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Transactions of the American Mathematical Society

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A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb{S}^n$

Authors: Mark S. Ashbaugh and Rafael D. Benguria
Journal: Trans. Amer. Math. Soc. 353 (2001), 1055-1087
MSC (1991): Primary 58G25; Secondary 35P15, 49Rxx, 33C55
Published electronically: November 8, 2000
MathSciNet review: 1707696
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Abstract: For a domain $\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\mathbb{S}^n$ we prove the optimal result $\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $\Omega^{\star}$, the symmetric rearrangement of $\Omega$ in $\mathbb{S}^n$, is a geodesic ball in $\mathbb{S}^n$ having the same $n$-volume as $\Omega$. We also show that $\lambda_2/\lambda_1$ for geodesic balls of geodesic radius $\theta_1$ less than or equal to $\pi/2$ is an increasing function of $\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $\theta_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $\theta_1=\pi/2$. Here $j_{\nu,k}$ denotes the $k$th positive zero of the Bessel function $J_{\nu}(t)$. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\mathbb{S}^n$ and having a fixed value of $\lambda_1$ the one with the maximal value of $\lambda_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $\lambda_2/\lambda_1$. Various other results for $\lambda_1$and $\lambda_2$ of geodesic balls in $\mathbb{S}^n$ are proved in the course of our work.

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Additional Information

Mark S. Ashbaugh
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211–0001

Rafael D. Benguria
Affiliation: Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

Keywords: Eigenvalues of the Laplacian, Dirichlet problem for domains on spheres, Payne--P\'{o}lya--Weinberger conjecture, Sperner's inequality, ratios of eigenvalues, isoperimetric inequalities for eigenvalues
Received by editor(s): January 6, 1999
Received by editor(s) in revised form: June 9, 1999
Published electronically: November 8, 2000
Additional Notes: The first author was partially supported by National Science Foundation (USA) grants DMS–9114162, INT–9123481, DMS–9500968, and DMS–9870156.
The second author was partially supported by FONDECYT (Chile) project number 196–0462, a Cátedra Presidencial en Ciencias (Chile), and a John Simon Guggenheim Memorial Foundation fellowship.
Article copyright: © Copyright 2000 American Mathematical Society