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arxiv: 1906.11791 · v1 · pith:H2RVWFPUnew · submitted 2019-06-27 · 🧮 math.AP

Continuity of the Free Boundary in a Problem involving the A-Laplacian

Samia Challal , Abdeslem Lyaghfouri This is my paper

Pith reviewed 2026-05-25 14:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords free boundaryA-Laplaciancontinuitytwo-dimensionalelliptic equations
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The pith

In a two-dimensional free boundary problem involving the A-Laplacian, the free boundary is locally the graph of a family of continuous functions.

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

The paper studies a free boundary problem in two space dimensions where the governing operator is the A-Laplacian. It establishes that the free boundary can be written, at least locally, as the graph of continuous functions. This supplies a basic regularity statement for the interface separating the regions where the solution satisfies different equations. A reader interested in elliptic free-boundary problems would see this as a first step toward understanding the geometry of the unknown boundary under a nonlinear divergence-structure operator.

Core claim

We show that the free boundary is represented locally by graphs of a family of continuous functions.

What carries the argument

The A-Laplacian operator together with the structural conditions that permit the application of two-dimensional free-boundary regularity techniques.

Load-bearing premise

The setting is two-dimensional and the A-Laplacian satisfies the structural conditions that make the regularity arguments work.

What would settle it

An explicit example, in two dimensions, of an A-Laplacian free-boundary problem whose free boundary fails to be the graph of a continuous function.

read the original abstract

In this paper we investigate a two dimensional free boundary problem involving the A-Laplacian. We show that the free boundary is represented locally by graphs of a family of continuous functions.

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 investigates a two-dimensional free boundary problem involving the A-Laplacian. It claims to show that the free boundary is represented locally by graphs of a family of continuous functions.

Significance. If the result holds under the stated structural conditions on the A-Laplacian, it would extend local graph regularity results for free boundaries from the standard Laplacian to a broader class of nonlinear operators in 2D, contributing to the regularity theory for obstacle-type or Bernoulli-type problems.

major comments (1)
  1. Abstract: the claim that the free boundary is represented locally by graphs of continuous functions is stated without any derivation, statement of structural assumptions on A, or indication of the 2D techniques employed; this prevents verification that the central claim is supported by the argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the claim that the free boundary is represented locally by graphs of continuous functions is stated without any derivation, statement of structural assumptions on A, or indication of the 2D techniques employed; this prevents verification that the central claim is supported by the argument.

    Authors: We agree that the abstract is brief and omits explicit mention of the structural assumptions on A as well as the 2D-specific techniques. In the revised manuscript we will expand the abstract to include a concise statement of the relevant assumptions on A (uniform ellipticity and the standard growth conditions) together with an indication that the argument relies on planar techniques. The detailed hypotheses and proof strategy already appear in the introduction and subsequent sections; the abstract revision will improve readability without altering the paper's content. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract states a result on local graph representation of the free boundary for a 2D A-Laplacian free boundary problem but contains no equations, parameter fits, self-citations, or ansatzes. No load-bearing step reduces by construction to its own inputs, and the derivation is presented as relying on standard 2D regularity techniques without visible self-referential structure. This is the normal case of a self-contained claim whose internal steps cannot be inspected for circularity from the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; ledger remains empty pending full text.

pith-pipeline@v0.9.0 · 5541 in / 973 out tokens · 22619 ms · 2026-05-25T14:24:48.924694+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 2 internal anchors

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