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.

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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.
Viene presentata una semplice dimostrazione del fatto che se $\Omega$ é un dominio limitato di $\mathbb{R}^{N}$, $N \ge 2$, convesso e simmetrico in $k$ direzioni ortogonali, $1 \le k \le N$, allora gli insiemi nodali delle autofunzioni del laplaciano corrispondenti agli autovalori $\lambda_{2}, \cdots ,\lambda_{k+1}$ hanno intersezione non vuota con la frontiera del dominio. Questo risultato era stato dimostrato da Payne nel caso $N = 2$ per la seconda autofunzione, e da altri autori nel caso di domini piani convessi, sempre per la seconda autofunzione. 
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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/

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