Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty }$ bounded planar domains

Abstract

Let $(\Omega ,g)$ be a piecewise-smooth, bounded convex domain in ${\mathbb{R}}^2$ and consider $L^2$-normalized Neumann eigenfunctions ${\varphi }_{\lambda }$ with eigenvalue ${\lambda }^2$ and $u_{\lambda }:={\varphi }_{\lambda }{|}_{\partial \Omega }$ the associated Dirichlet data (ie. boundary restriction of ${\varphi }_{\lambda }$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0\in \overline{\Omega },$ (including boundary corner points) and any $\delta \in [0,1),$ $$\Vert {\varphi }_h{\Vert }_{B(x_0,{\lambda }^{-\delta })\cap \Omega }=O({\lambda }^{-\delta /2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for $L^2$ restrictions of boundary eigenfunctions that are valid up to boundary corners. In particular, in Theorem \ref{dirichlet} we prove that for any {\em flat} boundary edge $\Gamma$ (possibly including corner points), the boundary restrictions $u_h:={\varphi }_h{|}_{\partial \Omega }$ satisfy the bounds $$\Vert u_{\lambda }{\Vert }_{L^2(\Gamma )}=O_{\epsilon }({\lambda }^{1/4+\epsilon }),$$ for any $\epsilon >0.$ The exponent $1/4$ is sharp and the result improves on the $O({\lambda }^{1/3})$ universal $L^2$-restriction bound for Neumann eigenfunctions due to Tataru \cite{Ta}. The $O({\lambda }^{1/4})$ -bound is also an extension to the boundary (including corner points) of well-known interior $L^2$ restriction bounds of Burq-Gerard-Tzvetkov \cite{BGT} along totally-geodesic hypersurfaces.

64 pages

References in corpus (1)

• Small scale quantum ergodicity in negatively curved manifolds