On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N

Lucio Damascelli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

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

Abstract

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We present a simple proof of the fact that if Ω is a bounded domain in R N , N ≥ 2 , which is convex and symmetric with respect to k orthogonal directions, 1 ≤ k ≤ N , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues λ 2 , ⋯ , λ 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 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
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  12. Zhang, Liqun, On the multiplicity of the second eigenvalue of Laplacian in R 2 . Comm. Anal. Geom., 3, 1995, 273-296. Zbl0849.35086MR1362653

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