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Exact Poincaré Constants in three-dimensional Annuli

Bernd Rummler, Michael Ruzicka, Gudrun Thäter

TL;DR

The paper derives exact and approximate Poincaré constants for 3d-annuli, linking them to the first eigenvalues of the Laplacian and Stokes operators under vanishing Dirichlet traces. It proves the scalar Poincaré constant is exactly c_p(A)=1/π for all gap sizes, while giving precise Stokes constants c_p,S(A) and exploring limiting regimes as A→0 and A→∞ via Green's functions and small-gap analysis. The authors establish that in the zero-gap and large-gap limits the eigenvalues converge to the unit-ball problem or to a radial one-dimensional limit, respectively, and provide both analytic arguments and numerical validations. Their results yield sharp bounds and asymptotics that improve understanding of stability and spectral properties in 3d-annuli, with practical implications for fluid-structure problems and related PDE analyses.

Abstract

We study 3d-annuli. In our non-dimensional setting each annulus $Ω_{\cal A}$ 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 Poincaré constants for scalar functions and calculate precise Poincaré 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 $Ω_σ^{*}$, 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 $Ω_σ^{*}$ with vanishing Dirichlet traces on $\partial Ω_σ^{*}$ is used to show that for $σ\,\to\,0$ 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 ${\cal A}\to\infty$.

Exact Poincaré Constants in three-dimensional Annuli

TL;DR

The paper derives exact and approximate Poincaré constants for 3d-annuli, linking them to the first eigenvalues of the Laplacian and Stokes operators under vanishing Dirichlet traces. It proves the scalar Poincaré constant is exactly c_p(A)=1/π for all gap sizes, while giving precise Stokes constants c_p,S(A) and exploring limiting regimes as A→0 and A→∞ via Green's functions and small-gap analysis. The authors establish that in the zero-gap and large-gap limits the eigenvalues converge to the unit-ball problem or to a radial one-dimensional limit, respectively, and provide both analytic arguments and numerical validations. Their results yield sharp bounds and asymptotics that improve understanding of stability and spectral properties in 3d-annuli, with practical implications for fluid-structure problems and related PDE analyses.

Abstract

We study 3d-annuli. In our non-dimensional setting each annulus is defined via two concentrical balls with radii and . For these geometries we provide the exact value for the Poincaré constants for scalar functions and calculate precise Poincaré 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 , the 3d-annuli are investigated - for comparison but also to provide limits for . In particular, the Green's function of the Laplacian on with vanishing Dirichlet traces on is used to show that for 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 .

Paper Structure

This paper contains 10 sections, 5 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical groundwork
  3. Available Bounds for the Poincaré constant
  4. Laplace and Stokes operators on 3d-annuli
  5. Eigenvalues and Eigenfunctions
  6. Investigation of the Limiting Cases
  7. The behaviour of the Laplace eigenvalues for $\boldsymbol{\sigma\,\to\,0}$
  8. The behaviour of the first Stokes eigenvalues for $\boldsymbol{\sigma\,\to\,0}$
  9. The behaviour of the first Laplace and Stokes eigenvalues for $\boldsymbol{{\cal A}\,\to\,{\infty}}$
  10. Results and Calculations

Key Result

Theorem 3

The Stokes operator ${\boldsymbol S}$ is positive and self-adjoint. Its inverse ${\boldsymbol S}^{-1}$ is injective, self-adjoint and compact.

Figures (3)

  • Figure 1: 3d-annulus with inner/outer radius in a Cartesian coordinate system
  • Figure 2: Analytical bounds of Poincaré constants and the constant value $\frac{1}{\pi}$ vs. $\mathcal{A}$
  • Figure 3: The constant value $c_{p}({\cal A})\,=\,\frac{1}{\pi}$ and calculated Stokes Poincaré constants $c_{p,S}({\cal A})$ as functions of $\mathcal{A}$

Theorems & Definitions (11)

  • proof
  • Definition 1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • ...and 1 more