On the nodal line of the second eigenfunction of the Laplacian over some concave domains in $\mathbb{R}^2$

TL;DR

This paper proves that for certain simply connected concave domains in the plane, the nodal line of the second Laplacian eigenfunction intersects the boundary exactly twice, revealing geometric properties of eigenfunctions.

Contribution

It establishes a precise boundary intersection property of the second eigenfunction's nodal line in specific concave planar domains, advancing understanding of eigenfunction behavior.

Findings

Nodal line intersects boundary at exactly two points

Results apply to certain simply connected concave domains

Enhances understanding of eigenfunction nodal sets in planar domains

Abstract

In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $\Omega$ in $\mathbb{R}^2$ must intersect the boundary $\partial\Omega$ at exactly two points.