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arxiv: 2604.07783 · v1 · submitted 2026-04-09 · 🧮 math.AP

Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth

Sun-Sig Byun , Hongsoo Kim This is my paper

Pith reviewed 2026-05-10 17:50 UTC · model grok-4.3

classification 🧮 math.AP
keywords Harnack inequalityanisotropic elliptic equationsnonstandard growthviscosity solutionsfully nonlinear operatorsintrinsic scalingdegenerate equationsmeasure estimates
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The pith

Viscosity solutions to anisotropic fully nonlinear elliptic equations with nonstandard growth satisfy intrinsic Harnack inequalities.

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

The paper proves Harnack inequalities for viscosity solutions of degenerate fully nonlinear anisotropic elliptic equations that exhibit nonstandard growth. These equations include examples like the degenerate anisotropic (p_i)-Laplacian. The approach adapts the sliding paraboloid method using anisotropic functions to obtain basic measure estimates. A novel barrier function is constructed to establish the doubling property. These elements are then combined with intrinsic geometry techniques to derive the intrinsic Harnack inequality when the exponents satisfy appropriate conditions.

Core claim

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations with non-standard growth conditions. Using an adapted sliding paraboloid method with anisotropic functions and a novel barrier function for the doubling property, combined with intrinsic geometry techniques, the intrinsic Harnack inequality is proved under suitable conditions on the exponents (p_i).

What carries the argument

The sliding paraboloid method adapted with anisotropic functions, together with a novel barrier function that yields the doubling property for intrinsic scaling.

Load-bearing premise

The sliding paraboloid method can be successfully adapted using suitably chosen anisotropic functions to derive the measure estimates, and the novel barrier function produces the necessary doubling property.

What would settle it

Construct a specific viscosity solution to the anisotropic (p_i)-Laplacian equation and check whether the ratio of supremum to infimum in intrinsic cylinders remains bounded as required by the Harnack inequality.

read the original abstract

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations exhibiting non-standard growth conditions. A primary example of such operators is the degenerate anisotropic $(p_i)$-Laplacian. Our approach relies on the sliding paraboloid method, adapted with suitably chosen anisotropic functions to derive the basic measure estimates. A central contribution of this work is the development of a doubling property, achieved through the explicit construction of a novel barrier function. By combining these tools with the intrinsic geometry techniques introduced in [DGV08, VV25], we prove the intrinsic Harnack inequality for this class of operators under appropriate conditions on the exponents $(p_i)$.

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 / 3 minor

Summary. The manuscript establishes Harnack inequalities for viscosity solutions of degenerate fully nonlinear anisotropic elliptic equations with nonstandard growth, with the degenerate anisotropic (p_i)-Laplacian as a primary example. The approach adapts the sliding paraboloid method using suitably chosen anisotropic functions to obtain basic measure estimates, constructs a novel barrier function to prove the doubling property, and combines these with intrinsic geometry techniques from [DGV08, VV25] to obtain the intrinsic Harnack inequality under appropriate conditions on the exponents (p_i).

Significance. If the central claims hold, the result meaningfully extends Harnack-type inequalities to anisotropic operators with nonstandard growth, a setting relevant to regularity theory for degenerate elliptic equations. The explicit construction of the novel anisotropic barrier function is a concrete contribution that strengthens the doubling property argument and may have broader applicability. The integration with existing intrinsic scaling methods from the cited references is a natural and effective step.

minor comments (3)
  1. The phrase 'appropriate conditions on the exponents (p_i)' appears in the abstract and introduction but is not stated explicitly in the main theorem; adding a precise range or set of inequalities for the p_i (e.g., in Theorem 1.1) would improve clarity and allow readers to check applicability immediately.
  2. The references [DGV08, VV25] are invoked for the intrinsic geometry techniques; ensure the bibliography provides complete bibliographic details (authors, title, journal, year, etc.) for these works.
  3. In the description of the barrier function construction, verify that all anisotropic scaling parameters are defined consistently with the nonstandard growth exponents before the doubling property is derived.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted. As the report contains no specific major comments, we have no individual points requiring point-by-point responses or revisions at this stage.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper adapts the sliding paraboloid method using anisotropic test functions to derive measure estimates, then explicitly constructs a novel barrier function to obtain the doubling property. These new tools are combined with intrinsic geometry techniques from the external references [DGV08, VV25] to reach the intrinsic Harnack inequality under stated conditions on the exponents. No step in the provided abstract or high-level argument reduces by definition or construction to its own inputs, fitted parameters, or a load-bearing self-citation chain; the central claims rest on independent constructions and standard adaptations rather than renaming or smuggling prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on the definition of viscosity solutions for fully nonlinear operators, the validity of the sliding paraboloid method in the anisotropic setting, and the existence of a barrier function that produces the doubling property; these are not derived from first principles in the abstract.

axioms (2)
  • domain assumption Viscosity solutions satisfy the comparison principle and maximum principle for the given class of degenerate anisotropic operators.
    Invoked implicitly when applying the sliding paraboloid method to obtain measure estimates.
  • domain assumption The exponents (p_i) satisfy conditions that allow the intrinsic scaling and the construction of the anisotropic barrier function.
    Stated as 'under appropriate conditions on the exponents (p_i)' but not specified in the abstract.
invented entities (1)
  • Novel anisotropic barrier function no independent evidence
    purpose: To establish the doubling property for the measure estimates.
    Explicitly constructed in the paper as a central contribution; no independent evidence outside the construction itself.

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