Exact Poincaré Constants in three-dimensional Annuli
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
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2025
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| _version_ | 1866911186594824192 |
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| author | Rummler, Bernd Ruzicka, Michael Thäter, Gudrun |
| author_facet | Rummler, Bernd Ruzicka, Michael Thäter, Gudrun |
| contents | 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 ${\cal A}\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact Poincaré Constants in three-dimensional Annuli Rummler, Bernd Ruzicka, Michael Thäter, Gudrun Analysis of PDEs 35J05, 35J08, 35Q35, 76D07, 76E06, 76M22 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 ${\cal A}\to\infty$. |
| title | Exact Poincaré Constants in three-dimensional Annuli |
| topic | Analysis of PDEs 35J05, 35J08, 35Q35, 76D07, 76E06, 76M22 |
| url | https://arxiv.org/abs/2506.13891 |