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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb {S}^n$
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by Mark S. Ashbaugh and Rafael D. Benguria PDF
Trans. Amer. Math. Soc. 353 (2001), 1055-1087 Request permission


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
  • Email:
  • Rafael D. Benguria
  • Affiliation: Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
  • MR Author ID: 34600
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 1055-1087
  • MSC (1991): Primary 58G25; Secondary 35P15, 49Rxx, 33C55
  • DOI:
  • MathSciNet review: 1707696