pith. sign in

arxiv: 2606.20099 · v1 · pith:4VKULHIWnew · submitted 2026-06-18 · 🧮 math.AP

On weak and viscosity solutions to a nonhomogeneous mixed local-nonlocal equation

R. Lakshmi , Sekhar Ghosh This is my paper

Pith reviewed 2026-06-26 16:34 UTC · model grok-4.3

classification 🧮 math.AP
keywords weak solutionsviscosity solutionsmixed local-nonlocalp-Laplace equationcomparison principlenonhomogeneous equationsupersolutions
0
0 comments X

The pith

For the nonhomogeneous mixed local-nonlocal p-Laplace equation, continuous weak supersolutions are viscosity supersolutions when 1 < p < ∞, and bounded viscosity supersolutions are weak supersolutions when p ≥ 2.

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

This paper examines how weak and viscosity solutions relate for a nonhomogeneous equation that mixes local and nonlocal p-Laplace terms. It first establishes a comparison principle for weak subsolutions and supersolutions under suitable conditions on the data. Using this principle, the authors show that continuous weak supersolutions qualify as viscosity supersolutions for any p greater than 1. They further prove the converse direction for p at least 2 when the solutions are bounded.

Core claim

The paper establishes the equivalence between continuous weak supersolutions and viscosity supersolutions for 1<p<∞, and between bounded viscosity supersolutions and weak supersolutions for p≥2, for the nonhomogeneous mixed local-nonlocal p-Laplace equation in bounded Lipschitz domains, by first deriving the comparison principle for weak solutions.

What carries the argument

The comparison principle for weak subsolutions and supersolutions, which transfers properties between weak and viscosity solution classes for the mixed operator.

If this is right

  • The comparison principle allows transferring regularity results from one solution class to the other.
  • Solutions can be studied indifferently in either framework under the given regularity assumptions.
  • The results apply to a range of p values, covering both subquadratic and superquadratic cases differently.

Where Pith is reading between the lines

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

  • This link between solution concepts could facilitate proving existence of solutions by viscosity methods and then verifying they are weak.
  • The results might extend to other mixed operators or different growth conditions on the nonhomogeneous term.
  • Similar equivalences could be explored for equations with different nonlocal kernels.

Load-bearing premise

The comparison principle holds for weak sub- and supersolutions under the given conditions on the nonhomogeneous term and the mixed operator.

What would settle it

A counterexample of a continuous weak supersolution that is not a viscosity supersolution for some 1 < p < ∞ would falsify the claim.

read the original abstract

This paper explores the relationship between weak and viscosity solutions to a nonhomogeneous mixed local and non-local $p$-Laplace equation in a bounded Lipschitz domain in $\mathbb{R}^N$. Under certain conditions, we derive the comparison principle for weak subsolutions and weak supersolutions to the problem. For $1<p<\infty$, we establish that continuous weak supersolutions to the problem are viscosity supersolutions, using the comparison principle. Furthermore, we show that bounded viscosity supersolutions are weak supersolutions for $p \geq 2$.

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

0 major / 2 minor

Summary. The paper studies the relationship between weak and viscosity solutions to a nonhomogeneous mixed local-nonlocal p-Laplace equation in a bounded Lipschitz domain in R^N. It derives a comparison principle for weak sub- and supersolutions under certain conditions on the nonhomogeneous term. Using this principle, it proves that for 1 < p < ∞ continuous weak supersolutions are viscosity supersolutions, and that for p ≥ 2 bounded viscosity supersolutions are weak supersolutions.

Significance. If the comparison principle and the ensuing equivalences hold, the work connects two standard notions of solutions for mixed local-nonlocal operators, which is relevant to regularity theory and existence questions in this setting. The p-range asymmetry is consistent with known integrability distinctions for p-Laplacian-type problems. The manuscript supplies the conditions and the proof of the comparison principle as stated in the abstract.

minor comments (2)
  1. [Abstract / Introduction] The abstract refers to 'certain conditions' on the nonhomogeneous term; the introduction or §2 should state these conditions explicitly (e.g., growth or integrability assumptions) so that the comparison principle can be checked without reading the full proof.
  2. [§1 or §2] Notation for the mixed operator (local plus nonlocal terms) should be fixed early and used consistently; any dependence on the nonhomogeneous term f should be indicated in the statement of the comparison principle.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The summary accurately captures the main results on the comparison principle and the equivalence between weak and viscosity supersolutions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins by establishing a comparison principle for weak sub- and supersolutions under explicit conditions on the nonhomogeneous term and mixed operator; this principle is then used to transfer regularity and obtain the two equivalences (continuous weak supersolutions are viscosity supersolutions for 1<p<∞; bounded viscosity supersolutions are weak supersolutions for p≥2). No equations reduce by construction to fitted parameters or prior self-citations, no ansatz is smuggled via self-reference, and the comparison principle is presented as derived rather than assumed from overlapping prior work. The logical chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard background results in viscosity solution theory for p-Laplace-type operators and on an unspecified comparison principle whose hypotheses are not stated.

axioms (1)
  • domain assumption Standard comparison principle for weak sub- and supersolutions holds under the problem's structural assumptions
    Invoked to pass from weak to viscosity supersolutions and vice versa

pith-pipeline@v0.9.1-grok · 5610 in / 1217 out tokens · 24529 ms · 2026-06-26T16:34:45.893421+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

  1. [1]

    Barrios, M

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

    1979

  2. [2]

    Biagi, D

    S. Biagi, D. Mugnai, E. Vecchi, A Brezis-Oswald approach for mixed local and nonlocal operators,Commun. Contemp. Math., 26(2), Paper No. 2250057, 28 pp., 2024

    2024

  3. [3]

    J. E. M. Braga, R. A. Leitão, J. E. L. Oliveira, Free boundary theory for singular/degenerate nonlinear equations with right hand side: a non-variational approach,Calc. Var. Partial Differential Equations, 59(2), Paper No. 86, 29 pp., 2020

    2020

  4. [4]

    Brezis.Functional analysis, Sobolev spaces and partial differential equations

    H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Springer, New York, 2011

    2011

  5. [5]

    Crandall, H

    M. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 27:1–67, 1992

    1992

  6. [6]

    L. M. Del Pezzo, R. Ferreira, J. D. Rossi, Eigenvalues for a combination between local and nonlocal𝑝-Laplacians, Fract. Calc. Appl. Anal., 22(5):1414–1436, 2019

    2019

  7. [7]

    Demengel, G

    F. Demengel, G. Demengel,Functional Spaces For The Theory Of Elliptic Partial Differential Equations, Springer, London; EDP Sciences, Les Ulis, XVIII, 465, 2012

    2012

  8. [8]

    Di Castro, T

    A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional𝑝-minimizers,Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(5):1279–129, 2016

    2016

  9. [9]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces,Bull. Sci. Math., 136(5):521–573, 2012

    2012

  10. [10]

    Y. Fang, V. D. Rădulescu, C. Zhang, Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation,Math. Ann., 388(3):2519–2559, 2024

    2024

  11. [11]

    Y. Fang, C. Zhang, On weak and viscosity solutions of nonlocal double phase equations,Int. Math. Res. Not. IMRN, 2023(5):3746–3789, 2023

    2023

  12. [12]

    Garain, J

    P. Garain, J. Kinnunen, On the regularity theory for mixed local and nonlocal quasilinear elliptic equations,Trans. Amer. Math. Soc., 375(8):5393–5423, 2022

    2022

  13. [13]

    Ghosh, and R

    S. Ghosh, and R. Lakshmi, C. Zhang,On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data, Preprint arXiv:2505.16461, 2025

    work page arXiv 2025

  14. [14]

    Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,Funkcial

    H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,Funkcial. Ekvac., 38(1):101–120, 1995

    1995

  15. [15]

    Ishii, G

    H. Ishii, G. Nakamura, A class of integral equations and approximation of𝑝-Laplace equations,Calc. Var. Partial Differential Equations, 37(3-4):485–522, 2010

    2010

  16. [16]

    Julin, P

    V. Julin, P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the𝑝-Laplace equation, Comm. Partial Differential Equations, 37(5):934–946, 2012

    2012

  17. [17]

    Juutinen, P

    P. Juutinen, P. Lindqvist, J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi- linear equation,SIAM J. Math. Anal., 33(3):699–717, 2001

    2001

  18. [18]

    Juutinen, T

    P. Juutinen, T. Lukkari, M. Parviainen Equivalence of viscosity and weak solutions for the𝑝(𝑥)-Laplacian,Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(6):1471–1487, 2010

    2010

  19. [19]

    Koike, A Beginner’s Guide to the Theory of Viscosity Solutions,MSJ Memoirs 13, Mathematical Society of Japan, Tokyo, 2004

    S. Koike, A Beginner’s Guide to the Theory of Viscosity Solutions,MSJ Memoirs 13, Mathematical Society of Japan, Tokyo, 2004

    2004

  20. [20]

    Korvenpää, T

    J. Korvenpää, T. Kuusi, E. Lindgren, Equivalence of solutions to fractional𝑝-Laplace type equations,J. Math. Pures Appl. (9), 132:1–26, 2019

    2019

  21. [21]

    Lakshmi, R

    R. Lakshmi, R. Kr. Giri, S. Ghosh, A weighted eigenvalue problem for mixed local and nonlocal operators with potential,Math. Nachr., 299(2):367–396, 2026

    2026

  22. [22]

    Lakshmi, S

    R. Lakshmi, S. Ghosh, Equivalence of weak and viscosity solutions to mixed local and nonlocal𝑝-Laplace equation, Z. Anal. Anwend., 2026.https://doi.org/10.4171/ZAA/1815

    work page doi:10.4171/zaa/1815 2026

  23. [23]

    Leoni,A First Course In Fractional Sobolev Spaces, American Mathematical Society, Providence, RI, XV, 586, 2023

    G. Leoni,A First Course In Fractional Sobolev Spaces, American Mathematical Society, Providence, RI, XV, 586, 2023

    2023

  24. [24]

    Lindqvist, Notes on the infinity Laplace equation, Springer, 2016

    P. Lindqvist, Notes on the infinity Laplace equation, Springer, 2016

    2016

  25. [25]

    Lindqvist, Notes on the stationary𝑝-Laplace equation, Springer, Berlin, 2019

    P. Lindqvist, Notes on the stationary𝑝-Laplace equation, Springer, Berlin, 2019

    2019

  26. [26]

    Lindqvist, J

    P. Lindqvist, J. Manfredi, Viscosity solutions of the evolutionary𝑝-Laplace equation,Differ. Integr. Equ., 20:1303– 1319, 2007. 18

    2007

  27. [27]

    Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations

    P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness,Comm. Partial Differential Equations, 8(11):1229–1276, 1983

    1983

  28. [28]

    Medina, P

    M. Medina, P. Ochoa, On viscosity and weak solutions for non-homogeneous𝑝-Laplace equations,Adv. Nonlinear Anal., 8(1):468–481, 2019

    2019

  29. [29]

    Medina, P

    M. Medina, P. Ochoa, Equivalence of solutions for non-homogeneous𝑝(𝑥)-Laplace equations,Math. Eng., 5(2), Paper No. 044, 19 pp., 2023

    2023

  30. [30]

    Shang, C

    B. Shang, C. Zhang, A strong maximum principle for mixed local and nonlocal𝑝-Laplace equations,Asymptot. Anal., 133(1-2):1–12, 2023

    2023

  31. [31]

    Siltakoski, Equivalence of viscosity and weak solutions for the normalized𝑝(𝑥)-Laplacian,Calc

    J. Siltakoski, Equivalence of viscosity and weak solutions for the normalized𝑝(𝑥)-Laplacian,Calc. Var. Partial Differential Equations, 57(4), Paper No. 95, 20 pp., 2018

    2018