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Nodal Sets of Laplacian Eigenfunctions with an Eigenvalue of Multiplicity 2

Andrew Lyons

TL;DR

This work analyzes the zero-set geometry of Laplacian eigenfunctions with a double eigenvalue on perturbed rectangular domains. Using Hadamard variation, it proves that a degenerate pair on a rectangle splits into two simple branches on small domain deformations under a nonzero boundary functional \Lambda_φ, and provides explicit descriptions of the nodal sets for both branches on the unperturbed rectangle and on the perturbed domain. For even k, the upper-branch nodal set exhibits a center crossing on the rectangle that becomes a hyperbola under perturbation, while odd k preserves three nodal domains with no interior crossing; the lower-branch nodal set remains disjoint curves separating the nodal domains. Across center, away from center, and near boundaries, the paper establishes precise local descriptions, regularity, and boundary-orthogonality, with detailed asymptotics in terms of the perturbation size \eta and aspect ratio N. These results advance understanding of spectral partitions and nodal-set stability under degeneracy, with implications for Courant sharpness and partition geometry in perturbed domains.

Abstract

We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets corresponding to an eigenvalue of multiplicity 2. We establish geometric properties, such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.

Nodal Sets of Laplacian Eigenfunctions with an Eigenvalue of Multiplicity 2

TL;DR

This work analyzes the zero-set geometry of Laplacian eigenfunctions with a double eigenvalue on perturbed rectangular domains. Using Hadamard variation, it proves that a degenerate pair on a rectangle splits into two simple branches on small domain deformations under a nonzero boundary functional \Lambda_φ, and provides explicit descriptions of the nodal sets for both branches on the unperturbed rectangle and on the perturbed domain. For even k, the upper-branch nodal set exhibits a center crossing on the rectangle that becomes a hyperbola under perturbation, while odd k preserves three nodal domains with no interior crossing; the lower-branch nodal set remains disjoint curves separating the nodal domains. Across center, away from center, and near boundaries, the paper establishes precise local descriptions, regularity, and boundary-orthogonality, with detailed asymptotics in terms of the perturbation size \eta and aspect ratio N. These results advance understanding of spectral partitions and nodal-set stability under degeneracy, with implications for Courant sharpness and partition geometry in perturbed domains.

Abstract

We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets corresponding to an eigenvalue of multiplicity 2. We establish geometric properties, such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.
Paper Structure (14 sections, 28 theorems, 216 equations, 6 figures)

This paper contains 14 sections, 28 theorems, 216 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Spectral partitions.
  3. Comprehensive description of the nodal set geometry.
  4. Structure of the nodal sets.
  5. Regularity of the nodal sets.
  6. Outline.
  7. Hadamard Variation Methods
  8. Proof of Theorem \ref{['prop:n=0']}
  9. Estimates on Eigenfunction Behavior
  10. Description of the Nodal Set Near the Center (k even)
  11. Description of the Nodal Set Away from the Center (k even)
  12. Description of the Nodal Set Near the Boundary (k even)
  13. The Eigenfunction along the Upper Branch (k odd)
  14. The Eigenfunction along the Lower Branch

Key Result

Theorem 1.3

Let $k\geq 3$ be an integer and $N>0$ be such that $k^2=3N^2+4$. Let $\phi \in\mathcal{A}$ as in Definition def:phi and consider $v_0,w_0$ as presented in (eq:onRect). There exists a constant $c_0\in\left(0,\frac{1}{2}\right)$, dependent only on $\Lambda_\phi$, such that the following holds: In each case, every intersection of the nodal set with the boundary $\partial R(N)$ is orthogonal.

Figures (6)

  • Figure 1: The Domain $\Omega_\phi(\eta, N)$.
  • Figure 2: Nodal Sets for $v_0$ when $k=6,7$; Nodal Set for $w_0$ when $k=8$.
  • Figure 3: Nodal Sets for $w_0$ with Boundary Function $\phi_1(y)$, $k=5,6,7$.
  • Figure 4: Nodal Sets for $w_0$ with Boundary Function $\phi_2(y)$, $k=5,6,7$.
  • Figure 5: The Subdomain $\widetilde{\Omega}$.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 50 more