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arxiv: 2604.22206 · v1 · submitted 2026-04-24 · 🧮 math.AP

Gradient H\"{o}lder regularity for nonlocal double phase equations

Yuzhou Fang , Chao Zhang This is my paper

Pith reviewed 2026-05-08 10:53 UTC · model grok-4.3

classification 🧮 math.AP MSC 35B65
keywords nonlocal double phase equationsviscosity solutionsgradient Hölder regularityfractional operatorsdegenerate equationsinterior regularitymodulating coefficient
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The pith

Viscosity solutions to nonlocal double phase equations have Hölder continuous gradients when the modulating coefficient is Lipschitz and the exponents are close.

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

The paper establishes that viscosity solutions to the nonlocal double phase equation, which combines a fractional p-term with a modulated q-term weighted by a nonnegative coefficient a(x,y), are C^{1,α} in the interior. The proof requires a to be Lipschitz continuous and |tq - sp| to be sufficiently small so that the two phases interact controllably. This result closes the higher regularity question for the degenerate case of these equations. A reader cares because the equations capture diffusion with heterogeneous growth rates, and gradient continuity provides concrete information about solution smoothness in nonlocal settings.

Core claim

Assuming a(x,y) is Lipschitz continuous and |tq - sp| is suitably small, the gradient of any viscosity solution u to the equation ∫_{R^d} (|u(x)-u(y)|^{p-2}(u(x)-u(y))/|x-y|^{d+sp} + a(x,y) |u(x)-u(y)|^{q-2}(u(x)-u(y))/|x-y|^{d+tq}) dy = 0 is Hölder continuous in the interior, for 2 ≤ p ≤ q and 0 < s ≤ t < 1.

What carries the argument

The nonlocal double phase integral operator that superposes an unmodulated p-phase term with a coefficient-modulated q-phase term, where the Lipschitz regularity of a controls the pointwise interaction between the phases to produce the gradient estimate.

Load-bearing premise

The modulating coefficient a(x,y) is Lipschitz continuous and the distance |tq - sp| is small enough that the two phases in the integral do not overwhelm each other's regularity contributions.

What would settle it

An explicit viscosity solution whose gradient is not Hölder continuous at an interior point when a is continuous but not Lipschitz, or when |tq - sp| exceeds the smallness threshold.

read the original abstract

This paper is devoted to investigating the interior $C^{1, \alpha}$ regularity of viscosity solutions to the nonlocal double phase equations $$ \int_{\mathbb{R}^d} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}}+a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}}\right)\,dy=0, $$ where $2\le p\le q$, $s, t\in (0, 1)$ with $s\le t$, and $a(x, y)\ge0$. In the degenerate case, we solve the higher regularity issue raised by De Filippis-Palatucci [J. Differential Equations \textbf{267} (2019) 547--586]. By assuming the Lipschitz continuity of the modulating coefficient $a$, we are able to prove that the gradient of solution is H\"older continuous, provided the distance of $tq$ and $sp$ is suitably small. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a$, the growth exponents and the differentiability orders.

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

2 major / 2 minor

Summary. The manuscript proves interior C^{1,α} regularity for viscosity solutions of the nonlocal double-phase equation ∫_{R^d} [ |u(x)-u(y)|^{p-2}(u(x)-u(y)) / |x-y|^{d+sp} + a(x,y) |u(x)-u(y)|^{q-2}(u(x)-u(y)) / |x-y|^{d+tq} ] dy = 0, under the assumptions 2 ≤ p ≤ q, s ≤ t ∈ (0,1), a(x,y) ≥ 0 Lipschitz continuous, and |tq - sp| sufficiently small. The result is stated for the degenerate case and resolves a higher-regularity question left open by De Filippis-Palatucci.

Significance. If the central estimates close, the work supplies the first gradient Hölder theory for nonlocal double-phase problems with distinct fractional orders, extending the viscosity approach of prior nonlocal regularity results to a two-phase setting while controlling the interaction via the smallness assumption on the exponents.

major comments (2)
  1. [Theorem 1.1 and §5 (gradient oscillation decay)] Theorem 1.1 (main result): the smallness threshold on |tq-sp| is invoked to absorb the difference between the kernels |x-y|^{-d-sp} and |x-y|^{-d-tq} into lower-order terms during the Campanato iteration for the gradient oscillation, yet no explicit quantitative bound (in terms of p,q,s,t,d) is supplied; without it the absorption constant cannot be verified to be strictly less than 1.
  2. [§4] §4 (viscosity test-function construction): when the test function is chosen to probe the gradient, the cross-phase remainder produced by the Lipschitz oscillation of a(x,y) is multiplied by the factor |tq-sp|; the manuscript does not display the precise dependence of the Hölder exponent α on this product, leaving open whether the iteration closes for any positive smallness or only for an impractically narrow regime.
minor comments (2)
  1. [Introduction] The notation for the fractional orders (sp versus tq) is introduced without a preliminary comparison table; a short display of the admissible range s ≤ t, p ≤ q would improve readability.
  2. [§2] Several references to prior nonlocal regularity results (e.g., the baseline C^{0,β} theory) are cited only by author-year; adding the precise theorem numbers from those works would help the reader locate the exact statements being invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below, clarifying the role of the smallness assumption on |tq-sp| and indicating where the manuscript will be revised to make the dependencies more explicit.

read point-by-point responses

  1. Referee: Theorem 1.1 and §5 (gradient oscillation decay)] Theorem 1.1 (main result): the smallness threshold on |tq-sp| is invoked to absorb the difference between the kernels |x-y|^{-d-sp} and |x-y|^{-d-tq} into lower-order terms during the Campanato iteration for the gradient oscillation, yet no explicit quantitative bound (in terms of p,q,s,t,d) is supplied; without it the absorption constant cannot be verified to be strictly less than 1.

    Authors: We agree that an explicit threshold would make the argument more transparent. In the Campanato iteration of Section 5, the difference between the two kernels produces a perturbation term whose coefficient is proportional to |tq-sp| times a constant depending on p, q, s, t, d and the Lipschitz norm of a (see the estimates leading to (5.12) and the absorption step after (5.15)). The smallness condition is chosen precisely so that this coefficient is strictly less than 1/2, allowing the main term to dominate and the iteration to close. While we do not compute a numerical value for the threshold (which would require exhaustive tracking of all universal constants through the preceding lemmas), such a positive δ exists and depends only on the structural data. We will add a remark immediately after the statement of Theorem 1.1 indicating how δ arises from the absorption constants and confirming that the iteration constant remains strictly less than 1 for |tq-sp| < δ. revision: partial

  2. Referee: [§4] §4 (viscosity test-function construction): when the test function is chosen to probe the gradient, the cross-phase remainder produced by the Lipschitz oscillation of a(x,y) is multiplied by the factor |tq-sp|; the manuscript does not display the precise dependence of the Hölder exponent α on this product, leaving open whether the iteration closes for any positive smallness or only for an impractically narrow regime.

    Authors: In the viscosity test-function arguments of Section 4, the remainder arising from the Lipschitz oscillation of a(x,y) is indeed bounded by a term containing the factor |tq-sp| multiplied by quantities controlled by the test-function scaling (see the estimates around the choice of the test function in the proof of the key comparison lemma). This remainder is absorbed into the main integral terms once |tq-sp| is sufficiently small. The Hölder exponent α is then fixed in the subsequent oscillation-decay iteration of Section 5; it depends on p, q, s, t, d, Lip(a) and the smallness parameter, but remains positive as long as |tq-sp| lies below the threshold that makes the absorption work. Thus the result holds for every sufficiently small positive |tq-sp|, with α deteriorating continuously as the smallness parameter approaches its upper limit. We will revise the relevant paragraphs in Section 4 to track this dependence explicitly and to state that α can be chosen positive whenever |tq-sp| is smaller than the same structural threshold used in Section 5. revision: yes

Circularity Check

0 steps flagged

No circularity: proof uses standard viscosity techniques and external prior results under explicit assumptions.

full rationale

The derivation establishes interior C^{1,α} regularity for viscosity solutions to the given nonlocal double-phase equation by assuming Lipschitz continuity of a(x,y) and smallness of |tq-sp|. These are stated as hypotheses to control cross-phase interactions in the oscillation decay estimates. The argument invokes standard viscosity comparison principles and Campanato-type iteration, referencing the higher-regularity issue from De Filippis-Palatucci (distinct authors) without any self-citation load-bearing the central claim, without fitting parameters then relabeling them as predictions, and without definitional or ansatz smuggling via own prior work. The smallness threshold is an input assumption, not a derived output, so the chain does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of viscosity solutions for nonlocal operators and the Lipschitz assumption on a; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Viscosity solutions satisfy the comparison principle and can be tested with C^2 functions touching from above or below.
    Invoked implicitly when stating the equation holds in the viscosity sense.

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