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Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces


Authors: A. R. Aithal and G. Santhanam
Journal: Trans. Amer. Math. Soc. 348 (1996), 3955-3965
MSC (1991): Primary 35P15, 58G25
MathSciNet review: 1363942
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Abstract: In this paper, we prove that for a bounded domain $\Omega $ in a rank-$1$ symmetric space, the first non-zero Neumann eigenvalue $\mu _{1}(\Omega )\leq \mu _{1}(B(r_{1}))$ where $B(r_{1})$ denotes the geodesic ball of radius $r_{1}$ such that

\begin{equation*}vol(\Omega )=vol(B(r_{1}))\end{equation*}

and equality holds iff $\Omega =B(r_{1})$. This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.


References [Enhancements On Off] (What's this?)

  • 1. M.S. Ashbaugh and R.D. Benguria, Sharp upper bound to the first non-zero eigenvalue for bounded domains in spaces of constant curvature, preprint.
  • 2. M. Berger, Lectures on geodesics in Riemannian geometry, Tata Institute of Fundamental Research Lectures on Mathematics, No. 33, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215258
  • 3. Lionel Bérard-Bergery and Jean-Pierre Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math. 26 (1982), no. 2, 181–200. MR 650387
  • 4. Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 3rd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR 507190
  • 5. S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987. MR 909697
  • 6. G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356. MR 0061749
  • 7. H. F. Weinberger, An isoperimetric inequality for the 𝑁-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633–636. MR 0079286

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Additional Information

A. R. Aithal
Affiliation: Department of Mathematics, University of Bombay, Vidyanagare, Bombay-400098, India
Email: aithal@mathbu.ernet.in

G. Santhanam
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400-005, India
Email: santhana@math.tifr.res.in

DOI: https://doi.org/10.1090/S0002-9947-96-01682-0
Keywords: Eigenvalue, centre of mass, Riemannian submersion
Received by editor(s): January 20, 1994
Article copyright: © Copyright 1996 American Mathematical Society