Dirichlet eigenfunction and heat kernel estimates on annular domains

Abstract

Abstract Motivated by Euclidean boxes, we consider “thin” annular domains of the form U = ( a , b ) × U 0 ⊆ R n $U=\left(a,b\right){\times}{U}_{0}\subseteq {\mathbb{R}}^{n}$ in polar coordinates, where the spherical base U 0 ⊆ S n − 1 ${U}_{0}\subseteq {\mathbb{S}}^{n-1}$ is an inner uniform domain. We show that, with respect to the measure φ U 2 ${\varphi }_{U}^{2}$ determined by the principal Dirichlet Laplacian eigenfunction φ U , such annular domains satisfy volume doubling and Poincaré inequalities uniformly over all locations and scales. This implies sharp Dirichlet heat kernel estimates expressed in terms of φ U . Our results hold uniformly over the collection of all annuli in R n ${\mathbb{R}}^{n}$ . We also give matching two-sided bounds for the first Dirichlet Laplacian eigenfunction and eigenvalue for some annular domains including annuli in R n ${\mathbb{R}}^{n}$ . Moreover, we prove eigenfunction inequalities for φ U under domain perturbations of U . The proofs of our main results utilize eigenfunction comparison techniques due to Lierl and the authors (“The Dirichlet heat kernel in inner uniform domains: Local results, compact domains and non-symmetric forms,” J. Funct. Anal. , vol. 266, no. 7, pp. 4189–4235, 2014; “Perturbing the principal Dirichlet eigenfunction,” arXiv preprint arXiv:2504.18783, 2025), small scale φ U 2 ${\varphi }_{U}^{2}$ -Poincaré inequalities (Lierl and Saloff-Coste, “The Dirichlet heat kernel in inner uniform domains: Local results, compact domains and non-symmetric forms,” J. Funct. Anal. , vol. 266, no. 7, pp. 4189–4235, 2014), as well as a discretization technique of Coulhon and Saloff-Coste (“Variétés riemanniennes isométriques à l’infini,” Rev. Mat. Iberoam. , vol. 11, pp. 687–726, 1995). Finally, our methods also imply uniform Neumann heat kernel estimates for thin annular domains.