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Asymptotics of Large Solutions of p-Laplace Equations on Cylinders Becoming Unbounded

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Pith's one-line read On long cylinders, p-Laplace solutions with either finite or blow-up boundary data converge locally in Sobolev norm, with gradient error at most $C/\ell^{1/p}$.

desk verdict A clean rate proof for a known convergence, but the large-solution theorem depends on an unpublished companion paper and needs a verifiable proof. read the letter →

arxiv 2505.23600 v1 pith:TWBIO53Y submitted 2025-05-29 math.AP

N. N. Dattatreya This is my paper
classification math.AP MSC 35B4435A0135J9235J6235J25
keywords p-Laplaceequationboundaryblow-upsolutionlargeinfinitecylinderasymptoticbehaviorSobolevconvergencerateKeller-Ossermanconditionquasilinearelliptic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to establish that sequences of quasilinear equations $\operatorname{div}(|\nabla u|^{p-2}\nabla u)=f(u)$ on finite cylinders $S_\ell=\ell\omega_1\times\omega_2$ expand, as $\ell\to\infty$, to the infinite cylinder $S=\mathbb{R}^m\times\omega_2$. The central claim is that the solutions $u_\ell$ converge locally, in the $W^{1,p}$ gradient norm, to a solution of the reduced cross-sectional problem on $\omega_2$, at an algebraic rate: on any compactly contained piece $\tilde S$ of $S$, $\|\nabla(u_\ell-u)\|_{L^p(\tilde S)}\le C/\ell^{1/p}$ for large $\ell$. The result is proved simultaneously for two boundary regimes, finite Dirichlet data and boundary blow-up (large) solutions. This matters because it quantifies how accurately a long but finite pipe or cavity can be modeled by the infinite-cylinder profile, and it extends earlier convergence statements, which held only for $p\ge2$ and without an explicit rate, to the full range $1

What carries the argument

The argument is carried by an energy estimate with a carefully chosen test function. One takes $\psi=\phi_\ell^p(u-u_\ell)$, where $\phi_\ell$ is a cut-off that equals 1 on the compact set $\tilde S$ and has gradient of size at most $c_1/\ell$ in the long directions, and uses the vector inequality $|x-y|^p\le c\,(|x|^{p-2}x-|y|^{p-2}y)\cdot(x-y)$ to make the $p$-Laplace difference coercive. The competing terms are controlled by local uniform bounds on $u_\ell$ and $\nabla u_\ell$ that come from comparing with a one-dimensional large solution supplied by the Keller-Osserman condition and from a standard gradient estimate. The possible loss of uniqueness for large solutions is handled separately: Theorem 4.2, whose proof for $2\le p<\infty$ is deferred to the cited manuscript [9] and for $1<p<2$ is said to follow the same lines, identifies the limit $u$ as a cross-sectional solution, after which the same rate estimate applies.

What would settle it

The claim would be refuted by a sequence of blow-up solutions on $S_\ell$, satisfying (A1), whose locally uniform limit (4.2) depends on the axial variable $X_1$ or fails to satisfy the weak formulation on $S$; a concrete numerical check would be to confirm that $\ell^{1/p}\|\nabla(u_\ell-u_\infty)\|_{L^p(\tilde S)}$ remains bounded for $f(s)=s^q$, $q>p-1$, $1<p<2$, as $\ell\to\infty$.

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Extended reading notes

Core claim

Under the Keller-Osserman condition (A1), if $u_\ell$ solves (1.2) on $S_\ell$ with either the Dirichlet data (1.3) or the boundary blow-up data (1.4), and $u$ denotes the corresponding solution of the cross-sectional problem (1.5), then for every $\tilde S\Subset S$ and all sufficiently large $\ell$, $\|\nabla(u_\ell-u)\|_{L^p(\tilde S)}\le C/\ell^{1/p}$, with $C$ independent of $\ell$. The same rate and essentially the same proof cover finite data, so the boundary condition does not enter the local estimate. In the large-solution case, when the uniqueness condition (A2) holds the limit is the unique cross-sectional large solution $u_\infty$; without (A2), Theorem 4.2 still identifies the locally uniform limit $u$ as a weak solution of the infinite-cylinder problem and shows it is independent of the axial variable, hence a solution of the cross-sectional problem.

Load-bearing premise

The argument assumes that the limit of the blow-up solutions is indeed a solution of the reduced cross-sectional problem, and the proof of that identification is delegated to an unpublished companion manuscript; if that identification fails or cannot be verified, the stated convergence rate for large solutions has no well-defined limiting solution to converge to.

Editorial extensions

If this is right

  • For any compact part of the infinite cylinder, the gradient error between a very long finite cylinder and the infinite profile is bounded by $C/\ell^{1/p}$, with the same form for finite and blow-up boundary data.
  • The convergence-rate result, previously available only for $p\ge2$ without an explicit rate, now covers the full quasilinear range $1<p<2$.
  • When the large solution is unique (condition (A2)), the limit of the blow-up solutions is exactly the cross-sectional large solution $u_\infty$; when uniqueness fails, the limit still solves the infinite-cylinder problem and is independent of the axial variable.
  • Because the estimate is local, the boundary data affect the solution only through the uniform bounds used in the proof, not through the rate $\ell^{-1/p}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension not explored in the paper would be to check whether the same $\ell^{-1/p}$ local gradient rate survives other lateral boundary conditions, such as Neumann or periodic data; the locality of the proof suggests it may, provided uniform bounds remain available.
  • If Theorem 4.2 were made self-contained, the large-solution rate would stand unconditionally; until then it inherits a dependence on an unpublished manuscript [9].
  • Only the $L^p$ gradient norm is controlled here, so local uniform pointwise rates are not a consequence; whether an $L^\infty$ version holds is left open.
  • The proof uses only the geometry of the cross-section through volumes and cut-off slopes, so one could try to extract explicit constants $C$ in terms of $\omega_1,\omega_2$ and $f$, which would be useful for numerical approximation of long domains.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies sequences of p-Laplace boundary-value problems on finite cylinders S_ℓ = ℓω1 × ω2 that expand to the infinite cylinder S = R^m × ω2. Two boundary regimes are considered: finite Dirichlet data and infinite (large) boundary data. The main theorems assert local convergence of solutions in the Sobolev norm to a solution of the cross-sectional problem on ω2, with an algebraic rate ||∇(u_ℓ − u)||_{L^p(\tilde S)} ≤ C/ℓ^{1/p}. The paper claims this rate for both finite and large data, for all 1 < p < ∞, and frames the large-data case as an addendum to the author's previous work [9]. Section 3 contains the rate proof for finite data and, under a uniqueness assumption, for large data; Section 4 addresses loss of uniqueness and attempts to identify the limit of large solutions with a cross-sectional solution, relying on an unpublished manuscript by the author.

Significance. If fully established, the results would give a clean unified convergence-rate statement for p-Laplace problems on growing cylinders, covering both finite and blow-up boundary data and all 1 < p < ∞. The finite-data rate (Theorem 1.1) is new, is proved in the paper, and is a useful quantitative complement to prior qualitative convergence results. The large-data portion, however, inherits its key identification step from the unpublished reference [9], and the claimed extension to 1 < p < 2 is not actually proved by the argument in Section 3. These gaps make the central large-solution claim unverifiable in the present form, although the underlying approach appears promising and the missing pieces are potentially restorable within the manuscript's scope.

major comments (3)
  1. [Section 4, Theorem 4.2] Theorem 4.2 is not proved in this paper. For 2 ≤ p < ∞ the proof says 'we refer to [9]', and for 1 < p < 2 it says 'the rest of the proof is similar to [9, Theorem 4.3]'. Reference [9] is an unpublished manuscript by the present author, with no preprint number or repository given. Since Theorem 1.2(1)–(3) for large solutions rely on Theorem 4.2 to define the limit u, to show that u solves (4.1), and to identify u with a cross-sectional solution, the headline rate in Theorem 1.2(3) is not independently verified. The author must either include a complete proof of Theorem 4.2 in this paper or supply a publicly verifiable reference.
  2. [Section 3, Eq. (3.1)] The vector inequality (3.1), |x − y|^p ≤ c (|x|^{p−2}x − |y|^{p−2}y)·(x − y), is valid for a universal constant only when p ≥ 2. For 1 < p < 2 it fails; for example, with x = t and y = t + 1 in R, the left side is about 1 while the right side behaves like (p − 1)t^{p−2}, which tends to 0 as t → ∞. Since (3.1) is the first step in the proof of both Theorem 1.1 and the rate part of Theorem 1.2, the claimed extension to 1 < p < 2 is not proved. The Section 4 argument for 1 < p < 2 establishes convergence, not the stated O(ℓ^{−1/p}) rate.
  3. [Section 4, Theorem 4.2(2)] Theorem 4.2(2) is internally inconsistent: it asserts that u is independent of X2, then defines v(X1) = u(X1,X2) and says that v solves (1.5)–(1.7), which are posed in the X2 variable. The intended statement must be that u is independent of X1 and that v(X2) = u(X1,X2) solves the cross-sectional problem. As written, the statement cannot be true, and this inconsistency also casts doubt on the identification argument that the theorem is supposed to provide.
minor comments (5)
  1. [Section 2, Proposition 2.5] The statement of Proposition 2.5 contains garbled notation: 'div (Q|∇u|)∇u)' should read 'div(|∇u|^{p−2}∇u)'.
  2. [Section 2, Proposition 2.3 proof] In the proof of Proposition 2.3, the phrase 'φ ∈ C∞_c(B_R(x0))' should refer to ψ, the test function, not to the one-dimensional profile φ.
  3. [Section 4, proof of Theorem 4.2] The expression '||u_ℓ − u||_{L^p} ℓ→0− − →0' is malformed; it should read '→ 0 as ℓ → ∞'.
  4. [References] Reference [9] is listed without journal, preprint number, or year. If the manuscript is available online, the full bibliographic data should be supplied; otherwise the proof cannot be checked.
  5. [Throughout] There are numerous typographical issues, including 'wich', 'GakkB otosho', and broken equation formatting such as 'u(x) − − → x∈S ∞'. A careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

The large-solution convergence at the core of Theorem 1.2 is not proved in this paper; Theorem 4.2 defers to the author's own unpublished [9], making the central claim self-citation load-bearing.

  1. self citation load bearing [Section 4, Theorem 4.2 (proof of Theorem 1.2(1)-(2))]
    "Theorem 4.2. Assume ((A1)) (1) u is a weak solution of (4.1). (2) The solution u is independent of X2 variable. if we denote v(X1) = u(X1, X2) for a fixed X2, then v solves (1.5) and (1.7). Proof. Case 1: 2 ≤ p < ∞ For this case we refer to [9]. Case 2: 1 < p <2: ... Rest of the proof is similar to [9, Theorem 4.3]."

    The central claim that large solutions converge to a cross-sectional solution is exactly Theorem 4.2: the limit u solves the infinite-cylinder equation and its slices solve the cross-sectional problem. This theorem is not proved here: for 2≤p<∞ 'we refer to [9]', and for 1<p<2 the key weak-convergence argument is 'similar to [9, Theorem 4.3]'. Reference [9] is an unpublished manuscript by the present author, with no venue or repository. The Section 3 rate proof then uses this imported u as the comparison solution ('The proof of the Theorem 1.2 is the same as the one in Section 3 with u∞ replaced by u'), so Theorem 1.2(3) inherits the same unverified self-citation. This is not a minor pointer; it is the load-bearing step.

  2. self citation load bearing [Section 2, Proposition 2.3 and Corollary 2.4]
    "We now establish bounds; proofs of which are present in [9]. The proof of the first precisely follow argument in [9, Proposition 3.3], and is outlined for completeness."

    Proposition 2.3 (local boundedness by a one-dimensional barrier) and the resulting Corollary 2.4 are the compactness inputs used to pass from the monotone pointwise limit u to a W^{1,p} limit and to control the rate proof. The paper states that the proofs are in [9] and only outlines the first; the barrier ϕ in the outline is the unique solution of a one-dimensional blow-up problem whose existence and uniqueness are also deferred to [9]. This makes a second load-bearing ingredient of the derivation a self-citation to the same unpublished manuscript.

full rationale

The finite-data result (Theorem 1.1) and the algebraic-rate estimate in Section 3 are, conditional on having a cross-sectional limit u, genuine new arguments and do not assume the target rate. That is why this is not a 8 or 10. However, the paper's headline large-solution convergence (Theorem 1.2(1)-(2)) is not proved in the paper: Theorem 4.2 delegates the existence of the limit solution and its identification with the cross-sectional problem to [9], an unpublished manuscript by the present author. The paper even says the current work 'extends the results discussed in [9]' and that the proof of Theorem 1.2 is 'the same as the one in Section 3 with u∞ replaced by u', so the central claim is inherited from the self-citation rather than derived. Proposition 2.3, used throughout, is likewise sourced to [9, Proposition 3.3] with only an outline. Reference [9] has no repository or independent verification. This is load-bearing self-citation and warrants a score of 6: partial circularity, since the rate proof and finite-data theorem carry independent content, but the core large-solution convergence story is outsourced. I also note, as a manuscript flaw that supports the verdict, that Theorem 4.2(2) is internally inconsistent as printed ('u is independent of X2' but v(X1)=u(X1,X2) is said to solve (1.5), a problem in the X2 variable), which further undermines the self-containedness of the limit identification.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. The load-bearing assumptions are the Keller-Osserman and uniqueness conditions, the unstated well-posedness of the finite problem on truncation ends, and the heavy reliance on the author's unpublished [9] for identifying the limit solution. The latter is the main source of circularity burden and correctness risk.

assumptions (6)
  • domain assumption (A1) Keller-Osserman condition: Ψ_p(r) < ∞ for all r > 0
    Used in Corollary 2.4 and Proposition 2.3 to bound u_ℓ locally via a one-dimensional large solution. The paper admits (Section 1) that (A1) is a technical requirement for Theorem 1.1 and could be dropped with an alternative bound.
  • domain assumption (A2) uniqueness condition: lim inf Ψ_p(t)/Ψ_p(ξ t) > 1 for ξ in (0,1)
    Assumed in Theorem 1.2(2) to identify the limit u with the unique cross-sectional large solution. Without it, the limit can depend on the subsequence or on X_1.
  • domain assumption Well-posedness of the finite boundary problem (1.2), (1.3)
    Existence and uniqueness of finite solutions are cited from [12], [13], [14], but the boundary condition (1.3) is ambiguous on the truncation ends ℓ∂ω_1 × ω_2 because g is defined only for X_2 ∈ ∂ω_2. This is an unflagged assumption in the statement of Theorem 1.1.
  • standard math Comparison principle (Proposition 2.2)
    Cited from Diaz-Letelier [11]. Used to compare u_ℓ with the one-dimensional large solution and to prove monotonicity of the sequence u_ℓ. This is standard but essential.
  • standard math Vector inequalities for the p-Laplacian, including (3.1) and the (|x|+|y|)^{p−2}|x−y|^2 bound
    Used throughout the energy estimates in Sections 3 and 4. These monotonicity inequalities are standard for the p-Laplacian for 1 < p < ∞ with p-dependent constants.
  • ad hoc to paper Identification result from [9] for 2 ≤ p < ∞ and its analogue for 1 < p < 2
    Theorem 4.2 is not self-contained. The p ≥ 2 case is referred to [9], and the 1 < p < 2 case is sketched 'similar to [9, Theorem 4.3]'. Since [9] is by the present author and not publicly available, this is an ad hoc reliance on an unverified source.

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Cite this review

Pith. "Pith review of Asymptotics of Large Solutions of p-Laplace Equations on Cylinders Becoming Unbounded." pith.science (2026). https://pith.science/paper/TWBIO53Y

@misc{pith2026250523600,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of Large Solutions of p-Laplace Equations on Cylinders Becoming Unbounded},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TWBIO53Y}},
  note         = {Machine review of arXiv:2505.23600}
}
abstract

In this article, we study the asymptotic behavior of large solutions for a quasi-linear equation involving the p-Laplacian, defined on a sequence of finite cylindrical domains converging to an infinite cylinder. We demonstrate that the sequence of solutions converges locally, in the Sobolev norm, to a solution of the corresponding cross-sectional problem. Moreover, we establish a convergence rate. As part of our analysis, we extend existing convergence results for the case $p\geq 2$, which previously lacked explicit convergence rates, to the range $1<p<2$. We additionally address solutions with finite Dirichlet boundary data within a unified framework and exhibit that this rate of convergence is independent of the boundary data.

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