Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
N. Lindemulder (TU Delft - Analysis)
E. Lorist (TU Delft - Analysis)
F.B. Roodenburg (TU Delft - Analysis)
M.C. Veraar (TU Delft - Analysis)
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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∞-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded C1,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.
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