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Pure and Applied Mathematics Quarterly
Volume 19 (2023)
Number 4
Special Issue in honor of Victor Guillemin
Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe
Lower bounds for Steklov eigenfunctions
Pages: 1873 – 1898
DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a7
Authors
Abstract
Let $(\Omega,g)$ be a compact, real analytic Riemannian manifold with real analytic boundary $\partial \Omega = M$. We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset \Omega^\circ$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3897008}{[\textrm{GT19}]}$
Keywords
Steklov, high energy asymptotics, lower bounds
2010 Mathematics Subject Classification
35P20, 58J50
J.G. is grateful to the EPSRC for support under Early Career Fellowship EP/V001760/1 and Standard Grant EP/V051636/1.
J.T. was partially supported by NSERC Discovery Grant # OGP0170280 and by the French National Research Agency project Gerasic-ANR-13-BS01-0007-0.
Received 12 January 2022
Received revised 28 April 2022
Accepted 3 May 2022
Published 20 November 2023