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

TL;DR

This paper demonstrates that the Laplace operator on domains with minimal boundary smoothness admits a bounded functional calculus on weighted Sobolev spaces, enabling maximal regularity results for heat equations in rough domains.

Contribution

It establishes the bounded $H^{ abla}$-functional calculus for the Laplacian on weighted Sobolev spaces on minimally smooth domains, extending parabolic PDE theory.

Findings

Bounded $H^{ abla}$-functional calculus on weighted Sobolev spaces.

Maximal regularity for heat equations on rough domains.

Trade-off between domain regularity and weight exponent.

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.