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arXiv · 2507.13478

Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains

Nick Lindemulder·Emiel Lorist·Floris Roodenburg·Mark Veraar
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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 $C^{1,\lambda}$-domains with $\lambda\in[0,1]$, 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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Nick Lindemulder, Emiel Lorist, Floris Roodenburg, Mark Veraar. 2025-07-17. Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains. https://doi.org/10.1016/j.jde.2025.113884

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