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arxiv: 2605.31018 · v1 · pith:35B7PPNMnew · submitted 2026-05-29 · 🧮 math.AP

Existence results for nonlinear cone degenerate Laplace equations

Hua Chen , Xiaochun Liu , Yawei Wei , Mengnan Zhang This is my paper

Pith reviewed 2026-06-28 21:46 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear elliptic equationscone degenerate Laplacianviscosity solutionsweak solutionsAlexandrov-Bakelman-Pucci estimateHölder estimatesexistence results
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The pith

Nonlinear cone degenerate Laplace equations admit viscosity solutions via ABP and Hölder estimates, with weak solutions following from equivalence.

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

This paper studies a class of non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian. The authors prove the Alexandrov-Bakelman-Pucci estimate together with Hölder continuity to establish existence of viscosity solutions. They then prove equivalence between weak solutions and viscosity solutions to obtain existence in the weak sense as well. These results apply to operators motivated by cone calculus and provide solvability in this degenerate setting.

Core claim

Under the structural assumptions on the nonlinearity, the cone degenerate Laplacian equation admits viscosity solutions whose existence follows from the Alexandrov-Bakelman-Pucci estimate and Hölder estimates. Equivalence between weak and viscosity solutions then yields existence of weak solutions.

What carries the argument

The cone degenerate Laplacian, a non-divergence elliptic operator with degeneracy modeled on cone calculus, which supports the ABP estimate and Hölder continuity.

Load-bearing premise

The nonlinearity and the cone degenerate operator satisfy structural assumptions that allow the ABP estimate to hold and the equivalence between weak and viscosity solutions to be valid.

What would settle it

An explicit nonlinearity and operator meeting the structural assumptions but possessing no viscosity solution would disprove the existence claim.

read the original abstract

This paper concerns a class of non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian, which is motivated by cone calculus. We establish the existence of viscosity solutions by proving the Alexandrov-Bakelman-Pucci and H\"older estimates. Furthermore, we obtain the existence of weak solutions by proving the equivalence between weak solutions and viscosity solutions.

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 studies non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian. It claims existence of viscosity solutions via proofs of the Alexandrov-Bakelman-Pucci estimate and Hölder continuity, and existence of weak solutions via an equivalence argument between weak and viscosity solutions under suitable structural assumptions on the nonlinearity.

Significance. If the ABP and Hölder estimates are established rigorously under the stated assumptions, the results would extend classical existence theory for elliptic equations to the cone-degenerate setting motivated by cone calculus. The weak-viscosity equivalence provides a useful bridge between solution concepts and strengthens the overall contribution.

minor comments (2)
  1. [Abstract] The abstract states the main results but does not list the precise structural assumptions on the nonlinearity or the explicit form of the cone degenerate operator; including these (or a reference to the relevant section) would improve readability.
  2. The manuscript would benefit from a dedicated section or subsection that collects all structural assumptions in one place before the statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims existence of viscosity solutions via ABP and Hölder estimates for a class of non-divergence nonlinear elliptic equations with cone degenerate Laplacian, followed by weak-viscosity equivalence. These steps rely on classical PDE estimates under structural assumptions rather than any self-definitional reduction, fitted inputs renamed as predictions, or load-bearing self-citations. The abstract and description provide no equations or derivations that collapse to inputs by construction, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

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Reference graph

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