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Pointwise Bounds for Steklov Eigenfunctions

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Abstract

Let \((\Omega ,g)\) be a compact, real-analytic Riemannian manifold with real-analytic boundary \(\partial \Omega .\) The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle \(S^*\partial \Omega .\) These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations \(Pu=0\) near the characteristic set \(\{\sigma (P)=0\}\).

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Acknowledgements

The authors would like to thank Iosif Polterovich and Steve Zelditch for their comments on an earlier version of this paper. Thanks also to Andras Vasy and Maciej Zworski for valuable suggestions. Finally, thanks to the anonymous referee for many helpful suggestions. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661. The research of J.T. was partially supported by NSERC Discovery Grant # OGP0170280 and an FRQNT Team Grant. J.T. was also supported by the French National Research Agency project Gerasic-ANR- 13-BS01-0007-0.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, USA

    Jeffrey Galkowski

  2. Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada

    John A. Toth

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  1. Jeffrey Galkowski
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  2. John A. Toth
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Correspondence to Jeffrey Galkowski.

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Galkowski, J., Toth, J.A. Pointwise Bounds for Steklov Eigenfunctions. J Geom Anal 29, 142–193 (2019). https://doi.org/10.1007/s12220-018-9984-7

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