On the Nodal set of a second Dirichlet eigenfunction in a doubly connected domain
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 863-873.

Ce papier étudie la géométrie de l’ensemble nodal de la seconde fonction propre du laplacien avec conditions de Dirichlet dans un domaine doublement connexe de forme $\Omega =D\setminus \overline{B}$. Les résultats obtenus sont utilisés dans un problème d’optimisation de la seconde valeur propre.

We investigate the geometry of the nodal set of a second eigenfunction of the Dirichlet Laplacian in a doubly connected Euclidean plane domain of the form $\Omega =D\setminus \overline{B}$ and obtain results of Payne’s type. For instance, we prove that when $D$ and $B$ are symmetric and convex with respect to a line, then the nodal set cannot enclose $B$. Moreover, if $\Omega$ has a second axis of symmetry, then the nodal line intersects both $\partial B$ and $\partial D$.

We also use these results in the optimization of the second eigenvalue for the problem of optimal placement of $B$ within $D$.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1585
Classification : 35P15,  49R50
Mots clés : Dirichlet Laplacian, Nodal set, second eigenfunction, extremal eigenvalue
@article{AFST_2018_6_27_4_863_0,
author = {Rola Kiwan},
title = {On the Nodal set of a second Dirichlet eigenfunction in a doubly connected domain},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {863--873},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 27},
number = {4},
year = {2018},
doi = {10.5802/afst.1585},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1585/}
}
Rola Kiwan. On the Nodal set of a second Dirichlet eigenfunction in a doubly connected domain. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 863-873. doi : 10.5802/afst.1585. https://afst.centre-mersenne.org/articles/10.5802/afst.1585/

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