Galkowski, J;
Toth, JA;
(2019)
Pointwise Bounds for Steklov Eigenfunctions.
Journal of Geometric Analysis
, 29
(1)
pp. 142-193.
10.1007/s12220-018-9984-7.
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Abstract
Let (Ω,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary ∂Ω. 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∗∂Ω. 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 {σ(P)=0}.
| Type: | Article |
|---|---|
| Title: | Pointwise Bounds for Steklov Eigenfunctions |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s12220-018-9984-7 |
| Publisher version: | https://doi.org/10.1007/s12220-018-9984-7 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Steklov eigenfunctions, FBI transform, Analytic microlocal analysis, Exponential weighted estimates |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10083900 |
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