pith. sign in

arxiv: 2605.22178 · v1 · pith:DXPDUGSKnew · submitted 2026-05-21 · 🧮 math.AP

Gradient estimates for pleft(cdotright)-harmonic differential forms

Anna Balci , Swarnendu Sil , Mikhail Surnachev This is my paper

Pith reviewed 2026-05-22 04:42 UTC · model grok-4.3

classification 🧮 math.AP
keywords p(·)-harmonic formsgradient estimateshigher integrabilityMeyers theoremHölder continuityvariable exponentsCoulomb gaugedifferential forms
0
0 comments X

The pith

p(·)-harmonic differential forms satisfy gradient Hölder continuity under Hölder continuous exponents and higher integrability under log-Hölder continuity.

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

This paper establishes gradient estimates for solutions to equations involving p(·)-harmonic differential forms that satisfy a Coulomb-type gauge condition. For exponents p(·) that are log-Hölder continuous, the gradients enjoy higher integrability in the sense of Meyers' theorem, improving on the natural integrability from the energy space. When p(·) is Hölder continuous, the gradients are themselves Hölder continuous. A sympathetic reader would care because these results allow the classical regularity theory for constant p to carry over to variable exponents, which is needed to model materials whose properties vary in space.

Core claim

For p(·)-harmonic differential forms subject to a Coulomb-type gauge condition, higher integrability estimates of Meyers type hold for the gradient when the variable exponent satisfies the log-Hölder continuity assumption. Under the stronger assumption that the exponent function is Hölder continuous, the gradient of the solutions is Hölder continuous.

What carries the argument

The p(·)-harmonic system for differential forms equipped with the Coulomb gauge condition, which enables the application of variable exponent techniques to obtain the regularity improvements.

If this is right

  • The results extend the classical regularity theory for constant-exponent p-harmonic systems to the variable-exponent setting.
  • These estimates are essential for modeling nonhomogeneous and anisotropic media.
  • Higher integrability holds beyond the natural energy space under log-Hölder continuity.
  • Gradient Hölder continuity follows from Hölder continuity of p(·).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the techniques generalize, similar higher integrability might hold for other classes of variable growth problems in PDEs.
  • The Coulomb gauge condition appears crucial for controlling the forms and could be tested in related geometric settings.
  • These estimates might inform the design of numerical methods for variable exponent problems by guaranteeing improved regularity.

Load-bearing premise

The variable exponent p(·) satisfies the log-Hölder continuity assumption, or the stronger Hölder continuity for proving gradient Hölder continuity.

What would settle it

Constructing a p(·)-harmonic differential form with a non-log-Hölder continuous exponent where the gradient fails to have the expected higher integrability would falsify the claim.

read the original abstract

In this paper, we establish gradient bounds for $p(\cdot)$-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-H\"older continuity assumption, we derive higher integrability estimates of Meyers type, ensuring improved regularity beyond the natural energy space. Furthermore, under the stronger assumption of H\"older continuity of the exponent function, we prove that the gradient of solutions exhibits H\"older continuity. These results extend classical regularity theory for constant-exponent $p$-harmonic systems to the variable-exponent setting, which is essential for modeling nonhomogeneous and anisotropic media.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript establishes gradient bounds for p(·)-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-Hölder continuity assumption, higher integrability estimates of Meyers type are derived, improving regularity beyond the natural energy space. Under the stronger assumption of Hölder continuity of the exponent function, the gradient of solutions is shown to be Hölder continuous. The work extends classical regularity theory for constant-exponent p-harmonic systems to the variable-exponent setting for modeling nonhomogeneous and anisotropic media.

Significance. If the central claims hold, the results would provide a useful extension of Meyers-type higher integrability and Schauder-type regularity to the variable-exponent setting for differential forms. The adaptation of reverse-Hölder inequalities and Gehring iteration, with explicit tracking of the continuity modulus of p(·), follows established techniques in the literature on variable-exponent problems and could support applications in inhomogeneous media.

major comments (1)
  1. §3 (higher integrability theorem): the statement should make explicit the dependence of the improved integrability exponent q on the log-Hölder constant of p(·) and the dimension; without this, the claim that the estimates are 'parameter-free' in the abstract is difficult to verify.
minor comments (3)
  1. §2.1: the definition of the Coulomb gauge condition for differential forms should include a brief remark on its uniqueness up to harmonic forms to clarify the setting.
  2. Introduction: add a sentence comparing the obtained Hölder exponent with the corresponding result for constant p to highlight the new dependence on the modulus of continuity of p(·).
  3. Notation: ensure consistent use of the Hodge star operator and exterior derivative throughout; a short table of symbols in §2 would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the higher integrability result. We address the point below and will incorporate the suggested clarification.

read point-by-point responses

  1. Referee: §3 (higher integrability theorem): the statement should make explicit the dependence of the improved integrability exponent q on the log-Hölder constant of p(·) and the dimension; without this, the claim that the estimates are 'parameter-free' in the abstract is difficult to verify.

    Authors: We agree that the dependence should be stated explicitly. In Theorem 3.1 the improved exponent q > 2 is determined by the dimension n, the log-Hölder constant of p(·), the bounds on p(·), and the structural constants appearing in the ellipticity and growth conditions of the p(·)-harmonic system. We will revise the theorem statement to record this dependence explicitly (including the explicit functional dependence on the log-Hölder modulus). The phrase 'parameter-free' in the abstract was meant to indicate that the estimates do not depend on the particular solution or on the size of the domain once the structural data are fixed; however, to prevent misunderstanding we will remove the phrase or replace it with a clearer formulation such as 'uniform higher integrability'. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adapts standard methods independently

full rationale

The paper's central claims rely on adapting classical reverse-Hölder inequalities, Gehring lemmas, and Schauder estimates to the variable-exponent setting for differential forms under a Coulomb gauge. The log-Hölder continuity assumption on p(·) is invoked only to secure the modular inequalities needed for Meyers-type higher integrability, while the stronger Hölder assumption is used solely for the final gradient Hölder continuity step; both track the continuity modulus in a manner consistent with prior literature on p(·)-Laplace systems and do not reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain remains self-contained against external benchmarks in variable-exponent Sobolev theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard functional-analytic assumptions for variable-exponent Sobolev spaces and the log-Hölder condition on p(·); no free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption Variable exponents p(·) are log-Hölder continuous
    Invoked to obtain Meyers-type higher integrability.
  • domain assumption Solutions satisfy a Coulomb-type gauge condition
    Used to control the differential forms.

pith-pipeline@v0.9.0 · 5624 in / 1145 out tokens · 29503 ms · 2026-05-22T04:42:25.177824+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    Russian J

    Zhikov, V.V.: On some variational problems. Russian J. Math. Phys.5(1), 105– 116 (1997)

    work page 1997

  2. [2]

    Zhikov, V.V.: Meyer-type estimates for solving the nonlinear Stokes system. Differ. Equ.33(1), 108–115 (1997)

    work page 1997

  3. [3]

    Differentsial’nye Uravneniya33(12), 1651–1660 (1997)

    Alkhutov, Y.A.: Harnack’s inequality and H¨ older continuity of solutions of non- linear elliptic equations with nonstandard growth conditions. Differentsial’nye Uravneniya33(12), 1651–1660 (1997)

    work page 1997

  4. [4]

    Acerbi, E., Mingione, G.: Regularity Results for a Class of Functionals with Non- Standard Growth. Arch. Rational Mech. Anal.156, 121–140 (2001)

    work page 2001

  5. [5]

    Acerbi, E., Mingione, G.: Gradient estimates for thep(x)-Laplacean system. J. Reine Angew. Math.584, 117–148 (2005)

    work page 2005

  6. [6]

    Acta Math

    Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(3-4), 219–240 (1977)

    work page 1977

  7. [7]

    Hamburger, C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math.431, 7–64 (1992)

    work page 1992

  8. [8]

    Springer, Berlin (1994)

    Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Oper- ators and Integral Functionals. Springer, Berlin (1994)

    work page 1994

  9. [9]

    Springer, Berlin (2000) 48

    R˚ uˇ ziˇ cka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000) 48

    work page 2000

  10. [10]

    Lecture Notes in Mathematics, vol

    Diening, L., Harjulehto, P., H¨ ast¨ o, P., R˚ uˇ ziˇ cka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)

    work page 2017

  11. [11]

    Applied and Numer- ical Harmonic Analysis

    Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Applied and Numer- ical Harmonic Analysis. Birkh¨ auser, Basel (2013). Foundations and harmonic analysis

    work page 2013

  12. [12]

    Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. (4)177, 37–115 (1999)

    work page 1999

  13. [13]

    PhD thesis, EPFL (2016)

    Sil, S.: Calculus of Variations for Differential Forms. PhD thesis, EPFL (2016). Thesis No. 7060

    work page 2016

  14. [14]

    Sil, S.: Calculus of variations: A differential form approach. Adv. Calc. Var.12(1), 57–84 (2019)

    work page 2019

  15. [15]

    Sil, S.: Nonlinear Stein theorem for differential forms. Calc. Var. Partial Differ- ential Equations58(4), 154 (2019)

    work page 2019

  16. [16]

    Discrete Contin

    Sengupta, B., Sil, S.: Morrey-Lorentz estimates for Hodge-type systems. Discrete Contin. Dyn. Syst.45(1), 334–360 (2025)

    work page 2025

  17. [17]

    Sil, S.: Regularity for elliptic systems of differential forms and applications. Calc. Var. Partial Differential Equations56(6), 56–172 (2017)

    work page 2017

  18. [18]

    Annales Henri Lebesgue7, 1457–1534 (2024) https://doi.org/10.5802/ahl.224

    Gaudin, A.: Hodge decompositions and maximal regularities for Hodge Laplacians in homogeneous function spaces on the half-space. Annales Henri Lebesgue7, 1457–1534 (2024) https://doi.org/10.5802/ahl.224

    work page doi:10.5802/ahl.224 2024

  19. [19]

    Preprint, arXiv:2511.19091

    Breit, D., Gaudin, A.: Optimal regularity results for the Stokes–Dirichlet problem. Preprint, arXiv:2511.19091. arXiv:2511.19091 [math.AP] (2025). https://arxiv. org/abs/2511.19091

    work page arXiv 2025

  20. [20]

    Calculus of Variations and Partial Differential Equations 63(3), 62–44 (2024)

    Balci, A.K., Surnachev, M.: The Lavrentiev phenomenon in calculus of variations with differential forms. Calculus of Variations and Partial Differential Equations 63(3), 62–44 (2024)

    work page 2024

  21. [21]

    Izvestiya: Mathematics (2026)

    Balci, A., Sil, S., Surnachev, M.: Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces. Izvestiya: Mathematics (2026). Preprint available at arXiv:2504.20772

    work page arXiv 2026

  22. [22]

    Yin, H.M.: On ap-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory. Q. Appl. Math.LIX, 47–66 (2001)

    work page 2001

  23. [23]

    The $p$-CurlCurl : Spaces, traces, coercivity and a Helmholtz decomposition in $L^p$

    Laforest, M.: Thep-CurlCurl: Spaces, Traces, Coercivity and a Helmholtz Decomposition inL p. arXiv preprint (2018) arXiv:1808.05976 49

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  24. [24]

    SIAM Journal on Numerical Analysis58(1), 460–491 (2020)

    Wan, A., Laforest, M.: A Posteriori Error Estimation for thep-Curl Problem. SIAM Journal on Numerical Analysis58(1), 460–491 (2020)

    work page 2020

  25. [25]

    Journal of Computational and Applied Mathematics334, 173–189 (2018)

    Choi, H., Kim, H.J., Laforest, M.: Relaxation Model for thep-Curl Problem with Stiffness. Journal of Computational and Applied Mathematics334, 173–189 (2018)

    work page 2018

  26. [26]

    Journal of Computational Physics378, 691–614 (2019)

    Law, Y.-M., Laforest, M.: A nonlinear relaxation formulation of the p-curl problem modelling high-temperature superconductors: A modified Yee’s scheme. Journal of Computational Physics378, 691–614 (2019)

    work page 2019

  27. [27]

    Progress in Nonlinear Differential Equations and their Applications, vol

    Csat´ o, G., Dacorogna, B., Kneuss, O.: The Pullback Equation for Differential Forms. Progress in Nonlinear Differential Equations and their Applications, vol

    work page

  28. [28]

    Birkh¨ auser/Springer, New York (2012)

    work page 2012

  29. [29]

    Charles B.: Multiple Integrals in the Calculus of Variations

    Morrey, J. Charles B.: Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, vol. 130. Springer, New York (1966)

    work page 1966

  30. [30]

    PhD thesis, EPFL (2012)

    Csat´ o, G.: Some Boundary Value Problems Involving Differential Forms. PhD thesis, EPFL (2012). Thesis No. 5414

    work page 2012

  31. [31]

    Russian Journal of Mathematical Physics3(2), 249–269 (1995)

    Zhikov, V.V.: On Lavrentiev Phenomenon. Russian Journal of Mathematical Physics3(2), 249–269 (1995)

    work page 1995

  32. [32]

    Bandyopadhyay, S., Dacorogna, B., Sil, S.: Calculus of variations with differential forms. J. Eur. Math. Soc. (JEMS)17(4), 1009–1039 (2015)

    work page 2015

  33. [33]

    Campanato, S.: Propriet` a di H¨ olderianit` a di alcune classi di funzioni. Ann. Sc. Norm. Sup. Pisa17, 175–188 (1963)

    work page 1963

  34. [34]

    Annals of Mathematics Studies, vol

    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)

    work page 1983

  35. [35]

    Giaquinta, M., Martinazzi, L.: An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edn. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 11. Edizioni della Normale, Pisa (2012)

    work page 2012

  36. [36]

    Archive for Rational Mechan- ics and Analysis39, 206–226 (1970)

    Kress, R.: Die Behandlung zweier Randwertprobleme f¨ ur die vektorielle Pois- songleichung nach einer Integralgleichungsmethode. Archive for Rational Mechan- ics and Analysis39, 206–226 (1970)

    work page 1970

  37. [37]

    Archive for Rational Mechanics and Analysis 47, 59–80 (1972) 50

    Kress, R.: Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebiger Dimension und beliebigen Ranges. Archive for Rational Mechanics and Analysis 47, 59–80 (1972) 50

    work page 1972

  38. [38]

    Bolik, J.: Zur L¨ osung potentialtheoretischer Randwertprobleme: A-priori- Absch¨ atzungen und Zerlegungss¨ atze f¨ ur Differentialformen. Ph.d. thesis, Univer- sit¨ at Bayreuth, Fakult¨ at f¨ ur Mathematik und Physik, Bayreuth (1996)

    work page 1996

  39. [39]

    Analysis17(2-3), 227–238 (1997)

    Bolik, J.: A priori estimates for differential forms with components inC 1,λ. Analysis17(2-3), 227–238 (1997)

    work page 1997

  40. [40]

    Weyl’s boundary value problems for differential forms

    Bolik, J.: H. Weyl’s boundary value problems for differential forms. Differential and Integral Equations14(8), 937–952 (2001)

    work page 2001

  41. [41]

    Analysis (Munich)24(2), 103–126 (2004)

    Bolik, J.: Boundary value problems for differential forms on compact Riemannian manifolds. Analysis (Munich)24(2), 103–126 (2004)

    work page 2004

  42. [42]

    Bolik, J.: Boundary value problems for differential forms on compact Riemannian manifolds. II. Analysis (Munich)27(4), 477–493 (2007)

    work page 2007

  43. [43]

    Diening, L., R˚ uˇ ziˇ cka, M.: Calderon-Zygmund operators on generalized Lebesgue spacesL p(·) and problems related to fluid dynamics. J. reine angew. Math.563, 197–220 (2003) https://doi.org/10.1515/crll.2003.081

    work page doi:10.1515/crll.2003.081 2003

  44. [44]

    Mathematische Zeitschrift 265, 297–320 (2010) https://doi.org/10.1007/s00209-009-0517-8

    Costabel, M., McIntosh, A.: On Bogovski˘ ı and regularized Poincar´ e integral oper- ators for de Rham complexes on Lipschitz domains. Mathematische Zeitschrift 265, 297–320 (2010) https://doi.org/10.1007/s00209-009-0517-8

    work page doi:10.1007/s00209-009-0517-8 2010

  45. [45]

    Communications on Pure and Applied Analysis7(6), 1295–1333 (2008) https: //doi.org/10.3934/cpaa.2008.7.1295

    Mitrea, D., Mitrea, M., Monniaux, S.: The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure and Applied Analysis7(6), 1295–1333 (2008) https: //doi.org/10.3934/cpaa.2008.7.1295

    work page doi:10.3934/cpaa.2008.7.1295 2008

  46. [46]

    Bojarski, B., Iwaniec, T.: Analytical foundations of the theory of quasiconformal mappings inR n. Ann. Acad. Sci. Fenn. Ser. A I Math.8(2), 257–324 (1983)

    work page 1983

  47. [47]

    In: Quasiconformal Mappings and Analysis, pp

    Iwaniec, T.: The Gehring Lemma. In: Quasiconformal Mappings and Analysis, pp. 181–204. Springer, New York (1998) 51

    work page 1998