Lower bounds for Steklov eigenfunctions

TL;DR

This paper establishes optimal $L^2$ lower bounds for Steklov eigenfunctions and their restrictions on interior hypersurfaces in analytic Riemannian manifolds, complementing previous upper bound results.

Contribution

It provides the first optimal $L^2$ lower bounds for Steklov eigenfunctions and their restrictions in a geometric neighborhood of the boundary.

Findings

Optimal $L^2$ lower bounds for eigenfunctions

Lower bounds for restrictions to interior hypersurfaces

Complementary to existing upper bound results

Abstract

Let $(\Omega,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial \Omega = M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset \Omega^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.