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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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.


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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: http://dx.doi.org/10.1090/S0002-9947-96-01682-0
PII: S 0002-9947(96)01682-0
Keywords: Eigenvalue, centre of mass, Riemannian submersion
Received by editor(s): January 20, 1994
Article copyright: © Copyright 1996 American Mathematical Society