Dirichlet eigenfunction and heat kernel estimates on annular domains

Abstract

Motivated by Euclidean boxes, we consider "thin" annular domains of the form $U=(a,b)\times U_0\subseteq \mathbb{R}^n$ in polar coordinates, where the spherical base $U_0\subseteq \mathbb{S}^{n-1}$ is an inner uniform domain. We show that, with respect to the measure $\varphi_U^2$ determined by the principal Dirichlet Laplacian eigenfunction $\varphi_U$, such annular domains satisfy volume doubling and Poincar\'e inequalities uniformly over all locations and scales. This implies sharp Dirichlet heat kernel estimates expressed in terms of $\varphi_U$. Our results hold uniformly over the collection of all annuli in $\mathbb{R}^n$. We also give matching two-sided bounds for the first Dirichlet Laplacian eigenfunction and eigenvalue for some annular domains including annuli in $\mathbb{R}^n$. Moreover, we prove eigenfunction inequalities for $\varphi_U$ under domain perturbations of $U$. The proofs of our main results utilize eigenfunction comparison techniques due to Lierl and the authors (arXiv:1210.4586, arXiv:2504.18783), small scale $\varphi_U^2$-Poincar\'e inequalities, as well as a discretization technique of Coulhon and Saloff-Coste. Finally, our methods also imply uniform Neumann heat kernel estimates for thin annular domains.