Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains

Abstract

Let $\Omega$ be a piecewise-smooth, bounded convex domain in ${\mathbb{R}}^2$ and consider $L^2$-normalized Neumann eigenfunctions ${\varphi }_{\lambda }$ with eigenvalue ${\lambda }^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0\in \overline{\Omega },$ (including boundary and corner points) and any $\delta \in [0,1),$ $$\Vert {\varphi }_{\lambda }{\Vert }_{B(x_0,{\lambda }^{-\delta })\cap \Omega }=O({\lambda }^{-\delta /2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument.

This is an expanded version of the first half of the preprint arXiv:2012.15237 [math.AP] by the same authors