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arxiv: 2505.15119 · v4 · pith:KAA44GEDnew · submitted 2025-05-21 · 🧮 math.AP

Lipschitz regularity for fully nonlinear elliptic equations with (p,q)-growth

Sun-Sig Byun , Hongsoo Kim This is my paper

Pith reviewed 2026-05-22 14:27 UTC · model grok-4.3

classification 🧮 math.AP
keywords Lipschitz regularityfully nonlinear elliptic equations(p,q)-growthdouble phase problemsinterior regularityglobal regularityHolder continuity
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The pith

Solutions to fully nonlinear elliptic equations with (p,q)-growth are locally and globally Lipschitz continuous when the gap q minus p is small enough.

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

The paper shows that solutions of fully nonlinear elliptic equations obeying a (p,q)-growth structure become Lipschitz continuous in the interior and up to the boundary whenever the difference q minus p stays below a threshold that depends on the data. The same small-gap hypothesis upgrades any given Holder continuous solution to Lipschitz regularity under a slightly stronger bound on the gap. A reader would care because Lipschitz continuity supplies uniform control on the gradient, which opens the door to further analysis of the solution's behavior and to numerical approximation in nonlinear models.

Core claim

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with (p,q)-growth. We prove that for a small gap q-p, a solution is locally or globally Lipschitz continuous. We also prove that a given Holder continuous solution is Lipschitz continuous under improved bounds for the gap. These gap conditions are similar to those required for the regularity of double phase problems in divergence form.

What carries the argument

The (p,q)-growth condition imposed on the fully nonlinear operator, which encodes ellipticity that scales between p and q powers and reduces to standard uniform ellipticity when p equals q.

If this is right

  • Interior Lipschitz continuity holds for solutions under the small-gap hypothesis.
  • Global Lipschitz continuity holds for solutions under the same small-gap hypothesis together with suitable boundary data.
  • Any Holder continuous solution upgrades to Lipschitz continuity when the gap satisfies a stricter bound than the basic smallness requirement.
  • The gap thresholds parallel those already known for double-phase variational problems.

Where Pith is reading between the lines

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

  • Techniques developed for double-phase problems in divergence form may transfer directly to the non-variational fully nonlinear setting.
  • It would be natural to ask whether the same small-gap condition also yields higher integrability or differentiability of the gradient.
  • The results suggest testing the sharpness of the gap bound by constructing examples in which solutions lose Lipschitz continuity exactly when q-p exceeds the derived threshold.

Load-bearing premise

The equation satisfies the structural (p,q)-growth condition and the gap between q and p is sufficiently small, with the precise smallness depending on the data.

What would settle it

An explicit example of a fully nonlinear equation obeying (p,q)-growth with q-p below the paper's threshold yet possessing a solution whose gradient is unbounded in the domain would disprove the claimed Lipschitz regularity.

read the original abstract

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with $(p,q)$-growth. We prove that for a small gap $q-p$, a solution is locally or globally Lipschitz continuous. We also prove that a given H\"older continuous solution is Lipschitz continuous under improved bounds for the gap. These gap conditions are similar to those required for the regularity of double phase problems in divergence form.

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 proves interior and global Lipschitz regularity for viscosity solutions of fully nonlinear elliptic equations satisfying (p,q)-growth structural conditions, provided the gap q-p is sufficiently small (with the smallness depending on the data and ellipticity constants). It further shows that any Hölder continuous solution is Lipschitz continuous under a stricter smallness requirement on the gap. The gap conditions are stated to be analogous to those appearing in the regularity theory for double-phase problems in divergence form.

Significance. If the proofs are correct, the work supplies a non-variational, viscosity-solution counterpart to the Lipschitz regularity results known for double-phase variational problems. This extends the scope of fully nonlinear elliptic theory to equations with non-standard growth and may be relevant for models with variable ellipticity or growth rates. The approach via oscillation control appears consistent with standard techniques in the field.

minor comments (3)
  1. [§1] §1 (Introduction): the comparison with double-phase regularity should include at least two or three specific references to the divergence-form literature so that the analogy is immediately verifiable by readers.
  2. [Theorem 1.1] Theorem 1.1 and Theorem 1.2: the dependence of the admissible gap q-p on the structural constants (λ,Λ,p,q, dimension) is described only qualitatively; an explicit (even if not optimal) bound would strengthen the statement.
  3. [Notation] Notation section: the symbol C is used for generic constants without always tracking which quantities it may depend on; a short remark on this convention at the beginning of the proofs would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly identifies the main contributions: interior and global Lipschitz regularity for viscosity solutions of fully nonlinear equations with (p,q)-growth when the gap q-p is sufficiently small, as well as the improved gap condition under which Hölder continuous solutions become Lipschitz. We appreciate the recognition that this provides a non-variational counterpart to known results for double-phase problems.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a direct proof of interior and global Lipschitz regularity for viscosity solutions of fully nonlinear equations under (p,q)-growth when the gap q-p is sufficiently small, using oscillation control and structural assumptions that parallel standard double-phase theory. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the gap smallness is an input hypothesis rather than an output, and the derivation remains self-contained against external benchmarks and the stated ellipticity conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

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