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$H^\infty$-calculus for the Dirichlet Laplacian on conical domains

Petru A. Cioica-Licht, Emiel Lorist, P. Tobias Werner

TL;DR

This work develops an operator-theoretic framework for the Dirichlet Laplacian on conical domains with mixed distance-based weights, proving a bounded $H^0\infty$-calculus of angle $0$ on appropriate weighted $L^p$-spaces and extending these results to wedges. The authors achieve this by extrapolating from the unweighted $L^2$ setting using refined Green function bounds tailored to cones, after conjugation to Gaussian-type bounds with respect to a weighted doubling measure. A precise description of the homogeneous domain is obtained via a cylindrical transformation and a sum-operator approach that reduces the elliptic problem on the cone to a tangential Laplace–Beltrami problem on the cross-section. As a key corollary, maximal $L^p$-regularity for the Poisson equation in Krylov–Kondratiev spaces is established, enabling sharp time-space regularity results with potential applications to SPDEs on non-smooth domains.

Abstract

We establish boundedness of the $H^\infty$-calculus for the Dirichlet Laplacian on conical domains in $\mathbb{R}^d$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip/edge of the cone/wedge. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on conical domains in appropriate weighted Sobolev spaces.

$H^\infty$-calculus for the Dirichlet Laplacian on conical domains

TL;DR

This work develops an operator-theoretic framework for the Dirichlet Laplacian on conical domains with mixed distance-based weights, proving a bounded -calculus of angle on appropriate weighted -spaces and extending these results to wedges. The authors achieve this by extrapolating from the unweighted setting using refined Green function bounds tailored to cones, after conjugation to Gaussian-type bounds with respect to a weighted doubling measure. A precise description of the homogeneous domain is obtained via a cylindrical transformation and a sum-operator approach that reduces the elliptic problem on the cone to a tangential Laplace–Beltrami problem on the cross-section. As a key corollary, maximal -regularity for the Poisson equation in Krylov–Kondratiev spaces is established, enabling sharp time-space regularity results with potential applications to SPDEs on non-smooth domains.

Abstract

We establish boundedness of the -calculus for the Dirichlet Laplacian on conical domains in and corresponding wedges on -spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip/edge of the cone/wedge. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal -regularity for the Poisson equation on conical domains in appropriate weighted Sobolev spaces.

Paper Structure

This paper contains 21 sections, 26 theorems, 205 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear operators and holomorphic functional calculus
  4. Spaces of homogeneous type, Muckenhoupt weights and extrapolation
  5. The underlying domain D
  6. Weighted Sobolev spaces
  7. Uniformly weighted Sobolev spaces
  8. Kondratiev-type weighted Sobolev spaces
  9. Krylov-Kondratiev type weighted Sobolev spaces
  10. The Dirichlet Laplacian on conical domains
  11. Boundedness of the H-calculus in unweighted L-spaces
  12. Extrapolation of the functional calculus
  13. The homogeneous Dirichlet Laplacian
  14. Proof of the main theorem
  15. Sharpness: The necessity of singular weights
  16. ...and 6 more sections

Key Result

Theorem A

Let $\mathcal{D}:=\mathcal{D}_\Omega$ be a cone of the form eq:D:intro with a domain $\Omega$ admitting a $C^2$-boundary and such that $\overline{\Omega}\subsetneq S^{d-1}$. Let $(\lambda_n)_{n\in\mathbb{N}}\subseteq (0,\infty)$ be the ordered sequence of eigenvalues of the Dirichlet Laplace--Beltra Let $1<p<\infty$ and let $\gamma,\nu\in\mathbb{R}$ be such that Define the Dirichlet Laplacian $\D

Figures (3)

  • Figure 1: The stereographic projection in $d=2$ with $\mathfrak{p}=(0,-1)$.
  • Figure 2: Illustration of $\mathcal{D},\widetilde{\mathcal{D}},$ and $\widehat{\mathcal{D}}$ with $d=2$.
  • Figure 3: Illustration of Lemma \ref{['lem:doubling']}\ref{['it:ball:windshield']} for $d=2$.

Theorems & Definitions (59)

  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 49 more