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

Let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$, under suitable conditions on $\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }$ .