On the nodal line of a second eigenfunction of the Laplacian-Dirichlet in some annular domains with dihedral symmetry

TL;DR

This paper investigates the behavior of the second eigenfunction of the Laplacian with Dirichlet boundary conditions in symmetric annular domains, proving that its nodal line must intersect the boundary under certain symmetry and derivative conditions.

Contribution

It establishes a new intersection property of the nodal line for second eigenfunctions in symmetric annular domains with dihedral symmetry, extending previous understanding of eigenfunction nodal sets.

Findings

Nodal line intersects the boundary under symmetry conditions

Results depend on the derivative with respect to angular coordinate

Provides conditions for nodal line behavior in symmetric domains

Abstract

Let $\Omega$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $\Omega$, under suitable conditions on $\frac{\partial}{\partial \theta}$ .