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September  2008, 7(5): 1193-1201. doi: 10.3934/cpaa.2008.7.1193

Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

Rola Kiwan 1, and  Ahmad El Soufi 1,

1. 

Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, F-37200 Tours, France, France

Received  November 2006 Revised  February 2008 Published  June 2008

We prove that among all doubly connected domains of $\mathbb R^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch [12] in dimension 2, and by Harrell, Kröger and Kurata [10] and Kesavan [13] in any dimension.
We also prove that the same result remains valid when the ambient space $\mathbb R^n$ is replaced by the standard sphere $\mathbb S^n$ or the hyperbolic space $\mathbb H^n$.
Keywords: Dirichlet Laplacian, spherical shell., extremal eigenvalue, obstacle, eigenvalues.
Mathematics Subject Classification: 35P15, 49R50, 58J5.
Citation: Rola Kiwan, Ahmad El Soufi. Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1193-1201. doi: 10.3934/cpaa.2008.7.1193
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