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

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