Exact Poincare Constants in n-dimensional Annuli

Abstract

We study $n$-dimensional annuli for $n\,\in\,\{2,\dots,N\}$ with $N\,<\,\infty$. We choose a non-dimensional setting such that for any fixed $n $ and given number ${\cal A}>0$ the annuli $Ω_{(n),\cal A}$ are defined as space between two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$ in ${ R}^{n}$. For these geometries we provide calculated (precise) Poincaré constants. These depend on ${\cal A}$ and the dimension $n$. Additionally we find a direct match of the Poincaré constants for solenoidal vector fields in ${R}^{n}$ and the Poincaré constants for scalar functions in ${ R}^{n+2}$ (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator. In addition we consider the limit ${\cal A}\,\to\,0$. In this context problems in domains $Ω_{(n),σ}^{*}$ are investigated. These domains enable us to use the Green's function of the Laplacian with vanishing Dirichlet traces to show that the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in ${ R}^{n}$. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.