Sharp Upper Bound for the First Non-zero Neumann Eigenvalue for Bounded Domains in Rank-1 Symmetric Spaces
نویسندگان
چکیده
In this paper, we prove that for a bounded domain Ω in a rank-1 symmetric space, the first non-zero Neumann eigenvalue μ1(Ω) ≤ μ1(B(r1)) where B(r1) denotes the geodesic ball of radius r1 such that vol(Ω) = vol(B(r1)) and equality holds iff Ω = B(r1). This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.
منابع مشابه
A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces
M |H| 2, where H is the mean curvature of the hypersurface M. These inequalities of Bleecker–Weiner and Reilly are also sharp for geodesic spheres in Rn. Since then, Reilly’s inequality has been extended to hypersurfaces in other simply connected space forms (see [7] and [8] for details and related results). While trying to understand these results, we noticed that one can obtain a similar shar...
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