Exact Poincar\'e Constants in three-dimensional Annuli

TL;DR

This paper derives exact Poincaré constants for scalar and vector fields in three-dimensional annuli, providing precise eigenvalues and exploring limits as the annulus parameters vary.

Contribution

It offers the first exact values for Poincaré constants in 3D annuli for scalar and solenoidal vector functions, including boundary conditions and eigenvalue analysis.

Findings

Exact Poincaré constants for scalar functions in 3D annuli.

Precise Poincaré constants for solenoidal vector fields.

Eigenvalue limits as annulus parameters tend to zero or infinity.

Abstract

We study 3d-annuli. In our non-dimensional setting each annulus ${\Omega}_{\cal A}$ is defined via two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$. For these geometries we provide the exact value for the Poincar\'e constants for scalar functions and calculate precise Poincar\'e constants for solenoidal vector fields (in both cases with vanishing Dirichlet traces on the boundary). For this we use the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. Additionally, corresponding problems in domains ${\Omega}_{\sigma}^{*}$, the 3d-annuli are investigated - for comparison but also to provide limits for ${\cal A}\,\to\,0$. In particular, the Green's function of the Laplacian on ${\Omega}_{\sigma}^{*}$ with vanishing Dirichlet traces on $\partial {\Omega}_{\sigma}^{*}$ is used to show that for ${\sigma}\,\to\,0$ the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.