right-hand side of the equality are nonnegative. It follows that

$$ 2\ell - 2 \geq \sum_{z \in S_b(u) \cap \Gamma_1} \rho(u, z) \geq m - 1, $$
so that

$$ (6.10) \qquad \dim U \leq (2\ell - 1). $$

Choosing $U = U(\lambda_k)$, inequality (6.10) yields the following result.

**Proposition 6.2** ([Nadi1987], Theorem 2). Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with smooth boundary. Let $\{\lambda_k, k \geq 1\}$ be the eigenvalues of the operator $-\Delta + V$ in $\Omega$, with Dirichlet or Robin condition. Then, for any $k \geq 1$,

$$ \mathrm{mult}(\lambda_k) \leq (2k - 1). $$

In view of Proposition 6.2, in order to prove Theorem 6.1, it suffices to show that the equalities $\dim U(\lambda_k) = (2k - 1)$ and $\dim U(\lambda_k) = (2k - 2)$ cannot occur for $k \geq 3$. This is the purpose of Sections 7 and 9 in which we revisit the arguments of [HoMN1999] for the three boundary conditions (3.4).

Before proceeding, we examine the case of the second eigenvalue ($k = 2$) which is of special interest. This is the purpose of the next subsection.

## 6.4. Multiplicity of the second eigenvalue and the nodal line conjecture.

When $k = 2$, Proposition 6.2 gives $\mathrm{mult}(\lambda_2) \leq 3$. A natural question, in view of Table 2.1, is whether this bound is sharp (depending on the boundary condition). As we shall see, this question is related to the so-called “nodal line conjecture”.

### 6.4.1. Nodal sets of second eigenfunctions.

We use the notation of Subsection 6.2, and write $\partial\Omega = \bigcup_{j=1}^q \Gamma_j$, with $q \geq 1$. Let $u \in U(\lambda_2)$ be any second eigenfunction. By Courant's theorem, $u$ has exactly two nodal domains. Euler's formula (6.8) yields

$$ (6.11) \qquad \left\{ \begin{array}{l} 0 = \kappa(u) - 2 = [b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1] + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \\ \qquad + \sum_{j \in J(u)} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right). \end{array} \right. $$

Since all the terms are the right hand side are nonnegative (use Corollary 3.8 and the definition of $J(u)$), we immediately deduce that

$$ (6.12) \qquad \left\{ \begin{array}{l} \mathcal{Z}(u) \cup \Gamma(u) \text{ is connected,} \\ \sum_{z \in S_i(u)} (\nu(u, z) - 2) = 0 \quad \text{i.e, } S_i(u) = \emptyset, \\ \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2 \quad \forall j \in J(u). \end{array} \right. $$

The structure of $\mathcal{Z}(u)$ depends on whether $J(u) = \emptyset$ or $J(u) \neq \emptyset$. We now consider the two simplest situations. The proofs of the following properties are clear.

**Property 6.3.** Assume that $\Omega$ is simply connected. Let $u$ be a second eigenfunction. Then, either $\mathcal{Z}(u)$ does not hit $\partial\Omega$ and $S(u) = \emptyset$, or $\mathcal{Z}(u)$ hits $\partial\Omega$, $S_i(u) = \emptyset$, and $\sum_{z \in S_b(u)} \rho(u, z) = 2$. More precisely, there are three distinct possibilities.

(a) If $J(u) = \emptyset$, then $\mathcal{Z}(u)$ is a nodal circle, ie a simple closed regular connected curve contained in $\Omega$, not touching $\partial\Omega$. This case is characterized by the fact that the function $\tilde{u}$ defined in (3.12) does not vanish on $\partial\Omega$.

(b) If $J(u) = \{1\}$ and $S_b(u) = \{y\}$ for some $y \in \partial\Omega$ with $\rho(u, y) = 2$, then $\mathcal{Z}(u)$ is a nodal loop at $y$, ie $\mathcal{Z}(u) \setminus \{y\}$ a simple regular connected curve contained in $\Omega$. This case is characterized by the fact that the function $\tilde{u}$ vanishes only at $y_1$ on $\partial\Omega$, and does not change sign.
(c) If $J(u) = \{1\}$ and $S_b(u) = \{y_1, y_2\}$ for some $y_1 \neq y_2 \in \partial\Omega$ with $\rho(u, y_1) = 1$, $\rho(u, y_2) = 1$, then $Z(u)$ is a nodal arc from $y_1$ to $y_2$, ie $Z(u) \setminus \{y_1, y_2\}$ is a simple regular connected arc contained in $\Omega$. This case is characterized by the fact that the function $\tilde{u}$ vanishes precisely at $y_1$ and $y_2$ on $\partial\Omega$, and changes sign at these points.

**Property 6.4.** Assume that $\Omega$ has one hole, and write $\partial\Omega = \Gamma_1 \cup \Gamma_2$. Let $u$ be a second eigenfunction. Then, either $Z(u)$ does not hit $\partial\Omega$ and $S(u) = \emptyset$, or $Z(u)$ hits $\partial\Omega$, $S_i(u) = \emptyset$, and $\sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2$ for all $j \in J(u)$. More precisely, there are three distinct cases (up to relabeling the two components of $\partial\Omega$).

(1) If $J(u) = \emptyset$, then $Z(u)$ is a nodal circle contained in $\Omega$, not touching $\partial\Omega$. This case is characterized by the fact that the function $\tilde{u}$ defined in (3.12) does not vanish on $\partial\Omega$.

(2) If $J(u) = \{1\}$, then either $S_b(u) = \{y\}$ for some $y \in \Gamma_1$ with $\rho(u, y_1) = 2$, or $S_b(u) = \{y_1, y_2\}$ for some $y_1 \neq y_2 \in \Gamma_1$ with $\rho(u, y_1) = \rho(u, y_2) = 1$. We have either a nodal loop at $y$, or a nodal arc from $y_1$ to $y_2$. This case is characterized by the fact that the function $\tilde{u}$ vanishes only at $y$ (without changing sign along $\Gamma_1$), or vanishes at $y_1$ and $y_2$ (and changes sign along $\Gamma_1$). In both subcases, $\tilde{u}$ does not vanish on $\Gamma_2$.

(3) If $J(u) = \{1, 2\}$, then $Z(u)$ hits both component $\Gamma_1$ and $\Gamma_2$ at one point with index 2, or at two distinct points of index 1. Furthermore, the components $\Gamma_1$ and $\Gamma_2$ are linked by two nodal arcs (possibly with one or two common boundary points).

**Remark 6.5.** It is not clear a priori whether the possible nodal patterns described in Property 6.3 or 6.4 are actually realized for some choice of domain $\Omega$ and potential $V$ (when $\Omega$ is convex and $V \equiv 0$, see [Ales1994]). Applying Lemmas 3.10 or 3.11, one can at least prescribe one or two boundary singular points.

(i) Assume that $\dim U(\lambda_2(-\Delta + V)) \ge 3$. If $\Omega$ is simply connected, then there exists an eigenfunction whose nodal set satisfies (b), resp. (c), in Properties 6.3. If $\Omega$ has one hole, then there exists an eigenfunction whose nodal set hits both $\Gamma_1$ and $\Gamma_2$.

(ii) Assume that $\dim U(\lambda_2(-\Delta + V)) = 2$. Then, there exists an eigenfunction whose nodal domain hits $\partial\Omega$.

Nodal sets and $\mathrm{mult}(\lambda_2)$ are known precisely in few circumstances only, either in very specific cases, or under additional assumptions on the domain (some convexity or symmetry conditions, see [Shen1988], [Putt1990], [Putt1991] in the simply connected case, and [Kiwa2018] for a convex domain with a convex sub-domain removed).

The following figures display some particular cases. The second (Dirichlet or Neumann) eigenvalue of an equilateral triangle with rounded corners has multiplicity two, with one symmetric and one antisymmetric eigenfunction.

The nodal domains of the symmetric eigenfunction appear in Figure 6.1 (left), see [BeHe2021t]. The second (Dirichlet or Neumann) eigenvalue of an ellipse is simple with nodal domains as in Figure 6.1 (center); this is a particular case of the domains described in [Shen1988], [Putt1990] and [Putt1991]. The nodal set of a second Dirichlet eigenvalue of $D \setminus B$, where $D, B$ are convex symmetric domains, has been studied in [Kiwa2018], see Figure 6.1 (right).

Numerical computations, playing with the position of the holes, give rise to some other patterns, see Figure 6.2.
<figure><img src="image_11.png" /><figcaption>FIGURE 6.1. Nodal patterns of second eigenfunctions</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 6.2. Nodal patterns of second eigenfunctions</figcaption></figure>

In view of the above remarks, the following questions are natural.

**Question 1:** Does there exist a second eigenfunction of $-\Delta + V$ whose nodal set is a nodal circle ?

**Question 2:** Does there exist a second eigenfunction of $-\Delta + V$ whose nodal set is a nodal loop at some $y \in \partial\Omega$ ?

6.4.2. *The nodal line conjecture.* For a simply connected domain $\Omega$, Pleijel [Plej1956, p. 546] observed that a second Neumann eigenfunction of $-\Delta$ cannot have a closed nodal line. The idea is to use the inequality $\lambda_2(D, -\Delta, n) \le \lambda_1(D, -\Delta, \mathfrak{d})$ between the second Neumann eigenvalue and the first Dirichlet eigenvalue of $-\Delta$ in a domain $D \subset \mathbb{R}^2$, and the monotonicity property of the first Dirichlet eigenvalue. Indeed, if there exists a second Neumann eigenfunction $u_2$ one of whose two nodal domains (say $D_1$) has the property that $\partial D_1 \cap \partial\Omega$ consists of isolated points (which occurs in particular when $\partial D_1 \subset \Omega$), then the restriction of $u_2$ to $D_1$ is the ground state of the Dirichlet problem in $D_1$, $\lambda_1(D_1, \mathfrak{d}) = \lambda_2(\Omega, n)$. By the strict domain monotonicity of the Dirichlet eigenvalues, we get

$$ \lambda_1(\Omega, \mathfrak{d}) < \lambda_1(D_1, \mathfrak{d}) = \lambda_2(\Omega, n), $$

hence a contradiction. This argument may fail when $\Omega$ as a hole because both nodal domains of $u$ may touch the boundary.

The inequality $\lambda_2(D, -\Delta, n) \le \lambda_1(D, -\Delta, \mathfrak{d})$ is due to Szegö (1954) when $\Omega \subset \mathbb{R}^2$ is simply connected and smooth. The strict inequality is proved in an earlier paper of Pólya (1952) which does apparently not use the assumption that the domain is
simply connected. It was later generalized to higher dimensions, and smooth enough domains (not necessarily simply connected) by Weinberger (1956), see [Payn1967], Theorem 3, p. 463. It has been extended to domains with $C^1$ boundary, see [Maz1991] in which Mazzeo revisits the earlier paper of L. Friedlander [Frie1991]. When $\partial D_1$ meets $\partial \Omega$ we actually need the inequality to hold for domains with piecewise $C^1$ boundary (see [ArMa2007, ArMa2012] for the Lipschitz case).

In view of Pleijel's observation, Payne conjectured that a second Dirichlet eigenfunction of $-\Delta$ cannot have a closed nodal line, see [Payn1967], Conjecture 5, p. 467. One can make a similar conjecture for second Robin eigenfunctions, and also consider domains in $\mathbb{R}^n$, $n \ge 3$, see [Four2001], [Ken2013] and their bibliographies.

**Remark 6.6.** As observed in [Liq1995] (end of Section 2, p. 277), if a simply connected domain satisfies the *nodal line conjecture*, then $\dim U(\lambda_2) \le 2$. This is an immediate consequence of Remark 6.5(ii).

We now consider the three boundary conditions, Dirichlet, Neumann and Robin separately.

### 6.4.3. Dirichlet boundary condition.
The following results are due to Lin and Ni.

* [LiNi1988, Theorem 3.6]: For all $n \ge 2$, there exists a radius $R_n$ and a nonzero smooth radial potential $V_n$, such that

$$ \mathrm{mult}(\lambda_2; B(R_n), -\Delta + V_n, \mathfrak{d}) = 1, $$

and the corresponding eigenfunction is radial, with nodal set a sphere in $B(R_n)$. Here, $B(R_n)$ is the ball of radius $R_n$ in $\mathbb{R}^n$.

* [LiNi1988, Theorem 3.8]: For all $n \ge 2$, there exists a radius $R_n$ and a nonzero smooth radial potential $V_n$, such that

$$ \mathrm{mult}(\lambda_2; B(R_n), -\Delta + V_n, \mathfrak{d}) = (n+1), $$

and there exists a radial second eigenfunction.

In dimension 2, the second assertion implies that the bound of the multiplicity $\mathrm{mult}(\lambda_2; \Omega, -\Delta + V, \mathfrak{d}) \le 3$ is sharp.

The following results are due to M. and T. Hoffmann-Ostenhof and Nadirashvili [HoHN1997, HoHN1998] (see for the second statement [HeHJ2020] for corrections and complements).

* [HoHN1998, Theorem 2.1] There exists $N_0$ and domains $D_{N,\varepsilon} \subset \mathbb{R}^2$ such that for all $N \ge N_0$, and $\varepsilon$ small enough, $\lambda_2(D_{N,\varepsilon}; -\Delta, \mathfrak{d})$ is simple, with a closed nodal set contained in $D_{N,\varepsilon}$. The domain $D_{N,\varepsilon}$ is homeomorphic to a disk minus $N$ points.

* [HoHN1998, Theorem 2.2]: For all $N \ge 3$, and $\varepsilon$ small enough, the domains $D_{N,\varepsilon} \subset \mathbb{R}^2$ satisfy

$$ \mathrm{mult}(\lambda_2; D_{N,\varepsilon}, -\Delta, \mathfrak{d}) = 3. $$

The second assertion implies the upper bound 3 for the second Dirichlet eigenvalue of $-\Delta$ is sharp for non simply connected domains.

See [DaGH2021] for a counterexample to the nodal line conjecture (for $-\Delta$) with six holes. However, counter-examples are still missing for domains with one or two holes.
The nodal line conjecture for $(\Omega, -\Delta)$ is known to be true when $\Omega$ is a bounded convex domain in $\mathbb{R}^2$: [Payn1973] and [Lin1987] (under additional symmetry assumptions), [Mela1992] (smooth convex domains) and [Ales1994] (general convex domains). This is also the case for domains which are convex in one direction only<fnref n="4" />

As a by-product of these results, we have the upper bound

$$ \mathrm{mult}(\lambda_2; \Omega, -\Delta, \mathfrak{d}) \le 2 \quad \text{for any convex bounded domain } \Omega. $$

This bound holds for domains which are convex in one direction and for domains which satisfy the nodal line conjecture (see Remark 6.6). This result supports the following conjecture.

**Conjecture 6.7.**

$$ \mathrm{mult}(\lambda_2; \Omega, -\Delta, \mathfrak{d}) \le 2 \quad \text{for any simply connected bounded domain } \Omega. $$

**6.4.4. Neumann boundary condition.** The discussion in Paragraph 6.4.2 shows that the nodal line conjecture holds for the Neumann Laplacian in any simply connected bounded regular domain in $\mathbb{R}^2$. Nadirashvili proved that the multiplicity of the second eigenvalue of a simply connected domain with nonpositive curvature is at most 2 and that this estimate is sharp, see [Nadi1987], Theorem 2 and Corollary 1. As a matter of fact, his proof also shows that the nodal line conjecture is true in such domains (for the Neumann condition).

**6.4.5. Robin boundary condition.** As observed by J.B. Kennedy<fnref n="5" /> in [Ken2011], the proof of the nodal line conjecture for the h-Robin boundary condition works in the same way as in the case of Neumann conditions provided that the following inequality holds,

$$ \lambda_2(h, \Omega) \le \lambda_1^D(\Omega). $$

Observing the monotonicity of the Robin problem with respect to $h$, we obtain the existence of some $h_\Omega > 0$ such that this inequality holds for $h \le h_\Omega$.

J.B. Kennedy also shows that, as in the Dirichlet case (which corresponds to $h = +\infty$), one can find examples of multiply connected domains for which counter examples to the nodal line conjecture can be constructed. One can also expect to construct examples for which the multiplicity is 3 (as in [HoHN1998] and [HeHJ2020]) but this is still open at the moment.

On the positive side, it is natural to ask if convexity is enough to ensure multiplicity at most 2 for every Robin parameter $h > 0$, following Lin's approach in [Lin1987]. This is still open at the moment. What we do know is that a sufficient condition on $\Omega$ is that nodal line conjecture holds (Remark 6.6).

**7. PLANE DOMAINS, THE ESTIMATE $\mathrm{mult}(\lambda_k) \le (2k-2)$ FOR $k \ge 3$**

The ultimate goal of this section is to prove that the inequality $\mathrm{mult}(\lambda_k) \le (2k-2)$ holds for $k \ge 3$, see Proposition 7.17. Note that the inequality is not true for $k = 1$ and $k = 2$. For this purpose, we provide detailed proofs of the statements in [HoMN1999, Section 2]. The general idea is to prove that an a priori upper bound

<footnote n="4">⁴See [Liq1995, Corollary 2.7]. Note however that the other results on the multiplicity presented in this paper are true under strong additional conditions only. We thank the author for clarifying this point.</footnote>

<footnote n="5">⁵We thank J.B. Kennedy for useful discussions around this problem.</footnote>
on the number of nodal domains of functions in a given subspace $U$ of eigenfunctions implies an upper bound on $\dim U$. Indeed, the bigger $\dim U$, the easier to construct eigenfunctions with prescribed high order singular points and, by Euler's formula, with more nodal domains.

We first introduce an abstract setting, to be used when $\Omega$ is multiply connected, and some notation to describe the nodal sets in a neighborhood of a boundary singular point. The following subsections are devoted to studying eigenfunctions with prescribed boundary singular points, and drawing consequences.

**7.1. An abstract setting.** When the domain $\Omega \subset \mathbb{R}^2$ is not simply connected, its boundary $\partial\Omega$ has $(q+1)$ components, with $q \ge 1$. One of them $\Gamma_1$, the “outer boundary”, bounds the unbounded component of $\mathbb{R}^2 \setminus \partial\Omega$. The other components, $\Gamma_j$, $j \ne 1$, are contained in the bounded component of $\mathbb{R}^2 \setminus \Gamma_1$. We consider the following equivalence relation in $\bar{\Omega}$:

$$ (7.1) \qquad x \sim y \text{ if and only if } x, y \in \Gamma_j \text{ for some } j \in \{2, \dots, (q+1)\}. $$

**Notation 7.1.** Let $\bar{\Omega}$ denote the quotient space $\bar{\Omega}/\sim$, where each $\Gamma_j$, $j \ne 1$, is identified to one point $\xi_j$ in $\check{\Omega}$ (see [Bona2009]). Define

$$ \Xi := \{\xi_2, \dots, \xi_{q+1}\}. $$

Generally speaking, $\check{A}$ will denote the image of the set $A \subset \bar{\Omega}$ under the projection map from $\bar{\Omega}$ to $\check{\Omega}$.

We also introduce $S_\Omega$, the quotient space in which each $\Gamma_j$, $j \ge 1$, is identified to a point.

**7.2. Describing nodal sets near a boundary singular point.** In the plane $\mathbb{R}^2$ with coordinates $(\xi_1, \xi_2)$, define the sets

$$ \begin{align} D_+(a) &= \{(\xi_1, \xi_2) \in \mathbb{R}^2 \mid \xi_1^2 + \xi_2^2 < a \text{ and } \xi_2 > 0\}, \tag{7.2} \\ \bar{D}_+(a) &= \{(\xi_1, \xi_2) \in \mathbb{R}^2 \mid \xi_1^2 + \xi_2^2 \le a \text{ and } \xi_2 \ge 0\}, \notag \\ S_+(a) &= \{(\xi_1, \xi_2) \in \mathbb{R}^2 \mid \xi_1^2 + \xi_2^2 = a \text{ and } \xi_2 \ge 0\}. \notag \end{align} $$

Fix $x \in \partial\Omega$, and choose a direct orthonormal frame $\{\vec{e}_1, \vec{e}_2\}$ at $x$ such that $\vec{e}_1$ is tangent to $\partial\Omega$ at $x$, and $\vec{e}_2$ is normal to $\partial\Omega$, pointing inwards. The vector $\vec{e}_1$ determines the orientation of $\partial\Omega$.

Let $u$ be an eigenfunction with boundary singular point $x$, and whose index at the point $x$ is $\rho(u, x)$ (the number of nodal arcs hitting the boundary at this point). In order to describe the nodal set of the eigenfunction $u$ near the point $x$, we choose conformal coordinates $(U, E)$, where $U$ is a neighborhood of $x$ in $\bar{\Omega}$, and $E$ a $C^\infty$ conformal map from $\bar{D}_+(1)$ onto $U$, such that $E(0) = x$ (see [YaZh2021], Section 2).

The equation $(-\Delta + V)u = \lambda u$ in $U$ is transformed into an equation of the form $(-\Delta + \tilde{V})\tilde{u} = \lambda \tilde{A} \tilde{u}$, where $\tilde{u} = u \circ E$, $\tilde{V}$ is a real valued $C^\infty$ potential, and $\tilde{A}$ a positive $C^\infty$ function. Furthermore, $\tilde{u}$ satisfies the Dirichlet or a Robin type boundary condition on $\bar{D}_+(1) \cap \{\xi_2 = 0\}$. The function $u$ does not vanish at infinite order at any point, see [DoFe1990a].

Denote by $(r, \omega)$ the polar coordinates associated with $(\xi_1, \xi_2)$. If $\mathrm{ord}(u, x) = p$, then $\mathrm{ord}(\tilde{u}, 0) = p$ as well. Applying the local structure theorem to $\tilde{u}$ at the point 0, and
for $r \le r_0$ with $r_0 < 1$ small enough, we can write

$$ (7.3) \qquad \begin{cases} \tilde{u}(r, \omega) = a_{u,x} r^p \sin(p\omega) + \mathcal{O}(r^{p+1}) & \text{(Dirichlet case)}, \\ \tilde{u}(r, \omega) = a_{u,x} r^p \cos(p\omega) + \mathcal{O}(r^{p+1}) & \text{(Robin case)}, \end{cases} $$

where $a_{u,x}$ is a nonzero scalar. The important point is that near the point 0, the nodal set of $\tilde{u}$ consists of $\rho(u_1, 0) = (p-1)$ (Dirichlet case), resp. $\rho(u_1, 0) = p$ (Robin case), nodal semi-arcs emanating from 0, tangentially to rays $\tilde{\omega}_j$, $1 \le j \le \rho(u_1, 0)$ at 0. These rays divide the half-circle $S_+(r_0)$ into equal arcs. More details are given in Appendix C.

Define the sets

$$ (7.4) \qquad \begin{aligned} B_+(x, r) &:= E(D_+(r)), \\ \bar{B}_+(x, r) &:= E(\bar{D}_+(r)), \\ C_+(x, r) &:= E(S_+(r)), \\ L_m &:= \{1, \dots, m\} \text{ for } m \ge 1. \end{aligned} $$

Since $\mathcal{Z}(u) \cap \bar{B}_+(x, r_0) = E(\mathcal{Z}(u_1) \cap \bar{D}_+(r_0))$, $\mathcal{Z}(u) \cap \bar{B}_+(x, r_0)$ consists of $\rho(u, x)$ semi-arcs emanating from $x$, tangentially to the rays $\omega_j$, $j \in L_{\rho(u,x)}$. The rays are labeled counter-clockwise, according to their angle with respect to the vector $\vec{e}_1$. The nodal semi-arc emanating from $x$ tangentially to the ray $\omega_j$ is denoted by $\delta_j$.

**7.3. Analysis of eigenfunctions with two prescribed boundary singular points.** We use the notation of Subsection 6.2. In this subsection, we assume that $U$ is a linear subspace of an eigenspace $U(\lambda)$ of (3.3), and that for some $\ell \ge 2$,

$$ \left\{ \begin{array}{l} \sup \{\kappa(u) \mid 0 \neq u \in U\} \le \ell \quad \text{and} \\ \dim U = (2\ell - 1). \end{array} \right. $$

For $x \neq y \in \Gamma_1$, we introduce the subspace

$$ U_{x,y} := \{u \in U \mid \rho(u,x) \ge (2\ell - 3) \text{ and } \rho(u,y) \ge 1\}. $$

According to Lemma 3.11, $U_{x,y} \neq \{0\}$. The purpose of this subsection is to investigate the properties of the functions $u \in U_{x,y}$ –precise order of vanishing, structure of their nodal sets– under the above assumptions on $U$.

### 7.3.1. $\Omega$ simply connected. Properties of $U_{x,y}$.

**Lemma 7.2.** Assume that $\Omega$ is simply connected. Let $U$ be a linear subspace of an eigenspace of (3.3) in $\Omega$, such that $\sup \{\kappa(u) \mid 0 \neq u \in U\} \le \ell$ for some $\ell \ge 2$, and $\dim U = (2\ell - 1)$. Let $x \neq y \in \Gamma_1$, and define

$$ U_{x,y} := \{u \in U \mid \rho(u,x) \ge (2\ell - 3) \text{ and } \rho(u,y) \ge 1\}. $$

Then,

(i) $\dim U_{x,y} = 1$ and, for any $0 \neq u \in U_{x,y}$;

(ii) $S_i(u) = \emptyset$ and $S_b(u) = \{x, y\}$;

(iii) $\rho(u, x) = (2\ell - 3)$ and $\rho(u, y) = 1$;

(iv) $\kappa(u) = \ell$;

(v) the set $\mathcal{Z}(u) \cup \partial\Omega$ is connected.

A generator of $U_{x,y}$ will be denoted by $u_{x,y}$ (defined up to scaling).
Proof. For simplicity, in the proof, we write $\nu(z)$ for $\nu(u, z)$, ....

The fact that $\dim U_{x,y} \ge 1$ follows from Lemma 3.11. In view of our assumptions, for any $0 \ne u \in U$, Euler's formula (6.9) gives,

$$ (7.5) \qquad 0 \ge \kappa(u) - \ell = (b_0(\mathcal{Z}(u) \cup \partial\Omega) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(z) - 2) \\ \qquad \qquad + \frac{1}{2} \sum_{z \in S_b(u), z \ne x, y} \rho(z) + \frac{1}{2} (\rho(x) + \rho(y) - 2\ell + 2). $$

Each term in the right-hand side of the equality being nonnegative, the inequality implies that each term is zero, thus proving Assertions (ii)-(v).

To prove the first assertion, assume that there exist two linearly independent functions $u_1$ and $u_2$ in $U$. By Assertion (ii) they both satisfy $\rho(u_i, x) = (2\ell - 3)$ and $\rho(u_i, y) = 1$. Apply Lemma 3.12 at the point $y$ to find a nontrivial linear combination $\tilde{u}$ of $u_1$ and $u_2$ such that $\rho(\tilde{u}, y) \ge 2$ contradicting Assertion (ii). $\square$

<figure><img src="image_13.png" /><figcaption>FIGURE 7.1. $\Omega$ simply connected: some possible nodal patterns for $u_{x,y}$</figcaption></figure>

**Remark 7.3.** The nodal patterns displayed in Figure 7.1 are valid for both the Dirichlet and Robin boundary conditions. Unless otherwise stated this remark applies to all figures of this section.

**Definition 7.4.** By a nodal pattern, we mean a nodal set, up to continuous deformations under which singular points may move, but neither appear nor disappear (singular points occur when two nodal arcs cross, or when a nodal arc hits the boundary).

7.3.2. $\Omega$ simply connected. Structure and combinatorial type of nodal sets in $U_{x,y}$. Fix $x \ne y \in \partial\Omega$. Under the assumptions of Lemma 7.2, the nodal set of an eigenfunction $u \in U_{x,y}$ can be described as follows, using the notation of Subsection 7.2. For $r_0$ small enough, in the neighborhood $B_+(x, r_0)$ of $x$ the nodal set $\mathcal{Z}(u)$ consists of $(2\ell - 3)$ nodal semi-arcs $\delta_j$ emanating from $x$, tangentially to the rays $\omega_j$, $1 \le j \le (2\ell - 3)$. Choosing any $j$, we follow the nodal semi-arc $\delta_j$ along $\mathcal{Z}(u)$, until we reach a singular point of $u$. Otherwise stated, we consider the connected component of $\mathcal{Z}(u) \setminus S(u)$ which contains the semi-arc $\delta_j$ (by abuse of notation we also denote this connected component by $\delta_j$). This is a nodal arc one of whose end points is $x$. Since $S(u) = S_b(u) = \{x, y\}$, the other end point is either $x$ or $y$. More precisely, in view of the general properties of nodal sets, we define a map

$$ (7.6) \qquad \tau_{x,y}^u : \{\downarrow\} \cup L_{(2k-3)} \to \{\downarrow\} \cup L_{(2k-3)} $$
as follows.

(i) There exists a unique element $a \in L_{(2k-3)}$ such that starting from $x$ along $\delta_a$, we reach the boundary $\partial\Omega$ at $y$. We let $\tau_{x,y}^u(\downarrow) = a$ and $\tau_{x,y}^u(a) = \downarrow$.

(ii) For $j \in L_{(2k-3)} \setminus \{a\}$, following $\delta_j$, we arrive back at $x$, along another nodal semi-arc, which we denote by $\delta_{\tau_{x,y}^u(j)}$; this semi-arc emanates from $x$ tangentially to the ray $\omega_{\tau_{x,y}^u(j)}$. This defines $\tau_{x,y}^u$ on $L_{(2k-3)} \setminus \{a\}$. The local structure theorem implies that for, $j \in L_{(2k-3)} \setminus \{a\}$, $\tau_{x,y}^u(j) \in L_{(2k-3)} \setminus \{a\}$, and $\tau_{x,y}^u(j) \neq j$.

Doing so, we have uniquely defined the map $\tau_{x,y}^u$ from $\{\downarrow\} \cup L_{(2\ell-3)}$ to itself, such that $(\tau_{x,y}^u)^2 = \text{Id}$, and $(\tau_{x,y}^u)(j) \neq j$.

The pair $\{\downarrow, a\}$ corresponds to the nodal arc $\delta_a$ from $x$ to $y$. For $j \in L_{(2k-3)} \setminus \{a\}$, the pair $\{j, \tau_{x,y}^u(j)\}$ corresponds to a loop $\gamma_{j,\tau_{x,y}^u(j)}$ at $x$. There are $(\ell-2)$ such loops. Since $\mathcal{S}(u) = \{x, y\}$ and $\rho(u, x) = 1$, these loops and arc do not intersect away from $x$. Since $\mathcal{Z}(u) \cup \partial\Omega$ is connected, the nodal set $\mathcal{Z}(u)$ is actually the union of these $(\ell-2)$ loops and arc. Otherwise stated, $\mathcal{Z}(u)$ is the wedge sum $B_{x,(\ell-2)}^y$ of the simple arc $\delta_{\tau_{x,y}^u(\downarrow)}$ from $x$ to $y$ with an $(\ell-2)$-bouquet of loops at $x$.

**Definition 7.5.** By analogy with Definition 5.4, we call the map $\tau_{x,y}^u$ the combinatorial type of the eigenfunction $u$ (or of the nodal set $\mathcal{Z}(u)$) with respect to the points $x$ and $y$.

We describe the map $\tau_{x,y}^u$ in matrix form as

$$ (7.7) \quad \tau_{x,y}^u = \begin{pmatrix} \downarrow & 1 & \dots & (a-1) & a & (a+1) & \dots & (2\ell-3) \\ a & \tau_{x,y}^u(1) & \dots & \tau_{x,y}^u(a-1) & \downarrow & \tau_{x,y}^u(a+1) & \dots & \tau_{x,y}^u(2\ell-3) \end{pmatrix}. $$

Figure 7.1 displays some possible nodal patterns (for $\ell = 5$, $\rho(x) = 7$, and $\rho(y) = 1$). The corresponding combinatorial types are given by

$$ \tau_A = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 1 & \downarrow & 3 & 2 & 5 & 4 & 7 & 6 \end{pmatrix}, \quad \tau_B = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 1 & \downarrow & 7 & 6 & 5 & 4 \end{pmatrix}, \quad \tau_C = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & \downarrow \end{pmatrix}. $$

### 7.3.3. $\Omega$ with one hole. Properties of $U_{x,y}$.

**Lemma 7.6.** Assume that $\Omega$ has one hole, with $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ is the outer boundary. Let $U$ be a linear subspace of an eigenspace of (3.3) in $\Omega$, such that $\sup\{\kappa(u) | 0 \neq u \in U\} \le \ell$ for some $\ell \ge 2$, and $\dim U = (2\ell-1)$. Let $x \neq y \in \Gamma_1$, and define

$$ U_{x,y} := \{u \in U \mid \rho(u, x) \ge (2\ell - 3) \text{ and } \rho(u, y) \ge 1\}. $$

Then,

(i) $\dim U_{x,y} = 1$.

(ii) For $0 \neq u \in U_{x,y}$, the following alternative holds,

◇ either $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 1$, in which case $\mathcal{Z}(u)$ hits $\Gamma_2$, $u$ has precisely two singular points on $\Gamma_2$ (counting multiplicities), $\sum_{z \in S_b(u) \cap \Gamma_2} \rho(u, z) = 2$, $S_b(u) \cap \Gamma_1 = \{x, y\}$ and $S_i(u) = \emptyset$;

◇ or $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 2$, in which case $\mathcal{Z}(u) \cap \Gamma_2 = \emptyset$, $S_b(u) \cap \Gamma_1 = \{x, y\}$, and $S_i(u) = \emptyset$;

(iii) $\rho(u, x) = (2\ell - 3)$ and $\rho(u, y) = 1$.

(iv) $\kappa(u) = \ell$.

A generator of $U_{x,y}$ will be denoted by $u_{x,y}$ (defined up to scaling).
Proof. The fact that dim $U_{x,y} \ge 1$ follows from Lemma 3.11. Since we now have $b_0(\partial\Omega) = 2$, Euler's formula (6.9) applied to $u$ gives

$$ (7.8) \qquad \begin{aligned} 0 \ge \kappa(u) - \ell &= (b_0(\mathcal{Z}(u) \cup \partial\Omega) - 2) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(z) - 2) \\ & \qquad + \frac{1}{2} \sum_{z \in S_b(u) \cap \Gamma_2} \rho(z) \\ & \qquad + \frac{1}{2} \sum_{\substack{y \in S_b(u) \cap \Gamma_1, \\ z \ne x, y}} \rho(z) + \frac{1}{2} (\rho(x) + \rho(y) - 2\ell + 2). \end{aligned} $$

Except for the term $(b_0(\mathcal{Z}(u) \cup \partial\Omega) - 2)$, all the terms in the right-hand side of the equality are nonnegative. This implies that

$$ 2 \ge b_0(\mathcal{Z}(u) \cup \partial\Omega) \ge 1, $$

and we have to examine two cases.

♦ If $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 1$, then the nodal set $\mathcal{Z}(u)$ must hit $\Gamma_2$. According to Proposition 4.22, the sum $\sum_{z \in S_b(u) \cap \Gamma_2} \rho(z)$ is an even integer, and we deduce from (7.8) that $\sum_{z \in S_b(u) \cap \Gamma_2} \rho(z) = 2$. This equality now implies that the other terms are zero, and hence that $\kappa(u) = \ell$.

♦ If $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 2$, then all the terms in the right-hand side of (7.8) vanish, and $\kappa(u) = \ell$.

This proves Assertions (ii)–(iv). As in the proof of Lemma 7.2, assuming that there are at least two linearly independent functions $u_1$ and $u_2$ in $U_{x,y}$, the first assertion follows from Assertion (iii) and Lemma 3.12. $\square$

### 7.3.4. $\Omega$ with one hole. Structure and combinatorial type of nodal sets in $U_{x,y}$. From a geometric point of view, once we have fixed $x \ne y \in \Gamma_1$ and under the assumptions of Lemma 7.6, either $\mathcal{Z}(u) \cap \Gamma_2 = \emptyset$ or $\mathcal{Z}(u) \cap \Gamma_2 = \{y_1, y_2\}$, possibly with $y_1 = y_2$. In the first case, we simply reproduce the description given in Paragraph 7.3.2. In the second case, the connectivity of $\mathcal{Z}(u) \cup \partial\Omega$ implies that one of the nodal arcs hitting $\Gamma_2$ also contains $x$. The other one contains either $x$ or $y$. More precisely, we choose any $j \in L_{(2\ell-3)}$ and follow the nodal semi-arc $\delta_j$ emanating from $x$ along $\mathcal{Z}(u)$, until we meet a singular point $z$ as described in Paragraph 7.3.2. Since $z \in S(u) = \{x, y, y_1, y_2\}$, there are two possibilities. If $z \in \{x, y\}$ the description is similar to the one in Paragraph 7.3.2. If $z \in \{y_1, y_2\}$, say $y_1$, we continue our path from $y_1$ to $y_2$ along $\Gamma_2$, and leave $\Gamma_2$ along the second nodal arc hitting $\Gamma_2$ at $y_2$ until we meet a singular point. We then either reach the point $x$ again or the point $y$. It then follows that the nodal set of $u \in U_{x,y}$ consists of $(\ell - 2)$ “generalized” nodal loops at $x$ (one of the loops may comprise some part of $\Gamma_2$), and a “generalized” simple arc from $x$ to $y$ (this arc may comprise a sub-arc from $y_1$ to $y_2$ on $\Gamma_2$). In the preceding description, the points $y_1$ and $y_2$ may coincide. These “generalized” loops and arc do not intersect away from $x$. Then, $\mathcal{Z}(u)$ is the wedge sum $B_{x,(\ell-2)}^y$ of an $(\ell - 2)$-bouquet of “generalized” loops at $x$, with a simple “generalized” arc from $x$ to $y$. We can then define the combinatorial type $\tau_{x,y}^u$ of $u$ with respect to the points $x$ and $y$ as we did in Paragraph 7.3.2, somehow ignoring $\Gamma_2$.

Projecting $\mathcal{Z}(u)$ to $\tilde{\Omega}$, we obtain a set $\tilde{\mathcal{Z}}(u) \subset \tilde{\Omega}$ which is the wedge sum $B_{\tilde{x},(\ell-2)}^{\tilde{y}}$ of an $(\ell - 2)$-bouquet of loops at $\tilde{x}$ with a simple arc from $\tilde{x}$ to $\tilde{y}$. One of the loops or the arc may contain the point $\xi_2$, the image of $\Gamma_2$ in $\tilde{\Omega}$. Since there are only two semi-arcs at $\xi_2$, this point is a regular point of the projected nodal partition $\tilde{D}_u$. The general picture is then similar to the picture in the simply connected case.
Figures 7.2 and 7.3 display some possible nodal patterns for $0 \neq u \in U_{x,y}$ when $\Omega$ has one hole ($\ell = 5$, $\rho(x) = 7$, and $\rho(y) = 1$).

<figure><img src="image_14.png" /><figcaption>FIGURE 7.2. $\Omega$ with one hole: some possible nodal patterns for $u_{x,y}$</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>FIGURE 7.3. $\Omega$ with one hole: some possible nodal patterns for $u_{x,y}$</figcaption></figure>

For the nodal patterns in Figures 7.2 and 7.3, we have the combinatorial types

$$ \tau_A^{7.2} = \tau_B^{7.3} = \tau_C^{7.3} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 1 & \downarrow & 7 & 6 & 5 & 4 \end{pmatrix}, $$

$$ \tau_C^{7.2} = \tau_A^{7.3} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 1 & \downarrow & 3 & 2 & 5 & 4 & 7 & 6 \end{pmatrix}, $$

$$ \tau_B^{7.2} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 2 & 1 & 4 & 3 & \downarrow & 7 & 6 \end{pmatrix}. $$

7.3.5. $\Omega$ with $k$ holes. Properties of $U_{x,y}$. In this case, $\partial\Omega$ has $(k+1)$ components, $\partial\Omega = \bigcup_{j=1}^{k+1} \Gamma_j$. Fix $x \neq y \in \Gamma_1$. With the notation of Subsection 6.2, we have the following lemma.

**Lemma 7.7.** Assume that $\Omega$ has $k$ holes, with $\partial\Omega = \bigcup_{j=1}^{k+1} \Gamma_j$. Let $U$ be an eigenspace of (3.3) in $\Omega$, such that, for some $\ell \geq 2$, $\sup\{\kappa(u) \mid 0 \neq u \in U\} \leq \ell$, and $\dim U = (2\ell - 1)$. Let $x \neq y \in \Gamma_1$, and define the subspace

$$ U_{x,y} := \{u \in U \mid \rho(u, x) \geq (2\ell - 3) \text{ and } \rho(u, y) \geq 1\}. $$

Then, $\dim U_{x,y} = 1$. Furthermore, for all $0 \neq u \in U_{x,y}$:
(i) The set $\mathcal{Z}(u) \cup \Gamma(u)$ is connected.

(ii) If $J(u) = \{1\}$, the only singular points of $u$ are the points $x$ and $y$, with $\rho(u, x) = (2\ell - 3)$ and $\rho(u, y) = 1$.

(iii) If $J(u) \neq \{1\}$, each component $\Gamma_j, j \in J(u)$, is hit by exactly two nodal arcs, the function $u$ has no interior singular point, and its only singular points on $\Gamma_1$ are $x$ and $y$, with $\rho(u, x) = (2\ell - 3)$ and $\rho(u, y) = 1$.

(iv) In all cases, $\kappa(u) = \ell$.

(v) In all cases, the nodal set of $u$ consists of $(\ell - 2)$ simple non-intersecting "generalized" nodal loops at $x$ (loops comprising nodal arcs, and possibly arcs contained in some boundary components $\Gamma_j, j \in J(u) \setminus \{1\}$), a simple nodal arc from $y$ to either $x$ (when $J(u) = \{1\}$) or to some inner component of $\partial\Omega$, a simple nodal arc from $x$ to some component $\Gamma_j, j \in J(u) \setminus \{1\}$, and possibly some nodal arcs joining components which meet $\mathcal{Z}(u)$. These nodal arcs can only intersect at $x$ or possibly on the components $\Gamma_j, j \in J(u) \setminus \{1\}$. In all cases, the point $x$ is joined to the point $y$ by a simple arc comprising nodal arcs and sub-arcs of the $\Gamma_j, j \in J(u) \setminus \{1\}$.

A generator of $U_{x,y}$ will be denoted by $u_{x,y}$ (defined up to scaling).

**Proof of Lemma 7.7.** With the assumptions of the lemma, Euler's formula (6.9) can be rewritten as,

$$ (7.9) \qquad 0 \ge \kappa(u) - \ell = (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(z) - 2) \\ \qquad \qquad + \sum_{j \in J(u), j \ne 1} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(z) - 2 \right) \\ \qquad \qquad + \frac{1}{2} \sum_{z \in S_b(u) \cap \Gamma_1, z \ne x, y} \rho(z) + \frac{1}{2} (\rho(x) + \rho(y) - 2\ell + 2). $$

In view of our assumptions, and Proposition 4.22, the terms in the right-hand side of (7.9) are all non-negative. In view of the left hand side of the inequality, they must all be zero. We now examine two cases.

◇ If $J(u) = \{1\}$, the second line is the right hand side disappears, the nodal set $\mathcal{Z}(u)$ only meets $\Gamma_1$, $b_0(\mathcal{Z}(u) \cup \Gamma_1) = 1$, the only singular points of the function $u$ are the points $x$ and $y$, and $\rho(x) = (2\ell - 3)$, $\rho(y) = 1$.

◇ If $J(u) \neq \{1\}$, all the terms in the right hand side must be zero: $b_0(\mathcal{Z}(u) \cup \Gamma(u)) = 1$, each component $\Gamma_j, j \in J(u) \setminus \{1\}$, is hit by precisely two nodal arcs of $\mathcal{Z}(u)$, $\rho(x) = (2\ell - 3)$, and $\rho(y) = 1$, and the function $u$ has no other singular point whether in the interior of $\Omega$ or on $\partial\Omega$. Furthermore, there is a simple nodal arc from $x$ to one of the components $\Gamma_j, j \in J(u)$, a simple nodal arc from $y$ to one of the components $\Gamma_j, j \in J(u)$, and there is a simple nodal arc, possibly comprising arcs contained in $\Gamma(u)$ joining $x$ to $y$. Finally, $\kappa(u) = \ell$. This proves assertions (i)–(v).

To prove the first assertion, assuming there are at least two linearly independent functions $u_1$ and $u_2$ in $U_{x,y}$, we can apply Lemma 3.12 as in the previous proofs, and construct yet another function $0 \neq \tilde{u}$ such that $\rho(\tilde{u}, x) \ge (2\ell - 3)$ and $\rho(\tilde{u}, y) \ge 2$, contradicting Assertions (ii). □

### 7.3.6. $\Omega$ with $k$ holes. Structure and combinatorial type of nodal sets in $U_{x,y}$. We can adapt the description of the nodal set $\mathcal{Z}(u)$, $u \in U_{x,y}$ given in Paragraph 7.3.4 to the present case (multiple components of $\partial\Omega$. The "generalized" loops or arc will then hit one or several components $\Gamma_j, j \in J(u) \setminus \{1\}$. We can also define the combinatorial type $\tau_{x,y}^u$ of $u$ with respect to the points $x$ and $y$.
<figure><img src="image_16.png" /><figcaption>FIGURE 7.4. $\Omega$ with two holes: some possible nodal patterns for $u_{x,y}$</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 7.5. $\Omega$ with two holes: some possible nodal patterns for $u_{x,y}$</figcaption></figure>

Figures 7.4 and 7.5 display possible nodal patterns for $0 \neq u \in U_{x,y}$ (in these examples, $l = 5$, $\rho(x) = 7$, and there are 3 loops). For these nodal patterns, we have the combinatorial types

$$ \tau_A^{7.4} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 1 & \downarrow & 3 & 2 & 7 & 6 & 5 & 4 \end{pmatrix}, $$

$$ \tau_B^{7.4} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 1 & \downarrow & 7 & 6 & 5 & 4 \end{pmatrix}, $$

$$ \tau_C^{7.4} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 4 & 3 & 2 & 1 & \downarrow & 7 & 6 \end{pmatrix}, $$

and

$$ \tau_A^{7.5} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 1 & \downarrow & 5 & 4 & 3 & 2 & 7 & 6 \end{pmatrix}, $$

$$ \tau_B^{7.5} = \tau_C^{7.5} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 1 & \downarrow & 5 & 4 & 7 & 6 \end{pmatrix}. $$

Lemma 7.7 can be reformulated in the abstract setting of Subsection 7.1 as follows. For any $0 \neq u \in U_{x,y}$, the projection $\tilde{Z}(u)$ of the nodal set $Z(u)$ consists of $(l-2)$ continuous simple loops at $\tilde{x}$ and a continuous simple curve from $\tilde{x}$ to $\tilde{y}$. The loops and curve only intersect at $\tilde{x}$ and may contain points in $\Xi$. If $\xi_j \in \tilde{Z}(u)$, there are
exactly two projected nodal semi-arcs at this point, and the point $\xi_j$ is a regular point of $\tilde{D}_u$. The set $\tilde{Z}(u) \subset \tilde{\Omega}$ is therefore the wedge sum $B_{\tilde{x},(\ell-2)}^{\tilde{y}}$ of an $(\ell-2)$-bouquet of loops at $\tilde{x}$, with a simple arc from $\tilde{x}$ to $\tilde{y}$. The loops or the arc may contain points in $\Xi$. This is illustrated in Figure 7.6.

<figure><img src="image_18.png" /><figcaption>FIGURE 7.6. Nodal patterns in $\Omega$ and their projections in $\tilde{\Omega}$</figcaption></figure>

**Remark 7.8.** From the point of view of Definition 4.2, the points in $\Xi$ are not singular points of $\tilde{D}_u$, the projection of the nodal partition $D_u$ of $u$.

**7.4. Analysis of eigenfunctions with one prescribed boundary singular point.** We use the notation of Subsection 6.2. In this subsection, we assume that $U$ is a linear subspace of an eigenspace $U(\lambda)$ of (3.3), and that for some $\ell \ge 2$,

$$ \begin{cases} \sup \{\kappa(u) \mid 0 \neq u \in U\} \le \ell & \text{and} \\ \dim U = (2\ell - 1). & \end{cases} $$

For $x \in \Gamma_1$, we introduce the subspaces

$$ (7.10) \qquad \left\{ \begin{array}{l} U_x^1 = \{u \in U \mid \rho(u, x) \ge (2\ell - 2)\}, \\ U_x^2 = \{u \in U \mid \rho(u, x) \ge (2\ell - 3)\}. \end{array} \right. $$

According to Lemma 3.11, $U_x^1 \neq \{0\}$. The purpose of this subsection is to investigate the properties of the functions $u \in U_x^1$ or $U_x^2$ –precise order of vanishing, structure of their nodal sets– under the above assumptions on $U$.

### 7.4.1. Properties of $U_x^1$ and $U_x^2$.

**Lemma 7.9.** Let $U$ be a linear subspace of an eigenspace of (3.3) in $\Omega$, with
$$ \sup \{\kappa(u) \mid 0 \neq u \in U\} \le \ell, \quad \text{and} \quad \dim U = (2\ell - 1). $$

Fix some $x \in \Gamma_1$. For the spaces $U_x^1$ and $U_x^2$ defined in (7.10), we have
(i) $\dim U_x^1 = 1$, $\dim U_x^2 = 2$ and,

(ii) for any $0 \neq u \in U_x^2$,

$$
(7.11) \quad \left\{
\begin{array}{ll}
\kappa(u) = \ell \quad \text{and} \quad \mathcal{Z}(u) \cup \Gamma(u) \text{ is connected,} \\
S_i(u) = \emptyset, \\
\sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2 \quad \text{for all } j \in J(u) \setminus \{1\}, \\
\sum_{z \in S_b(u) \cap \Gamma_1} \rho(u, z) = (2\ell - 2) \quad \text{and, more precisely,} \\
\quad \text{(i) either } \rho(u, x) = (2\ell - 2) \quad \text{and } S_b(u) \cap \Gamma_1 = \{x\}, \\
\quad \text{(ii) or } \rho(u, x) = (2\ell - 3), \exists y_u \in \Gamma_1 \setminus \{x\} \text{ with } \rho(u, y_u) = 1, \\
\quad \text{and } S_b(u) \cap \Gamma_1 = \{x, y_u\}.
\end{array}
\right.
$$

Proof. Assume that $\partial\Omega$ has $(q+1)$ components, $\Gamma_1, \dots, \Gamma_{q+1}$, with $x \in \Gamma_1$.

Clearly, $\{0\} \neq U_x^1 \subset U_x^2$. Take any $0 \neq u \in U_x^2$. Euler's formula (6.9) can be rewritten as

$$
(7.12) \qquad
\begin{aligned}
0 \ge \kappa(u) - \ell &= (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(z) - 2) \\
&\quad + \sum_{i \in J(u), i \ne 1} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_i} \rho(z) - 2 \right) \\
&\quad + \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_1} \rho(z) - 2\ell + 2 \right).
\end{aligned}
$$

The first $|J(u)| + 2$ terms in the right-hand side of the equality are nonnegative. Since $\sum_{z \in S_b(u) \cap \Gamma_1} \rho(z)$ is even, and larger than or equal to $(2\ell - 3)$, the last term is nonnegative also. In view of the first inequality, the four terms must vanish. This proves the relations (7.11).

◇ *Proof that* $\dim U_x^1 = 1$. Assume that this is not the case. Then, there exist two linearly independent functions $u_1, u_2$ in $U_x^1$ such that $\rho(u_i, x) = (2\ell - 2)$. By Lemma 3.12, there would exist a nontrivial linear combination $u$ such that $u \in U_x^1$ and $\rho(u, x) \ge (2\ell - 1)$, a contradiction with (7.11).

◇ *Proof that* $\dim U_x^2 = 2$. Choose some $0 \ne v_1 \in U_x^1$. Clearly $v_1 \in U_x^2$. On the other-hand, given any $y \in \Gamma_1 \setminus \{x\}$, Lemma 7.2 provides a function $u_{x,y}$ belonging to $U_x^2$, not to $U_x^1$, and hence $\dim U_x^2 \ge 2$. Choose $0 \ne v_2 \in U_x^2$ orthogonal to $v_1$. Since $\dim U_x^1 = 1$, the function $v_2$ satisfies $\rho(v_2, x) = (2\ell - 3)$, and by Proposition 4.22, there must exist some $y_2 \in \Gamma$ such that $\rho(v_2, y_2) \ge 1$. By Lemma 7.7, $\rho(v_2, y_2) = 1$ and $v_2 \in U_{x,y_2}$. The subspace $U_x^{1,\perp} := \{u \in U_x^2 \mid u \perp u_1\}$ has dimension at least one. Assume that $\dim U_x^2 \ge 3$. Then $\dim U_x^{1,\perp} \ge 2$, and we can find two linearly independent functions $u_1, u_2 \in U_x^{1,\perp}$ such that $\rho(u_i, x) = (2\ell - 3)$. By Lemma 3.12, there exists a linear combination $u \in U_x^{1,\perp}$ such that $\rho(u, x) \ge (2\ell - 2)$, a contradiction. $\square$

**Remark 7.10.** Up to scaling, there is a uniquely defined orthogonal basis $\{v_1, v_2\}$ of $U_x^2$, with $v_1 \in U_x^1$, $v_2 \in U_x^{1,\perp}$, and a uniquely defined $y_2 \in \Gamma_1 \setminus \{x\}$ such that $\rho(v_2, y_2) = 1$. In view of Lemma 3.14, we can choose $v_1$ such that $\tilde{v}_1 > 0$ on $\Gamma_1 \setminus \{x\}$, and $v_2$ such that $\tilde{v}_2 > 0$ on the arc from $x$ to $y_2$ moving counter-clockwise on $\Gamma_1$.
<figure><img src="image_19.png" /><figcaption>FIGURE 7.7. Some possible nodal patterns for $0 \neq u \in U_x^1$</figcaption></figure>

7.4.2. *Structure and combinatorial type of nodal sets in $U_x^1$ and $U_x^2$.*

◇ Relations (7.11) and an analysis as in Subsection 7.3, show that the nodal set of any $0 \neq u \in U_x^1$ consists of $(\ell-1)$ “generalized” nodal loops at the point $x$, and that these loops do not intersect away from $x$. In the abstract setting of Subsection 7.1, for any $0 \neq u \in U_x^1$, the projection $\tilde{Z}(u)$ of the nodal set $Z(u)$ consists of $(\ell-1)$ continuous loops at $\tilde{x}$. The loops only intersect at $\tilde{x}$, and may contain points in $\Xi$. If $\xi_j \in \tilde{Z}(u)$, there are exactly two projected nodal semi-arcs at this point. It follows that $\xi_j$ is a regular point of $\tilde{D}_u$. The set $\tilde{Z}(u)$ is an $(\ell-1)$-bouquet of loops $B_{\tilde{x},(2\ell-2)}$ at $\tilde{x}$.

Adapting the description given in Paragraph 7.3.2, for $0 \neq u \in U_x^1$, we define the combinatorial type $\tau_x^u$ of the nodal set $Z(u)$ with respect to $x$. This is a map from $L_{(2\ell-2)}$ to itself.

Some possible nodal patterns for $u \in U_x^1$ are displayed in Figure 7.7, where $\ell = 5$, and $\rho(x) = 8$. The corresponding combinatorial types are

$$ \tau_A^{7.7} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 1 & 4 & 3 & 8 & 7 & 6 & 5 \end{pmatrix}, $$

$$ \tau_B^{7.7} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 3 & 2 & 1 & 8 & 7 & 6 & 5 \end{pmatrix}, $$

$$ \tau_C^{7.7} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 8 & 3 & 2 & 5 & 4 & 7 & 6 & 1 \end{pmatrix}. $$

◇ If $u \in U_x^2$ and $u \notin U_x^1$, there exists a unique $y_u \in \Gamma_1$, such that $y_u \neq x$ and $S_b(u) \cap \Gamma_1 = \{x, y_u\}$, with $\rho(u, x) = (2\ell-3)$, $\rho(u, y_u) = 1$. Furthermore, $U_{x,u_u} = [u]$. The nodal set $Z(u)$ and its combinatorial type $\tau_{x,y_u}^u$ are described in Paragraph 7.3.5. Projecting $Z(u)$ to $\tilde{\Omega}$, $\tilde{Z}(u)$ is the wedge sum $B_{\tilde{x},(\ell-2)}^{\tilde{y}_u}$ of a simple arc from $\tilde{x}$ to $\tilde{y}_u$ with an $(\ell-2)$-bouquet of loops at $\tilde{x}$.

7.5. **Application of the previous analysis.** Fix some $x \in \Gamma_1$. We now apply the analysis of Subsections 7.3 and 7.4 to investigate the limits $u_{x,y} \in U_x^2 \setminus U_x^1$, when $y$ tends to $x$ on $\Gamma_1$, clockwise or anti-clockwise. The notation are the same as in Subsection 7.3.

We choose a basis $\{v_1, v_2\}$ of $U_x^2$ as described in Remark 7.10. In particular, $\rho(v_1, x) = (2\ell-2)$, $v_1 \perp v_2$ in $L^2(\Omega)$, $\rho(v_2, x) = (2\ell-3)$, there exists $y_2 \in \Gamma_1 \setminus \{x\}$ such
that $\rho(v_2, y_2) = 1$, and $S(v_2) \cap \Gamma_1 = \{x, y_2\}$. Recall the definition of the functions $\check{v}_i$ on $\partial\Omega$,

$$ (7.13) \qquad \check{v}_i := \begin{cases} \partial_\nu v_i & \text{in the Dirichlet case,} \\ v_i|_{\partial\Omega} & \text{in the Robin case.} \end{cases} $$

According to Lemma 3.14, the function $\check{v}_1$ vanishes only at $x$ and does not change sign on $\Gamma_1$. The function $\check{v}_2$ does not vanish on $\Gamma_1 \setminus \{x, y_2\}$, and changes sign when crossing $y_2$ and $x$ along $\Gamma_1$.

Let $\gamma : [0, 2\pi] \to \Gamma_1$ be a parametrization such that $\gamma(0) = \gamma(2\pi) = x$. Given any $y \in \Gamma_1 \setminus \{x\}$, there exists a function $u_{x,y}$ which satisfies (7.11)(ii), and this function is uniquely defined up to multiplication by a nonzero scalar. In the Dirichlet case, this function is characterized by the fact that $\check{u}_{x,y} = \partial_\nu u_{x,y}|_{\Gamma_1}$ only vanishes at $x$ and $y$. In the Robin case, it is characterized by the fact that $\check{u}_{x,y} = u_{x,y}|_{\Gamma_1}$ only vanishes at $x$ and $y$. Up to a constant factor, we may choose

$$ (7.14) \qquad u_{x,y} = a(y) v_1 + b(y) v_2, $$

with

$$ (7.15) \qquad \begin{cases} a(y) = -\check{v}_2(y) (\check{v}_1^2(y) + \check{v}_2^2(y))^{-\frac{1}{2}}, \\ b(y) = \check{v}_1(y) (\check{v}_1^2(y) + \check{v}_2^2(y))^{-\frac{1}{2}}, \end{cases} $$

where $\check{v}_1, \check{v}_2$ are defined in (7.13).

Then, there exists a unique $\theta(y) \in (0, \pi)$ such that $\cos(\theta(y)) = a(y)$ and $\sin(\theta(y)) = b(y)$ (this is because $\check{v}_1$ is positive on $\Gamma_1 \setminus \{x\}$). Defining

$$ (7.16) \qquad w_{\theta} = \cos \theta v_1 + \sin \theta v_2, $$

we have $u_{x,y} = w_{\theta(y)}$. Conversely, according to the proof of Lemma 7.9, any function $w_\theta$ has exactly two singular points on $\Gamma_1$, the point $x$ and some other point $y_\theta \neq x$. Note that the point $y$ determines the eigenfunction $u_{x,y}$ uniquely (up to scaling) and vice versa. It follows that we have a continuous, bijective map $(0, 2\pi) \ni t \mapsto \theta(\gamma(t)) \in (0, \pi)$. This map is strictly monotone (we can assume that it is increasing), and provides a diffeomorphism from $(0, 2\pi)$ to $(0, \pi)$, with limits 0 and $\pi$ respectively. Otherwise stated, the function $u_{x,\gamma(t)}$ defined in (7.14) tends to $v_1$ when $t$ tends to 0 and to $-v_1$ when $t$ tends to $2\pi$. There exists $t_2$ such that $\gamma(t_2) = y_2$, and hence $\theta(y_2) = \frac{\pi}{2}$. We have proved the following property.

**Property 7.11.** The function $u_{x,y}$ defined in (7.14) tends to $v_1$ when $y \neq x$ tends to $x$ clockwise, and to $-v_1$ when $y \neq x$ tends to $x$ counter-clockwise.

7.6. **Proof that** mult$(\lambda_k) \le (2k-2)$ **for all** $k \ge 3$, **assuming** $\Omega$ **to be simply connected.** In this subsection, we work with the family of functions $\{w_\theta | \theta \in [0, \pi]\}$ introduced in (7.16), and we assume that $\partial\Omega = \Gamma_1$.

7.6.1. **Preparation.** In view of Proposition 6.2, and reasoning by contradiction, we assume that $\dim U(\lambda_k) = (2k-1)$. By Courant's theorem, we have

$$ \sup \{\kappa(u) \mid 0 \neq u \in U\} \le k. $$

We can apply Lemma 7.9 with $\ell = k$ and $U := U(\lambda_k)$.

In the arguments below we keep the notation of Lemma 7.9 and its proof (with $\ell = k$). We fix a basis $\{v_1, v_2\}$ of $U_x^2$ as described at the beginning of Subsection 7.5,
and the direct frame {$\vec{e}_1, \vec{e}_2$} such that $\vec{e}_1$ is tangent to $\partial\Omega$ at $x$, and $\vec{e}_2$ is normal to $\partial\Omega$, pointing inwards.

7.6.2. *Structure and combinatorial types for $v_1$ and $v_2$. We apply Subsection 7.2 with some $r_0$ small enough. In the neighborhood $B_+(x, r_0)$ of $x$, the nodal set $\mathcal{Z}(v_1)$ consists of $(2\ell - 2)$ nodal semi-arcs $\delta_{1,j}$ emanating from $x$ tangentially to rays $\omega_{1,j}, j \in L_{(2k-2)}$; the nodal set $\mathcal{Z}(v_2)$ consists of $(2\ell - 3)$ nodal semi-arcs $\delta_{2,j}$ emanating from $x$ tangentially to rays $\omega_{2,j}, j \in L_{(2k-3)}$.

The combinatorial type of the function $v_1 \in U_x^1$ with respect to $x$ is defined in Subsection 7.4. This is a map

$$ (7.17) \qquad \begin{array}{l} \tau_x^{v_1} : L_{(2k-2)} \to L_{(2k-2)} \text{ such that} \\ \tau_x^{v_1}(j) \neq j \text{ and } (\tau_x^{v_1})^2(j) = j, \text{ for all } j \in L_{(2k-2)}. \end{array} $$

The nodal set $\mathcal{Z}(v_1)$ is a $(k-1)$-bouquet of loops at $x$ described by the map $\tau_x^{v_1}$.

The combinatorial type $\tau_{x,y_2}^{v_2}$ of the function $v_2 \in U_x^{1,\perp}$ with respect to $x$ and $y_2$ is defined in Subsection 7.4. Recall that it is described as a map

$$ (7.18) \qquad \left\{ \begin{array}{l} \tau_{x,y_2}^{v_2} : \{\downarrow\} \cup L_{(2k-3)} \to \{\downarrow\} \cup L_{(2k-3)} \text{ such that} \\ \tau_{x,y_2}^{v_2}(\downarrow) := a \in L_{(2k-3)} \text{ and } \tau_{x,y_2}^{v_2}(a) = \downarrow, \\ \tau_{x,y_2}^{v_2}(j) \neq j \text{ and } (\tau_{x,y_2}^{v_2})^2(j) = j, \text{ for all } j \in L_{(2k-3)} \setminus \{a\}. \end{array} \right. $$

Here, $\tau_{x,y_2}^{v_2}(\downarrow)$ is the element $a \in L_{(2k-3)}$ such that the semi-arc $\delta_a$ of $\mathcal{Z}(v_2)$ which emanates from $x$ tangentially to $\omega_{2,a}$ eventually hits $\Gamma_1$ at the point $y_2$. For $a \neq j \in L_{(2k-3)}$, the pairs $(j, \tau_{x,y_2}^{v_2}(j))$ describe the loops of $\mathcal{Z}(v_2)$ at the point $x$, so that $\mathcal{Z}(v_2)$ is the wedge sum of the nodal arc $\delta_a$ with an $(k-2)$-bouquet of loops at $x$, described by the map $\tau_{x,y_2}^{v_2}$.

Since $\Omega$ is simply connected, the arc $\delta_a$ separates $\Omega$ into two connected components $\Omega_{a,+}$ (on the right side of $\delta_a$), and $\Omega_{a,-}$ (on the left side of $\delta_a$). There are three cases to consider, $a = 1, 1 < a < (2k-3)$, and $a = (2k-3)$. The following properties follow easily from looking at the local structure of $\mathcal{Z}(v_2)$ at $x$.

### Properties 7.12.

(i) If $a = 1$, the component $\Omega_{a,+}$ does not contain any nodal arc, and the rays $\omega_{2,j}$, $2 \le j \le (2k-3)$ point inside $\Omega_{a,-}$.

(ii) If $1 < a < (2k-3)$, the rays $\omega_{2,j}$, $1 \le j \le (a-1)$ point inside $\Omega_{a,+}$; the rays $\omega_{2,j}$, $(a+1) \le j \le (2k-3)$ point inside $\Omega_{a,-}$.

(iii) If $a = (2k-3)$, the rays $\omega_{2,j}$, $1 \le j \le (2k-4)$ point inside $\Omega_{a,+}$; the component $\Omega_{a,-}$ does not contain any nodal arc.

If the ray $\omega_{2,j}$ points inside $\Omega_{a,+}$, the whole nodal arc $\delta_j$ of $\mathcal{Z}(v_2)$ is contained in $\Omega_{a,+}$, and so does the corresponding loop $\gamma_{j,\tau_{x,y_2}^{v_2}(j)}$. There is an analogous statement for $\Omega_{a,-}$.

This means that

$$ (7.19) \qquad \left\{ \begin{array}{l} a = \tau_{x,y_2}^{v_2}(\downarrow) \in L_{(2k-3)} \quad \text{is odd,} \\ \tau_{x,y_2}^{v_2}(\{1, \dots, (a-1)\}) \subset \{1, \dots, (a-1)\}, \\ \tau_{x,y_2}^{v_2}(\{(a+1), \dots, (2k-3)\}) \subset \{(a+1), \dots, (2k-3)\}. \end{array} \right. $$
More precisely, define

$$ (7.20) \quad \left\{ \begin{array}{ll} K_{a,+} := \{1, \dots, (a-1)\}, & \text{with } K_{a,+} = \emptyset \text{ if } a=1, \\ K_{a,-} := \{(a+1), \dots, (2k-3)\}, & \text{with } K_{a,-} = \emptyset \text{ if } a=(2k-3), \\ a_{(+)} := \frac{a-1}{2}. & \end{array} \right. $$

Then, the set $K_{a,+}$ corresponds to $a_{(+)}$ loops in the component $\Omega_{a,+}$ of $\Omega \setminus \delta_a$, and these loops divide $\Omega_{a,+}$ into $(a_{(+)}+1)$ nodal domains of $v_2$. The set $K_{a,-}$ corresponds to $a_{(-)} := (k-2-a_{(+)})$ loops in the component $\Omega_{a,-}$ of $\Omega \setminus \delta_a$, and these loops divide $\Omega_{a,-}$ into $a_{(-)} + 1 = (k-1-a_{(+)})$ nodal domains of $v_2$, so that we recover the fact that $v_2$ has $k$ nodal domains.

Otherwise stated the nodal set $\mathcal{Z}(v_2)$ consists of the wedge sum of the nodal arc $\delta_a$ with two bouquets of loops, one contained in $\Omega_{a,+}$ and corresponding to $\tau_{x,y_2}^{v_2}|_{K_{a,+}}$, another contained in $\Omega_{a,-}$ and corresponding to $\tau_{x,y_2}^{v_2}|_{K_{a,-}}$. One of these bouquets may be empty (when $a=1$ or $a=(2k-3)$).

For $\theta \in (0, \pi)$, let $w_\theta := \cos \theta v_1 + \sin \theta v_2$. Then $S_b(w_\theta) = \{x, y_\theta\}$. The nodal set $\mathcal{Z}(w_\theta)$ has a structure similar to the structure of $\mathcal{Z}(v_2)$:

* one nodal arc $\delta_{a_\theta, \theta}$ emanating from $x$ tangentially to some ray $\omega_{2,a_\theta}$, and hitting the boundary $\Gamma_1$ at the point $y_\theta \neq x$; $a_\theta$ is odd, and $a_\theta = \tau_{x,y_\theta}^{w_\theta}(\downarrow)$;

* loops at $x$, on either sides of $\delta_{a_\theta, \theta}$, described by the restriction of the combinatorial type $\tau_{x,y_\theta}^{w_\theta}$ to $L_{(2k-3)} \setminus \{a_\theta\}$.

Note: In the above description, the nodal arc is denoted by $\delta_{a_\theta, \theta}$ because it does not only depend on $a_\theta$. This point will be needed later on.

**Lemma 7.13.** *Recall the notation $a = \tau_{x,y_2}^{v_2}(\downarrow)$ and $a_\theta = \tau_{x,y_\theta}^{w_\theta}(\downarrow)$. For all $\theta \in (0, \pi)$, we have $a_\theta = a$, and $\tau_{x,y_\theta}^{w_\theta} = \tau_{x,y_2}^{v_2}$, i.e., the combinatorial type of the nodal set $\mathcal{Z}(w_\theta)$ is the same as the combinatorial type of $\mathcal{Z}(v_2)$.*

*Proof of Lemma 7.13.* We consider local conformal coordinates as in Subsection 7.2. The proof of the local structure theorem shows that one can choose a uniform $r_0$ with respect to $\theta$. We use polar coordinates $(r, \omega)$ associated with $(\xi_1, \xi_2)$ in $\mathbb{R}^2$, and we write $E(r, \omega)$ for $E(r \cos \omega, r \sin \omega)$.

To prove that $a_\theta$ is constant, by connectedness, it suffices to prove that it is locally constant:

*For all $\theta \in (0, \pi)$ there exists $\varepsilon_\theta > 0$ such that $a_{\theta'} = a_\theta$ for $|\theta' - \theta| < \varepsilon_\theta$.*

Assume, by contradiction, that this is not the case. Then, there exists $\theta_0 \in (0, \pi)$ such that for all $n \ge 1$, there exists $\theta_n$ with $|\theta_n - \theta_0| < \frac{1}{n}$ and $a_{\theta_n} \neq a_{\theta_0} =: a_0$. Since $a_\theta$ can only take finitely many values in $L_{(2k-3)}$, taking a subsequence if necessary, we can assume that $a_{\theta_n} \equiv a_1 \neq a_0$. By the local structure theorem, there exists a uniform $r_0 > 0$ (depending on $\theta_0$) such that the nodal arc $\delta_{a_1, \theta_n}$ intersects the set $C_+(x, r_0)$ at the point $z_n := E(r_0, \tilde{\omega}_{a_1}(r_0, \theta_n))$, where the function $\tilde{\omega}_{a_1}(r, \theta)$ is smooth in a neighborhood of $(r_0, \theta_0)$ (with the notation of Appendix B). The arcs $\delta_{a_1, \theta_n} \cap \Omega \setminus B(x, r_0)$ are compact and connected, and we can find a subsequence which converges in the Hausdorff distance to some compact connected set $\bar{\delta}$ which contains the point $z_0 = E(r_0, \tilde{\omega}_{a_1}(r_0, \theta_0))$ and the point $y_{\theta_0} = \lim_{n \to \infty} y_{\theta_n}$ at the boundary. The set $\bar{\delta}$ is also contained in $\mathcal{Z}(w_{\theta_0})$ because $w_{\theta_n}$ tends to $w_{\theta_0}$ uniformly. Since $a_1 \neq a_0$, we have a contradiction.
Since $a_{\theta} \equiv a$ in $(0, \pi)$, in order to prove that $\tau_{\theta} := \tau_{x,y_{\theta}}^{w_{\theta}}$ does not depend on $\theta$, it suffices to show that its restrictions to the sets $K_{a,+}$ and $K_{a,-}$ are locally constant in $\theta$. We give the proof for $K_{a,+}$ in the case $a > 1$. The other cases are similar. Reasoning by contradiction, we assume that there exists $\theta_0 \in (0, \pi)$ such that, for all $n \ge 1$, there exist $\theta_n$, $|\theta_n - \theta_0| < \frac{1}{n}$, and $j_n \in K_{a,+}$, such that $\tau_{\theta_n}(j_n) \ne \tau_{\theta_0}(j_n)$. Since $K_{a,+}$ is finite, taking subsequences if necessary, we may assume that $j_n \equiv b$ and $\tau_{\theta_n}(b) \equiv c$ for some $b, c \in K_{a,+}$ with $c \ne \tau_{\theta_0}(b)$. Since $\theta_n$ is close to $\theta_0 \in (0, \pi)$ we have a uniform structure theorem, and we can reason as in the proof of Property 5.6 to conclude. $\square$

Once we are given the basis $\{v_1, v_2\}$, we have the associated odd integer $a := \tau^{v_2}(\downarrow) \in L_{(2k-3)}$, where $\tau^{v_2} := \tau_{x,y_2}^{v_2}$ is the combinatorial type of $v_2$. For all $\theta \in (0, \pi)$, the nodal set $\mathcal{Z}(w_{\theta})$ has the same combinatorial type $\tau^{v_2}$. In particular, it contains a single simple nodal arc $\delta_{a,\theta}$, emanating from $x$ tangentially to the ray $\omega_{2,a}$, and hitting the boundary $\Gamma_1 = \partial\Omega$ at the point $y_{\theta}$.

Call $\Omega_{\theta,+}$ the component of $\Omega \setminus \delta_{a,\theta}$ with semi-tangent at $x$ the vector $\vec{e}_1$, and $\Omega_{\theta,-}$ the other component, with semi-tangent at $x$ the vector $-\vec{e}_1$. The component $\Omega_{\theta,+}$ contains the $a_{(+)}$ loops corresponding to the set $K_{a,+}$. These loops bound $(a_{(+)} + 1)$ nodal domains of $w_{\theta}$ which can be labeled from 1 to $(a_{(+)} + 1)$. The component $\Omega_{\theta,-}$ contains the $a_{(-)} = (k - a_{(+)} - 2)$ loops corresponding to the set $K_{a,-}$. These loops bound $(k - a_{(+)} - 1)$ nodal domains of $w_{\theta}$ which can be labeled from $(a_{(+)} + 2)$ to $k$.

### 7.6.3. Labeling and comparing nodal domains.

The eigenfunctions we consider in this subsection all have $k$ nodal domains, and a boundary singular point $x$ with index $(2k-2)$ or $(2k-3)$. Let $w$ be such a function, and $n_w := \rho(w, x) \in \{(2k-3), (2k-2)\}$ its index at $x$. For $r_0$ small enough, the local structure theorem applied to $w$ at $x$, implies that $\mathcal{Z}(w) \cap C_+(x, r_0)$ consists of $n_w$ points which determine $(n_w + 1)$ intervals on $C_+(x, r_0)$, each contained in one nodal domain. Note that every nodal domain actually hits at least one of these intervals. This is indeed the case because any nodal domain is bounded by nodal arcs, and the nodal set $\mathcal{Z}(w)$ consists of loops at $x$, and possibly a nodal arc from $x$ to the boundary. These intervals are labeled $I_1^w, \dots, I_{n_w+1}^w$, in a counter-clockwise manner along $C_+(x, r_0)$.

A labeling of the nodal domains of $w$ is a set $\mathcal{J} := \{j_1, \dots, j_k\}$ in bijection with the set of nodal domains of $w$. To any labelling $\mathcal{J}$ of the nodal domains of $w$, we associate a word of length $(n_w + 1)$, $m_{\mathcal{J},w} = \ell_{\mathcal{J},1} \cdots \ell_{\mathcal{J},(n_w+1)}$. The letter $\ell_{\mathcal{J},j}$ of $m_{\mathcal{J},w}$ is the label, in the labeling $\mathcal{J}$, of the nodal domain of $w$ which contains the interval $I_j$. Since eigenfunctions change sign across a nodal line, two consecutive letters in a word $m_{\mathcal{J},w}$ are distinct elements of $\mathcal{J}$. Different labelings $\mathcal{J}_1$ and $\mathcal{J}_2$ of the nodal domains of $w$ might give rise to different words. To compare nodal sets, we introduce the following invariant which does actually not depend of the labeling

$$ (7.21) \qquad \min \{j \mid j \ge 2 \text{ and } \ell_{\mathcal{J},j} = \ell_{\mathcal{J},1}\}. $$

Note that the inf does exist because the number $k$ of nodal domains is less than the number $n_w$ of intervals.

To label the nodal domains of the family of eigenfunctions $w_{\theta}$, we use the following convention. For a given $\theta \in (0, \pi)$, the local structure theorem for the nodal set $\mathcal{Z}(w_{\theta})$ at $x$ implies that for $r_0$ small enough, the $(2k-3)$ distinct points in $C_+(x, r_0) \cap \Omega \cap \mathcal{Z}(w_{\theta})$ determine $(2k-2)$ intervals, each contained in a nodal domain. The first
$a_{(+)}$ intervals are contained in $\Omega_{\theta,+}$, the remaining ones in $\Omega_{\theta,-}$. Working anti-clockwise, the nodal domain which contains the first interval is labeled 1. The nodal domain which contains the second interval is labeled 2. If the nodal domain which contains the 3rd interval has not yet been labeled, we label it 3, if it has already been labeled, we keep its label. We call this labeling $J_{\theta}$. Doing so, the nodal domains contained in $\Omega_{a,+}$ meet one of the first $a$ intervals, and they are labeled from 1 to $(a_{(+)} + 1)$. The remaining nodal domains are contained in $\Omega_{a,-}$, meet the next $(2k - 2 - a)$ intervals, and are labeled from $(a_{(+)} + 2)$ to $k$.

**Lemma 7.14.** The labeling $J_{\theta}$ of nodal domains is completely determined by the map $\tau^{v_2}$, and conversely. It does not depend on $\theta \in (0, \pi)$, and we denote it by $J = \{1, \dots, k\}$.

*Proof.* See Appendix D. $\square$

To the function $w_{\theta}$, we associate the word $m_{\theta} := m_{J_{\theta},w_{\theta}}$. This word has length $(2k - 2)$. Each letter of the word $m_{\theta}$ is an element of the set $J = \{1, \dots, k\}$. More precisely, the word $m_{\theta}$ has the form

$$m_{\theta} = 1p_{+}1(a_{(+)} + 2)p_{-}(a_{(+)} + 2),$$

where $p_{+}$ is a word of length $|p_{+}| = (a - 2)$ with letters in $\{1, \dots, (a_{(+)} + 1)\} \subset J$, and $p_{-}$ a word of length $|p_{-}| = (2k - a - 4)$ with letters in $\{(a_{(+)} + 2), \dots, k\} \subset J$. The word $1p_{+}1$ corresponds to the labeling of the nodal domains contained in $\Omega_{\theta,+}$; the word $(a_{(+)} + 2)p_{-}(a_{(+)} + 2)$ corresponds to the labeling of the nodal domains contained in $\Omega_{\theta,-}$.

7.6.4. The rotating function argument. Letting $\theta$ tend to zero, the nodal arc $\delta_{a,\theta}$ tends to a loop in the nodal set of $w_0 = \lim_{\theta \to 0} w_{\theta}$. The location of the nodal domains of $w_0$ with respect to $C_+(x, r_0)$ is described by the word of length $(2k - 1)$, $m^{(0)} := (a_{(+)} + 2)m_{\theta}$. When $\theta$ tends to $\pi$, the nodal arc $\delta_{a,\theta}$ tends to a loop in the nodal set of $w_{\pi} = \lim_{\theta \to \pi} w_{\theta}$, and the location of the nodal domains of $w_{\pi}$ with respect to $C_+(x, r_0)$ is described by the word of length $(2k - 1)$, $m^{(\pi)} := m_{\theta}1$.

To compare these words, we look at the invariant given in (7.21), ie, at the first position at which the first letter of the word is repeated. For the word $m^{(0)}$ this position must be $4+|p_+|$. For the word $m^{(1)}$ this position is $2+|p_+|$. These positions being different the nodal patterns must be different, and this contradicts the fact that $w_0 = v_1 = -w_{\pi}$.

This contradiction completes the proof that $\dim U(\lambda_k) = (2k - 1)$ cannot occur when $\Omega$ is simply connected.

The above proof is illustrated by the following figures. In Figure 7.8, $m_{\theta} = 121343$, $m^{(\pi)} = 1213431$, $m^{(0)} = 3121343$, $p_{+} = 2$, and $p_{-} = 5$. In Figure 7.9, $m_{\theta} = 121314$, $m^{(\pi)} = 1213141$, $m^{(0)} = 4121314$, $p_{+} = 213$, and $p_{-}$ is empty.
<figure><img src="image_20.png" /><figcaption>FIGURE 7.8</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>FIGURE 7.9</figcaption></figure>

**Remark 7.15.** Instead of looking at the labeling of nodal domains to reach a contradiction, we can look directly at the maps $\tau$. The map $\tau^{v_2} := \tau_{x,y_2}^{v_2}$ is given by

$$ \tau^{v_2} = \begin{pmatrix} \downarrow & 1 & \dots & (a-1) & a & (a+1) & \dots & (2k-3) \\ a & \tau^{v_2}(1) & \dots & \tau^{v_2}(a-1) & \downarrow & \tau^{v_2}(a+1) & \dots & \tau^{v_2}(2k-3) \end{pmatrix}, $$

where the first row represents the set $\{\downarrow\} \cup L_{(2k-3)}$, the second row its image under the map $\tau^{v_2}$, where $a \in L_{(2k-3)}$ is the label of the nodal arc $\delta_a^{w_\theta}$ from $x$ to $y_\theta$.

When $\theta$ tends to 0, the map $\tau^{v_2}$ yields a map $\tau^{(0)}$ associated with the nodal set of $\lim_{\theta \to 0} w_\theta$. This map given by

$$ \tau^{(0)} = \begin{pmatrix} 0 & 1 & \dots & (a-1) & a & (a+1) & \dots & (2k-3) \\ a & \tau^{v_2}(1) & \dots & \tau^{v_2}(a-1) & 0 & \tau^{v_2}(a+1) & \dots & \tau^{v_2}(2k-3) \end{pmatrix}, $$

where the first row represents a set with $(2k-2)$ elements labeled from 0 to $(2k-3)$, and the second row its image under $\tau^{(0)}$.

When $\theta$ tends to $\pi$, the map $\tau^{v_2}$ yields a map $\tau^{(\pi)}$ associated with the nodal set of $\lim_{\theta \to \pi} w_\theta$. This map is given by

$$ \tau^{(\pi)} = \begin{pmatrix} 1 & \dots & (a-1) & a & (a+1) & \dots & (2k-3) & (2k-2) \\ \tau^{v_2}(1) & \dots & \tau^{v_2}(a-1) & (2k-2) & \tau^{v_2}(a+1) & \dots & \tau^{v_2}(2k-3) & a \end{pmatrix} $$

where the first row is $L_{(2k-3)}$.
To compare these maps without relabeling the source set, we look at an invariant, namely $\max_j \{|\tau(j) - j|\}$, for both $\tau^{(0)}$ and $\tau^{(\pi)}$, so that we have to compare the numbers

$$ \max \{a, (2k - 4 - a)\} \text{ and } \max \{(a - 2), (2k - 2 - a)\}. $$

These numbers are different unless $a = (k - 1)$. In this case they are equal to

$$ (k - 1) = \max \{(k - 1), (k - 3)\} = \max \{(k - 3), (k - 1)\}. $$

When $a = (k - 1)$, we relabel the source set of $\tau^{(0)}$ to $\{1, \dots, (2k - 2)\}$ and we find that $\tau^{(0)}(1) = k$ and $1 < \tau^{(\pi)}(1) \le (k - 2)$. This shows that the maps are different.

In any case the combinatorial patterns of the limit nodal sets are different which contradicts the fact that $w_0 = v_1 = -w_\pi$.

**Remarks 7.16.**

(i) Assume that $1 \le a \le (2k - 3)$. Then, $a \frac{\pi}{m} < a \frac{\pi}{m-1} < (a + 1) \frac{\pi}{m}$. Since $a$ is odd, there exists some $q$, $1 \le q \le a$ such that $\tau_1(q) > a$. This implies that the nodal arc $\delta_a$ of $Z(v_1)$ intersects the loop $\gamma_{q,\tau_1(q)}^{(1)}$ at at least one point $z_a$.

(ii) Although $a_\theta$ does not depend on $\theta$, the nodal semi-arc of $Z(w_\theta)$ emanating from $x$ tangentially to $\omega_{2,a}$ does depend on $\theta$, and hence denoted by $\delta_{a,\theta}$.

(iii) When $\theta$ varies, the arcs $\delta_{a,\theta}$ all pass through any point $z_a$ given in (i). Such points are common zeros of the family of functions $w_\theta$.

### 7.6.5. Proof that $\mathrm{mult}(\lambda_k) \le (2k - 2)$ for $k \ge 3$, general case.

Under the assumption that $\dim U(\lambda_k) = (2k - 1)$, the arguments in the simply connected case only use Euler's formula applied to nodal partitions $D_u$, Jordan's separation theorem, the structure of the nodal sets $Z(v_1)$ (a bouquet of loops at $x$) and $Z(v_2)$ (the wedge sum of an arc from $x$ to the boundary $\Gamma_1$ with one or two bouquets of loops at $x$), and the fact that the combinatorial type $\tau_{x,a_\theta,w_\theta}$ of the nodal sets $Z(w_\theta)$ is constant for $\theta \in (0, \pi)$.

In the general case, Euler's formula leads to a similar structure for the nodal sets $Z(v_1)$, $Z(v_2)$, and $Z(w_\theta)$, with “generalized” loops and arcs. We can now look at the projection of these sets to $\tilde{\Omega}$ as in Section 7.1. As observed in Remark 7.8, the only singular points of the projected sets $\tilde{Z}(u)$ (or partitions $\tilde{D}_u$), $u \in \{v_2, w_\theta\}$, are the points $\tilde{x}$ and $\tilde{y}_\theta$, and the combinatorial type of $\tilde{Z}(w_\theta)$ is constant for $\theta \in (0, \pi)$. Since $\tilde{\Omega}$ is simply connected, we can now apply the same arguments as in the simply connected case.

We summarize Section 7 in the following proposition.

**Proposition 7.17 ([HoMN1999])**. Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with smooth boundary (no assumption on the topology of $\Omega$). Let $\{\lambda_k, k \ge 1\}$ be the eigenvalues of the operator $-\Delta + V$ in $\Omega$, with Dirichlet or Robin boundary condition (where $V$ is a real valued $C^\infty$ function). Then,

$$ \mathrm{mult}(\lambda_k) \le (2k - 2) \quad \text{for } k \ge 3. $$

In [HoMN1999, Theorem B, p. 1172], the authors state that the preceding bound can be improved to $\mathrm{mult}(\lambda_k) \le (2k - 3)$ for all $k \ge 3$. For the proof, they propose the same strategy as for the proof of Proposition 7.17: assume that $\mathrm{mult}(\lambda_k) = (2k - 2)$, construct $\lambda_k$-eigenfunctions with prescribed singular points, analyze their nodal sets, and derive a contradiction. However, we have so far not succeeded in writing down
complete details for the arguments in [HoMN1999], even when $\Omega$ is simply connected (see [Berd2018, Section 4]). In Sections 8 and 9, we provide some properties of $\lambda_k$-eigenfunctions under the assumption that $\mathrm{mult}(\lambda_k) = (2k - 2)$. In Section 10, we indicate the main steps of the proof sketched in [HoMN1999, Section 3], and we point out where we are stuck.

## 8. PLANE DOMAINS, PROPERTIES OF $\lambda_k$-EIGENFUNCTIONS ASSUMING THAT $\mathrm{mult}(\lambda_k) = (2k - 2)$

The purpose of this section is to derive properties of $\lambda_k$-eigenfunctions assuming that the upper bound in Proposition 7.17 is achieved. Throughout this section, we assume that

$$ (8.1) \qquad k \ge 3 \quad \text{and} \quad \dim U(\lambda_k) = (2k-2). $$

We do not yet make any assumption on the topology of $\Omega$. For simplicity, we denote $U(\lambda_k)$ by $U$ and we use the notation of Subsection 6.2. According to Courant's nodal domain theorem, $\sup \{\kappa(u) \mid u \in U\} \le k$, so that $\ell = k$.

For any $0 \ne u \in U$, (6.8) becomes,

$$ (8.2) \qquad \begin{cases} 0 \ge \kappa(u) - k = [b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1] + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \\ + \sum_{j \in J(u)} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right) - (k-2). \end{cases} $$

In this section, we analyze eigenfunctions with prescribed singular points, under the assumption (8.1)

### 8.1. Eigenfunctions with one prescribed interior singular point.
For $x \in \Omega$, define the subspace

$$ (8.3) \qquad U_x := \{u \in U \mid \nu(u, x) \ge 2k-2\}. $$

In view of the assumption (8.1), Lemma 3.9 implies that $U_x \ne \{0\}$.

The following lemma, appears as Lemma 3.7 in [HoMN1999, p. 1183].

#### 8.1.1. Properties of $U_x$.

**Lemma 8.1.** Let $x \in \Omega$ and $U := U(\lambda_k)$. Assume that $\dim U = (2k - 2)$. The subspace $U_x$ defined in (8.3) has the following properties.

(i) The dimension of $U_x$ is 1.

(ii) For all $0 \ne u \in U_x$,

$$ (8.4) \qquad \left\{ \begin{array}{l} \kappa(u) = k, \\ \mathcal{Z}(u) \cup \Gamma(u) \text{ is connected,} \\ S_i(u) = \{x\} \quad \text{and} \quad \nu(u, x) = 2(k-1), \\ \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2 \quad \text{for all} \ j \in J(u). \end{array} \right. $$

(iii) If $u_x$ is a generator of $U_x$, the map $\Omega \ni x \mapsto [u_x] \in \mathbb{P}(U)$ is $C^\infty$.

*Proof.* We already know that that $\dim U_x \ge 1$.

*Proof of Assertion (ii).*

✓ We first assume that $\Omega$ is simply connected.
The assumptions of the lemma and (8.2) imply that

$$ (8.5) \qquad 0 \ge \kappa(u) - k = (b_0(\mathcal{Z}(u) \cup \partial\Omega) - 2) + \frac{1}{2} \sum_{z \in S_i(u), z \ne x} (\nu(u, z) - 2) \\ \qquad \qquad + \frac{1}{2} (\nu(u, x) - 2k + 2) + \frac{1}{2} \sum_{z \in S_b(u)} \rho(u, z). $$

The terms in the right-hand side are nonnegative, except possibly the first one. The inequality implies that $b_0(\mathcal{Z}(u) \cup \partial\Omega) \le 2$. We now consider two cases.

◇ If $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 2$, the terms in the right-hand side are nonnegative, with a nonpositive sum. They must all vanish: $\kappa(u) = k$, $S_i(u) = \{x\}$, $S_b(u) = \emptyset$, and $\nu(u, x) = 2(k-1)$. In this case, $\mathcal{Z}(u) \cap \partial\Omega = \emptyset$ and $\mathcal{Z}(u)$ is connected.

◇ If $b_0(\mathcal{Z}(u) \cup \partial\Omega) = 1$, the nodal set $\mathcal{Z}(u)$ must hit $\partial\Omega$, which implies that $\sum_{z \in S_b(u)} \rho(z) \ge 2$ (use Proposition 4.22). Re-arranging the inequality, we conclude that $\kappa(u) = k$, $S_i(u) = \{x\}$, $\nu(u, x) = 2(k-1)$, and $\sum_{z \in S_b(u)} \rho(z) = 2$.

✓ Assume now that $\partial\Omega = \cup_{j=1}^q \Gamma_j$. Let $u$ be a function such that $\nu(u, x) \ge 2(k-1)$. With the notation of Subsection 6.2, the inequality (8.2) becomes

$$ (8.6) \qquad 0 \ge \kappa(u) - k = (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u), z \ne x} (\nu(u, z) - 2) \\ \qquad \qquad + \sum_{j \in J(u)} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right) + \frac{1}{2} (\nu(u, x) - 2k + 2). $$

The terms in the right-hand side of (8.6) are nonnegative and their sum is nonpositive. They must all vanish:

$$ (8.7) \qquad \begin{cases} \kappa(u) = k \quad \text{and} \quad b_0(\mathcal{Z}(u) \cup \Gamma(u)) = 1, \\ S_i(u) = \{x\} \quad \text{and} \quad \nu(u, x) = 2(k-1), \\ \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2, \quad \text{for all } j \in J(u). \end{cases} $$

We have proved Assertion (ii).

*Proof of Assertion (i).* Lemma 3.12 implies that $\dim U_x \le 2$. Assume by contradiction that $\dim U_x = 2$. We again use a *rotating function argument* similar to the one used in Subsection 5.2.

As in Proposition 5.2, one can choose a basis $\{v_1, v_2\}$ of $U_x$ such that, in local polar coordinates centered at $x$,

$$ \left\{ \begin{array}{l} v_1 = r^{k-1} \sin((k-1)\omega) + \mathcal{O}(r^k), \\ v_2 = r^{k-1} \cos((k-1)\omega) + \mathcal{O}(r^k). \end{array} \right. $$

Introducing the family of functions

$$ w_{\theta} = \cos((k-1)\theta) v_1 - \sin((k-1)\theta) v_2, $$

and letting $\theta$ tend to 0 or $\frac{\pi}{(k-1)}$, one can follow the arguments given in the proofs of Properties 5.5 and 5.6 to reach a contradiction.

*Proof of Assertion (iii).* The function $u_x$ is characterized by the vanishing of the derivatives of order less than or equal to $(k-2)$. Taking a basis for $U$, this is equivalent to writing a linear system. Since the function $u_x$ is uniquely determined up to scaling, this linear system has constant rank, and we have a locally defined
smooth map $x \mapsto u_x$ in the neighborhood of any given point $x_0$. This means that we have a smooth map $x \mapsto [u_x]$ from $\Omega$ into the projective space of $U$. $\square$

8.1.2. *Structure and combinatorial type of nodal sets in $U_x$.* Using Assertion (ii), one can describe the possible nodal patterns for a generator $u_x$ of $U_x$. If $\partial\Omega = \Gamma_1$ (i.e., $\Omega$ is simply connected), there are two cases.

(1) Either $Z(u_x)$ consists of $(k-1)$ loops at $x$ which do not intersect away from $x$, and do not hit $\Gamma_1$.

(2) Or $\mathcal{Z}(u_x)$ it consists of
    * $(k-2)$ loops at $x$ which do not hit the boundary, and
    * two arcs emanating from $x$ and hitting $\Gamma_1$ at points $y_1 \neq y_2$, such that $\rho(u_x, y_i) = 1$ or, possibly, at one point $y$, with $\rho(u_x, y) = 2$.

Furthermore, the loops at $x$ and the arcs from $x$ to the boundary are pairwise disjoint away from $x$. In this case, we have a “generalized” nodal loop at $x$ which consists of the two arcs, and a portion of the boundary.

When $\partial\Omega = \Gamma_1 \cup \dots \cup \Gamma_q$ ($q \ge 2$, i.e. $\Omega$ not simply connected) the nodal set $\mathcal{Z}(u_x)$ consists of arcs and loops which can be “generalized arcs or loops” in the sense that they may include some arcs contained in one or several of the components $\Gamma_j, j \in J(u)$. Such arcs and loops become “ordinary” arcs and loops once projected to $\tilde{\Omega}$ as in Subsection 7.1, see Remark 7.8. Viewing $\Omega$ as a domain on the sphere, we can also identify each component of $\partial\Omega$ to a point, and consider the quotient space $S_\Omega$, see Notation 7.1. The nodal sets projected from $\Omega$ to $S_\Omega$ then consist of $(k-1)$ simple loops at $\tilde{x}$, intersecting only at $\tilde{x}$, as in the closed case, Subsection 5.2.

<figure><img src="image_22.png" /><figcaption>FIGURE 8.1. $\Omega$ simply connected, some nodal patterns for $u_x$</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 8.2. $\Omega$ with one hole, some nodal patterns for $u_x$</figcaption></figure>

Figures 8.1 and 8.2 display some possible nodal patterns for $u_x$.
**Remark 8.2.** The nodal patterns displayed in Figure 8.1 and 8.2 are valid for both the Dirichlet and Robin boundary conditions. Unless otherwise stated this remark applies to all figures of this section.

## 8.2. Eigenfunctions with one prescribed boundary singular point.

For $y \in \Gamma_1$, define the subspace

$$ (8.8) \qquad W_y := \{u \in U \mid \rho(u, y) \ge 2k - 3\}. $$

In view of the assumption (8.1), Lemma 3.10 implies that $W_y \neq \{0\}$.

Lemmas 8.3 and 8.4 below expand and explain the lemma stated in [HoMN1999, §3.2] and their Lemma 3.3. In particular, Lemma 8.4, Assertion (iv), explains and proves the statement “the nodal pattern of $w_y$ is constant along each connected components of $\Gamma_{1,(2\ell-3)}$”, and Assertion (iii) states the continuity of the map $y \mapsto z(y)$ (see [HoMN1999, p. 1179], last sentence in Lemma 3.3, and second sentence after the same lemma).

### 8.2.1. Properties of $W_y$.

**Lemma 8.3.** Let $y \in \Gamma_1$ and $U := U(\lambda_k)$. Assume that $\dim U = 2(k-1)$. The subspace $W_y$ defined in (8.8) has the following properties.

(i) The dimension of $W_y$ is 1.

(ii) For all $0 \neq u \in W_y$,

$$ (8.9) \qquad \left\{ \begin{array}{ll} \kappa(u) = k \quad \text{and} \quad \mathcal{Z}(u) \cup \Gamma(u) \text{ is connected,} \\ S_i(u) = \emptyset, \\ \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) = 2 \quad \text{for all } j \in J(u), \quad j \neq 1, \\ \sum_{z \in S_b(u) \cap \Gamma_1} \rho(u, z) = 2k-2. \end{array} \right. $$

Furthermore,

$$ (8.10) \qquad \left\{ \begin{array}{ll} \text{either} & \rho(u, y) = 2k-2 \quad \text{and} \quad S_b(u) \cap \Gamma_1 = \{y\}, \\ \text{or} & \rho(u, y) = 2k-3 \quad \text{and} \quad S_b(u) \cap \Gamma_1 = \{y, z(y)\}, \\ & \text{for some } z(y) \in \Gamma_1, z(y) \neq y, \text{ with } \rho(u, z(y)) = 1. \end{array} \right. $$

(iii) If $w_y$ denotes a generator of $W_y$, then the map $\Gamma_1 \ni y \mapsto [w_y] \in \mathbb{P}(U)$ is $C^\infty$.

*Proof.* We already know that $\dim W_y \ge 1$.

*Proof of Assertion (ii).*

✓ Case 1: $\Omega$ is simply connected.

Choose a function $0 \neq u \in W_y$, and apply the inequality (8.2) to obtain,

$$ (8.11) \qquad \begin{aligned} 0 \ge \kappa(u) - k = & (b_0(\mathcal{Z}(u) \cup \partial\Omega) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \\ & + \frac{1}{2} \left( \sum_{z \in S_b(u)} \rho(u, z) - 2k + 2 \right). \end{aligned} $$

Since $\rho(u, y) \ge 2k-3$, Proposition 4.22 implies that the last term in (8.11) is nonnegative; all the terms in the right-hand side are nonnegative, with nonpositive sum, and hence they must all vanish. This proves (8.9) in the special case $\partial\Omega = \Gamma_1$. Looking at the two possible cases, $\rho(u, y) = (2k-2)$ or $(2k-3)$, we obtain (8.10).
✓ Case 2: $\partial\Omega = \cup_{j=1}^q \Gamma_j$, with $y \in \Gamma_1$. Let $0 \neq u \in W_y$. With the notation of Subsection 6.2, we can write

$$ 
\begin{align} 
0 \geq \kappa(u) - k &= (b_0(\mathcal{Z}(u) \cup \Gamma(u)) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \tag{8.12} \\ 
&\quad + \sum_{j \in J(u), j \neq 1} \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_j} \rho(u, z) - 2 \right) \nonumber \\ 
&\quad + \frac{1}{2} \left( \sum_{z \in S_b(u) \cap \Gamma_1} \rho(u, z) - 2k + 2 \right). \nonumber 
\end{align} 
$$

The definition of $J(u)$, and Proposition 4.22, imply that the terms in the right-hand side of (8.12) are nonnegative, with a nonpositive sum. They must all be zero and hence, Equations (8.9) and (8.10) hold. We have proved Assertion (ii).

*Proof of Assertion (i).* We already know that $\dim W_y \geq 1$. Assume that there exist at least two linearly independent functions $w_1, w_2$ in $W_y$. By (8.9), we have $2k-3 \leq \rho(w_1, y) \leq 2k-2$. If $\rho(w_1, y) = \rho(w_2, y) = 2k-3$, by Lemma 3.12 there exists some linear combination $w$ of $w_1$ and $w_2$ such that $\rho(w, y) = 2k-2$. This function $w$ must satisfy (8.10) and hence, by Lemma 3.12, is uniquely defined (up to scaling). If $\rho(w_1, y) = 2k-2$, then we must have $\rho(w_2, y) = 2k-3$ since $w_1$ is uniquely defined. Any other function in $W_y$ must be a linear combination of $w_1$ and $w_2$. It follows that $\dim W_y \leq 2$.

Assume that $\dim W_y = 2$, and choose a basis $\{w_1, w_2\}$ of $W_y$, with $\rho(w_1, y) = 2k-2$, $\rho(w_2, y) = 2k-3$, and let $y_2$ be the unique other critical zero of $w_2$ on $\partial\Omega$. We have to consider two cases, depending on whether $U$ is a Dirichlet or Robin eigenspace, and we can use a *rotating function argument* as in Subsections 7.5 and 7.6 to conclude that $\dim W_y = 1$. The claim is proved. This completes the proof of Assertion (i). ✓

*Proof of Assertion (iii).* Using a basis for the eigenspace $U$, the condition that a generator $w_y$ of $W_y$ satisfies $\rho(w, y) \geq 2k-3$ is equivalent to a linear system. Since $\dim W_y = 1$, this linear system has constant rank when $y$ varies and we can deduce the existence of a smooth map $y \mapsto [w_y]$ from $\partial\Omega$ into the projective space $\mathbb{P}(U)$ of the eigenspace $U$, where the generator $w_y$ of $W_y$ satisfies (8.9) and (8.10). □

### 8.2.2. Structure and combinatorial type of nodal sets in $W_y$. Using Assertion (ii) of the lemma, one can describe the possible nodal patterns of a generator $w_y$ of $W_y$, as we did in Paragraph 8.1.2. For example, in the simply connected case, if $\rho(w_y, y) = (2k-3)$, the nodal set $\mathcal{Z}(w_y)$ consists of $(k-2)$ simple loops at $y$, and a simple arc from $y$ to $z(y) \in \Gamma_1$; if $\rho(w_y, y) = (2k-2)$, the nodal set $\mathcal{Z}(w_y)$ consists of $(k-1)$ simple loops at $y$. The loops and the arc do not intersect away from $y$. Figures 8.3 and 8.4 display some possible nodal patterns. Figure 8.5 displays the nodal pattern in Figure 8.4(A) projected to $\tilde{\Omega}$, or viewed on $\mathbb{S}^2$. In view of Lemma 8.3, define the following subsets of $\Gamma_1$:

$$ 
(8.13) \qquad \left\{ \begin{array}{l} \Gamma_{1,(2k-3)} := \{y \in \Gamma_1 \mid \rho(w_y, y) = 2k-3\}, \\ \Gamma_{1,(2k-2)} := \{y \in \Gamma_1 \mid \rho(w_y, y) = 2k-2\}. \end{array} \right. 
$$

Let $p \in \{(2k-3), (2k-2)\}$. For a given $y \in \Gamma_1$, we apply Subsection 7.2 to a generator $w_y$ of $W_y$.
<figure><img src="image_24.png" /><figcaption>FIGURE 8.3. Ω simply connected, some nodal patterns for $w_y$</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>FIGURE 8.4. Ω with one hole, some nodal patterns for $w_y$</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE 8.5. Figure 8.4 viewed in $\tilde{\Omega}$ (left) or in $S^2$ (right)</figcaption></figure>

For both Dirichlet or Robin boundary condition, in a neighborhood of $y$, the nodal set $\mathcal{Z}(w_y)$ consists of $p$ nodal semi-arcs. As in Paragraph 7.6.2, for a given $j \in L_p$, we follow the nodal semi-arc emanating from $y$ tangentially to the rays $\omega_j$ along $\mathcal{Z}(w_y)$. There are two cases.

* Case $p = (2k-2)$, i.e. $y \in \Gamma_{1,(2k-2)}$. According to Lemma 8.3, we eventually arrive back at $y$ along a nodal semi-arc emanating from another ray which we denote by $\omega_{\tau_y^W(j)}$. This uniquely defines a map $\tau_y^W : L_{(2k-2)} \to L_{(2k-2)}$, such that $\tau_y^W(j) \neq j$
and $(\tau_y^W)^2 = \text{Id}$. In this case, the nodal set $\mathcal{Z}(w_y)$ consists of a $(k-1)$-bouquet of loops at $y$, denoted by $\gamma_{j, \tau_y^W(j)}^{w_y}$, $j \in L_{(2k-2)}$.

◇ Case $p = (2k-3)$, i.e. $y \in \Gamma_{1,(2k-3)}$. We define a map $\tau_y^W$ from $\{\downarrow\} \cup L_{(2k-3)}$ to itself as follows. According to Lemma 8.3, there exists a unique $a \in L_{(2k-3)}$ (depending on $y$) such that the nodal semi-arc emanating from $y$ tangentially to the ray $\omega_a$ eventually hits $\Gamma_1$ at some $z(y) \neq y$. We let $\tau_y^W(a) = \downarrow$ and $\tau_y^W(\downarrow) = a$. For $j \neq a$, following the nodal semi-arc, denoted by $\delta_a$, emanating from $y$ tangentially to $\omega_j$ along $\mathcal{Z}(w_y)$, we will eventually arrive back at $y$ along another ray denoted by $\omega_{\tau_y^W(j)}$. This uniquely defines a map $\tau_y^W$ from $\{\downarrow\} \cup L_{(2k-3)}$ to itself such that $\tau_y^W(j) \neq j$ and $(\tau_y^W)^2 = \text{Id}$. In this case, the nodal set $\mathcal{Z}(w_y)$ is the wedge sum of the arc $\delta_a$ from $y$ to $z(y)$ and a $(k-2)$-bouquet of loops denoted by $\gamma j, \tau_y^W(j)$.

In both cases, the loops and arc might be “generalized” in the sense that they may contain sub-arcs of $\partial\Omega \setminus \Gamma_1$ (when $\Omega$ is not simply connected); they can then be viewed as loops and arc when projected to $\tilde{\Omega}$. When $\Omega$ is simply connected and $y \in \Gamma_{1,(2k-3)}$, the $(k-2)$-bouquet of loops actually consists of one or two bouquets of loops, one in each connected component of $\Omega \setminus \delta_{\alpha(y)}$.

As in Paragraph 7.6.2, we define the map $\tau_y^W$ as the combinatorial type of the nodal set $\mathcal{Z}(w_y)$ at $y$, when $y \in \Gamma_{1,(2k-2)}$, resp. $y \in \Gamma_{1,(2k-3)}$. The source-set of $\tau_y^W$ is either $L_{(2k-2)}$ or $\{\downarrow\} \cup L_{(2k-3)}$.

### 8.2.3. Properties of $\Gamma_{1,(2k-3)}$ and $\Gamma_{1,(2k-2)}$.

**Lemma 8.4.** Let $\Omega \subset \mathbb{R}^2$ be any smooth bounded domain, and $U := U(\lambda_k)$. Assume that $\dim U = (2k-2)$. Then, the following properties hold.

(i) The sets $\Gamma_{1,(2k-3)}$ and $\Gamma_{1,(2k-2)}$ are disjoint and

$$ \Gamma_1 = \Gamma_{1,(2k-3)} \sqcup \Gamma_{1,(2k-2)}. $$

(ii) The set $\Gamma_{1,(2k-3)}$ is open in $\Gamma_1$; the set $\Gamma_{1,(2k-2)}$ is finite.

(iii) The map $\Gamma_{1,(2k-3)} \ni y \mapsto z(y) \in \Gamma_1$, where $z(y)$ is defined in (8.10) is continuous in $\Gamma_{1,(2k-3)}$, and for any $\eta \in \Gamma_{1,(2k-2)} \lim_{y \to \eta, y \in \Gamma_{1,(2k-3)}} z(y) = \eta$.

(iv) The combinatorial type of $w_y$ is constant in any component of $\Gamma_{1,(2k-3)}$ in the sense that the maps $y \mapsto \tau_y^W(\downarrow)$ and $y \mapsto \tau_y^W$ are constant in each connected component of $\Gamma_{1,(2k-3)}$.

*Proof.* Assertion (i) follows from Lemma 8.3.

*Proof of Assertion (ii).* The first part follows from the fact that $\rho(w_y, y) = (2k-3)$ is an open condition on the vanishing of derivatives, see Lemma 3.14.

For the second part, let $y_0 \in \Gamma_{1,(2k-2)}$. By a conformal change of coordinates (see for example Section 2 in [YaZh2021] and Paragraph 7.2), we can assume that a neighborhood of the point $y_0$ in $\bar{\Omega}$ is sent to the half-disk $\{\xi_1^2 + \xi_2^2 < 1, \xi_2 \ge 0\}$, with $y_0$ sent to $(0,0)$. By abuse of notation, we identify $\Gamma_1$ to $(-1,1) \times \{0\}$ in this neighborhood of $y_0$. Let $t \in (-1,1)$ be a point corresponding to some $y \in \Gamma_1$ close to $y_0$.

According to Lemma 8.3, in a neighborhood of $(0,0)$ in $\Gamma_1$, we have a $C^\infty$ family of functions $w_t := w_{(t,0)} \in W_{(t,0)}$ in the form

$$ (8.14) \qquad w_t = \sum_{j=1}^{2k-2} a_j(t) \phi_j, $$
where {$\phi_j, 1 \le j \le 2k-2$} is a basis of $U$, and $\rho(w_t, (t, 0)) = (2k-3)$ if $(t, 0) \in \Gamma_{1,(2k-3)}$, or $\rho(w_t, (t, 0)) = (2k-2)$ if $(t, 0) \in \Gamma_{1,(2k-2)}$. Let

$$q := \begin{cases} (2k-2) & \text{in the Dirichlet case,} \\ (2k-3) & \text{in the Robin case,} \end{cases}$$

We write Taylor's formula for $w_t$ at the point $(t, 0)$ in the coordinates $(\xi_1, \xi_2)$,

$$(8.15) \qquad \left\{ \begin{aligned} w_t(\xi_1, \xi_2) &= \sum_{|\alpha|=q} \frac{1}{\alpha!} \partial^\alpha w_t(t, 0) (\xi_1 - t, \xi_2)^\alpha \\ &\quad + \sum_{|\alpha|=q+1} \frac{1}{\alpha!} \partial^\alpha w_t(t, 0) (\xi_1 - t, \xi_2)^\alpha \\ &\quad + \sum_{|\beta|=q+2} R_\beta(t; \xi_1, \xi_2) (\xi_1 - t, \xi_2)^\beta, \end{aligned} \right.$$

where

$$R_\beta(t; \xi_1, \xi_2) = \frac{q+2}{\beta!} \int_0^1 (1-s)^{q+1} \partial^\beta w_t(t+s(\xi_1-t), \xi_2) \, ds.$$

It follows from (8.14) that the functions $\partial^\alpha w_t(t, 0)$ are $C^\infty$ in $t \in (-1, 1)$, and that the functions $R_\beta(t; \xi_1, \xi_2)$ are $C^\infty$ in $(-1, 1) \times \{\xi_1^2 + \xi_2^2 < 1, \xi_2 \ge 0\}$. Note that the first sum vanishes identically if $(t, 0) \in \Gamma_{1,(2k-2)}$.

We can rewrite (8.15) as

$$(8.16) \qquad \left\{ \begin{aligned} w_t(\xi_1, \xi_2) &= \sum_{|\alpha|=q} A_{0,\alpha}(t) (\xi_1 - t, \xi_2)^\alpha \\ &\quad + \sum_{|\alpha|=q+1} A_{1,\alpha}(t) (\xi_1 - t, \xi_2)^\alpha \\ &\quad + \sum_{|\beta|=q+2} R_\beta(t; \xi_1, \xi_2) (\xi_1 - t, \xi_2)^\beta, \end{aligned} \right.$$

where the functions $A_{0,\alpha}$ and $A_{1,\alpha}$ are $C^\infty$. Call $p_0(t, \xi_1, \xi_2)$ (resp. $p_1(t, \xi_1, \xi_2)$) the polynomial of degree $q$ (resp. $(q+1)$), with coefficients $A_{0,\alpha}$ (resp. $A_{1,\alpha}$).

To prove that the set $\Gamma_{1,(2k-2)}$ is finite, it suffices to prove that it is discrete. Assume that the point $y_0$ is not isolated. Then, there exists a sequence $\{t_n\}$ tending to zero, such that $w_{t_n}$ satisfies $\rho(w_{t_n}, (t_n, 0)) = (2k-2)$ for all $n$, so that $p_0(t_n, \xi_1, \xi_2)$ is the zero polynomial. According to Remark B.1, the homogeneous polynomials $p_0$ and $p_1$ are harmonic. Define

$$\left\{ \begin{aligned} \varpi_n(\xi_1, \xi_2) &:= t_n^{-1} (w_{t_n}(\xi_1, \xi_2) - w_0(\xi_1, \xi_2)) \\ &= t_n^{-1} (p_1(t_n; \xi_1 - t, \xi_2) - p_1(0; \xi_1, \xi_2)) \\ &\quad + \sum_{|\beta|=q+2} t_n^{-1} \left( R_\beta(t_n; \xi_1 - t, \xi_2) (\xi_1 - t, \xi_2)^\beta - R_\beta(0; \xi_1, \xi_2) (\xi_1, \xi_2)^\beta \right). \end{aligned} \right.$$

The functions $\varpi_n$ belong to $U$. When $t_n$ tends to 0, they converge to the function $\varpi \in U$ given by

$$(8.17) \qquad \left\{ \begin{aligned} \varpi(\xi_1, \xi_2) &= \sum_{|\alpha|=q+1} A'_{1,\alpha}(0) (\xi_1, \xi_2)^\alpha \\ &\quad - \sum_{|\alpha|=q+1} \alpha_1 A_{1,\alpha}(0) (\xi_1, \xi_2)^{\alpha-(1,0)} \\ &\quad + \sum_{|\beta|=q+2} \partial_t B_\beta(0; \xi_1, \xi_2) (\xi_1, \xi_2)^\beta \\ &\quad - \sum_{|\beta|=q+2} \beta_1 B_\beta(0; \xi_1, \xi_2) (\xi_1, \xi_2)^{\beta-(1,0)} \end{aligned} \right.$$

The harmonic polynomial $p_1(0; \xi_1, \xi_2)$ is nonzero because it corresponds to the leading term in $w_0$. Hence, its derivative $\partial_{\xi_1} p_1$ is not zero (otherwise, being harmonic, $p_1$ would be a polynomial in $\xi_2$, of degree one, contradicting our assumption that $k \ge 3$). Looking at the order of vanishing of the terms in the right-hand side, the second line in (8.17) shows that $w \ne 0$, and that $\rho(w, y_0) = (2k-2)$, while $\rho(w_{y_0}, y_0) = (2k-3)$. This contradicts the fact that $\dim W_{y_0} = 1$. This proves that
the point $y_0$ is isolated in $\Gamma_{1,(2k-2)}$, and since this is true for any point in $\Gamma_{1,(2k-2)}$, this set is discrete, hence finite. This proves Assertion (ii). ✓

*Proof of Assertion (iii).* Consider a component $C$ of $\Gamma_{1,(2k-3)}$. Define

$$ (8.18) \qquad \check{w}_y := \begin{cases} \partial_\nu w_y & \text{in the Dirichlet case,} \\ w_y|_{\partial\Omega} & \text{in the Robin case.} \end{cases} $$

Let $y \in C$, and let $\{y_n\} \subset C$ be a sequence such that $y_n$ converges to $y$, so that $w_n := w_{y_n}$ converges to $w := w_y$ (uniformly in the $C^p$ topology for $p$ large enough, Lemma 8.3). Since $\check{w}_n$ converges uniformly to $\check{w}$, and since $\check{w}$ changes sign at $z(y)$, it follows that $z(y_n)$ belongs to some neighborhood of $z(y)$, and that $z(y_n)$ tends to $z(y)$. This proves that $y \to z(y)$ is continuous in $C$. We now investigate the behaviour of $z(y)$ when $y$ tends to $\partial C$ (when $C \neq \Gamma_1$). Write $C$ as an open arc $(\eta_-, \eta_+) \subset \Gamma_1$, and assume that $\{y_n\} \subset C$, with $y_n$ tending to some $\eta \in \{\eta_-, \eta_+\} \subset \Gamma_{1,(2k-2)}$. Choose a subsequence of $\{z(y_n)\}$ which converges to some $z$. Since $w_n$ tends to $w_\eta$, with $\eta \in \Gamma_{1,(2k-2)}$, we conclude that $\check{w}_\eta(z) = 0$, and hence that $z = \eta$ since $\eta$ is the unique zero of $\check{w}_\eta$ in $\Gamma_1$. ✓

*Proof of Assertion (iv).* Let $C$ be a component of $\partial\Gamma_{1,(2k-3)}$. Define $\alpha(y) := \tau_y^W(\downarrow) \in L_{(2k-3)}$. Assume that the map $y \mapsto \alpha(y)$ is not locally constant. There exists $y \in C$ and a sequence $y_n$ tending to $y$ in $C$ such that $\alpha(y_n) \neq \alpha(y) := a$. Since the map $\alpha$ takes finitely many values, after taking a subsequence if necessary, we may assume that $\alpha(y_n) = b \neq a$. Let $w_n := w_{y_n}$. The nodal arc of $\mathcal{Z}(w_n)$ which emanates from $y_n$ tangentially to the ray $\omega_b$ hits the boundary at the point $z_n := z(y_n)$. Since the sequence $\{w_n\}$ converges to $w_y$ in the $C^p$ topology for any fixed $p$, taking subsequences if necessary, we may assume that $\mathcal{Z}(w_n)$ converges to $\mathcal{Z}(w_y)$ in the Hausdorff distance, and that $\{z_n\}$ converges to $z(y)$. On the other-hand, we can apply the local structure theorem to the functions $w_n$ in a neighborhood of $y$: the arcs emanating from $y_n$ intersect a circle of radius $\varepsilon$ (with $\varepsilon$ independent of $n$), at points $x_{n,j}, 1 \le j \le 2k-3$ and these points converge to the corresponding points $x_j, 1 \le j \le 2k-3$, for the function $w_y$. To prove that $\varepsilon$ can be taken independent of $n$ we use the fact that, for any fixed $m$, the derivatives of $w_n$ of order less than or equal to $m$ converge uniformly to the corresponding derivatives of $w_y$ so that the remainder term in Taylor's formula can be controlled independently of $n$, see the proof of the local structure theorem in Appendix C, Remark C.1. The arc in $\mathcal{Z}(w_n)$ between $x_{n,b}$ and $z_n$ must tend in the Hausdorff distance to the arc in $\mathcal{Z}(w_y)$ between $x_b$ and $y$, and we get a contradiction since $b \neq a$. It follows that the map $\alpha$ is locally constant, hence constant, on the component $C$. Since the map $y \mapsto \tau_y^W(\downarrow)$ is constant on $C$, the set $L_{(2k-3)} \setminus \{\tau_y^W(\downarrow)\}$ is constant, and it suffices to look at the restriction of $\tau_y^W$ to this set. To prove that the map $y \mapsto \tau_{y,\alpha}$ is locally constant on $C$ we reproduce the arguments in the proof of Property 5.6 or Lemma 7.13. □

**Remark 8.5.** Whether $\Gamma_{1,(2k-2)}$ can be empty will be discussed in Section 10.

## 9. SIMPLY CONNECTED PLANAR DOMAINS, PROPERTIES OF $\lambda_k$-EIGENFUNCTIONS ASSUMING THAT $\mathrm{mult}(\lambda_k) = (2k-2)$

The purpose of this section is to derive further properties of $\lambda_k$-eigenfunctions assuming the upper bound in Proposition 7.17 is achieved. Through out this section, we assume that
$$ (9.1) \qquad \left\{ \begin{array}{l} \Omega \text{ is simply connected,} \\ k \ge 3 \text{ and } \dim U(\lambda_k) = (2k-2). \end{array} \right. $$

In this section, to simplify the notation, we write $\Gamma$ instead of $\Gamma_1$, $\Gamma_{(2k-3)}$, instead of $\Gamma_{1,(2k-3)}$, and $\Gamma_{(2k-2)}$ instead of $\Gamma_{1,(2k-2)}$.

**Remark 9.1.** The assumption that $\Omega$ is simply connected is motivated by Berdnikov's argument ([Berd2018], Section 4) showing why the last step in the proof of the upper bound $\dim U(\lambda_k) \le (2k-3)$ given in [HoMN1999] is incomplete in the non simply connected case. This assumption also makes the proofs of the following lemmas simpler. It would be worthwhile determining when it is actually necessary.

9.1. Boundary behaviour of the map $x \mapsto u_x$. The following lemma expands Lemma 3.8 in [HoMN1999, p. 1184].

**Lemma 9.2.** Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded domain and $U := U(\lambda_k)$ for some $k \ge 3$. Assume that $\Omega$ is simply connected, and $\dim U = 2k-2$. Let $\{x_n\} \subset \Omega$ be a sequence converging to some $y \in \Gamma$. Let $\{u_n\}$ be a corresponding sequence of eigenfunctions, with $u_n := u_{x_n} \in U_{x_n} \cap \mathbb{S}(U)$.

(i) If $w$ is a limit point of $\{u_n\}$, then $w \in W_y$. In particular, the continuous maps

$$ \left\{ \begin{array}{l} \Omega \ni x \mapsto [u_x] \text{ of Lemma 8.1, and} \\ \Gamma \ni y \mapsto [w_y] \text{ of Lemma 8.3} \end{array} \right. $$

give rise to a continuous map $x \mapsto [\bar{u}_x]$ from $\bar{\Omega}$ into $\mathbb{P}(U)$.

(ii) The point $y$ belongs to $\Gamma_{(2k-3)}$ if and only if, for $n$ large enough, $S_b(u_n) = \{y_n, z(y_n)\}$ with $y_n \to y$, and $z(y_n) \to z \ne y$.

(iii) The point $y$ belongs to $\Gamma_{(2k-2)}$ if and only if there exists an infinite subsequence $\{u_{s(n)}\}$ such that $S_b(u_{s(n)}) = \emptyset$, or an infinite subsequence $\{u_{s(n)}\}$ such that $S_b(u_{s(n)}) \ne \emptyset$, and the points in $S_b(u_{s(n)})$ converge to $y$.

**Remark 9.3.** Since $\Omega$ is simply connected, we can lift the $C^\infty$ map $\Omega \ni x \to [u_x] \in \mathbb{P}(U)$ to a $C^\infty$ map $x \to u_x \in \mathbb{S}(U)$. Let $h_{x,(k-1)}(u_x)$ be the first nonzero term in the Taylor expansion of $u_x$ at the point $x$ (this is a harmonic polynomial of degree $(k-1)$). Then, the map $x \to h_{x,(k-1)}(u_x)$ is smooth and, by Assertion (i), extends continuously to $\bar{\Omega}$. Unfortunately, this extension is not so interesting because $\lim_{x \to y \in \Gamma} h_{x,(k-1)}(u_x) = 0$ since $u_x$ tends to $w_y$ and $h_{y,(k-1)}(w_y) = 0$. See also the final comment in [BeNP2016]. We will mainly use Assertions (ii) and (iii).

*Proof of Lemma 9.2.* We divide the proof into several steps labeled (A), (B), ... for later cross reference.

(A) To the sequence of interior points, $\{x_n\} \subset \Omega$, we associate a sequence $\{u_n = u_{x_n}\}$ in the sphere $\mathbb{S}(U)$ (Lemma 8.1). Taking a subsequence if necessary, we may assume that $\{u_n\}$ converges to some $w \in \mathbb{S}(U)$. Then, the convergence is uniform in $C^m$ for any fixed $m \ge 0$. Since $\nu(u_n, x_n) = 2(k-1)$, or equivalently $\mathrm{ord}(u_n, x_n) = (k-1)$, with $k \ge 3$, and since the convergence is uniform, we have $\mathrm{ord}(w, y) \ge (k-1) \ge 2$. Define $p := \mathrm{ord}(w, y)$, $q := \rho(w, y)$.

By Lemma 3.15, the (sub)sequence $\{\mathcal{Z}(u_n)\}$ converges to $\mathcal{Z}(w)$ in the Hausdorff distance. This in particular implies that the set $\mathcal{Z}(w)$ is connected.