Pointwise Bounds for Steklov Eigenfunctions

TL;DR

This paper establishes exponential decay bounds for Steklov eigenfunctions inside real-analytic manifolds, providing sharp decay rates at the boundary using microlocal analysis and FBI transform estimates.

Contribution

It introduces sharp decay estimates for Steklov eigenfunctions in terms of eigenvalues, combining Poisson representation with microlocal concentration techniques.

Findings

Steklov eigenfunctions decay exponentially into the interior.

Decay rate at the boundary is sharp to first order.

Microlocal estimates are derived from FBI transform analysis.

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 $S^*\partial \Omega.$ 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 $\{\sigma(P)=0\}$.