Pointwise Bounds for Steklov Eigenfunctions

Abstract

Let $(\Omega ,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial \Omega .$ 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 $S^{\ast }\partial \Omega .$ 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 $\{\sigma (P)=0\}$.

arXiv comment1 figure, 42 pages