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Gradient continuity for $p$-Laplacian obstacle problems under mean oscillation conditions

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Pith's one-line read Mean oscillation replaces Dini continuity for gradient regularity

desk verdict Genuine extension of C¹-regularity for p-Laplacian obstacle problems to DMO conditions; one gap in the 1<p<2 argument needs fixing read the letter →

arxiv 2607.07018 v1 pith:N22I5NR6 submitted 2026-07-08 math.AP

Sungjin Lee , Jihoon Ok This is my paper · ORCID
classification math.AP
keywords meanobstacleoscillationconditioncontinuitydatadinigradient
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the gradient of a solution to a p-Laplacian obstacle problem is continuous provided the data — the coefficient matrix, the source term, and the obstacle — satisfy a Dini mean oscillation condition rather than the classical Dini continuity condition. The obstacle problem is a variational inequality: one seeks a function u that stays above a given obstacle ψ and satisfies a nonlinear divergence-form equation involving a coefficient matrix A and a source F. The p-Laplacian structure, governed by the exponent 1 < p < ∞, makes the equation nonlinear in the gradient. The authors show that what matters for gradient continuity is not pointwise continuity of the data (the classical Dini condition) but rather that the mean oscillation of the data over small balls, raised to an appropriate power, is integrable in a Dini sense. The proof proceeds by comparing the solution u to auxiliary functions v solving frozen-coefficient equations on nested balls, deriving sharp L¹ comparison estimates that track the mean oscillation of the data, and then running an iteration that controls the excess decay of Du across scales. The key mechanism is that the DMO condition, combined with Calderón–Zygmund L^q estimates for q > p, provides enough control at each scale to close the iteration and force the gradient excess to zero.

What carries the argument

The argument uses a two-step comparison: first compare the solution u to an intermediate function w solving a frozen-coefficient obstacle problem, then compare w to a function v solving a homogeneous frozen-coefficient equation. The comparison estimates (Lemma 3.2, Corollary 3.3, Lemma 3.4) bound the L¹ difference of Du and Dv in terms of ω_data(r, x₀), the local mean oscillation of the data. The C^{1,α} excess decay estimate for the frozen-coefficient equation (Lemma 2.9) provides decay at each scale. An iteration (Lemma 4.2) then controls the excess E_j across dyadic scales, and the Dini condition on ω_data ensures the accumulated oscillation errors are summable, forcing Du to be locally L

What would settle it

A counterexample would be a p-Laplacian obstacle problem where the coefficient matrix, source, and obstacle satisfy the DMO condition of Condition 2.3 but the gradient Du fails to be continuous — for instance, if Du is locally unbounded at some point despite the Dini mean oscillation integrability. Alternatively, if the cited Calderón–Zygmund estimates or the excess decay estimate for the frozen-coefficient equation fail for obstacle problems with VMO coefficients, the main comparison estimates would not hold.

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Extended reading notes

Core claim

The gradient Du of a solution to the p-Laplacian obstacle problem is continuous whenever the coefficient matrix A is DMO_{2+κ} for some κ > 0, and the nonlinear quantities |Dψ|^{p-2}Dψ and F satisfy a (p'/2)-DMO_{p'} condition (for p > 2) or DMO_{2+κ} condition (for 1 < p ≤ 2), with the combined mean oscillation function ω_data satisfying a Dini integrability condition. This replaces the classical assumption of Dini continuity of the data with a strictly weaker mean oscillation condition, extending the linear theory (p = 2) to the nonlinear p-Laplacian setting with obstacles.

Load-bearing premise

The proof relies on Calderón–Zygmund L^q estimates for the variational inequality with VMO coefficients and on a recent C^{1,α} excess decay estimate for the frozen-coefficient p-Laplacian, both cited from external results. If either of these estimates does not hold under the stated conditions, the comparison and iteration arguments would not close.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 9 minor

Summary. This paper establishes C^1-regularity (gradient continuity) for solutions to obstacle problems associated with p-Laplacian type equations with a coefficient matrix A, source term F, and obstacle ψ. The main result (Theorem 1.1) replaces the classical Dini continuity assumption on the data with a Dini mean oscillation (DMO) type condition (Condition 2.3). The proof proceeds by deriving L^1-comparison estimates between the solution u and solutions to frozen-coefficient problems (Section 3), then using an iteration argument (Section 4) to obtain local boundedness of Du and finally continuity. The framework adapts techniques from [20] (for non-obstacle nonlinear equations) and [32] (for obstacle problems with Dini-continuous data) to the DMO setting.

Significance. The result is a genuine extension of known C^1-regularity theory: it weakens Dini continuity to a mean oscillation condition for obstacle problems, which is the natural borderline for gradient continuity. The adaptation of the sharp L^1-comparison and iteration framework from [20] to the variational inequality setting is non-trivial, as the obstacle introduces additional comparison terms (the I_2 estimate in Lemma 3.2). The result also extends the source-term regularity in [27] for 1<p≤2. The paper is well-motivated and the proof structure is clear.

major comments (1)
  1. Lemma 3.2, I_2 estimate for 1<p<2 (p. 10): The authors state 'we use the inequality 0<2-p<p and assume that (2-p)(2+κ)/κ > p without loss of generality.' This assumption is not WLOG. The exponent q=(2-p)(2+κ)/κ is already fixed in (3.1) and used in the Calderón–Zygmund estimate (2.9) with t=1/p. The condition q>p translates to κ < 2(2-p)/(2(p-1)). For p close to 2 (e.g., p=1.9), this requires κ < 0.111, while Condition 2.3 only requires κ>0 to be arbitrary. If data satisfies the Dini condition with some κ_0 ≥ 2(2-p)/(2(p-1)), the proof as written does not apply. The authors need to either (a) justify that κ can always be chosen small enough without loss of generality (e.g., by noting that DMO_{2+κ} implies DMO_{2+κ'} for κ'<κ, so one can always reduce κ), or (b) handle the case q≤p separately. Without this, the comparison estimate (3.9) and Corollary 3.3 may not hold for all admissible κ
minor comments (9)
  1. p. 3, line below (1.3): 'Yonung's inequality' should be 'Young's inequality'. The same typo appears in the proof of Lemma 3.1.
  2. p. 7, proof of Lemma 3.1: 'qusai-triangle' should be 'quasi-triangle'.
  3. p. 10, I_2 estimate for 1<p<2: The phrase 'without loss generality' is missing 'of'.
  4. Lemma 2.9 cites [1, Theorem 4.1], which is a very recent preprint. The authors should verify that the result in [1] applies exactly as stated (particularly the L^1 excess decay formulation in (2.12)) and clarify the dependency.
  5. Condition 2.3 (p. 4): The definition of ω_data uses different exponents for p>2 and 1<p≤2. A brief remark explaining why the exponent p'/2 appears for p>2 but 2+κ for 1<p≤2 would help readers understand the structure of the condition.
  6. Remark 2.4 (p. 5): The computation showing that θ_p-Dini continuity of Dψ implies p'/2-DMO_{p'} for p>2 is sketched but not fully detailed. Adding one or two lines of explanation would make the comparison with [32] more transparent.
  7. p. 12, Lemma 3.4: The statement says 'there exists C_3' but the proof uses C_3 in the final estimate without explicitly tracking its dependence. The dependence on n, p, Λ should be stated explicitly.
  8. Section 4, proof of Theorem 1.1: The 'Claim' on p. 18 and its proof involve a case analysis with four sub-cases (i)-(iv). The logic is correct but dense; a brief roadmap sentence before the case analysis would improve readability.
  9. p. 17, proof of Proposition 4.3: The definition of M involves δ^{-2n}/(3ε_0^3), and the choice ε=ε_0=2^{-n-2} is made. It would help to explicitly note that with this choice, the conditions (4.2)–(4.3) are satisfied, since this is used in the iteration.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; self-citations are methodological, not load-bearing

full rationale

The paper's central claim (Theorem 1.1: DMO conditions on data imply C^1-regularity for p-Laplacian obstacle problems) is not forced by any self-citation or definitional reduction. The two self-citations are: (1) [20] (co-authored by Ok), which provides the iteration framework and sharp L^1 comparison estimates — this is a methodological tool adapted to the obstacle setting, not a result that assumes the conclusion; (2) [32] (sole-authored by Ok), which is the prior Dini-continuity result being generalized — the current paper weakens the assumptions from Dini continuity to DMO, and the DMO condition is demonstrably weaker (Remark 2.2(v) gives explicit examples of DMO functions that are not Dini continuous). The comparison estimates in Section 3 are derived from scratch using structure inequalities (2.2)-(2.7) and the frozen-coefficient equation. The external citations [5, 7] (CZ estimates), [1] (C^{1,α} excess decay), and [27] (Kuusi-Mingione, different authors) provide independent mathematical inputs. No step in the derivation chain reduces to its own inputs by construction. The skeptic's concern about the WLOG assumption on q in the I_2 estimate is a correctness issue, not a circularity issue. Score 1 reflects the presence of self-citations that are not load-bearing for the logical structure of the proof.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new mathematical objects or entities. All function spaces (DMO_q, θ-DMO_q, L^{n,1}), operators (p-Laplacian with coefficients), and solution concepts (variational inequality solutions) are standard. The mean oscillation function ω_data is a composite of standard quantities. The free parameters (κ, q, δ, ϵ) are all determined by the structure of the problem and prior lemmas, not fitted to data.

free parameters (4)
  • κ = any κ > 0
    Introduced in Condition 2.3 as an existence parameter: A must be DMO_{2+κ} for some κ > 0. Not fitted to data but chosen to make the Hölder exponents in the CZ estimates work. The specific value is arbitrary as long as it is positive.
  • q = q = max(p(2+κ)/κ, (2-p)(2+κ)/κ)
    Defined in (3.1) as the integrability exponent for the CZ estimates, chosen to match the mean oscillation exponents. Determined by p and κ, not independently free.
  • δ = chosen so that 64C₀C₁δ^α ≤ ϵ
    The geometric ratio in the iteration scheme, chosen in (4.1) to control the excess decay. Depends on ϵ and the constants C₀, C₁, α from prior lemmas.
  • ϵ = ϵ ∈ (0, ϵ₀], ϵ₀ = 2^{-n-2}
    Controls the smallness in the iteration; set to ϵ₀ = 2^{-n-2} for the local boundedness proof and varied for the continuity proof.
assumptions (6)
  • domain assumption Uniform ellipticity of A: Λ⁻¹|ξ|² ≤ ⟨A(x)ξ,ξ⟩ ≤ Λ|ξ|²
    Standard ellipticity condition (1.2); assumed throughout. Not specific to this paper.
  • standard math Calderón–Zygmund Lᵠ-estimates for variational inequalities with VMO coefficients (Lemma 2.7, citing [5, Theorem 1.5] and [7, Theorem 2.5])
    The CZ estimate (2.9) is the quantitative engine for the comparison estimates. It is cited from published works [5, 7] and is a standard result in the field, though its application here requires the Dini condition on mean oscillation.
  • domain assumption C¹,α excess decay for frozen-coefficient p-Laplacian (Lemma 2.9, citing [1, Theorem 4.1])
    The L¹ excess decay estimate (2.12) is cited from a 2026 preprint by Antonini [1]. This is a recent improvement of classical C¹,α-regularity. If this result does not hold as stated, the iteration in Section 4 fails.
  • standard math Comparison principle for the frozen-coefficient obstacle problem: w ≥ ψ (Lemma 3.2, citing [7, Lemma 3.5])
    Used to ensure w ∈ A_ψ so that w can be used as a test function in the variational inequality. Cited from [7].
  • standard math Reverse Hölder inequality for solutions to the variational inequality (Lemma 2.6, citing [32, Theorem 3.1])
    Used to control Du in Lᵖ(1+σ) and feed into the CZ estimates. Cited from the second author's prior work [32].
  • ad hoc to paper Assumption that (2−p)(2+κ)/κ > p in the I₂ estimate for 1 < p < 2
    Stated as 'without loss of generality' in the proof of Lemma 3.2, but no justification is given for why this can be assumed. This affects the comparison estimate for the obstacle term when 1 < p < 2.

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Pith. "Pith review of Gradient continuity for $p$-Laplacian obstacle problems under mean oscillation conditions." pith.science (2026). https://pith.science/paper/N22I5NR6

@misc{pith2026260707018,
  author       = {Pith},
  title        = {Pith review of: Gradient continuity for $p$-Laplacian obstacle problems under mean oscillation conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N22I5NR6}},
  note         = {Machine review of arXiv:2607.07018}
}
abstract

We establish the $C^1$-regularity of solutions to the obstacle problems associated with $p$-Laplacian type equations, where $1<p<\infty$. Specifically, we prove that the gradient of the solution is continuous under a Dini mean oscillation ($\mathsf{DMO}$) type condition on the data, which includes the coefficient matrix, the source term, and the obstacle function. This result relaxes the classical Dini continuity assumption on the data to a more general mean oscillation condition.

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [32]

    Mediterr

    Ok, Jihoon.,Gradient continuity for nonlinear obstacle problems. Mediterr. J. Math.14(2017), no. 1, Paper No. 16, 24 pp. GRADIENT CONTINUITY FORp-LAPLACIAN OBSTACLE PROBLEMS UNDER MEAN OSCILLATION CONDITIONS 21 (S. Lee) Department ofMathematics, SogangUniversity, 35 Baekbeom-ro, Mapo-gu, Seoul04107, Republic ofKorea Email address:sungjinlee@sogang.ac.kr (...

    work page 2017

  2. [20]

    arXiv preprint arXiv:2504.02159, (2025)

    H ¨ast¨o, Peter; Lee, Mikyoung; Ok, Jihoon.,Mean oscillation conditions for nonlinear equation and regularity results. arXiv preprint arXiv:2504.02159, (2025)

    work page arXiv 2025

  3. [1]

    arXiv preprint arXiv:2601.07140 (2026)

    Antonini, Carlo Alberto.,Local and global C 1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type. arXiv preprint arXiv:2601.07140 (2026)

    work page arXiv 2026

  4. [27]

    Kuusi, Tuomo; Mingione, Giuseppe.,A nonlinear Stein theorem. Calc. Var. Partial Differential Equa- tions51(2014), no. 1-2, 45–86

    work page 2014

  5. [2]

    Differential Equations367(2023), 415–450

    Baroni, Paolo.,Gradient continuity for p(x)-Laplacian systems under minimal conditions on the exponent, J. Differential Equations367(2023), 415–450

    work page 2023

  6. [3]

    Baroni, Paolo.,A new condition ensuring gradient continuity for minimizers of non-autonomous func- tionals with mild phase transition. Calc. Var. Partial Differential Equations64(2025), no. 4, Paper No. 112, 30 pp

    work page 2025

  7. [4]

    Baroni, Paolo; Kuusi, Tuomo; Mingione, Giuseppe.,Borderline gradient continuity of minima. J. Fixed Point Theory Appl.15(2014), no. 2, 537–575

    work page 2014

  8. [5]

    B ¨ogelein, Verena; Duzaar, Frank; Mingione, Giuseppe,Degenerate problems with irregular obstacles. J. Reine Angew. Math.650(2011), 107–160. 20 S. LEE AND J. OK

    work page 2011

Show all 32 references
  1. [6]

    Burch, Charles C.,The Dini condition and regularity of weak solutions of elliptic equations. J. Differential Equations30(1978), no. 3, 308–323

    1978

  2. [7]

    Byun, Sun-Sig; Cho, Yumi; Wang, Lihe.,Calder´ on-Zygmund theory for nonlinear elliptic problems with irregular obstacles. J. Funct. Anal.263(2012), no. 10, 3117–3143

    2012

  3. [8]

    Choe, Hi Jun.,A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems. Arch. Rational Mech. Anal.114(1991), no. 4, 383–394

    1991

  4. [9]

    Choe, Hi Jun; Lewis, John L.,On the obstacle problem for quasilinear elliptic equations of p Laplacian type. SIAM J. Math. Anal.22(1991), no. 3, 623–638

    1991

  5. [10]

    Nonlinear Anal

    DiBenedetto, E.,C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(1983), no. 8, 827–850

    1983

  6. [11]

    Forum Math.20(2008), no

    Diening, Lars; Ettwein, Frank.,Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math.20(2008), no. 3, 523–556

    2008

  7. [12]

    Ann.370(2018), no

    Dong, Hongjie; Escauriaza, Luis; Kim, Seick.,On C 1, C 2, and weak type-(1,1)estimates for linear elliptic operators: part IIMath. Ann.370(2018), no. 1-2, 447–489

    2018

  8. [13]

    Dong, Hongjie; Escauriaza, Luis; Kim, Seick.,On C 1/2,1, C1,2, and C 0,0 estimates for linear parabolic operatorsJ. Evol. Equ.21(2021), no. 4, 4641–4702

    2021

  9. [14]

    Dong, Hongjie; Kim, Seick.,On C 1, C 2, and weak type-(1,1)estimates for linear elliptic operators. Comm. Partial Differential Equations42(2017), no. 3, 417–435

    2017

  10. [15]

    Dong, Hongjie; Kim, Seick.,Green’s function for nondivergence elliptic operators in two dimensions. SIAM J. Math. Anal.53(2021), no. 4, 4637–4656

    2021

  11. [16]

    Duzaar, Frank; Mingione, Giuseppe.,Gradient estimates via linear and nonlinear potentials. J. Funct. Anal.259(2010), no. 11, 2961–2998

    2010

  12. [17]

    Evans, Lawrence C.,A new proof of local C 1,α regularity for solutions of certain degenerate elliptic p.d.e. J. Differential Equations45(1982), no. 3, 356–373

    1982

  13. [18]

    Nonlinear Anal

    Fuchs, Martin.,H¨ older continuity of the gradient for degenerate variational inequalities. Nonlinear Anal. 15(1990), no. 1, 85–100

    1990

  14. [19]

    Hartman, Philip; Wintner, Aurel.,On uniform Dini conditions in the theory of linear partial differential equations of elliptic type. Amer. J. Math.77(1955), 329–354

    1955

  15. [21]

    Potential Anal.52(2020), no

    Hwang, Sukjung; Kim, Seick.,Green’s function for second order elliptic equations in non-divergence form. Potential Anal.52(2020), no. 1, 27–39

    2020

  16. [22]

    Jin, Tianling; Maz’ya, Vladimir; Van Schaftingen, Jean.,Pathological solutions to elliptic problems in divergence form with continuous coefficients. C. R. Math. Acad. Sci. Paris347(2009), no. 13-14, 773–778

    2009

  17. [23]

    Kinnunen, Juha; Zhou, Shulin.,A local estimate for nonlinear equations with discontinuous coefficients. Comm. Partial Differential Equations24(1999), no. 11-12, 2043–2068

    1999

  18. [24]

    Differential Integral Equations14(2001), no

    Kinnunen, Juha; Zhou, Shulin.,A boundary estimate for nonlinear equations with discontinuous coeffi- cients. Differential Integral Equations14(2001), no. 4, 475–492

    2001

  19. [25]

    Kuusi, Tuomo; Mingione, Giuseppe.,New perturbation methods for nonlinear parabolic problems. J. Math. Pures Appl. (9)98(2012), no. 4, 390–427

    2012

  20. [26]

    Kuusi, Tuomo; Mingione, Giuseppe.,Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal.207(2013), no. 1, 215–246

    2013

  21. [28]

    Mingione, Giuseppe.,Gradient potential estimates, J. Eur. Math. Soc. (JEMS)13(2011), no. 2, 459–486

    2011

  22. [29]

    Chinese Ann

    Li, Yanyan.,On the C 1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients. Chinese Ann. Math. Ser. B38(2017), no. 2, 489–496

    2017

  23. [30]

    Nonlinear Anal

    Ok, Jihoon.,Gradient continuity for p(·)-Laplace systems. Nonlinear Anal. bf 141 (2016), 139–166

    2016

  24. [31]

    Ok, Jihoon.,C 1 -regularity for minima of functionals with p(x)-growth. J. Fixed Point Theory Appl.19 (2017), no. 4, 2697–2731

    2017

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