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On the Nodal set of a second Dirichlet eigenfunction in a doubly connected domain
Rola Kiwan1
1 American University in Dubai, P.O.Box 28282, Dubai, United Arab Emirates
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 4, pp. 863-873.

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 Ω=D∖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 Ω has a second axis of symmetry, then the nodal line intersects both ∂B and ∂D.

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

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 Ω=D∖B ¯. Les résultats obtenus sont utilisés dans un problème d’optimisation de la seconde valeur propre.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1585
Classification: 35P15, 49R50
Keywords: Dirichlet Laplacian, Nodal set, second eigenfunction, extremal eigenvalue

Rola Kiwan 1

1 American University in Dubai, P.O.Box 28282, Dubai, United Arab Emirates
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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, Serie 6, Volume 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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