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

Nick Lindemulder, Emiel Lorist, Floris Roodenburg, Mark Veraar

TL;DR

The paper develops a robust framework for the Laplacian on rough bounded domains by proving a bounded H∞-calculus on inhomogeneous weighted Sobolev spaces with weights given by the distance to the boundary. A key methodological advance is a careful perturbation of the half-space calculus via Dahlberg–Kenig–Stein type pullbacks, combined with a localisation scheme that handles minimal boundary regularity. The main contributions include fractional-domain characterisations for Dirichlet and Neumann Laplacians on the half-space, the extension of the H∞-calculus to special and bounded domains, and the derivation of maximal Lq-regularity and Riesz transform bounds in the weighted setting. The results significantly extend well-posedness and regularity theory for parabolic problems to rough domains and provide a solid foundation for related SPDE analyses in weighted spaces, including vector-valued settings under the UMD assumption.

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.

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

TL;DR

The paper develops a robust framework for the Laplacian on rough bounded domains by proving a bounded H∞-calculus on inhomogeneous weighted Sobolev spaces with weights given by the distance to the boundary. A key methodological advance is a careful perturbation of the half-space calculus via Dahlberg–Kenig–Stein type pullbacks, combined with a localisation scheme that handles minimal boundary regularity. The main contributions include fractional-domain characterisations for Dirichlet and Neumann Laplacians on the half-space, the extension of the H∞-calculus to special and bounded domains, and the derivation of maximal Lq-regularity and Riesz transform bounds in the weighted setting. The results significantly extend well-posedness and regularity theory for parabolic problems to rough domains and provide a solid foundation for related SPDE analyses in weighted spaces, including vector-valued settings under the UMD assumption.

Abstract

We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded -functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded -domains with , 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.

Paper Structure

This paper contains 27 sections, 45 theorems, 238 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Holomorphic functional calculus
  5. Definitions
  6. Fractional domains
  7. Perturbation of the holomorphic functional calculus
  8. The UMD property
  9. Domains
  10. Weighted Sobolev spaces and trace characterisations
  11. Trace characterisations for weighted Sobolev spaces on the half-space
  12. Trace characterisations for weighted Sobolev spaces on special domains
  13. Trace characterisations for weighted Sobolev spaces on bounded domains
  14. Complex interpolation of weighted Sobolev spaces
  15. Fractional domains of the Laplacian on the half-space
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $p\in(1,\infty)$, $k\in\mathbb{N}_0$, $\lambda\in [0,1]$ and $\gamma\in (-1, 2p-1)\setminus\{p-1\}$. Furthermore, suppose that and $\mathcal{O}$ is a bounded $C^{1,\lambda}$-domain. Then for all $\mu\geq 0$ the operator has a bounded $H^{\infty}$-calculus of angle zero.

Figures (1)

  • Figure 1: The spaces $W^{k,p}(\mathcal{O}, w^{\partial\mathcal{O}}_{\alpha})$ where $\mu-\Delta_{\operatorname{Dir}}$ and $\mu-\Delta_{\operatorname{Neu}}$ as in Theorems \ref{['thm:introDir']} and \ref{['thm:introNeu']} (with $\alpha=\gamma+kp$ and $\alpha=\gamma+(k-1)p$, respectively) admit a bounded $H^{\infty}$-calculus.

Theorems & Definitions (94)

  • Theorem 1.1: $H^{\infty}$-calculus for the Dirichlet Laplacian
  • Theorem 1.2: $H^{\infty}$-calculus for the Neumann Laplacian
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: HNVW24
  • Proposition 2.4: HNVW24
  • Theorem 2.5: HNVW24
  • Theorem 2.6: HNVW24
  • Remark 2.7
  • Definition 2.8
  • ...and 84 more