Pith. sign in

REVIEW 3 minor 47 references

Strong comparison principle and symmetry results for the fractional $p$-Laplacian

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A strong comparison principle for the fractional p-Laplacian yields symmetry of positive solutions.

desk verdict The paper proves a strong comparison principle for the fractional p-Laplacian at p>2 and derives symmetry results from it, plus a supporting regularity statement. read the letter →

arxiv 2606.08559 v1 pith:OAOZZYUE submitted 2026-06-07 math.AP

Anup Biswas , Subhajit Roy , Aniket Sen This is my paper · ORCID
classification math.AP
keywords fractionalp-LaplacianstrongcomparisonprinciplesymmetryresultsDirichletboundaryconditionsnonlocalellipticequationsregularityofsolutions
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 establishes a strong comparison principle for the equation (-Δ_p)^s u = f(u) where f is locally Lipschitz, in bounded domains of dimension at least 2. This principle is proved in a fairly general setting and then applied directly to positive C^1 solutions that satisfy Dirichlet boundary conditions, producing symmetry results. The work also verifies that such solutions are indeed C^1 when p lies in the interval from 2 up to 2/(1-s). A sympathetic reader would care because the comparison principle supplies a tool that converts local touching information into global ordering of solutions, which in turn forces symmetry without additional assumptions on the domain shape beyond boundedness.

What carries the argument

The strong comparison principle for the fractional p-Laplacian operator, which upgrades pointwise touching of subsolutions and supersolutions into strict inequality in the interior.

What would settle it

A positive C^1 solution to the equation in a ball that satisfies the Dirichlet condition yet fails to be radially symmetric would show the symmetry claim is false.

Watch

Extended reading notes

Core claim

We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive C^1 solutions satisfying Dirichlet boundary conditions. We also show that the C^1 regularity assumption is indeed satisfied for p in [2, 2/(1-s)).

Load-bearing premise

The nonlinearity f is locally Lipschitz continuous.

Editorial extensions

If this is right

  • Positive C^1 solutions with Dirichlet data are symmetric.
  • The C^1 regularity of solutions holds for all p between 2 and 2/(1-s).
  • The comparison principle applies to the equation in any bounded domain.
  • Symmetry conclusions follow once two solutions touch at an interior point.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same comparison device could be tested on equations with nonlocal right-hand sides beyond locally Lipschitz f.
  • Symmetry might be used to reduce the problem to an ODE on radial profiles for numerical checks.
  • The principle may allow comparison between solutions on different domains when one is contained in the other.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper studies the equation (-Δ_p)^s u = f(u) in a bounded domain Ω ⊂ ℝ^n (n ≥ 2, p > 2) with f locally Lipschitz. It establishes a strong comparison principle in a general setting and applies it to obtain symmetry results for positive C^1 solutions satisfying Dirichlet boundary conditions. It also proves that the C^1 regularity assumption holds for p ∈ [2, 2/(1-s)).

Significance. If the proofs are correct, the work supplies a useful strong comparison principle and symmetry results for the fractional p-Laplacian, extending classical local results to the nonlocal setting. The self-contained regularity statement for the stated p-range strengthens the applicability of the symmetry theorems to the class of solutions considered.

minor comments (3)
  1. [Abstract] Abstract: the statement lists p > 2 while the regularity result includes the endpoint p = 2; clarify whether the comparison principle and symmetry results are intended to hold at p = 2 or only for p > 2.
  2. [Introduction] The introduction should explicitly recall the precise definition of the fractional p-Laplacian operator used throughout (including the normalization constant and the range of s) to avoid any ambiguity for readers.
  3. [Section 2] Notation for the space of admissible functions (e.g., the precise Sobolev or Hölder space in which the comparison principle is stated) appears only after the main theorem; move a brief definition to the setup section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and the recommendation of minor revision. The report does not list any major comments requiring a point-by-point response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper presents a direct mathematical proof of a strong comparison principle for the fractional p-Laplacian under the stated local Lipschitz condition on f, followed by symmetry results via moving planes and a self-contained regularity argument establishing C^1 solutions for p in [2, 2/(1-s)). No load-bearing steps reduce to self-citations, fitted parameters renamed as predictions, or definitions that presuppose the target result. The derivation chain relies on operator properties and standard analysis techniques that are independent of the paper's own outputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper rests on standard properties of the fractional p-Laplacian operator and the local Lipschitz condition on f; no free parameters or invented entities are introduced.

assumptions (2)
  • standard math The fractional p-Laplacian operator satisfies the standard comparison and maximum principles in the nonlocal setting.
    Invoked as background for the strong comparison principle.
  • domain assumption f is locally Lipschitz continuous.
    Stated in the equation setup; required for the comparison result.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strong comparison principle and symmetry results for the fractional $p$-Laplacian." pith.science (2026). https://pith.science/paper/OAOZZYUE

@misc{pith2026260608559,
  author       = {Pith},
  title        = {Pith review of: Strong comparison principle and symmetry results for the fractional $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OAOZZYUE}},
  note         = {Machine review of arXiv:2606.08559}
}
abstract

In this article, we study the equation $$ (-\Delta_p)^s u = f(u) $$ in a bounded domain $\Omega\subset \mathbb{R}^n$, where $n\geq 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.

Figures

Figures reproduced from arXiv: 2606.08559 by the authors.

Figure 1
Figure 1. Visualization of the cone as k Ñ 8, where ξk is a point on the line joining xk and yk, and e˜k is the unit vector along xk ´ yk. To derive a contradiction from (5.7) we show that, for r sufficiently small, lim sup kÑ8 1 δk J1,k ă 0. (5.8) Estimate (5.8) actually follows from [13, Theorem 2.3]. We add a proof here for the convenience of reading. Applying mean-value theorem on gptq “ t ´ n`sp 2 we note that 1 δk ˆ 1 |… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 1 canonical work pages

  1. [1]

    Badiale and E

    M. Badiale and E. Naban. A note on radiality of solutions ofp-Laplacian, Applicable Analysis 52.1-4, 35–43, 1994

    1994

  2. [2]

    Barrios and M

    B. Barrios and M. Medina. Equivalence of weak and viscosity solutions in fractional non-homogeneous problems. Math. Ann., 381(3-4): 1979–2012, 2021

    1979

  3. [3]

    Barrios, Montoro, L

    B. Barrios, Montoro, L. and B. Sciunzi. On the moving plane method for nonlocal problems in bounded domains. J. Anal. Math. 135, 37–57, 2018

    2018

  4. [4]

    Biswas and M

    A. Biswas and M. Modasiya. Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications. J. Anal. Math. 156, 47–81, 2025

    2025

  5. [5]

    Biswas and A

    A. Biswas and A. Sen. Improved Hölder regularity of fractionalpp, qq-Poisson equation with regular data, 2025

    2025

  6. [6]

    Biswas and E

    A. Biswas and E. Topp. Lipschitz regularity of fractionalp-Laplacian,Annals of PDE11, no. 27, 2025

    2025

  7. [7]

    Bögelein, F

    V. Bögelein, F. Duzaar, N. Liao, G. Molica Bisci, and R. Servadei. Regularity for the fractionalp-Laplace equation,J. Func. Anal.289(9), 2025

    2025

  8. [8]

    Bögelein, F

    V. Bögelein, F. Duzaar, N. Liao, G. Molica Bisci, and R. Servadei. Gradient regularity ofps, pq-harmonic func- tions,Calc. Var. Partial Differential Equations, to appear, arXiv:2409.02012, 2024

    work page arXiv 2024

Show all 47 references
  1. [9]

    Brasco, E

    L. Brasco, E. Lindgren, and A. Schikorra. Higher Hölder regularity for the fractional p-Laplacian in the su- perquadratic case,Adv. Math.338, 782–846, 2018

    2018

  2. [10]

    F. Brock. Continuous rearrangement and symmetry of solutions of elliptic problems. Proc. Indian Acad. Sci. (Math. Sci.) 110, 157–204, 2000

    2000

  3. [11]

    Caffarelli and L

    L. Caffarelli and L. Silvestre. Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62, 597–638, 2009 STRONG COMPARISON PRINCIPLE AND SYMMETRY 37

    2009

  4. [12]

    Chaker and L

    J. Chaker and L. Silvestre. Coercivity estimates for integro-differential operators. Calc. Var. Partial Differential Equations 59, no. 4, Paper No. 106, 20 pp., 2020

    2020

  5. [13]

    W. X. Chen, and C. M. Li. Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335, 735–758, 2018

    2018

  6. [14]

    Colasuonno, F

    F. Colasuonno, F. Ferrari, P. Gervasio, and A. Quarteroni. Some evaluations of the fractional -Laplace operator on radial functions, Mathematics in Engineering 5(1): 1-23, 2023

    2023

  7. [15]

    Cuesta and P

    M. Cuesta and P. Takáč. A Strong Comparison Principle for the Dirichletp-Laplacian Proceedings of the Confer- ence on Reaction diffusion systems (Trieste, 1995), 79–87, Lecture Notes in Pure and Appl. Math., 194, Dekker, New York, 1998

    1995

  8. [16]

    Damascelli

    L. Damascelli. Comparison theorems for some quasilinear degenerate elliptic operators and applications to sym- metry and monotonicity results, Ann. Inst. H. Poincaré. Analyse non linéaire, 15(4), 493–516, 1998

    1998

  9. [17]

    Damascelli and F

    L. Damascelli and F. Pacella. Monotonicity and symmetry of solutions ofp-Laplace equations,1ăpă2; via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (4), 689–707, 1998

    1998

  10. [18]

    Damascelli and F

    L. Damascelli and F. Pacella. Monotonicity and symmetry results forp-Laplace equations and applications, Adv. Differential Equations 5 (7-9), 1179–1200, 2000

    2000

  11. [19]

    Damascelli and B

    L. Damascelli and B. Sciunzi. Regularity, monotonicity and symmetry of positive solutions ofm-Laplace equa- tions, J. Differential Equations 206, 483–515, 2004

    2004

  12. [20]

    Damascelli and B

    L. Damascelli and B. Sciunzi. Harnack inequalities, maximum and comparison principles, and regularity of positive solutions ofm-Laplace equations, Calc. Var. Partial Differential Equations 25, no. 2, 139–159, 2006

    2006

  13. [21]

    Di Castro, T

    A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities,J. Funct. Anal.267 (6), 1807–1836, 2014

    2014

  14. [22]

    Di Castro, T

    A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractionalp-minimizers,Ann. Inst. H. Poincaré Anal. Non Linéaire33 , 1279–1299, 2016

    2016

  15. [23]

    Dyda and M

    B. Dyda and M. Kassmann. Regularity estimates for elliptic nonlocal operators, Anal. PDE 13(2), 317–370, 2020

    2020

  16. [24]

    Felmer and W

    P. Felmer and W. Wang. Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math. 61. 2014, 1350023

    2014

  17. [25]

    Garain and E

    P. Garain and E. Lindgren. Higher Hölder regularity for the fractionalp-Laplace equation in the subquadratic case.Math. Annalen390, 5753–5792, 2024

    2024

  18. [26]

    Gidas, W

    B. Gidas, W. M. Ni, and L. Nirenberg. Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, 209–243, 1979

    1979

  19. [27]

    E. Giusti. Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge, 2003

    2003

  20. [28]

    Giovagnoli, D

    D. Giovagnoli, D. Jesus, and L. Silvestre.C1`α regularity for fractionalp-harmonic functions. ArXiv, 2025

    2025

  21. [29]

    Grossi, S

    M. Grossi, S. Kesavan, F. Pacella, and M. Ramaswamy. Symmetry of positive solutions of some nonlinear equations, Topol. Methods Nonlinear Anal. 12 (1), 47–59, 1998

    1998

  22. [30]

    Guedda and L

    M. Guedda and L. Véron. Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13, no. 8, 879–902, 1989

    1989

  23. [31]

    Hurri-Syrjänen and Antti V

    R. Hurri-Syrjänen and Antti V. Vähäkangas. On fractional Poincaré inequalities, J. Anal. Math., 120, 85–104, 2013

    2013

  24. [32]

    Iannizzotto, S

    A. Iannizzotto, S. Mosconi, and N. S. Papageorgiou. On the logistic equation for the fractional p-Laplacian. Math. Nachr. 296, 1451–1468, 2023

    2023

  25. [33]

    Iannizzotto, S

    A. Iannizzotto, S. Mosconi, and M. Squassina. Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32,1353–1392, 2016

    2016

  26. [34]

    A.Iannizzotto, S.Mosconi, andM.Squassina.Fineboundaryregularityforthedegeneratefractionalp-Laplacian, J. Funct. Anal.279, no. 8, 108659, 54 pp., 2020

    2020

  27. [35]

    Iwaniec and J

    T. Iwaniec and J. Manfredi. Regularity ofp-harmonic functions on the plane, Rev. Mat. Iberoam. 5, 1–19, 1989

    1989

  28. [36]

    S. Jarohs. Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Stud., 18(4), 691–704, 2018

    2018

  29. [37]

    S.KichenassamyandJ.Smoller.Ontheexistenceofradialsolutionsofquasi-linearellipticequations, Nonlinearity 3, 677–694, 1990

    1990

  30. [38]

    Korvenpää, T

    J. Korvenpää, T. Kuusi and E Lindgren. Equivalence of solutions to fractionalp-Laplace type equations,J. Math. Pures Appl.132, 1–26, 2019

    2019

  31. [39]

    Korvenpää, T

    J. Korvenpää, T. Kuusi, and G. Palatucci. Hölder continuity up to the boundary for a class of fractional obstacle problems,Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.27, no. 3, 355–367, 2016

    2016

  32. [40]

    Kuusi, G

    T. Kuusi, G. Mingione, and Y. Sire. Nonlocal equations with measure data, Comm. Math. Phys. 337, 1317–1368, 2015

    2015

  33. [41]

    Kuusi, G

    T. Kuusi, G. Mingione, and Y. Sire. Nonlocal self-improving properties, Anal. PDE 8(1), 57–114, 2015

    2015

  34. [42]

    G. Leoni. A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, 2009 38 ANUP BISW AS, SUBHAJIT ROY, AND ANIKET SEN

    2009

  35. [43]

    Lucia and S

    M. Lucia and S. Prashant. Strong comparison principle for solutions of quasilinear equations, Proc. Amer. Math. Soc., 132, no. 4, 1005–1011, 2004

    2004

  36. [44]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136, no. 5, 521–573, 2012

    2012

  37. [45]

    R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970

    1970

  38. [46]

    Roselli and B

    P. Roselli and B. Sciunzi. A strong comparison principle for thep-Laplacian, Proc. Amer. Math. Soc. 135(10), 3217–3224, 2007

    2007

  39. [47]

    Serrin, A symmetry problem in potential theory, Arch

    J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304–318, 1971 Indian Institute of Science Education and Research-Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, INDIA. Email address:anup@iiserpune.ac.in, subhajit.roy@iiserpune.ac.in, aniket.s...

    1971

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.