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

TL;DR

This paper establishes a small-scale non-concentration estimate for Neumann eigenfunctions in piecewise-smooth convex planar domains, showing their mass distribution does not overly concentrate at small scales near any point.

Contribution

It provides the first small-scale non-concentration estimate for Neumann eigenfunctions in piecewise-smooth convex domains, including boundary and corner points.

Findings

Eigenfunctions do not concentrate mass at scales smaller than $ ext{constant} imes ext{diameter}$

The estimate applies uniformly to all points in the domain, including boundary and corners

The proof combines stationary vector field methods with small scale induction

Abstract

Let $\Omega$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ 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 $\delta \in [0,1),$ $$ \| \phi_\lambda \|_{B(x_0,\lambda^{-\delta})\cap \Omega} = O(\lambda^{-\delta/2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument.