Dirichlet eigenfunction and heat kernel estimates on annular domains
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arXiv
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2025
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| author | Chao, Brian Saloff-Coste, Laurent |
| author_facet | Chao, Brian Saloff-Coste, Laurent |
| contents | 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 $φ_U^2$ 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 $\mathbb{R}^n$. 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 $φ_U$ 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 $φ_U^2$-Poincaré 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirichlet eigenfunction and heat kernel estimates on annular domains Chao, Brian Saloff-Coste, Laurent Analysis of PDEs Probability Spectral Theory Primary 35B20, 35B51, 35J05, 35K08, 35P15, Secondary 35J25, 60J60, 60J65 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 $φ_U^2$ 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 $\mathbb{R}^n$. 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 $φ_U$ 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 $φ_U^2$-Poincaré 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. |
| title | Dirichlet eigenfunction and heat kernel estimates on annular domains |
| topic | Analysis of PDEs Probability Spectral Theory Primary 35B20, 35B51, 35J05, 35K08, 35P15, Secondary 35J25, 60J60, 60J65 |
| url | https://arxiv.org/abs/2510.17091 |