Non-concentration estimates for Laplace eigenfunctions on compact $C^{\infty}$ manifolds with boundary
Let $Ω$ be an $n$-dimensional compact Riemannian manifold $(n \geq 3)$ with $C^\infty$ boundary, and consider $L^2$-normalized eigenfunctions $ - Δϕ_λ = λ^2 ϕ_λ$ 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 $Ω$ it follows that for {\em any} $x_0 \in \overlineΩ$ (including boundary points) and for all $μ\geq C_Ω λ^{-1}$ with sufficiently large constant $C_Ω >0,$ \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 $\| ϕ_λ \|_{L^\infty(Ω)} = O(λ^{\frac{n-1}{2}})$ for Dirichlet or Neumann eigenfunctions proved by Grieser in \cite{Gr} are then an immediate consequence of Theorems \ref{thm1} and \ref{thm2}.