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

Let $\mathrm{\Omega }$ be an $n$-dimensional compact Riemannian manifold $(n\ge 3)$ with $C^{\mathrm{\infty }}$ boundary, and consider $L^2$-normalized eigenfunctions $-\mathrm{\Delta }{\phi }_{\lambda }={\lambda }^2{\phi }_{\lambda }$ 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 $\mathrm{\Omega }$ it follows that for {\em any} $x_0\in \bar{\mathrm{\Omega }}$ (including boundary points) and for all $\mu \ge C_{\mathrm{\Omega }}{\lambda }^{-1}$ with sufficiently large constant $C_{\mathrm{\Omega }}>0,$ \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 $\mathrm{\parallel }{\phi }_{\lambda }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}=O({\lambda }^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.