Dirichlet eigenfunction and heat kernel estimates on annular domains

TL;DR

This work identifies thin annular domains $U=(a,b)\times U_0$ with inner uniform base $U_0$ as a robust setting where the weighted measure $\varphi_U^2dx$ yields volume doubling and Poincaré inequalities uniformly in location and scale, enabling sharp Dirichlet heat kernel estimates expressed through $\varphi_U$. The analysis hinges on a separable representation $\varphi_U(r,\theta)\approx \varphi_I(r)\varphi_{U_0}(\theta)$ in the thin regime, combined with 1D radial eigenvalue problems and known VD/PI results on bases; this leads to two-sided bounds for $\lambda(U)$ and explicit caricature expressions for $\varphi_U$. The authors extend these results to stability under domain perturbations, obtain Neumann heat kernel analogues, and provide a detailed perturbation theory for annuli and for Euclidean boxes. As a complement, a counterexample demonstrates the limits of uniform VD without structural restrictions, highlighting the necessity of inner uniformity-type conditions. Overall, the paper advances a precise, quantitative framework for spectral and heat kernel control on thin domains, with broad implications for geometric analysis and stochastic processes on 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 Poincaré 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 $\varphi_U^2$-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.

Figures (5)

• Figure 1: Thin bounded Euclidean domains $U_{\varepsilon,L}$ parametrized by length parameters $\varepsilon>0$ and $L>0$. These domains are expected to satisfy volume doubling and Poincaré inequalities with respect to the weighted measure $\varphi_{U_{\varepsilon,L}}^2 dx$, uniformly in the regime $\varepsilon\to 0$ and/or $L\to \infty$.
• Figure 2: Depicted above are domains in $\mathbb{R}^2$ of the form $U=(1,b)\times U_0$, where the dashed line represents the angular base $U_0\subseteq \mathbb{S}^{1}$. Our work addresses $(\varphi_U^2\text{-VD})$, $(\varphi_U^2\text{-PI})$, and (\ref{['HKE']}) for "thin" domains in $\mathbb{R}^n$ generated by an inner uniform base $U_0$, with $\text{diam}(U_0)$ uniformly bounded below (e.g. the left figure). The middle figure (with $b$ bounded away from $1$ and $\infty$, and with $\textup{diam}(U_0)\asymp 1$) is inner uniform, hence it is covered by results of lierllsc; our methods apply in this case as well. When $U_0$ has small diameter (e.g. the rightmost figure), $U$ is not inner uniform; our results do not apply uniformly in the regime where $\text{diam}(U_0)\to 0$. On the other hand, in the regime where $\text{diam}(U_0)$ is uniformly bounded below and $b\to \infty$, the rightmost figure is still inner uniform (see Page 129 gyryalsc), and the results of lierllsc again apply.
• Figure 3: The annular domains $A$ and $B$ from Example \ref{['unitcircle']}. When these two domains become sufficiently close to each other, the behavior of $\varphi_U$ for any arbitrary domain $U$ with $A\subseteq U\subseteq B$ "stabilizes" in the sense of (\ref{['AB1']}) and (\ref{['AB2']}).
• Figure 4: The spherical triangles $T$ and $T_{\eta}$, and a subregion $K\subseteq T$ at distance $\asymp \eta$ away from one side of $T$. The triangle $T_{\eta}$ is the largest spherical triangle depicted above, i.e. the entirety of the gray shape. The triangle $T$ is $T_{\eta}$ with the portion from $\theta=-\eta$ to $\theta =0$ removed.
• Figure 5: Examples of domains $U_{\alpha}$ parametrized by an angle $\alpha>0$ which conjecturally fail to be $\varphi_{U_{\alpha}}^2$-volume doubling uniformly as $\alpha\to 0$. Depicted on the right is a bounded cone with opening angle $\alpha$, more explicitly $U_{\alpha}=\{x^2+y^2<\tan(\alpha/2)^2z^2,0<z<1\}\subseteq \mathbb{R}^3$.

Theorems & Definitions (78)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Example 1.8
• Theorem 1.9
• Theorem 1.10