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arXiv · 1611.05363

Pointwise Bounds for Steklov Eigenfunctions

Jeffrey Galkowski·John A. Toth
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Abstract

Let $(Ω,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial Ω.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions 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 $S^*\partial Ω.$ 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 $\{σ(P)=0\}$.

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Jeffrey Galkowski, John A. Toth. 2018-01-22. Pointwise Bounds for Steklov Eigenfunctions. https://arxiv.org/abs/1611.05363

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