Bay Area Microlocal Analysis Seminar: Pointwise Bounds for Steklov Eigenfunctions

Seminar | February 6 | 4:10-5 p.m. | 740 Evans Hall

 Jeff Galkowski, Stanford and McGill

 Department of Mathematics

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 and proving a conjecture of Hislop and Lutzer. The 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\}$. This talk is based on joint work with John Toth.