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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $\mathrm{\Omega }$ of order $\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1\mathrm{/}m}$, or 100% of the $L^2$ mass of ${\psi }_{\lambda }$ concentrates in a boundary layer of width $\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1\mathrm{/}m}$, as $\mathrm{\mid }\lambda \mathrm{\mid }\to +\mathrm{\infty }$.