A one point non-concentration estimate for Laplace eigenfunctions on polygons

TL;DR

This paper establishes a quantitative estimate on how Laplace eigenfunctions distribute their mass near points in polygonal domains, showing they cannot fully concentrate at a single point, using commutator techniques.

Contribution

It provides the first non-concentration estimate for eigenfunctions on polygons, extending previous results from triangles and simplices to more general polygonal domains.

Findings

Eigenfunctions cannot fully concentrate at a single point.

The estimate relates eigenfunction mass to the distance from boundary faces.

The method uses commutator techniques from prior work on triangles.

Abstract

In this paper we consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions. The main result is a quantitative estimate on the $L^2$ mass of eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face. In particular, eigenfunctions cannot concentrate completely at any one single point. The technique of proof is to use the commutator ideas from the recent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.