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

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 $\frac{\partial}{\partial \theta}$ .