Counting Nodal Lines Which Touch the Boundary of an Analytic Domain

TL;DR

This paper establishes that the number of boundary zeros and critical points of eigenfunctions in analytic domains grows linearly with the eigenvalue parameter, providing bounds on boundary touching nodal lines.

Contribution

It proves linear bounds on boundary zeros and critical points of eigenfunctions in analytic domains, linking boundary behavior to eigenvalue growth.

Findings

Number of boundary zeros is O(λ).

Number of boundary critical points is O(λ).

Number of boundary-touching nodal lines is O(λ).

Abstract

We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\lambda)$. It follows that the number of nodal lines of $\phi_{\lambda}$ (components of the nodal set) which touch the boundary is of order $\lambda$. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.