# On the nodal set of the second eigenfunction of the laplacian in symmetric domains in ${\mathbb{R}}^{N}$

• Volume: 11, Issue: 3, page 175-181
• ISSN: 1120-6330

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## Abstract

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We present a simple proof of the fact that if $\mathrm{\Omega }$ 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 ${\lambda }_{2},\mathrm{\cdots },{\lambda }_{k+1}$ 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.

## How to cite

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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. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.3 (2000): 175-181. <http://eudml.org/doc/252373>.

@article{Damascelli2000,
abstract = {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.},
author = {Damascelli, Lucio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Second eigenfunction; Nodal set; Maximum principle; second eigenfunction; Maximum principle.},
language = {eng},
month = {9},
number = {3},
pages = {175-181},
publisher = {Accademia Nazionale dei Lincei},
title = {On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $\mathbb\{R\}^\{N\}$},
url = {http://eudml.org/doc/252373},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Damascelli, Lucio
TI - On the nodal set of the second eigenfunction of the laplacian in symmetric domains in $\mathbb{R}^{N}$
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/9//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 3
SP - 175
EP - 181
AB - 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.
LA - eng
KW - Second eigenfunction; Nodal set; Maximum principle; second eigenfunction; Maximum principle.
UR - http://eudml.org/doc/252373
ER -

## References

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7. Melas, A.D., On the nodal line of the second eigenfunction of the laplacian in ${\mathbb{R}}^{2}$. J. Diff. Geom., 35, 1992, 255-263. Zbl0769.58056MR1152231
8. Payne, L.E., Isoperimetric inequalities and applications. SIAM Review, 9, 1967, 453-488. Zbl0154.12602MR218975
9. Payne, L.E., On two conjectures in the fixed membrane eigenvalue problem. Z. Angew. Math. Phys., 24, 1973, 721-729. Zbl0272.35058MR333487
10. Pólya, G. - Szegö, G., Isoperimetric Inequalities in Mathematical Physics. Annals of Math. Studies, 27, Princeton University Press, Princeton, NJ1951. Zbl0044.38301
11. Yau, S.T., Problem section, seminar on differential geometry. Annals of Math. Studies, 102, Princeton University Press, Princeton, NJ1982, 669-706. Zbl0471.00020MR645762
12. Zhang, Liqun, On the multiplicity of the second eigenvalue of Laplacian in ${\mathbb{R}}^{2}$. Comm. Anal. Geom., 3, 1995, 273-296. Zbl0849.35086MR1362653

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