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

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

2007-12-12

arxiv.org/abs/0712.2033v1

Communications on Pures and Applied Analysis 7, 5 (2008) 1193 -- 1201

Mathematics

Metric Geometry

To appear in Communications in Pure and Applied Analysis

Scientific paper

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 in dimension 2, and by Harrell, Kr\"oger and Kurata and Kesavan 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 $\mathbb{H}^n$ .

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