Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

Abstract

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 0 and 1 . First, we show that the Steklov eigenvalues can be divided into two families (λ ± m ) 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 → ∞. When we add an asymmetric perturbation of the metric to a symmetric warped product, we observe in almost all cases 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 0 , and the other half on the other connected component 1 as m → ∞. RésuméNous étudions les valeurs propres et les vecteurs propres de Steklov sur des variétés riemanniennes (M, g) dont le bord ∂ M est constitué de deux composantes connexes distinctes 0 et 1 . Dans un premier temps, nous montrons que les valeurs propres de Steklov peuvent se diviser en deux familles (λ ± m ) m≥0 satisfaisant des asymptotiques précises lorsque m → ∞. Ensuite, nous considérons les fonctions propres de Steklov associées qui sont, par définition, les extensions harmoniques des fonctions propres de l'opérateur Dirichlet à Neumann. Dans le cas de variétés produits tordus symétriques, nous montrons que les fonctions propres de Steklov sont exponentiellement localisées sur tout le bord ∂ M lorsque m tend vers l'infini. Par contre, lorsque nous ajoutons à ce dernier cas une perturbation de la métrique B François Nicoleau

Semi‐classical edge states for the Robin Laplacian

“…Although not written explicitly, the computations by Gendron can also lead to the same result (but for a particular case). This has been developed in the recent work [6].…”

“…Examining the case of the annulus, Ω = {𝑥 ∈ ℝ 2 ∶ 𝑟 0 < |𝑥| < 1}, we observe that the constant 𝛼 and the distance function 𝑑(𝑥, Γ) in (1.6) are non-optimal. The example of the annulus suggest the optimal decay rate is achieved with 𝛼 ≈ 1 and a distance function dΓ that depends on the curvature of the boundary (see [10, section 1.1.3] and [6]).…”

“…Examining the case of the annulus, $$\Omega = \lbrace x\in \mathbb {R}^2\nobreakspace :\nobreakspace r_0&lt;|x|&lt;1\rbrace$$, we observe that the constant α and the distance function $$d(x,\Gamma )$$ in (1.6) are non‐optimal. The example of the annulus suggest the optimal decay rate is achieved with $$\alpha \approx 1$$ and a distance function $$ \hat{d}_{\Gamma }$$ that depends on the curvature of the boundary (see [10, section 1.1.3] and [6]).…”

Semi-classical edge states for the Robin Laplacian

“…The example of the annulus suggest the optimal decay rate is achieved with α ≈ 1 and a distance function dΓ that depends on the curvature of the boundary (see [10,Sec. 1.1.3] and [6]).…”

“…Although not written explicitly, the computations by G. Gendron can also lead to the same result (but for a particular case). This has been developed in the recent work [6].…”