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

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}.