2021/03/25 by Thierry Daudé, Bernard Helffer, Daudé, Thierry +3 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2103.13889
openalex publication_date 2021/03/25 · openalex created_date 2021/03/29 · openalex updated_date 2026/07/28
This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds (M,g) whose boundary ∂ M consists in two distinct connected components Γ0 and Γ1. First, we show that the Steklov eigenvalues can be divided into two families (λm^±)m ≥ 0 which satisfy accurate asymptotics as m → ∞. Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary ∂ M as m → ∞. Whenever we add an asymmetric perturbation to a symmetric warped product, we observe a flea on the elephant effect. Roughly speaking, we prove that "half" the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say Γ0, and the other half on the other connected component Γ1 as m → ∞.