caa a22 4500 397533195 CHVBK 20180308164657.0 cr unu---uuuuu 161202e199510 xx s 000 0 eng 10.1112/jlms/52.2.402 doi (NATIONALLICENCE)oxford-10.1112/jlms/52.2.402 Sharp Upper Bound to the First Nonzero Neumann Eigenvalue for Bounded Domains in Spaces of Constant Curvature [Elektronische Daten] [Mark S. Ashbaugh, Rafael D. Benguria] The main result of this paper is that for a domain Ω contained in a hemisphere of the n-dimensional sphere Sn the first nonzero Neumann eigenvalue µ1(Ω) is less than or equal to the first nonzero Neumann eigenvalue µ1(D) where D is a geodesic ball in Sn of the same measure as Ω. Equality occurs if and only if Ω is isometric to D. This result generalizes old results of Szegö and Weinberger which gave the corresponding upper bound for µ1(Ω) in the Euclidean case, and a result of Chavel for domains in Sn which restricted Ω to lie in a geodesic ball of radius 14π when n = 2 and to even smaller geodesic balls for larger n. The techniques used are analogous to those for our recent proof of the Payne-Pólya-Weinberger conjecture: rearrangement inequalities and properties of special functions are the key elements. The general approach is a direct extension of Weinberger's for domains in Rn. © The London Mathematical Society Notes and Papers nationallicence Ashbaugh Mark S. Department of Mathematics, University of Missouri Columbia, Missouri 65211, USA E-mail: mathmsa@mizzoul.missouri.edu Fax number: 314-882-1869 aut Benguria Rafael D. Facultad de Física, P. Universida Católica de Chile Avda. Vicuña Mackenna 4860, Casilla 306, Santiago 22, Chile E-mail: rbenguri@lascar.puc.cl aut Journal of the London Mathematical Society Oxford University Press 52/2(1995-10), 402-416 0024-6107 52:2<402 1995 52 jlms https://doi.org/10.1112/jlms/52.2.402 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1112/jlms/52.2.402 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ashbaugh Mark S. Department of Mathematics, University of Missouri Columbia, Missouri 65211, USA E-mail: mathmsa@mizzoul.missouri.edu Fax number: 314-882-1869 aut NATIONALLICENCE 700 1- Benguria Rafael D. Facultad de Física, P. Universida Católica de Chile Avda. Vicuña Mackenna 4860, Casilla 306, Santiago 22, Chile E-mail: rbenguri@lascar.puc.cl aut NATIONALLICENCE 773 0- Journal of the London Mathematical Society Oxford University Press 52/2(1995-10), 402-416 0024-6107 52:2<402 1995 52 jlms Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford