$H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913810003001344 |
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| author | Kunstmann, Peer Christian |
| author_facet | Kunstmann, Peer Christian |
| contents | We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $Ω\subseteq\mathbb{R}^d$ that are uniformly $C^{2,1}$. Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $H^\infty$-calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $H^\infty$-calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $L^p$-regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains Kunstmann, Peer Christian Analysis of PDEs 35Q30, 76D07, 47A60, 47D06 We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $Ω\subseteq\mathbb{R}^d$ that are uniformly $C^{2,1}$. Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $H^\infty$-calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $H^\infty$-calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $L^p$-regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains. |
| title | $H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains |
| topic | Analysis of PDEs 35Q30, 76D07, 47A60, 47D06 |
| url | https://arxiv.org/abs/2504.18895 |