On the inner radius of the nonvanishing set for eigenfunctions of complex elliptic operators

TL;DR

This paper investigates the zero set structure of eigenfunctions of complex elliptic operators, establishing bounds on the inner radius of their nonvanishing regions or showing mass concentration near boundaries as eigenvalues grow large.

Contribution

It provides new quantitative bounds on the inner radius of the nonvanishing set or mass concentration for eigenfunctions of complex elliptic operators, extending understanding of their zero set geometry.

Findings

Eigenfunctions have a lower bound on the inner radius of their nonvanishing set proportional to ||^{-1/m}.

Alternatively, the eigenfunctions' L^2 mass concentrates near the boundary within a layer of width ||^{-1/m}.

Results hold for solutions of complex elliptic PDEs with constant coefficients in any open set.

Abstract

Let $\Omega\subset\mathbb{R}^d$ be any open set. We consider solutions of $H\psi_\lambda=\lambda \psi_\lambda$, $\lambda\in\mathbb{C}$, where $H$ is an $m$th order complex constant-coefficient elliptic partial differential operator. We prove that either the eigenfunctions satisfy a lower bound on the inner radius of the complement of the zero set of $\psi_\lambda$ in $\Omega$ of order $|\lambda|^{-1/m}$, or 100% of the $L^2$ mass of $\psi_\lambda$ concentrates in a boundary layer of width $|\lambda|^{-1/m}$, as $|\lambda|\to+\infty$.