Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the... 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial M$$\end{document} consists in two distinct connected components Γ0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _0$$\end{document} and Γ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _1$$\end{document}. First, we show that the Steklov eigenvalues can be divided into two families (λm±)m≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\lambda _m^\pm )_{m \ge 0}$$\end{document} which satisfy accurate asymptotics as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}. 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial M$$\end{document} as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}. 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 Γ0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _0$$\end{document}, and the other half on the other connected component Γ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _1$$\end{document} as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales mathématiques du Québec Springer Journals

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

Loading next page...
 
/lp/springer-journals/exponential-localization-of-steklov-eigenfunctions-on-warped-product-5Hkj1tceJS

References (30)

Publisher
Springer Journals
Copyright
Copyright © Fondation Carl-Herz and Springer Nature Switzerland AG 2021
ISSN
2195-4755
eISSN
2195-4763
DOI
10.1007/s40316-021-00185-3
Publisher site
See Article on Publisher Site

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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial M$$\end{document} consists in two distinct connected components Γ0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _0$$\end{document} and Γ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _1$$\end{document}. First, we show that the Steklov eigenvalues can be divided into two families (λm±)m≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\lambda _m^\pm )_{m \ge 0}$$\end{document} which satisfy accurate asymptotics as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}. 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial M$$\end{document} as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}. 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 Γ0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _0$$\end{document}, and the other half on the other connected component Γ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _1$$\end{document} as m→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m \rightarrow \infty $$\end{document}.

Journal

Annales mathématiques du QuébecSpringer Journals

Published: Oct 1, 2023

Keywords: Steklov eigenfunctions; Warped product manifolds; Tunnel effect; 35P15; 58J50

There are no references for this article.