pith. sign in

arxiv: 2606.22854 · v1 · pith:QVKDMMTOnew · submitted 2026-06-22 · 🧮 math.AP

Comparison Results for a class of Neumann Problems of the p-Laplace Equation on Riemannian Manifolds

Wentao Liu , Anqiang Zhu This is my paper

Pith reviewed 2026-06-26 08:00 UTC · model grok-4.3

classification 🧮 math.AP
keywords Neumann problemsp-Laplace equationRiemannian manifoldsRicci curvatureTalenti comparisonLorentz spacesspherical symmetrizationcomparison principles
0
0 comments X

The pith

Neumann boundary value problems for the p-Laplace equation admit stronger comparison principles on manifolds with nonnegative Ricci curvature.

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

This paper considers Neumann boundary value problems for the p-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, the authors derive Talenti-type comparison results in Lorentz spaces. They show that the Neumann setting requires weaker constraints than the Robin case, which leads to stronger comparison principles. These findings matter because they provide improved tools for analyzing solutions to nonlinear partial differential equations on curved geometries.

Core claim

We consider Neumann boundary value problems for the p-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, we derive Talenti-type comparison results in Lorentz spaces. We further show that, in contrast to the Robin case, the Neumann setting admits weaker constraints, which yields stronger comparison principles.

What carries the argument

Spherical symmetrization for obtaining Talenti-type comparison results in Lorentz spaces.

If this is right

  • Comparison principles hold with weaker conditions than for Robin problems.
  • Talenti-type results are obtained in Lorentz spaces.
  • Results apply to manifolds with nonnegative Ricci curvature.

Where Pith is reading between the lines

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

  • The stronger principles could improve existence proofs for such boundary value problems.
  • Similar techniques might apply to other boundary conditions or operators on manifolds.
  • These comparisons may connect to isoperimetric problems in geometric analysis.

Load-bearing premise

Spherical symmetrization applies under appropriate constraints on manifolds with nonnegative Ricci curvature.

What would settle it

A specific manifold with nonnegative Ricci curvature where the Talenti-type comparison for the Neumann p-Laplace problem fails to hold.

read the original abstract

We consider Neumann boundary value problems for the $p$-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, we derive Talenti-type comparison results in Lorentz spaces. We further show that, in contrast to the Robin case, the Neumann setting admits weaker constraints, which yields stronger comparison principles.

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 manuscript derives Talenti-type comparison results in Lorentz spaces for Neumann boundary-value problems for the p-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Spherical symmetrization is applied under appropriate constraints; the authors argue that the Neumann condition permits weaker constraints than the corresponding Robin problem, thereby yielding stronger comparison principles.

Significance. If the derivations hold, the work extends rearrangement techniques and comparison principles to the Neumann setting on manifolds, relaxing the constraints relative to the Robin case. This distinction, if rigorously established, would be a useful contribution to geometric PDE theory, particularly for symmetry and a priori estimates in Lorentz spaces.

minor comments (2)
  1. The abstract refers to 'appropriate constraints' without naming them; the introduction should state these constraints explicitly (e.g., any curvature or volume-growth hypotheses beyond Ric ≥ 0) so that the claimed weakening relative to the Robin case can be verified at a glance.
  2. Notation for the Lorentz-space norms and the symmetrized functions should be introduced once in a dedicated preliminary section and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on Talenti-type comparison results for Neumann p-Laplace problems on manifolds with nonnegative Ricci curvature, and for recommending minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on applying spherical symmetrization to obtain Talenti-type comparisons in Lorentz spaces for the Neumann p-Laplace problem under Ric ≥ 0 and appropriate constraints. These steps invoke standard rearrangement techniques and isoperimetric inequalities that are externally established and independent of the paper's fitted values or prior self-citations. The claimed weaker constraints relative to the Robin case follow directly from the boundary condition without any reduction to self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citation chains. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption of nonnegative Ricci curvature and the existence of appropriate constraints allowing spherical symmetrization; no free parameters or invented entities are evident from the abstract.

axioms (2)
  • domain assumption Riemannian manifolds with nonnegative Ricci curvature
    Explicitly stated as the geometric setting in the abstract.
  • domain assumption Spherical symmetrization applies under appropriate constraints to yield Talenti-type results in Lorentz spaces
    Invoked as the method to derive the comparison results.

pith-pipeline@v0.9.1-grok · 5574 in / 1199 out tokens · 18474 ms · 2026-06-26T08:00:50.238214+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

25 extracted references · 1 linked inside Pith

  1. [1]

    Sharp geometric in- equalities for closed hypersurfaces in manifolds with nonnegative ricci curvature

    Virginia Agostiniani, Mattia Fogagnolo, and Lorenzo Mazzieri. Sharp geometric in- equalities for closed hypersurfaces in manifolds with nonnegative ricci curvature. arXiv preprint arXiv:1812.05022 , 2018

    Pith/arXiv arXiv 2018

  2. [2]

    A talenti comparison result for solutions to elliptic problems with robin boundary conditions

    Angelo Alvino, Carlo Nitsch, and Cristina Trombetti. A talenti comparison result for solutions to elliptic problems with robin boundary conditions. Communications on Pure and Applied Mathematics , 76(3):585–603, 2023

    2023

  3. [3]

    Comparison results for solutions to p-laplace equations with robin boundary conditions

    Vincenzo Amato, Andrea Gentile, and Alba Lia Masiello. Comparison results for solutions to p-laplace equations with robin boundary conditions. Annali di Matematica Pura ed Applicata (1923-) , 201(3):1189–1212, 2022

    1923

  4. [4]

    Sobolev inequalities in manifolds with nonnegative curvature

    Simon Brendle. Sobolev inequalities in manifolds with nonnegative curvature. Com- munications on Pure and Applied Mathematics , 76(9):2192–2218, 2023

    2023

  5. [5]

    Recent developments in the theory of lorentz spaces and weighted inequalities

    María J Carro, José A Raposo, and Javier Soria. Recent developments in the theory of lorentz spaces and weighted inequalities. 2007

    2007

  6. [6]

    Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower ricci curvature bounds

    Fabio Cavalletti and Andrea Mondino. Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower ricci curvature bounds. Inventiones mathematicae , 208(3):803–849, 2017

    2017

  7. [7]

    A talenti comparison result for a class of neumann boundary value problems

    Antonio Celentano, Carlo Nitsch, and Cristina Trombetti. A talenti comparison result for a class of neumann boundary value problems. Nonlinear Analysis , 261:113864, 2025

    2025

  8. [8]

    Talenti’s comparison theorem for poisson equation and applications on riemannian manifold with nonnegative ricci curvature

    Daguang Chen and Haizhong Li. Talenti’s comparison theorem for poisson equation and applications on riemannian manifold with nonnegative ricci curvature. arXiv preprint arXiv:2104.05568 , 2021

    arXiv 2021

  9. [9]

    Comparison results for solutions of pois- son equations with robin boundary on complete riemannian manifolds

    Daguang Chen, Haizhong Li, and Yilun Wei. Comparison results for solutions of pois- son equations with robin boundary on complete riemannian manifolds. International Journal of Mathematics , 34(08):2350045, 2023. 25

    2023

  10. [10]

    Talenti comparison results and rigidity results for anisotropic p-laplacian operator with robin boundary conditions

    Lu Chen and Yabo Yang. Talenti comparison results and rigidity results for anisotropic p-laplacian operator with robin boundary conditions. Advanced Nonlinear Studies , 25(4):1047–1078, 2025

    2025

  11. [11]

    Comparison results, exit time moments, and eigenvalues on riemannian manifolds with a lower ricci curvature bound

    Don Colladay, Jeffrey J Langford, and Patrick McDonald. Comparison results, exit time moments, and eigenvalues on riemannian manifolds with a lower ricci curvature bound. The Journal of Geometric Analysis , 28(4):3906–3927, 2018

    2018

  12. [12]

    Neumann problems and steiner symmetrization

    Vincenzo Ferone and Anna Mercaldo. Neumann problems and steiner symmetrization. Communications in Partial Differential Equations , 30(10):1537–1553, 2005

    2005

  13. [13]

    Metric structures for Riemannian and non-Riemannian spaces

    Mikhail Gromov. Metric structures for Riemannian and non-Riemannian spaces . Springer, 2007

    2007

  14. [14]

    Convex and discrete geometry

    Peter M Gruber. Convex and discrete geometry . Springer, 2007

    2007

  15. [15]

    Some remarks on orlicz-lorentz spaces

    Anna Kamińska. Some remarks on orlicz-lorentz spaces. Mathematische Nachrichten , 147(1):29–38, 1990

    1990

  16. [16]

    Rearrangements and convexity of level sets in PDE

    Bernhard Kawohl. Rearrangements and convexity of level sets in PDE . Springer, 2006

    2006

  17. [17]

    Symmetrization and applications, volume 3

    Srinivasan Kesavan. Symmetrization and applications, volume 3. world scientific, 2006

    2006

  18. [18]

    Needle decompositions in Riemannian geometry, volume 249

    Bo’az Klartag. Needle decompositions in Riemannian geometry, volume 249. American Mathematical Society, 2017

    2017

  19. [19]

    Neumann comparison theorems in elliptic pdes

    Jeffrey J Langford. Neumann comparison theorems in elliptic pdes. Potential Analysis, 43(3):415–459, 2015

    2015

  20. [20]

    Symmetrization of poisson’s equation with neumann boundary conditions

    Jeffrey J Langford. Symmetrization of poisson’s equation with neumann boundary conditions. ANNALI SCUOLA NORMALE SUPERIORE-CLASSE DI SCIENZE , pages 1025–1063, 2015

    2015

  21. [21]

    Symmetrization in neumann problems

    Carla Maderna, Sandro Salsa, and C Pucci. Symmetrization in neumann problems. Applicable Analysis, 9(4):247–256, 1979

    1979

  22. [22]

    A talenti-type comparison theorem for rcd (k, n) spaces and applications

    Andrea Mondino and Mattia Vedovato. A talenti-type comparison theorem for rcd (k, n) spaces and applications. Calculus of Variations and Partial Differential Equations , 60(4):157, 2021

    2021

  23. [23]

    Convex bodies: the Brunn–Minkowski theory , volume 151

    Rolf Schneider. Convex bodies: the Brunn–Minkowski theory , volume 151. Cambridge university press, 2013

    2013

  24. [24]

    Elliptic equations and rearrangements

    Giorgio Talenti. Elliptic equations and rearrangements. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze , 3(4):697–718, 1976. 26

    1976

  25. [25]

    Symmetrization for fractional neumann problems

    Bruno Volzone. Symmetrization for fractional neumann problems. Nonlinear Analysis: Theory, Methods & Applications , 147:1–25, 2016. 27

    2016