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

Abstract

Let $\hat{I}\text{c◯}$ be an $n$-dimensional compact Riemannian manifold $(n\ge 3)$ with $C^{\infty }$ boundary, and consider $L^2$-normalized eigenfunctions $-\hat{I}”\ddot{I}{•}_{\hat{I}}»=\hat{I}{»}^2\ddot{I}{•}_{\hat{I}}»$ 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 $\hat{I}\text{c◯}$ it follows that for {\em any} $x_0\in \overline{\hat{I}}\text{c◯}$ (including boundary points) and for all $\hat{I}¼\ge C_{\hat{I}}\text{c◯}\hat{I}{»}^{-1}$ with sufficiently large constant $C_{\hat{I}}\text{c◯}>0,$ \begin{equation} \label{nonconbdy} \| ϕ_λ\|_{B(x_0,μ)\cap Ω}^2 = O(μ). \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} \| ϕ_λ\|_{L^\infty(Ω)} \leq C λ^{\frac{n}{2}} \cdot \Big( \sup_{x \in Ω} \| ϕ_λ \|_{L^2( B(x,λ^{-1}) \cap Ω)} \Big). \end{equation} The sharp sup bounds $\Vert \ddot{I}{•}_{\hat{I}}»{\Vert }_{L^{\infty }(\hat{I}\text{c◯})}=O(\hat{I}{»}^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.

This version fixes a mistake in the first version, and as a consequence the authors assume higher regularity of the boundary