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

2025/07/17 by Lindemulder, Nick, Lorist, Emiel, Roodenburg, Floris +1 · 2 citations
#46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47A60 #Secondary: 35K20

paper · doi:10.48550/arxiv.2507.13478

Abstract

We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded H^∞-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∈[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.

Citations

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Cited by

• Trace theory for parabolic boundary value problems with rough boundary conditions
• H^∞-calculus for the Dirichlet Laplacian on conical domains

Related

• The heat equation with rough boundary conditions and holomorphic\n functional calculus
• Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space
• Trace theory for parabolic boundary value problems with rough boundary conditions
• H^∞-calculus for the Dirichlet Laplacian on conical domains
• Regularity and well posedness for the Laplace operator on polyhedral domains