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arxiv: 1907.01392 · v1 · pith:7LRUVTXKnew · submitted 2019-07-02 · 🧮 math.AP

Variational approach to the asymptotic mean-value property for the p-Laplacian on Carnot groups

Tomasz Adamowicz , Antoni Kijowski , Andrea Pinamonti , Ben Warhurst This is my paper

Pith reviewed 2026-05-25 11:06 UTC · model grok-4.3

classification 🧮 math.AP
keywords p-LaplacianCarnot groupsviscosity solutionsasymptotic mean-value propertynormalized p-Laplacianvariational methods
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The pith

Continuous viscosity solutions of the normalized p-Laplacian on Carnot groups satisfy an asymptotic mean-value property.

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

The paper establishes an asymptotic characterization of continuous viscosity solutions to the normalized p-Laplacian equation on any Carnot group. It derives this characterization through a variational approach that connects the differential equation to an integral mean-value condition as the radius shrinks to zero. A sympathetic reader would care because the result removes the need for extra regularity hypotheses that appear in some Euclidean settings and applies uniformly across the family of stratified groups. This supplies a concrete way to recognize solutions by their averaging behavior rather than by direct differentiation.

Core claim

A continuous function u on a Carnot group G satisfies the normalized p-Laplacian equation Δ_{p G}^N u = 0 in the viscosity sense if and only if it obeys the corresponding asymptotic mean-value property.

What carries the argument

The variational approach that converts the viscosity definition of the normalized p-Laplacian into the asymptotic mean-value limit.

If this is right

  • The asymptotic mean-value property holds for every continuous viscosity solution when 1 < p ≤ ∞.
  • The property is valid in every Carnot group without further structural restrictions.
  • Solutions can be identified by their local averaging behavior at arbitrarily small scales.
  • The characterization works directly from the viscosity definition and does not require differentiability.

Where Pith is reading between the lines

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

  • The same variational link might be tested on other nonlinear operators defined on stratified groups.
  • Mean-value-based numerical schemes could be examined for boundary-value problems on Carnot groups.
  • Comparison principles or uniqueness statements might be re-proved using only the asymptotic averaging condition.

Load-bearing premise

The functions under study must be continuous and satisfy the normalized p-Laplacian in the viscosity sense on the Carnot group.

What would settle it

A concrete counterexample would be any continuous function on a Carnot group that solves the normalized p-Laplacian in the viscosity sense at some point but fails to satisfy the asymptotic mean-value equality for that operator at the same point.

read the original abstract

Let $1<p \leq \infty$. We provide an asymptotic characterization of continuous viscosity solutions $u$ of the normalized $p$-Laplacian $\Delta_{p\,\mathbb{G}}^N u=0$ in any Carnot group $\mathbb{G}$.

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

1 major / 0 minor

Summary. The manuscript claims to establish, via a variational approach, an asymptotic mean-value characterization for continuous viscosity solutions of the normalized p-Laplacian Δ_{p G}^N u = 0 on an arbitrary Carnot group G, for 1 < p ≤ ∞.

Significance. If the central claim is proved with explicit error estimates and without hidden regularity assumptions, the result would extend the Euclidean asymptotic mean-value property to the stratified nilpotent setting, strengthening the viscosity theory for subelliptic nonlinear operators. The variational method, if parameter-free and reproducible, would be a methodological contribution.

major comments (1)
  1. [Abstract] The abstract states the characterization but supplies neither the precise form of the asymptotic mean-value formula, the definition of the variational functional, nor any error estimate; without these the central claim cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The single major comment concerns the level of detail in the abstract. We respond point by point below.

read point-by-point responses

  1. Referee: [Abstract] The abstract states the characterization but supplies neither the precise form of the asymptotic mean-value formula, the definition of the variational functional, nor any error estimate; without these the central claim cannot be verified.

    Authors: The abstract is deliberately concise, as is standard for research articles. The precise asymptotic mean-value formula appears in the statement of Theorem 1.1, the variational functional is introduced and analyzed in Section 2, and the error estimates are obtained explicitly in the proofs of Theorems 3.1 and 4.2. The central claim is therefore fully stated and proved in the body of the manuscript; the abstract serves only as a high-level summary. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract states a direct claim of providing an asymptotic characterization for continuous viscosity solutions of the normalized p-Laplacian on Carnot groups via a variational approach. No derivation chain, equations, or self-citations are supplied in the available text that would allow any of the enumerated circularity patterns to be exhibited. The result is presented as an extension to a standard setting without reducing any prediction or uniqueness statement to a fitted input or prior self-citation by construction. The paper is therefore self-contained on the basis of the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified.

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discussion (0)

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Reference graph

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