Non-concentration estimates for Laplace eigenfunctions on compact $C^{\infty}$ manifolds with boundary

TL;DR

This paper extends interior nonconcentration bounds of Laplace eigenfunctions to the boundary of smooth compact manifolds, providing new estimates that lead to sharp sup bounds consistent with known results.

Contribution

It introduces boundary nonconcentration estimates for eigenfunctions on manifolds with boundary using stationary methods, extending previous interior results.

Findings

Boundary nonconcentration bounds for eigenfunctions

Extension of Sogge's sup bound result to manifolds with boundary

Derivation of sharp sup bounds matching known optimal estimates

Abstract

Let $\Omega$ be an $n$-dimensional compact Riemannian manifold $(n \geq 3)$ with $C^\infty$ boundary, and consider $L^2$-normalized eigenfunctions $ - \Delta \phi_{\lambda} = \lambda^2 \phi_\lambda$ with Dirichlet or Neumann boundary conditions . In this note, we extend well-known interior nonconcentration bounds up to the boundary. Specifically, in Theorem \ref{thm1}, using purely stationary local methods, we prove that for such $\Omega$ it follows that for {\em any} $x_0 \in \overline{\Omega}$ (including boundary points) and for all $\mu \geq C_{\Omega} \lambda^{-1}$ with sufficiently large constant $C_{\Omega} >0,$ \begin{equation} \label{nonconbdy} \| \phi_\lambda \|_{B(x_0,\mu)\cap \Omega}^2 = O(\mu). \end{equation} In Theorem \ref{thm2} we extend a result of Sogge \cite{So} to manifolds with smooth boundary and show that \begin{equation} \label{SUPBD} \| \phi_\lambda \|_{L^\infty(\Omega)} \leq C \lambda^{\frac{n}{2}} \cdot \Big( \sup_{x \in \Omega} \| \phi_{\lambda} \|_{L^2( B(x,\lambda^{-1}) \cap \Omega )} \Big). \end{equation} The sharp sup bounds $\| \phi_{\lambda} \|_{L^\infty(\Omega)} = O(\lambda^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.