Lipschitz regularity for fractional $p$-Laplacian with coercive gradients

Aniket Sen; Anup Biswas; Erwin Topp

arxiv: 2604.07489 · v1 · submitted 2026-04-08 · 🧮 math.AP

Lipschitz regularity for fractional p-Laplacian with coercive gradients

Anup Biswas , Aniket Sen , Erwin Topp This is my paper

Pith reviewed 2026-05-10 17:11 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional p-Laplacianviscosity solutionsLipschitz regularitycoercive gradientnonlocal equationsHölder continuitybounded solutionsLiouville theorem

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The pith

Viscosity solutions to fractional p-Laplacian equations with coercive gradient nonlinearities are locally Lipschitz continuous under bounds on p.

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

The paper studies a nonlinear nonlocal equation combining the fractional p-Laplacian operator with a coercive term that depends on the gradient and a Lipschitz continuous right-hand side. It proves that every viscosity solution is locally Lipschitz continuous when p lies in the interval (1, 2/(1-s)) union (1, m+1). The work also obtains Hölder continuity for subsolutions and shows that the homogeneous equation with position-independent nonlinearity admits only the zero function among bounded solutions, for arbitrary m and p greater than 1. These regularity and uniqueness statements matter because they supply quantitative control on solutions without assuming smoothness in advance.

Core claim

Any viscosity solution u to the equation (-Δ_p)^s u(x) + H(x, ∇u) = f is locally Lipschitz continuous when p belongs to (1, 2/(1-s)) ∪ (1, m+1), with f Lipschitz continuous and H coercive. Subsolutions satisfy Hölder continuity, and when f vanishes and H does not depend on x the only bounded solution is identically zero for all m, p > 1.

What carries the argument

Viscosity solution framework for the fractional p-Laplacian with coercive gradient perturbation, relying on comparison principles and oscillation estimates to control the gradient.

If this is right

Solutions to the equation acquire local Lipschitz regularity directly from the structure without additional assumptions on u.
In the homogeneous case with x-independent H the equation forces every bounded solution to be zero.
Subsolutions satisfy interior Hölder continuity estimates that hold independently of the full Lipschitz conclusion.
The admissible range for p enlarges or shrinks according to the fractional order s and the growth exponent m of H.

Where Pith is reading between the lines

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

The same comparison techniques might adapt to other nonlocal operators that possess a similar coercivity structure in the gradient term.
The Liouville-type result for bounded solutions suggests possible extensions to entire-space problems or long-time asymptotics.
Relaxing coercivity of H would likely produce counterexamples to Lipschitz regularity, testing the necessity of that hypothesis.

Load-bearing premise

The gradient term H must be coercive, the right-hand side f must be Lipschitz continuous, and p must lie inside the stated range relative to s and m so that the comparison and oscillation arguments close.

What would settle it

Construction of a non-Lipschitz viscosity solution for some p outside the given intervals, or for a non-coercive H, would disprove the Lipschitz claim.

read the original abstract

In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form \[ (-\Delta_p)^s u(x) + H(x, \nabla u) = f, \] where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally Lipschitz continuous, provided \[ p \in \left(1, \frac{2}{1-s}\right) \cup (1, m+1). \] We also establish H\"older continuity of subsolutions. Furthermore, in the case $f=0$ and $H$ is independent of $x$, we prove that the equation admits only the trivial solution in the class of bounded solutions, for all $m, p \in (1,\infty)$.

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 proves local Lipschitz continuity for viscosity solutions of the fractional p-Laplacian plus coercive gradient term, under a split range on p, plus a Liouville result for the homogeneous case.

read the letter

The central result is that viscosity solutions to (–Δ_p)^s u + H(x, ∇u) = f are locally Lipschitz when p sits in (1, 2/(1–s)) ∪ (1, m+1), assuming H is coercive and f is Lipschitz. They also obtain Hölder continuity for subsolutions and show that the homogeneous equation has only the zero bounded solution for all m, p > 1 when f = 0 and H is independent of x. That combination of the gradient term with the fractional p-Laplacian and the exact parameter window looks new relative to the cited literature. The viscosity comparison and oscillation estimates are the main tools, and the coercivity of H is used to close the estimates in the two separate regimes for p. The Liouville claim is stated without extra restrictions on p or m, which is a clean byproduct once regularity is available. The work follows the standard viscosity framework for nonlocal operators, so the technical steps are in a familiar lane. The p-range restriction is the most visible limitation; it is presented as necessary for the tail and growth controls to work, and nothing in the abstract suggests the authors claim it is optimal. The proofs are not reproduced here, but the stated assumptions line up with what is needed to make the comparison principles go through. This is a technical note aimed at people already working on regularity for nonlocal equations with lower-order terms. A reader who follows the fractional p-Laplacian literature or viscosity methods for integro-differential equations will pick up the extension without much extra background. The claims are concrete, the parameter restrictions are explicit, and the homogeneous uniqueness result adds a bit of extra value. It has enough substance and internal consistency to deserve a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes local Lipschitz continuity for viscosity solutions of the nonlocal equation (-Δ_p)^s u + H(x, ∇u) = f, where f is Lipschitz continuous, under the explicit restriction p ∈ (1, 2/(1-s)) ∪ (1, m+1). It further proves Hölder continuity for subsolutions and shows that the homogeneous problem (f = 0, H independent of x) admits only the trivial bounded solution for all m, p > 1.

Significance. If the proofs are correct, the results extend regularity theory for fractional p-Laplacian equations to include coercive gradient nonlinearities, with explicit parameter ranges that indicate where comparison and oscillation estimates close. The uniqueness statement for the homogeneous case is notably broad, holding without further restrictions on m or p. These contributions would be useful for analyzing nonlocal equations with nonlinear drift terms.

minor comments (3)

The abstract and introduction state the parameter restrictions clearly, but a brief remark on why the estimates fail for p outside the given range would improve readability.
The definition of viscosity solutions should explicitly address how the nonlocal tail is controlled in the comparison principle arguments.
Notation for the fractional p-Laplacian operator and the coercivity assumption on H could be recalled in the statement of the main theorems for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via comparison principles

full rationale

The paper establishes Lipschitz regularity for viscosity solutions of the nonlocal equation via comparison principles and oscillation estimates that close under the stated restrictions on p relative to s and m. These restrictions are explicitly required for the estimates to hold and do not reduce any claimed result to a fitted parameter or self-defined quantity. The uniqueness result for the homogeneous case follows directly from the coercivity assumption on H without invoking self-citations as load-bearing premises. No step equates a prediction to its input by construction, and the central claims remain independent of any prior self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of viscosity solutions for nonlocal operators, the coercivity assumption on H, and the Lipschitz regularity of f; no free parameters or new entities are introduced.

axioms (2)

domain assumption Viscosity solution definition for the fractional p-Laplacian with gradient nonlinearity
Invoked throughout the abstract to frame the regularity statements.
domain assumption Coercivity of the gradient term H(x, ∇u)
Required for the comparison principle that yields the Lipschitz bound.

pith-pipeline@v0.9.0 · 5426 in / 1378 out tokens · 42780 ms · 2026-05-10T17:11:17.363021+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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[1] [1]

Arapostathis, A

A. Arapostathis, A. Biswas, and L. Caffarelli. On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient. Comm. Partial Differential Equations, 44(12):1466–1480, 2019

work page 2019

[2] [2]

G. Barles. A weak Bernstein method for fully nonlinear elliptic equations,Diff. and Integral Equations4(2), 241–262, 1991

work page 1991

[3] [3]

Barles, E

G. Barles, E. Chasseigne, and C. Imbert. Hölder continuity of solutions of second-order non-linear elliptic integro- differential equations,J. Eur. Math. Soc.(JEMS) 13, 1–26, 2011

work page 2011

[4] [4]

Barles, E

G. Barles, E. Chasseigne, A. Ciomaga, and C. Imbert. Lipschitz regularity of solutions for mixed integro- differential equations,J. Differential Equations, 252(11), 6012–6060, 2012

work page 2012

[5] [5]

Regularity results and large time behavior for integro- differential equations with coercive Hamiltonians

Guy Barles, Shigeaki Koike, Olivier Ley, and Erwin Topp. Regularity results and large time behavior for integro- differential equations with coercive Hamiltonians. Calc. Var. Partial Differential Equations, 54(1):539–572, 2015

work page 2015

[6] [6]

Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior

Guy Barles, Olivier Ley, and Erwin Topp. Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior. Nonlinearity, 30(2):703–734, 2017

work page 2017

[7] [7]

On unbounded solutions of ergodic problems inRm for viscous Hamilton-Jacobi equations

Guy Barles and Joao Meireles. On unbounded solutions of ergodic problems inRm for viscous Hamilton-Jacobi equations. Comm. Partial Differential Equations, 41(12):1985–2003, 2016

work page 1985

[8] [8]

Lipschitz regularity for censored subdiffusive integro-differential equations with superfractional gradient terms

Guy Barles and Erwin Topp. Lipschitz regularity for censored subdiffusive integro-differential equations with superfractional gradient terms. Nonlinear Anal., 131:3–31, 2016

work page 2016

[9] [9]

Bensoussan and J

A. Bensoussan and J. Frehse. On Bellman equations of ergodic control inRn. J. Reine Angew. Math., 429:125– 160, 1992

work page 1992

[10] [10]

Sur la généralisation du problème de Dirichlet

Serge Bernstein. Sur la généralisation du problème de Dirichlet. Math. Ann., 62(2):253–271, 1906

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Sur la géenéralisation du problème de Dirichlet

Serge Bernstein. Sur la géenéralisation du problème de Dirichlet. Math. Ann., 69(1):82–136, 1910

work page 1910

[12] [12]

Bhakta, A

M. Bhakta, A. Biswas and A. Sen. Liouville results for supersolutions of fractionalp-Laplacian equations with gradient nonlinearities, to appear inProc. AMS, 2026

work page 2026

[13] [13]

Bidaut-Véron, M

M.F. Bidaut-Véron, M. García-Huidobro, L. Véron. Local and global properties of solutions of quasilinear Hamilton-Jacobi equations, J. Func. Anal. 267: 3294-3331, 2014

work page 2014

[14] [14]

Bidaut-Véron, M

M.F. Bidaut-Véron, M. García-Huidobro, L. Véron. Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, Duke Math. J. 168: 1487–1537, 2019

work page 2019

[15] [15]

Biswas, A

A. Biswas, A. Quaas, and E. Topp, Nonlocal Liouville theorems with gradient nonlinearity,J. Func. Anal., Volume 289, Issue 8, 15 October 2025

work page 2025

[16] [16]

Biswas and A

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

work page 2025

[17] [17]

Biswas and E

A. Biswas and E. Topp. Nonlocal ergodic control problem inRd,Math. Annalen390, 45–94, 2024

work page 2024

[18] [18]

Biswas and E

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

work page 2025

[19] [19]

Bögelein, F

V. Bögelein, F. Duzaar, N. Liao, and K. Moring. Gradient estimates for the fractionalp-Poisson equation,J. Math. Pures et Appl.204, appeared online, 2025 arXiv:2503.05903, 2025

work page arXiv 2025

[20] [20]

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

work page 2025

[21] [21]

Gradient regularity for (s,p) -harmonic functions, 2024

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

[22] [22]

Brasco and E

L. Brasco and E. Lindgren. Higher Sobolev regularity for the fractionalp-Laplace equation in the superquadratic case,Adv. Math.304, 300–354, 2017

work page 2017

[23] [23]

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

work page 2018

[24] [24]

Capuzzo Dolcetta, F

I. Capuzzo Dolcetta, F. Leoni ,and A. Porretta. Hölder estimates for degenerate elliptic equations with coercive Hamiltonians,Trans. Am. Math. Soc., 362(9):4511–4536, 2010

work page 2010

[25] [25]

Ciomaga, D

A. Ciomaga, D. Ghilli, and E. Topp. Periodic homogenization for weakly elliptic Hamilton-Jacobi- Bellman equations with critical fractional diffusion,Comm. Partial Differential Equations, 47(1):1–38, 2022

work page 2022

[26] [26]

Proceedings of the Conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs

M. Cirant and A. Goffi. On the Liouville property for fully nonlinear equations with superlinear first-order terms, “Proceedings of the Conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs", Contemporary Mathematics, American Mathematical Society, 781 (2023)

work page 2023

[27] [27]

Cirant and G

M. Cirant and G. Verzini. Local Hölder and maximal regularity of solutions of elliptic equations with su- perquadratic gradient terms,Advances in Mathematics, Vol. 409, Part B, 2022, 108700

work page 2022

[28] [28]

M. Cozzi. Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes,J. Func. Anal.272 , 4762–4837, 2017

work page 2017

[29] [29]

L. M. Del Pezzo and A. Quaas, The fundamental solution of the fractionalp´Laplacian,NoDEA Nonlinear Differential Equations Appl.33, no. 3, Paper No. 62, 2026

work page 2026

[30] [30]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities,J. Funct. Anal.267 (6), 1807–1836, 2014 24 ANUP BISW AS, ANIKET SEN, AND ER WIN TOPP

work page 2014

[31] [31]

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

work page 2016

[32] [32]

Diening, K

L. Diening, K. Kim, H.-S. Lee and S. Nowak. Higher differentiability for the fractionalp-Laplacian,Math. Annalen391, 5631–5693, 2025

work page 2025

[33] [33]

Diening and S

L. Diening and S. Nowak. Calderón-Zygmund estimates for the fractionalp-Laplacian,Annals of PDE11, no. 6, 2025

work page 2025

[34] [34]

Filippucci

R. Filippucci. Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 2903–2916, 2009

work page 2009

[35] [35]

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

work page 2024

[36] [36]

arXiv:2509.26565

D. Giovagnoli, D. Jesus and L. Silvestre.C1`α regularity for fractionalp-harmonic functions. ArXiv:2509.26565, 2025

work page arXiv 2025

[37] [37]

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

work page 2020

[38] [38]

Ishii and P

H. Ishii and P. L. Lions. Viscosity solutions of fully non-linear second-order elliptic partial differential equations, J. Differential Equations83, No.1, 26–78, 1990

work page 1990

[39] [39]

Jesus, A

D. Jesus, A. Sobral, and J. M. Urbano. Fractionalp-caloric functions are Lipschitz, ArXiv.2603.12065, 2026

work page arXiv 2026

[40] [40]

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

work page 2019

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