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

5 pages, 1 figure

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