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On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains
Martin Costabel, Alan McIntosh
TL;DR
The paper addresses regularity and cohomology for de Rham complexes by studying regularized Poincaré-type and Bogovskiĭ-type integral operators on differential forms. It proves pseudodifferential and function-space mapping properties, then applies them on starlike and bounded Lipschitz domains to obtain regularity results and smooth cohomology representatives.
Problem
The paper studies regularity and cohomology of de Rham complexes on starlike and bounded Lipschitz domains using integral operators acting on differential forms.
Method
The authors analyze regularized Poincaré-type and Bogovskiĭ-type operators through pseudodifferential calculus, continuity estimates, support properties, and finite coverings by starlike domains.
Results
The operators regularize differential forms by one degree of smoothness across Sobolev, Besov, and Triebel–Lizorkin scales, supporting de Rham regularity on bounded Lipschitz domains.
Takeaways & Limitations
The resulting cohomology spaces can be represented by finite-dimensional spaces of C ∞ differential forms independently of the regularity index s.
Takeaways & Limitations
Some continuity conclusions require domains starlike with respect to a ball, and the integral representation does not immediately show that smooth inputs produce smooth outputs.
Abstract
from arXiv · showhide
We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiĭ's integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincaré-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincaré-type operators) and with full Dirichlet boundary conditions (using Bogovskiĭ-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.
1 Introduction
The paper develops regularized Poincaré-type and Bogovskiĭ-type integral operators for differential forms, establishing their mapping properties and consequences for de Rham complexes on starlike and Lipschitz domains.
- Background: Bogovskiĭ’s operator solves div u = f under the zero-integral condition and preserves Sobolev regularity and support on suitable domains.Its solvability extends to large classes of domains, including bounded Lipschitz domains, with no loss of regularity.
- Operator properties: The regularized Poincaré operator preserves polynomial coefficients and local domain of influence while mapping W s,p(Ω) boundedly into W s+1,p(Ω).The result holds for all s ∈ R and 1 < p < ∞ when Ω is starlike with respect to a ball.
- Operator properties: The Bogovskiĭ-type and regularized Poincaré-type operators are classical pseudodifferential operators of order −1.Their symbols lie in the Hörmander class S−1 1,0(Rn), yielding boundedness across Hölder, Hardy, Sobolev, Besov, and Triebel–Lizorkin spaces.
- De Rham complexes: The operators provide regularity results for the exterior derivative on bounded Lipschitz domains with compact-support or no-boundary-condition spaces, without restricting s.The associated cohomology spaces admit finite-dimensional representatives with C ∞ coefficients independent of the regularity index.
- Bounded Lipschitz domains: Finite coverings by domains starlike with respect to balls extend the operator-based regularity analysis from local geometric settings to bounded Lipschitz domains.This construction also yields finite-dimensional spaces independent of the degree of regularity.
2 Notation and definitions
The paper sets notation for function spaces, differential forms, exterior differentiation, de Rham complexes, and Lipschitz or starlike domains. It also records the compact-support and extended complexes used later to analyze exactness and cohomology.
- Domains: A bounded Lipschitz domain is locally described below graphs of Lipschitz functions, while starlikeness with respect to a set requires each point’s convex hull with that set to remain inside the domain.Every bounded Lipschitz domain is described as a finite union of domains starlike with respect to open balls.
- Function spaces: The paper uses restrictions of Sobolev-type distributions to Ω and distributions on R^n with support in Ω as distinct function-space settings.For bounded Lipschitz domains, these spaces have standard relationships with smooth functions, extensions, closures, and duality.
- De Rham complexes: The exterior derivative satisfies d∘d=0 and generates de Rham complexes with and without boundary conditions, including compact-support and extended variants.The extended complex begins with constant functions, while compact-support formulations use the corresponding integral functional at the opposite endpoint.
- Exactness and cohomology: For bounded Lipschitz domains, the extended complexes are exact for starlike domains and the ordinary complexes have finite-dimensional cohomology independent of the regularity index.The stated result applies to both the compact-support and no-boundary-condition settings as specified in the paper.
- Algebraic operations: The exterior algebra operations used on forms include the Euclidean inner product, Hodge star, exterior product, and contraction with a vector.In R^3, these operations correspond to familiar scalar, dot-product, and cross-product identities.
3 The Bogovski˘ı and Poincar´e integral operators
The paper develops regularized Poincaré- and Bogovskiĭ-type integral operators on differential forms and analyzes their algebraic, support, smoothness, and continuity properties. The operators are pseudodifferential of order −1 and yield regularity mappings across broad function spaces on suitable domains.
- Operator properties: The regularized Poincaré operator maps polynomial-coefficient forms to polynomial-coefficient forms and depends only on the starlike hull of the support region and evaluation point.On domains starlike with respect to the ball B, this gives a local domain-of-influence property.
- Support and smoothness: Support properties imply that compactly supported smooth inputs produce compactly supported smooth outputs in the relevant starlike settings, while Bogovskiĭ-type operators require separate care for smoothness.The Bogovskiĭ representation involves an unbounded integration interval, so smoothness is not immediate from the formula alone.
- Homotopy relations: The operators satisfy homotopy or anticommutation relations with the exterior derivative, established directly for Poincaré-type operators and by duality for Bogovskiĭ-type operators.The relations remain valid across the broader function-space settings considered in the paper.
- Pseudodifferential structure: The operators R_ℓ and T_ℓ are pseudodifferential operators on R^n of order −1 with symbols in the Hörmander class S^−1.This analytic characterization supplies the continuity results used for Sobolev, Besov, and Triebel–Lizorkin spaces.
- Continuity: For a bounded domain starlike with respect to a ball containing supp θ, the operators define bounded mappings that raise regularity by one for every s∈R.The Sobolev result extends to Besov and Triebel–Lizorkin scales, with the stated parameter ranges.
4 Regularity of the de Rham complex
The paper constructs Poincaré- and Bogovskiĭ-type operators that regularize differential forms and uses them to establish de Rham regularity on starlike and bounded Lipschitz domains. It also shows polynomial preservation and finite-dimensional, smooth representatives for the resulting cohomology spaces.
- 4.1 Starlike domains: For domains starlike with respect to a ball, closed differential forms admit primitives with one additional Sobolev derivative for every s ∈R.The Poincaré- and Bogovskiĭ-type operators provide the primitives in the two boundary-condition settings.
- 4.2 Differential forms with polynomial coefficients: The Poincaré-type operator Rℓ maps polynomial differential forms to polynomial differential forms while retaining the stated domain and support framework.This property motivates finite-dimensional polynomial subcomplexes for finite element analysis.
- 4.3 Bounded Lipschitz domains: The operators Rℓ and Tℓ are pseudodifferential operators of order −1, while Kℓ and Lℓ are infinitely smoothing operators on Rn.The homotopy identities are dRℓu + Rℓ+1du = u − Kℓu and dTℓu + Tℓ+1du = u − Lℓu.
- 4.3 Bounded Lipschitz domains: On bounded Lipschitz domains, closed forms in Hs decompose as u = dv + w with v ∈ Hs+1 and a finite-dimensional cohomology component w.The estimate ∥v∥Hs+1 + ∥w∥Hs ≤ Cs∥u∥Hs holds, and the cohomology dimension is independent of s.
- 4.3 Bounded Lipschitz domains: The cohomology dimensions are bℓ without boundary conditions and ˜bℓ with compact support, and these spaces are isomorphic to spaces represented by C ∞ differential forms.The two Betti-number sequences can differ, as illustrated by the torus example.