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

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 $|\lambda |\to +\infty$.

9 pages, no figures