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

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 equal...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1996-10, Vol.348 (10), p.3955-3965
Main Authors: Aithal, A. R., Santhanam, G.
Format: Article
Language:English
Subjects:
Center of mass
Coordinate systems
Curvature
Decreasing functions
Eigenfunctions
Eigenvalues
Laplacians
Mathematical theorems
Sine function
Symmetry
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-96-01682-0