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arxiv: 1907.03392 · v1 · pith:BVUAK6CInew · submitted 2019-07-08 · 🧮 math.AP

Regularity results for a class of non-autonomous obstacle problems with (p,q)-growth

Cristiana De Filippis This is my paper

Pith reviewed 2026-05-25 01:28 UTC · model grok-4.3

classification 🧮 math.AP
keywords regularityobstacle problems(p,q)-growthnon-autonomousvariational problemsPDEpartial regularity
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The pith

Solutions to non-autonomous obstacle problems with (p,q)-growth are regular under suitable assumptions on the integrands, exponents, and obstacle.

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

The paper establishes regularity results for solutions to obstacle problems where the energy functional is non-autonomous and satisfies (p,q)-growth. This growth allows the integrand to behave with different powers at different points, matching models that arise in materials with inhomogeneous properties. The work is structured so that its conclusions apply to the principal examples already studied in the literature. A reader would care because regularity properties determine whether solutions can be used for further analysis or numerical approximation without uncontrolled singularities.

Core claim

Under suitable assumptions, the analysis covers the main models available in the literature for regularity in non-autonomous obstacle problems with (p,q)-growth.

What carries the argument

The (p,q)-growth condition on the non-autonomous integrand, which supplies the upper and lower bounds needed to close the regularity arguments.

If this is right

  • The regularity statements apply directly to the principal models already present in the literature.
  • The same conclusions hold once the growth exponents p and q and the obstacle meet the listed conditions.
  • Non-autonomous dependence does not require a separate theory from the autonomous case under the given hypotheses.

Where Pith is reading between the lines

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

  • The framework may extend to integrands whose growth oscillates between more than two exponents.
  • Numerical schemes for these obstacle problems could exploit the regularity to obtain error estimates.
  • Related free-boundary problems might inherit the same regularity once the obstacle is removed.

Load-bearing premise

The suitable assumptions on the non-autonomous integrands, the growth exponents, and the obstacle that are needed for the regularity conclusions to hold.

What would settle it

An explicit non-autonomous integrand obeying (p,q)-growth together with an admissible obstacle for which a solution fails to satisfy the claimed regularity.

read the original abstract

We study some regularity issues for solutions of non-autonomous obstacle problems with $(p,q)$-growth. Under suitable assumptions, our analysis covers the main models available in the literature.

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 examines regularity properties for solutions to non-autonomous obstacle problems involving integrands with (p,q)-growth. It asserts that, under suitable assumptions on the integrands, growth exponents, and the obstacle, the results encompass the primary models in the literature.

Significance. If the stated regularity conclusions hold under the paper's assumptions, the work offers a unified treatment of regularity for obstacle problems with non-autonomous (p,q)-growth, potentially consolidating several models from the literature. This is a standard but useful contribution in the calculus of variations; no machine-checked proofs, reproducible code, or parameter-free derivations are indicated.

minor comments (3)
  1. [Abstract] Abstract: the qualifier 'under suitable assumptions' is too vague to convey the scope; the abstract should briefly list the key structural assumptions (e.g., on the modulus of continuity of the non-autonomous term or on the obstacle) that enable the coverage claim.
  2. [Introduction] Introduction (or §2): when claiming coverage of 'the main models available in the literature,' each cited model should be paired with an explicit verification that its integrand satisfies the paper's hypotheses; otherwise the coverage statement remains formal.
  3. Notation: the distinction between the growth exponents p and q and the variable exponents p(x), q(x) should be made uniform throughout; inconsistent use of parentheses versus subscripts appears in several places.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. The report provides a concise summary of the work but does not raise any specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a theoretical analysis of regularity for obstacle problems in PDEs, with the central claim explicitly qualified by 'under suitable assumptions' and limited to covering main models in the literature. The abstract contains no equations, derivations, or parameter-fitting steps. No load-bearing steps reduce by construction to inputs, self-citations, or ansatzes; the structure is a standard qualified theorem statement without self-referential reduction. This matches the default expectation for non-circular mathematical papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all content is treated as standard background in PDE regularity theory.

axioms (1)
  • standard math Standard functional-analytic setting of Sobolev spaces and variational inequalities
    Implicit in any obstacle-problem regularity paper.

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