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Title: An upper bound for the first nonzero Steklov eigenvalue
Let (Mn,g) be a complete simply connectedn-dimensional Riemannian manifold with curvature bounds Sectg≤ κ for κ ≤ 0 and Ricg≥ (n− 1)KgforK≤ 0. We prove that for any bounded domain Ω ⊂Mnwith diameterdand Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ1(Ω) ≤Cσ1(Ω*), where σ1(Ω) and σ1(Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, andC=C(n, κ,K,d) is an explicit constant. When κ =K, we haveC= 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.  more » « less
Award ID(s):
2405257 2316659
PAR ID:
10616521
Author(s) / Creator(s):
; ;
Publisher / Repository:
EDP Sciences
Date Published:
Journal Name:
ESAIM: Control, Optimisation and Calculus of Variations
Volume:
31
ISSN:
1292-8119
Page Range / eLocation ID:
5
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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