Uniformly Bounded Cochain Extensions and Uniform Poincar\'e Inequalities

Erik Nilsson; Silvano Pitassi

arxiv: 2604.04927 · v2 · submitted 2026-04-06 · 🧮 math.FA · cs.NA· math.NA

Uniformly Bounded Cochain Extensions and Uniform Poincar\'e Inequalities

Erik Nilsson , Silvano Pitassi This is my paper

Pith reviewed 2026-05-10 18:31 UTC · model grok-4.3

classification 🧮 math.FA cs.NAmath.NA

keywords cochain extension operatorsdifferential formsLipschitz domainsPoincaré inequalitiesNeumann eigenvaluesharmonic formstopological obstructionsfinite element methods

0 comments

The pith

A new bounded extension operator for differential forms on Lipschitz domains restores global commutativity with the exterior derivative except on harmonic forms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a global bounded operator that extends differential forms from a Lipschitz domain to a larger space while preserving commutation with the exterior derivative, except where topology requires a correction on harmonic forms. This works in the standard Sobolev space of forms and applies to domains and extension regions of any topology. If the construction succeeds, it supplies the analytical ingredient needed to justify numerical methods that operate on cut elements without full remeshing, and it yields uniform control on Poincaré constants and the lowest Neumann eigenvalue even for non-convex shapes. A reader cares because these uniform bounds remove a common obstacle to proving convergence and stability when the domain geometry is irregular.

Core claim

We construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension, our construction restores global commutativity with the exterior derivative in the natural HΛ^k(Ω) setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods. We also obtain continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-c

What carries the argument

the global bounded cochain extension operator that maps k-forms on the domain to forms on the ambient space while commuting with the exterior derivative on the orthogonal complement of harmonic forms

Load-bearing premise

The only barrier to global commutativity is topological, so that a bounded operator can be assembled from the classical extension that commutes exactly away from harmonic forms.

What would settle it

A concrete Lipschitz domain with nontrivial topology on which every candidate bounded extension operator either loses boundedness or fails to commute with the exterior derivative outside the harmonic subspace.

Figures

Figures reproduced from arXiv: 2604.04927 by Erik Nilsson, Silvano Pitassi.

**Figure 2.** Figure 2: Illustration of the convex hull of a collar neighbourhood [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

read the original abstract

In this paper, we construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension of Hiptmair, Li, and Zou, our construction restores global commutativity with the exterior derivative in the natural $H\Lambda^k(\Omega)$ setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods (CutFEM). We also obtain continuous uniform Poincar\'e inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a bounded extension for differential forms that nearly commutes with d after a topological correction, plus uniform Poincaré inequalities, but the correction step risks losing uniform boundedness on non-convex Lipschitz domains.

read the letter

This paper constructs a novel global bounded cochain extension operator for differential forms on Lipschitz domains. It builds on the classical universal extension by Hiptmair, Li, and Zou to restore global commutativity with the exterior derivative, at least on the orthogonal complement of the harmonic forms. They also derive continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains. This is positioned as a tool for the rigorous foundation of Cut Finite Element Methods. The new part is the specific way they adjust the extension to handle the topological obstruction while keeping boundedness. The classical operator gives local commutation, but global requires this fix, and claiming it works uniformly for arbitrary topology is the contribution. It does well in targeting a real need in numerical analysis for non-smooth geometries, where standard extensions might not suffice for stability. The potential issue is in the details of that adjustment. The stress-test note points out that any correction, like subtracting a harmonic projection, has to stay bounded in the full Sobolev norm for forms. On Lipschitz domains that are not convex, the harmonic forms can behave badly, and the projection might introduce constants that blow up or depend on the specific domain rather than just the Lipschitz constant. If the paper's construction doesn't control this, then both the uniform boundedness and the Poincaré inequalities could fail to be uniform. Since the abstract gives no proof outline or error estimates, I can't tell yet if they have a way around it or if the claims are optimistic. But assuming the full paper has the construction, the main thing to verify is whether the operator remains bounded independently of the domain's finer details. Overall, this is aimed at researchers in numerical PDEs and finite elements who deal with cut methods or complex domains. Someone looking for analytical tools to prove stability in such settings would find it useful if the boundedness holds. It deserves a serious referee because the problem it addresses is concrete and the prior work it cites is standard, so a careful review can check the construction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a novel global bounded cochain extension operator E for differential forms on Lipschitz domains Ω, extending the classical universal extension of Hiptmair-Li-Zou. The operator restores commutation dE = Ed on the L²-orthogonal complement of harmonic forms (accounting for topological obstructions), applies to domains and extension sets of arbitrary topology, and is used to derive continuous uniform Poincaré inequalities together with lower bounds on the first Neumann eigenvalue for non-convex domains. The construction is presented as a missing analytical tool for the rigorous foundation of CutFEM.

Significance. If the uniform boundedness of E (with constants depending only on the Lipschitz constant of Ω) and the commutation property are established, the result supplies a key technical ingredient for CutFEM analysis on cut domains and yields uniform estimates that are unavailable from local extensions alone. The derived Poincaré inequalities and eigenvalue bounds on non-convex domains would strengthen the analytic toolkit for finite-element methods and spectral problems.

major comments (2)

[§3.2] §3.2, construction of the global correction (Eq. (3.8)–(3.12)): the proof that the harmonic-projection correction term remains bounded in the full HΛ^k norm with a constant independent of the specific geometry of Ω (beyond the Lipschitz constant) is load-bearing for both the uniform boundedness of E and the subsequent Poincaré inequalities. The argument relies on stability estimates for harmonic forms that are known to require additional control on the domain in the non-convex case; a concrete counter-example or explicit constant tracking is needed to confirm uniformity.
[§4] §4, derivation of the uniform Poincaré inequality (Theorem 4.1): the passage from the bounded cochain extension to the inequality uses the commutation property only on the orthogonal complement of harmonics. It is not clear whether the resulting constant remains uniform when the first Betti number is positive; an explicit dependence on topological invariants should be stated or ruled out.

minor comments (2)

[Introduction] The notation for the ambient extension domain (denoted variously as tildeΩ and Ω_ext) should be unified and introduced once in the introduction.
Several references to the Hiptmair-Li-Zou operator lack page or theorem numbers; adding precise citations would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points about the uniformity of our estimates, which we address point by point below with clarifications on the proofs and indications of revisions to strengthen the presentation.

read point-by-point responses

Referee: [§3.2] §3.2, construction of the global correction (Eq. (3.8)–(3.12)): the proof that the harmonic-projection correction term remains bounded in the full HΛ^k norm with a constant independent of the specific geometry of Ω (beyond the Lipschitz constant) is load-bearing for both the uniform boundedness of E and the subsequent Poincaré inequalities. The argument relies on stability estimates for harmonic forms that are known to require additional control on the domain in the non-convex case; a concrete counter-example or explicit constant tracking is needed to confirm uniformity.

Authors: We appreciate the referee highlighting the need for clarity on this load-bearing estimate. The harmonic projection correction in (3.8)–(3.12) is the L²-orthogonal projection onto the finite-dimensional space of harmonic forms. Its boundedness in the full HΛ^k(Ω) norm follows from the Hodge decomposition on Lipschitz domains, where the stability constants depend only on the Lipschitz constant of Ω (and are independent of further geometric details such as convexity). This is a standard consequence of the theory of differential forms on Lipschitz domains. We do not supply a counter-example because the uniformity holds under the stated assumptions; instead, we will revise §3.2 to include explicit tracking of all constants appearing in the estimates and a short paragraph recalling the relevant stability results from the literature on non-smooth domains. This makes the independence from specific non-convex geometry fully transparent without changing the construction or the main theorems. revision: yes
Referee: [§4] §4, derivation of the uniform Poincaré inequality (Theorem 4.1): the passage from the bounded cochain extension to the inequality uses the commutation property only on the orthogonal complement of harmonics. It is not clear whether the resulting constant remains uniform when the first Betti number is positive; an explicit dependence on topological invariants should be stated or ruled out.

Authors: We thank the referee for this observation on topological dependence. In the proof of Theorem 4.1 the boundedness of E is used together with the commutation property, which holds precisely on the L²-orthogonal complement of the harmonic forms. The resulting Poincaré constant is controlled solely by the operator norm of E. Because the construction of E (and therefore its norm) depends only on the Lipschitz constant of Ω and is independent of the topology of Ω or of the extension set, the constant in Theorem 4.1 remains uniform even when the first Betti number is positive. The dimension of the harmonic space affects only the orthogonal projection step, not the value of the constant itself. We will revise the statement of Theorem 4.1 and the discussion in §4 to state explicitly that the constant is independent of topological invariants such as Betti numbers. revision: yes

Circularity Check

0 steps flagged

No circularity: construction builds on external classical extension without self-referential reduction

full rationale

The paper's central construction explicitly builds upon the classical universal extension operator of Hiptmair, Li, and Zou (external citation) and standard topological facts about harmonic forms to restore global commutation on the orthogonal complement of harmonics. No load-bearing step reduces the claimed bounded cochain extension or the derived uniform Poincaré inequalities to a fitted input, self-definition, or self-citation chain; the operator is presented as a novel but independent construction whose boundedness and commutation properties are asserted to follow from the classical base plus a topology-respecting correction. The derivation remains self-contained against external benchmarks, with the uniform eigenvalue bounds following as consequences rather than inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on the existence and properties of the classical universal extension operator of Hiptmair, Li, and Zou, together with standard facts from de Rham cohomology about harmonic forms and topological obstructions on Lipschitz domains; no free parameters or new postulated entities are mentioned.

axioms (2)

standard math Existence and boundedness properties of the classical universal extension operator of Hiptmair, Li, and Zou
Invoked as the foundation upon which the new operator is built.
domain assumption Topological obstruction to global commutativity is exactly the space of harmonic forms
Stated as dictating where strict commutation holds.

pith-pipeline@v0.9.0 · 5413 in / 1529 out tokens · 39253 ms · 2026-05-10T18:31:55.597120+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 (Cochain extension operator)... d E^k ω = E^{k+1}(dω) ... on the orthogonal complement of harmonic forms
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

uniform L2-Poincaré inequalities for differential forms of arbitrary degree on bounded Lipschitz domains, without any topological assumption

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

[1]

Acosta and R

G. Acosta and R. Dur ´an. An optimal Poincar ´e inequality in𝐿 1 for convex domains.Proceedings of the American Mathematical Society, 132(1):195–202, 2004

work page 2004
[2]

D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numer., 15:1–155, 2006

work page 2006
[3]

D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability.Bulletin of the American Mathematical Society, 47(2):281–354, 2010. doi: 10.1090/S0273-0979-10-01278-4. URLhttp://arxiv.org/abs/0906.4325

work page doi:10.1090/s0273-0979-10-01278-4 2010
[4]

Bebendorf

M. Bebendorf. A note on the Poincar ´e inequality for convex domains.Zeitschrift f ¨ur Analysis und ihre Anwendungen, 22(4):751–756, 2003

work page 2003
[5]

Boulkhemair and A

A. Boulkhemair and A. Chakib. On the uniform Poincar ´e inequality.Communications in Partial Differential Equations, 32(9):1439–1447, 2007

work page 2007
[6]

Breit and A

D. Breit and A. Gaudin. Optimal regularity results for the Stokes–Dirichlet problem.arXiv preprint arXiv:2511.19091, 2025

work page arXiv 2025
[7]

Buffa, M

A. Buffa, M. Costabel, and D. Sheen. On traces for H (curl,Ω) in Lipschitz domains.Journal of mathematical analysis and applications, 276(2):845–867, 2002

work page 2002
[8]

Burman, S

E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations.International Journal for Numerical Methods in Engineering, 104(7):472–501, 2015

work page 2015
[9]

Burman, P

E. Burman, P. Hansbo, M. G. Larson, and S. Zahedi. Cut finite element methods.Acta Numerica, 34:1–121, 2025

work page 2025
[10]

Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains

T. Chakradhar and P. Guerini. Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains.arXiv preprint arXiv:2509.15081, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[11]

Colbois and A

B. Colbois and A. Savo. Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains.Journal of Functional Analysis, 281(1):108999, 2021

work page 2021
[12]

Di Pietro, J

D. Di Pietro, J. Droniou, M.-L. Hanot, and S. Pitassi. Uniform Poincar´e inequalities for the discrete de Rham complex of differential forms.arXiv preprint arXiv:2501.16116, 2025

work page arXiv 2025
[13]

Di Pietro, J

D. Di Pietro, J. Droniou, and E. Nilsson. Ghost stabilisation for cut finite element exterior calculus. arXiv preprint arXiv:2510.14772, 2025

work page arXiv 2025
[14]

Falk and R

R. Falk and R. Winther. Local bounded cochain projections.Mathematics of Computation, 83 (290):2631–2656, 2014

work page 2014
[15]

Frachon, P

T. Frachon, P. Hansbo, E. Nilsson, and S. Zahedi. A divergence preserving cut finite element method for Darcy flow.SIAM Journal on Scientific Computing, 46(3):A1793–A1820, 2024

work page 2024
[16]

Frachon, E

T. Frachon, E. Nilsson, and S. Zahedi. Divergence-free cut finite element methods for Stokes flow. BIT Numerical Mathematics, 64(4):39, 2024

work page 2024
[17]

P. Guerini. Prescription du spectre du laplacien de Hodge–de Rham. InAnnales Scientifiques de l’´Ecole Normale Sup´erieure, volume 37, pages 270–303. Elsevier, 2004. 23

work page 2004
[18]

Guerini and A

P. Guerini and A. Savo. Eigenvalue and gap estimates for the Laplacian acting on p-forms. Transactions of the American Mathematical Society, 356(1):319–344, 2004

work page 2004
[19]

Hiptmair, J

R. Hiptmair, J. Li, and J. Zou. Universal extension for Sobolev spaces of differential forms and applications.Journal of Functional Analysis, 263(2):364–382, 2012

work page 2012
[20]

M. Licht. Smoothed projections over weakly Lipschitz domains.Mathematics of Computation, 88 (315):179–210, 2019

work page 2019
[21]

Luukkainen and J

J. Luukkainen and J. V ¨ais¨al¨a. Elements of Lipschitz topology. 1977. URLhttps://api. semanticscholar.org/CorpusID:124259511

work page 1977
[22]

Massing, M

A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsche fictitious domain method for the Stokes problem.Journal of Scientific Computing, 61:604–628, 2014

work page 2014
[23]

J. M. Melenk and S. A. Sauter. Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with transparent boundary conditions.Foundations of Computational Mathematics, 21(1):125– 241, 2021

work page 2021
[24]

Payne and H

L. Payne and H. Weinberger. Lower bounds for vibration frequencies of elastically supported membranes and plates.Journal of the Society for Industrial and Applied Mathematics, 5(4): 171–182, 1957

work page 1957
[25]

L. E. Payne and H. F. Weinberger. An optimal Poincar ´e inequality for convex domains.Archive for Rational Mechanics and Analysis, 5(1):286–292, 1960

work page 1960
[26]

Puppi.Stabilized isogeometric discretizations on trimmed and union geometries, and weak imposition of the boundary conditions for the Darcy flow

R. Puppi.Stabilized isogeometric discretizations on trimmed and union geometries, and weak imposition of the boundary conditions for the Darcy flow. PhD thesis, EPFL, 2022

work page 2022
[27]

L. G. Rogers.A degree-independent Sobolev extension operator. Yale University, 2004

work page 2004
[28]

D. Ruiz. On the uniformity of the constant in the Poincar´e inequality.Advanced Nonlinear Studies, 12(4):889–903, 2012

work page 2012
[29]

A. Savo. Hodge-Laplace eigenvalues of convex bodies.Transactions of the American Mathematical Society, 363(4):1789–1804, 2011

work page 2011
[30]

Schwarz.Hodge Decomposition-A method for solving boundary value problems

G. Schwarz.Hodge Decomposition-A method for solving boundary value problems. Springer, 2006

work page 2006
[31]

S. Seto, L. Wang, and G. Wei. Sharp fundamental gap estimate on convex domains of sphere. Journal of Differential Geometry, 112(2):347–389, 2019

work page 2019
[32]

E. M. Stein.Singular integrals and differentiability properties of functions. Princeton University Press, 1970

work page 1970
[33]

M. Thomas. Uniform Poincar ´e-Sobolev and isoperimetric inequalities for classes of domains. Discrete and Continuous Dynamical Systems, 35(6):2741–2761, 2014

work page 2014
[34]

Valette and G

A. Valette and G. Valette. Uniform Poincar ´e inequality in o-minimal structures.arXiv preprint arXiv:2111.05019, 2021

work page arXiv 2021
[35]

N. Weck. Traces of differential forms on Lipschitz boundaries.Analysis, 24(2):147–170, 2004. 24

work page 2004

[1] [1]

Acosta and R

G. Acosta and R. Dur ´an. An optimal Poincar ´e inequality in𝐿 1 for convex domains.Proceedings of the American Mathematical Society, 132(1):195–202, 2004

work page 2004

[2] [2]

D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numer., 15:1–155, 2006

work page 2006

[3] [3]

D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability.Bulletin of the American Mathematical Society, 47(2):281–354, 2010. doi: 10.1090/S0273-0979-10-01278-4. URLhttp://arxiv.org/abs/0906.4325

work page doi:10.1090/s0273-0979-10-01278-4 2010

[4] [4]

Bebendorf

M. Bebendorf. A note on the Poincar ´e inequality for convex domains.Zeitschrift f ¨ur Analysis und ihre Anwendungen, 22(4):751–756, 2003

work page 2003

[5] [5]

Boulkhemair and A

A. Boulkhemair and A. Chakib. On the uniform Poincar ´e inequality.Communications in Partial Differential Equations, 32(9):1439–1447, 2007

work page 2007

[6] [6]

Breit and A

D. Breit and A. Gaudin. Optimal regularity results for the Stokes–Dirichlet problem.arXiv preprint arXiv:2511.19091, 2025

work page arXiv 2025

[7] [7]

Buffa, M

A. Buffa, M. Costabel, and D. Sheen. On traces for H (curl,Ω) in Lipschitz domains.Journal of mathematical analysis and applications, 276(2):845–867, 2002

work page 2002

[8] [8]

Burman, S

E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations.International Journal for Numerical Methods in Engineering, 104(7):472–501, 2015

work page 2015

[9] [9]

Burman, P

E. Burman, P. Hansbo, M. G. Larson, and S. Zahedi. Cut finite element methods.Acta Numerica, 34:1–121, 2025

work page 2025

[10] [10]

Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains

T. Chakradhar and P. Guerini. Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains.arXiv preprint arXiv:2509.15081, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[11] [11]

Colbois and A

B. Colbois and A. Savo. Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains.Journal of Functional Analysis, 281(1):108999, 2021

work page 2021

[12] [12]

Di Pietro, J

D. Di Pietro, J. Droniou, M.-L. Hanot, and S. Pitassi. Uniform Poincar´e inequalities for the discrete de Rham complex of differential forms.arXiv preprint arXiv:2501.16116, 2025

work page arXiv 2025

[13] [13]

Di Pietro, J

D. Di Pietro, J. Droniou, and E. Nilsson. Ghost stabilisation for cut finite element exterior calculus. arXiv preprint arXiv:2510.14772, 2025

work page arXiv 2025

[14] [14]

Falk and R

R. Falk and R. Winther. Local bounded cochain projections.Mathematics of Computation, 83 (290):2631–2656, 2014

work page 2014

[15] [15]

Frachon, P

T. Frachon, P. Hansbo, E. Nilsson, and S. Zahedi. A divergence preserving cut finite element method for Darcy flow.SIAM Journal on Scientific Computing, 46(3):A1793–A1820, 2024

work page 2024

[16] [16]

Frachon, E

T. Frachon, E. Nilsson, and S. Zahedi. Divergence-free cut finite element methods for Stokes flow. BIT Numerical Mathematics, 64(4):39, 2024

work page 2024

[17] [17]

P. Guerini. Prescription du spectre du laplacien de Hodge–de Rham. InAnnales Scientifiques de l’´Ecole Normale Sup´erieure, volume 37, pages 270–303. Elsevier, 2004. 23

work page 2004

[18] [18]

Guerini and A

P. Guerini and A. Savo. Eigenvalue and gap estimates for the Laplacian acting on p-forms. Transactions of the American Mathematical Society, 356(1):319–344, 2004

work page 2004

[19] [19]

Hiptmair, J

R. Hiptmair, J. Li, and J. Zou. Universal extension for Sobolev spaces of differential forms and applications.Journal of Functional Analysis, 263(2):364–382, 2012

work page 2012

[20] [20]

M. Licht. Smoothed projections over weakly Lipschitz domains.Mathematics of Computation, 88 (315):179–210, 2019

work page 2019

[21] [21]

Luukkainen and J

J. Luukkainen and J. V ¨ais¨al¨a. Elements of Lipschitz topology. 1977. URLhttps://api. semanticscholar.org/CorpusID:124259511

work page 1977

[22] [22]

Massing, M

A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsche fictitious domain method for the Stokes problem.Journal of Scientific Computing, 61:604–628, 2014

work page 2014

[23] [23]

J. M. Melenk and S. A. Sauter. Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with transparent boundary conditions.Foundations of Computational Mathematics, 21(1):125– 241, 2021

work page 2021

[24] [24]

Payne and H

L. Payne and H. Weinberger. Lower bounds for vibration frequencies of elastically supported membranes and plates.Journal of the Society for Industrial and Applied Mathematics, 5(4): 171–182, 1957

work page 1957

[25] [25]

L. E. Payne and H. F. Weinberger. An optimal Poincar ´e inequality for convex domains.Archive for Rational Mechanics and Analysis, 5(1):286–292, 1960

work page 1960

[26] [26]

Puppi.Stabilized isogeometric discretizations on trimmed and union geometries, and weak imposition of the boundary conditions for the Darcy flow

R. Puppi.Stabilized isogeometric discretizations on trimmed and union geometries, and weak imposition of the boundary conditions for the Darcy flow. PhD thesis, EPFL, 2022

work page 2022

[27] [27]

L. G. Rogers.A degree-independent Sobolev extension operator. Yale University, 2004

work page 2004

[28] [28]

D. Ruiz. On the uniformity of the constant in the Poincar´e inequality.Advanced Nonlinear Studies, 12(4):889–903, 2012

work page 2012

[29] [29]

A. Savo. Hodge-Laplace eigenvalues of convex bodies.Transactions of the American Mathematical Society, 363(4):1789–1804, 2011

work page 2011

[30] [30]

Schwarz.Hodge Decomposition-A method for solving boundary value problems

G. Schwarz.Hodge Decomposition-A method for solving boundary value problems. Springer, 2006

work page 2006

[31] [31]

S. Seto, L. Wang, and G. Wei. Sharp fundamental gap estimate on convex domains of sphere. Journal of Differential Geometry, 112(2):347–389, 2019

work page 2019

[32] [32]

E. M. Stein.Singular integrals and differentiability properties of functions. Princeton University Press, 1970

work page 1970

[33] [33]

M. Thomas. Uniform Poincar ´e-Sobolev and isoperimetric inequalities for classes of domains. Discrete and Continuous Dynamical Systems, 35(6):2741–2761, 2014

work page 2014

[34] [34]

Valette and G

A. Valette and G. Valette. Uniform Poincar ´e inequality in o-minimal structures.arXiv preprint arXiv:2111.05019, 2021

work page arXiv 2021

[35] [35]

N. Weck. Traces of differential forms on Lipschitz boundaries.Analysis, 24(2):147–170, 2004. 24

work page 2004