Geodesic
On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $\mathbb{R}^{N}$
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 3, pp. 175-181

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We present a simple proof of the fact that if $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N \ge 2$, which is convex and symmetric with respect to $k$ orthogonal directions, $1 \le k \le N$, then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues $\lambda_{2}, \cdots ,\lambda_{k+1}$ must intersect the boundary. This result was proved by Payne in the case $N = 2$ for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction. 
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     title = {On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $\mathbb{R}^{N}$},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {175--181},
     publisher = {mathdoc},
     volume = {Ser. 9, 11},
     number = {3},
     year = {2000},
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     mrnumber = {MR1841691},
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     url = {http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_3_a3/}
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Damascelli, Lucio. On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $\mathbb{R}^{N}$. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 3, pp. 175-181. http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_3_a3/