Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
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,...
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| Main Authors | , , , |
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| Format | Journal Article |
| Language | English |
| Published |
17.07.2025
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| Online Access | Get full text |
| DOI | 10.48550/arxiv.2507.13478 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Roodenburg, Floris Lindemulder, Nick Lorist, Emiel Veraar, Mark |
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| BackLink | https://doi.org/10.48550/arXiv.2507.13478$$DView paper in arXiv |
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| Snippet | We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded... |
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| SubjectTerms | Mathematics - Analysis of PDEs Mathematics - Functional Analysis |
| Title | Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains |
| URI | https://arxiv.org/abs/2507.13478 |
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