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arxiv: 2606.10590 · v1 · pith:B4ALZP6Znew · submitted 2026-06-09 · 🧮 math.AP

Higher integrability for parabolic double phase equations with an improved gap bound

Abhrojyoti Sen , Jarkko Siltakoski This is my paper

Pith reviewed 2026-06-27 12:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords parabolic double phase equationshigher integrabilityHölder continuous solutionsgap conditionslanted Steklov averageZ^κ classweak solutions
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The pith

Hölder continuous weak solutions to parabolic double phase equations have higher integrable gradients under a relaxed gap bound on p and q.

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

The paper establishes higher integrability of the gradient for Hölder continuous weak solutions to the parabolic double phase equation under the gap condition 2 ≤ p ≤ q ≤ p + qκ/(q - 2γ). This holds when the coefficient a belongs to the class Z^κ(Ω_T) that controls its growth away from its zero set and satisfies a mild almost increasing condition. The proof relies on a new mollification called the slanted Steklov average. A reader would care because the bound is purely parabolic and relaxes previous restrictions, extending the range of exponents for which such integrability results apply.

Core claim

We prove a local higher integrability result for the gradient of Hölder continuous weak solutions to the parabolic double phase equation ∂_t u - div(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du) = 0 in Ω_T. For u ∈ C^{0,γ,γ/q}_loc(Ω_T) with γ ∈ [0,1), the result holds under the gap bound 2 ≤ p ≤ q ≤ p + qκ/(q - 2γ) when a ∈ Z^κ(Ω_T) satisfies a mild almost increasing condition, via the slanted Steklov average.

What carries the argument

The slanted Steklov average, a modified mollification introduced to handle the almost increasing condition on a, combined with the one-sided pointwise bound from membership in Z^κ(Ω_T) that controls growth away from the zero set.

If this is right

  • The gradient Du gains local integrability in some L^r space with r > p.
  • The gap bound is strictly parabolic in character and stricter than the elliptic Lavrentiev gap.
  • Higher integrability holds for a larger set of exponent pairs than under prior gap restrictions.
  • The result applies directly to equations modeling two-phase growth phenomena in the parabolic setting.

Where Pith is reading between the lines

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

  • The slanted Steklov average could apply to other parabolic problems with almost increasing coefficients.
  • The relaxed gap may allow extensions of gradient Hölder continuity results to wider exponent ranges.
  • Similar techniques might adapt to inhomogeneous right-hand sides or systems of parabolic double phase type.

Load-bearing premise

The coefficient a must belong to Z^κ(Ω_T) and satisfy a mild almost increasing condition to close the higher integrability argument after averaging.

What would settle it

A concrete counterexample consisting of a Hölder continuous weak solution with a in Z^κ(Ω_T) where q exceeds p + qκ/(q - 2γ) and |Du| fails to gain higher integrability.

read the original abstract

We prove a local higher integrability result for the gradient of H\"older continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } \Omega_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^{\kappa}(\Omega_T)$ for some $\kappa \in (0,\infty)$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the H\"older continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u \in C^{0,\gamma,\gamma/q}_{\mathrm{loc}}(\Omega_T)$ with $\gamma \in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{q\kappa}{q - 2\gamma}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

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 paper proves a local higher integrability result for |Du| of Hölder continuous weak solutions u ∈ C^{0,γ,γ/q}_loc(Ω_T) to the parabolic double phase equation ∂_t u - div(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du) = 0. Under the assumptions that a ∈ Z^κ(Ω_T) satisfies a mild almost-increasing condition, the result holds for the relaxed gap 2 ≤ p ≤ q ≤ p + qκ/(q - 2γ). The proof introduces a new slanted Steklov averaging procedure to handle the almost-increasing condition on a and closes a reverse-Hölder inequality via the one-sided pointwise bound encoded in Z^κ.

Significance. If the central argument is complete, the result improves the admissible gap between p and q for parabolic double-phase higher integrability beyond what follows from the Lavrentiev gap, and the slanted Steklov construction supplies a new technical tool for coefficients with controlled growth away from their zero sets. The class Z^κ is shown to contain Hölder functions, which broadens applicability.

major comments (2)
  1. [Section containing the definition and application of the slanted Steklov average (likely the section deriving the revers] The proof that the slanted Steklov average of a coefficient a ∈ Z^κ(Ω_T) remains in Z^κ(Ω_T) with a constant independent of the averaging parameters is not supplied. Because the reverse-Hölder inequality is closed precisely by invoking the one-sided bound from Z^κ on the averaged coefficient, any deterioration of the constant would prevent the gap (G) from being attained uniformly; this step is therefore load-bearing for the main theorem.
  2. [Main theorem statement and its proof] The statement that the gap (G) is attained under the given assumptions on a requires an explicit dependence of the higher-integrability exponent on the Z^κ constant; without a quantitative tracking of this constant through the averaging step, it is unclear whether the conclusion remains uniform when the mollification scale tends to zero.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the precise definition of the class Z^κ(Ω_T) (one-sided pointwise bound) rather than only describing its properties.
  2. [Proof sections] Notation for the parabolic cylinder and the Hölder space C^{0,γ,γ/q} should be recalled or referenced at the first use in the proof sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of the result. We address the major comments point by point below. Both points identify places where the manuscript can be strengthened by making certain arguments fully explicit; we will revise accordingly.

read point-by-point responses

  1. Referee: The proof that the slanted Steklov average of a coefficient a ∈ Z^κ(Ω_T) remains in Z^κ(Ω_T) with a constant independent of the averaging parameters is not supplied. Because the reverse-Hölder inequality is closed precisely by invoking the one-sided bound from Z^κ on the averaged coefficient, any deterioration of the constant would prevent the gap (G) from being attained uniformly; this step is therefore load-bearing for the main theorem.

    Authors: We agree that this verification is essential and was only sketched rather than fully proved. In the revised manuscript we will insert a dedicated lemma immediately after the definition of the slanted Steklov average, proving that if a belongs to Z^κ with constant C then its slanted average also belongs to Z^κ with a constant depending only on C, κ, p, q and the structural constants, but independent of the averaging radii. This will guarantee that the one-sided bound used to close the reverse-Hölder inequality remains uniform. revision: yes

  2. Referee: The statement that the gap (G) is attained under the given assumptions on a requires an explicit dependence of the higher-integrability exponent on the Z^κ constant; without a quantitative tracking of this constant through the averaging step, it is unclear whether the conclusion remains uniform when the mollification scale tends to zero.

    Authors: We concur that quantitative dependence should be recorded. We will add a short proposition (or a remark following the application of Gehring’s lemma) that explicitly tracks the higher-integrability exponent δ in terms of p, q, γ, κ and the Z^κ-constant C. The argument will show that δ > 0 whenever the gap (G) holds and that the constants remain controlled as the mollification scale tends to zero, thereby confirming uniformity of the conclusion. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on external estimates and new mollification

full rationale

The paper establishes higher integrability for solutions to the parabolic double phase equation under the stated gap condition (G) by applying slanted Steklov averages to preserve the almost-increasing property and invoking membership in Z^κ(Ω_T) to close the reverse-Hölder inequality. This chain relies on standard analytic tools (Steklov averaging, Gehring lemma variants) and the definition of the coefficient class, none of which reduce by construction to the target integrability conclusion or gap bound. No fitted parameters are renamed as predictions, no self-citations serve as load-bearing uniqueness theorems, and the gap relation is derived from the parabolic structure rather than being tautological with the inputs. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on membership of a in the class Z^κ together with the almost-increasing condition on a and the Hölder regularity of u; the slanted Steklov average is introduced to handle the monotonicity assumption.

axioms (3)
  • domain assumption a belongs to Z^κ(Ω_T) for some κ > 0, satisfying a one-sided pointwise bound controlling growth away from its zero set
    Invoked in the abstract to close the higher integrability argument
  • domain assumption a satisfies a mild almost increasing condition
    Motivates the introduction of the slanted Steklov average
  • domain assumption u is locally Hölder continuous with exponents γ and γ/q
    Required for the main result to hold under gap (G)
invented entities (1)
  • slanted Steklov average no independent evidence
    purpose: New mollification operator accommodating the almost increasing condition on a
    Introduced to handle the monotonicity assumption on a

pith-pipeline@v0.9.1-grok · 5814 in / 1559 out tokens · 23995 ms · 2026-06-27T12:41:51.011546+00:00 · methodology

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