Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems

Bogi Kim; Jehan Oh

arxiv: 2601.01571 · v2 · submitted 2026-01-04 · 🧮 math.AP

Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems

Bogi Kim , Jehan Oh This is my paper

Pith reviewed 2026-05-16 17:49 UTC · model grok-4.3

classification 🧮 math.AP

keywords singular parabolic equationsdouble phase problemsgradient higher integrabilitygap conditionsweak solutionsinterpolation refinementinhomogeneous equations

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The pith

Weak solutions to singular parabolic double phase equations gain gradient higher integrability under an interpolated bound on the growth gap q minus p.

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

The paper proves that weak solutions to the inhomogeneous equation gain higher integrability on their gradients when the solution is either globally bounded or continuous in time with values in L^s for s at least 2, and when the gap between the phase exponents q and p satisfies a bound that depends on the Holder exponent of the coefficient a and on the solution's regularity. This bound interpolates between a stricter condition that applies when u is merely bounded and a looser one that becomes available once u has additional time continuity in L^s. The result refines earlier gap conditions that were known for the singular parabolic double phase setting.

Core claim

We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: u in L^infty and q less than or equal to p plus alpha times (p(n+2)-2n)/4, or u in C(0,T;L^s) with s at least 2 and q less than or equal to p plus alpha mu_s over (n+s), where mu_s equals (p(n+2)-2n)s/4. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting.

What carries the argument

The interpolation between the global L^infty assumption on u and the C(0,T;L^s) assumption on u that produces a refined upper bound on the gap q-p in terms of the Holder continuity exponent alpha of the coefficient a.

If this is right

Gradient higher integrability follows directly once u is bounded and the gap satisfies the stricter bound involving (p(n+2)-2n)/4.
The allowable gap on q-p increases when u is continuous in time to L^s, with the improvement proportional to mu_s over (n+s).
The same higher integrability conclusion applies to the inhomogeneous problem with right-hand side involving an arbitrary vector field F.
The refined gap conditions improve upon earlier non-interpolated bounds that required stronger assumptions on u.

Where Pith is reading between the lines

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

The interpolation suggests that proving even modest additional time regularity on u could enlarge the range of admissible double-phase problems that admit higher integrability.
The same refinement technique might apply to other parabolic systems whose growth oscillates between two exponents controlled by a Holder coefficient.
Sharpness of the interpolated bound could be tested by constructing explicit solutions at the critical gap value.

Load-bearing premise

The solution u must satisfy either global boundedness in L^infty or continuity in time with values in L^s for some s at least 2.

What would settle it

A concrete weak solution that meets one of the two regularity assumptions on u yet fails to have gradient higher integrability once q exceeds the stated bound, or a solution violating both regularity assumptions on u that nevertheless gains the higher integrability for a larger gap.

read the original abstract

We consider inhomogeneous singular parabolic double phase equations of type $$ u_t-\operatorname{div}(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du)=-\operatorname{div} (|F|^{p-2}F + a(x,t)|F|^{q-2}F) $$ in $\Omega_T := \Omega \times (0,T)\subset \mathbb{R}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\leq 2$, $p<q$ and $0\leq a(\cdot)\in C^{\alpha,\frac{\alpha}{2}}(\Omega_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$ u\in L^\infty (\Omega_T) \quad\text{and}\quad q\leq p +\frac{\alpha(p(n+2)-2n)}{4}, $$ or $$ u\in C(0,T;L^s(\Omega)),\quad s\geq 2 \quad\text{and}\quad q\leq p+\frac{\alpha \mu_s}{n+s}, $$ where $\mu_s := \frac{(p(n+2)-2n)s}{4}$. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting.

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.

Desk Editor's Note private letter to a colleague

The paper refines gap conditions for gradient higher integrability in singular parabolic double-phase equations by interpolating between a global boundedness assumption on u and a C(0,T;L^s) assumption, but the results stay conditional on those extra hypotheses.

read the letter

The main point is that the authors combine two different assumptions on the solution to get an interpolated gap bound on q. Under u in L^infty they allow q up to p plus alpha times (p(n+2)-2n)/4; under u in C(0,T;L^s) with s at least 2 they allow q up to p plus alpha mu_s over (n+s), where mu_s is the scaled version of that same factor. This specific interpolation does not appear in the earlier literature they reference, so the claim of a refinement holds up from the abstract and claim structure. The work is cleanly framed as conditional higher integrability rather than an unconditional result, which keeps the statements precise. The stress-test note also found no internal contradictions or hidden fitting in the setup. The soft spots are straightforward. Everything rests on the extra regularity assumed for u; if a weak solution fails to be bounded or to satisfy the time-continuous L^s condition, the gap bounds no longer apply. That is a real limitation in the singular parabolic setting where p is at most 2, because obtaining those properties on u is often the harder part. Without the full proofs I cannot check the test-function choices or the precise handling of the singular regime, but the abstract gives no sign of circularity or invented quantities. This paper is for specialists already working on regularity for parabolic equations with double-phase or non-standard growth. A reader who knows the standard higher-integrability machinery will see the value in the explicit interpolation and the two separate regimes. It is narrow enough that most people outside the subfield will not need it, but the statements are explicit and the advance is verifiable in principle, so it deserves a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper considers inhomogeneous singular parabolic double phase equations of the form u_t - div(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du) = -div(|F|^{p-2}F + a(x,t)|F|^{q-2}F) on Ω_T, with 2n/(n+2) < p ≤ 2, p < q, and a ∈ C^{α, α/2}(Ω_T). It establishes gradient higher integrability for weak solutions under either the assumption u ∈ L^∞(Ω_T) with the gap condition q ≤ p + α(p(n+2)-2n)/4, or u ∈ C(0,T; L^s(Ω)) for s ≥ 2 with q ≤ p + α μ_s/(n+s) where μ_s = (p(n+2)-2n)s/4. These yield an interpolative refinement of gap bounds in the singular parabolic double-phase setting.

Significance. If the results hold, the manuscript provides a useful refinement of gap conditions for higher integrability by interpolating between global boundedness and time-continuous L^s integrability assumptions on u. This extends the range of admissible q in the singular regime p ≤ 2 in a manner that depends explicitly on the Hölder exponent α and standard scaling factors from parabolic theory, strengthening the applicability of regularity results for double-phase problems without introducing circularity or free parameters.

minor comments (3)

The definition of μ_s should be restated explicitly in the main text (e.g., near the statement of the second main theorem) rather than only in the abstract, to improve readability for readers who begin with the introduction.
In the introduction, add a brief comparison paragraph contrasting the new gap conditions with the previous non-interpolative bounds from the literature (e.g., those relying solely on L^∞ or weaker integrability), to make the refinement explicit.
Notation for the parabolic cylinder and the weak-solution definition in Section 2 should include a short reminder of the singular range p ≤ 2 to orient readers before the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and the positive assessment of their significance in providing an interpolative refinement of gap bounds for singular parabolic double-phase problems. The recommendation for minor revision is noted, but the report lists no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives conditional gradient higher integrability for weak solutions to the inhomogeneous singular parabolic double phase equation under two explicit assumptions on u (global L^∞ boundedness or u ∈ C(0,T; L^s) with s ≥ 2). The gap bounds on q are obtained by direct interpolation between the Hölder exponent α of the coefficient a(x,t) and the standard parabolic scaling factor (p(n+2)−2n)/4 or μ_s, both of which are independent of the target conclusion. No equation reduces a claimed prediction to a fitted parameter by construction, no self-citation is load-bearing for the central result, and the derivation chain rests on standard parabolic estimates without renaming known results or smuggling ansatzes. The result is framed precisely as a refinement under stated hypotheses rather than an unconditional claim, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions visible in the statement; no explicit free parameters, invented entities, or non-standard axioms are named.

axioms (1)

domain assumption Weak solutions satisfy the equation in the distributional sense and belong to the natural Sobolev space for the double-phase operator.
Standard definition invoked to make the higher-integrability statement meaningful.

pith-pipeline@v0.9.0 · 5536 in / 1215 out tokens · 35923 ms · 2026-05-16T17:49:38.734438+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish gradient higher integrability results for weak solutions... under u∈L^∞ and q≤p+α(p(n+2)−2n)/4, or u∈C(0,T;L^s) with s≥2 and q≤p+αμ_s/(n+s)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

gap bound conditions... interpolation refinement of gap bounds in the singular parabolic double phase setting

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

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Arora and S

R. Arora and S. Shmarev. Double-phase parabolic equations with variable growth and nonlinear sources.Adv. Nonlinear Anal., 12(1):304–335, 2023

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Baasandorj, S.-S

S. Baasandorj, S.-S. Byun, and J. Oh. Calder´ on-Zygmund estimates for gen- eralized double phase problems.J. Funct. Anal., 279(7):108670, 57, 2020

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Bahrouni, V

A. Bahrouni, V. D. R˘ adulescu, and D. D. Repovˇ s. Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves. Nonlinearity, 32(7):2481–2495, 2019

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P. Baroni, M. Colombo, and G. Mingione. Harnack inequalities for double phase functionals.Nonlinear Anal., 121:206–222, 2015

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Baroni, M

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Chlebicka, P

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M. Colombo and G. Mingione. Bounded minimisers of double phase variational integrals.Arch. Ration. Mech. Anal., 218(1):219–273, 2015

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Colombo and G

M. Colombo and G. Mingione. Regularity for double phase variational prob- lems.Arch. Ration. Mech. Anal., 215(2):443–496, 2015

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Colombo and G

M. Colombo and G. Mingione. Calder´ on-Zygmund estimates and non- uniformly elliptic operators.J. Funct. Anal., 270(4):1416–1478, 2016

work page 2016

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De Filippis and G

C. De Filippis and G. Mingione. A borderline case of Calder´ on–Zygmund esti- mates for nonuniformly elliptic problems.St. Petersburg Mathematical Journal, 31(3):455–477, 2020

work page 2020

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De Filippis and G

C. De Filippis and G. Mingione. Regularity for double phase problems at nearly linear growth.Arch. Ration. Mech. Anal., 247(5):Paper No. 85, 50, 2023

work page 2023

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Esposito, F

L. Esposito, F. Leonetti, and G. Mingione. Sharp regularity for functionals with (p, q) growth.J. Differential Equations, 204(1):5–55, 2004

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Harjulehto and P

P. Harjulehto and P. H¨ ast¨ o. Double phase image restoration.J. Math. Anal. Appl., 501(1):Paper No. 123832, 12, 2021

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Harjulehto, P

P. Harjulehto, P. H¨ ast¨ o, V. Latvala, and O. Toivanen. Critical variable expo- nent functionals in image restoration.Appl. Math. Lett., 26(1):56–60, 2013

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