On the nodal set of the second eigenfunction of the Laplacian in symmetric domains \(\mathbb R^N \). (Q1851413)

A previous conjecture concerning the eigenfunctions of the Laplacian in a domain \(\Omega\in\mathbb R^N \) is considered: if \(\Omega\) is connected and symmetric with respect to \(k\) orthogonal directions, then the nodal set of the eigenfunctions must intersect the boundary. The proof is given by using some symmetry properties of the eigenfunctions and the maximum principle. For \(N=2,\,k=1\) the result was proved in \textit{L. E. Payne} [Z. Angew. Math. Phys. 24, 721--729 (1973; Zbl 0272.35058)]. Some resuls concerning the periodic solutions of the Laplacian in symmetric domains are given in \textit{J. F. Bourgat} [ Rapport I.R.I.A., Roquencourt, France (1978)].