Work

An upper bound for the first nonzero Steklov eigenvalue

Year: 2024

Type: article

Abstract: Let (Mn, g) be a complete simply connected n-dimensional Riemannian manifold with curvature bounds Sectg ≤ κ for κ ≤ 0 and Ricg ≥ (n − 1)Kg for K ≤ 0. We prove that for any bounded domain Ω ⊂ Mn with diameter d and 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, and C = C(n, κ, K, d) is an explicit constant. When κ = K, we have C = 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.

Source: ESAIM Control Optimisation and Calculus of Variations

DOI: 10.1051/cocv/2024088

Authors: Xiaolong Li, Kui Wang, Haotian Wu

Institutions: Wichita State University, Soochow University, The University of Sydney

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funders: National Science Foundation, Government of Jiangsu Province, Australian Research Council