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. 
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/
@article{RLIN_2000_9_11_3_a3,
     author = {Damascelli, Lucio},
     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},
     year = {2000},
     volume = {Ser. 9, 11},
     number = {3},
     zbl = {1042.35036},
     mrnumber = {MR1841691},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_3_a3/}
}
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