Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911466537353216 |
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| author | Lindemulder, Nick Lorist, Emiel Roodenburg, Floris Veraar, Mark |
| author_facet | Lindemulder, Nick Lorist, Emiel Roodenburg, Floris Veraar, Mark |
| contents | We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,λ}$-domains with $λ\in[0,1]$, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13478 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains Lindemulder, Nick Lorist, Emiel Roodenburg, Floris Veraar, Mark Analysis of PDEs Functional Analysis Primary: 47A60, Secondary: 35K20, 46E35 We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,λ}$-domains with $λ\in[0,1]$, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable. |
| title | Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains |
| topic | Analysis of PDEs Functional Analysis Primary: 47A60, Secondary: 35K20, 46E35 |
| url | https://arxiv.org/abs/2507.13478 |