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

Jeffrey Galkowski (Department of Mathematics. University College London, United Kingdom)

John A. Toth (Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada)

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

Full Text (PDF format)

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