{"id":"59d749f4-cb8a-442f-ab46-4ac36ad3f702","arxiv_id":"2505.23600","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On cylinders S_ℓ = ℓω_1 × ω_2, solutions u_ℓ converge locally in gradient L^p norm to the cross-sectional solution at rate at least C ℓ^{-1/p}.","lead":"This paper proves that solutions of a nonlinear p-Laplace equation on finite cylinders that grow longer converge to a solution on the cross-section, and it gives the speed of that convergence. The result covers both standard and boundary-blow-up data and extends earlier work to the range 1 < p < 2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rate in Theorem 1.2(3) for large solutions depends on convergence to the cross-sectional limit supplied solely by unpublished self-cited [9]; Theorem 4.2 is not self-contained, so the central claim lacks a verifiable proof.","rationale":"The reader's weakest assumption is precisely the reliance on unpublished [9] for Theorem 4.2, and I agree that this is the load-bearing gap. The rate estimate itself, once a limit solution u is assumed, is a fairly direct computation; the missing piece is the existence and identification of that limit solution for large solutions. The paper's own Section 4 sketch confirms this: it is titled 'Loss of uniqueness,' and its Theorem 4.2 is where the convergence to the cross-sectional problem is established, but the proof defers to [9] in both cases. The variable typo in Theorem 4.2(2) reinforces that this part of the manuscript is not finished. No internal contradiction appears in the Section 3 bookkeeping, but that section presupposes the limit profile u is a solution. Therefore, the appropriate verdict remains CONDITIONAL: the central claim would be solid if Theorem 4.2 were supplied with a complete, independently verifiable proof; otherwise Theorem 1.2(3) is not established. No change to the reader's verdict is needed.","tokens_in":8776,"tokens_out":15736,"duration_ms":137247,"concrete_test":"Obtain a public version of [9] (or a complete proof of Theorem 4.2 in a revision) and verify that it establishes, under exactly the assumptions here (1<p<∞, f nondecreasing continuous with (A1), Sℓ=ℓω1×ω2, boundary blow-up), that the monotone limit u is a weak solution of the infinite-cylinder problem and is independent of the longitudinal variable; if no such proof is available, Theorem 1.2(3) is unsubstantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline result, Theorem 1.2(3), gives an algebraic rate for large solutions on growing cylinders converging to a cross-sectional solution. The rate proof in Section 3 assumes a limit function u that solves the infinite-cylinder problem and, via Theorem 1.2(2), that u coincides with a cross-sectional large solution u∞. Both ingredients come from Theorem 4.2. But Theorem 4.2 is not actually proved in this paper: for 2≤p<∞ the proof is 'we refer to [9]'; for 1<p<2 it is '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. Passing to the limit in the degenerate p-Laplacian across growing domains, and proving the limit is independent of the longitudinal variable, is exactly the hard core of the large-solution result, yet it is outsourced. Moreover, Theorem 4.2(2) contains a variable inconsistency: it states 'u is independent of X2' but then says v(X1)=u(X1,X2) solves (1.5), which is posed in X2; the intended statement must be independence of X1. Without Theorem 4.2, the monotone limit u is only known to exist by boundedness and monotonicity—not to solve the cross-sectional problem—so the abstract's claim of convergence in the Sobolev norm to a cross-sectional solution is unsupported. This is a verifiability gap in the central claim, not a stylistic issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9104,"tokens_out":10074,"duration_ms":91436,"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":[{"comment":"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.","section":"Section 4, Theorem 4.2"},{"comment":"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.","section":"Section 3, Eq. (3.1)"},{"comment":"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.","section":"Section 4, Theorem 4.2(2)"}],"minor_comments":[{"comment":"The statement of Proposition 2.5 contains garbled notation: 'div (Q|∇u|)∇u)' should read 'div(|∇u|^{p−2}∇u)'.","section":"Section 2, Proposition 2.5"},{"comment":"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.","section":"Section 2, Proposition 2.3 proof"},{"comment":"The expression '||u_ℓ − u||_{L^p} ℓ→0− − →0' is malformed; it should read '→ 0 as ℓ → ∞'.","section":"Section 4, proof of Theorem 4.2"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's core new contribution is the finite-data rate, which appears sound given the estimates in Sections 2–3. The large-data rate is the advertised headline result, but its proof is not self-contained and relies on an unpublished self-citation. The discrepancy between the claimed range 1 < p < 2 and the true domain of validity of (3.1) is a substantive technical gap that the authors should address head-on. I would be willing to review a revised version that includes a complete proof of Theorem 4.2 or an alternative self-contained argument, and that either proves the rate for 1 < p < 2 or restricts the claim accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about arXiv:2505.23600. The rate result is real: for finite Dirichlet data and for large solutions, the author proves a local L^p gradient convergence rate of O(ℓ^{-1/p}) for p-Laplace problems on cylinders S_ℓ = ℓω_1 × ω_2, extending a convergence result previously known for p≥2 without a rate. The Section 3 argument is a clean cut-off and vector-inequality estimate with explicit constants. That part is worth reading.\n\nThe second thing is fragile. The large-solution half depends entirely on Theorem 4.2, which identifies the pointwise limit u of u_ℓ as a solution of the cross-sectional problem (1.5). This theorem is not proved here. For 2≤p<∞ the proof is a reference to [9], an unpublished manuscript by the same author; for 1<p<2 it says the rest is similar to [9, Theorem 4.3]. The rate proof in Section 3 then takes that u as given. So the headline claim of Theorem 1.2(3) is only as solid as an inaccessible citation. That is a verifiability gap, not a minor omission. A referee cannot check the core convergence step.\n\nTwo smaller issues. Theorem 4.2(2) says u is independent of X2 but then defines v(X1)=u(X1,X2) solving (1.5), which is a problem in X2; the intended statement is clearly independence of X1. And the finite boundary data (1.3) is under-specified: g is introduced on ∂S with X2∈∂ω2, but then imposed on the caps ℓ∂ω1×ω2 where g is not defined. The author needs to clarify that g is extended to all of ω2.\n\nThe finite-data Theorem 1.1 is nearly self-contained: the limit is the cross-sectional solution u_∞, and the rate proof goes through with standard existence and regularity. If the boundary condition wording is fixed, that half stands alone.\n\nWho should read this? Specialists in quasilinear elliptic asymptotics who want the rate statement or are following [9]. It is a modest addendum, not a breakthrough, and the author says as much.\n\nMy recommendation: send it to peer review, but the referee should require the author to either prove Theorem 4.2 in full or provide a publicly available version of [9] (e.g., arXiv). Without that, the large-solution theorem is unverifiable. The finite-data part can be accepted on its own merit after the boundary data is clarified.","headline":"A clean rate proof for a known convergence, but the large-solution theorem depends on an unpublished companion paper and needs a verifiable proof.","tokens_in":9663,"tokens_out":9306,"would_cite":false,"duration_ms":83920,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B44","35A01","35J92","35J62","35J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"On long cylinders, p-Laplace solutions with either finite or blow-up boundary data converge locally in Sobolev norm, with gradient error at most $C/\\ell^{1/p}$.","keywords":["p-Laplace equation","boundary blow-up solution","large solution","infinite cylinder","asymptotic behavior","Sobolev convergence rate","Keller-Osserman condition","quasilinear elliptic equation"],"falsifier":"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 $\\ell\\to\\infty$.","tokens_in":8542,"feed_emoji":"📐","tokens_out":13038,"duration_ms":124644,"temperature":0.7,"pith_summary":"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 $1<p<2$.","feed_headline":"Long-cylinder p-Laplace solutions converge locally at rate ℓ^{-1/p}","feed_subtitle":"Same local gradient bound for finite and blow-up boundary data, now proved for 1<p<2.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\ell^{-1/p}$."],"supporting_citations":[{"why":"Supplies the identification of the limiting blow-up solution with a cross-sectional solution in Theorem 4.2; the current paper defers to it for $2\\le p<\\infty$ and says the $1<p<2$ case follows similarly.","marker":"[9]"},{"why":"Provides the comparison principle (Proposition 2.2) used to convert one-dimensional large-solution bounds into pointwise bounds for $u_\\ell$.","marker":"[11]"},{"why":"Supplies the vector inequality (3.1) that makes the $p$-Laplace difference coercive in the energy estimate.","marker":"[19]"},{"why":"Introduced the Keller-Osserman condition that ensures the one-dimensional comparison function in Proposition 2.3 exists.","marker":"[18]"},{"why":"Together with [18], the origin of the Keller-Osserman condition and the existence theory for boundary blow-up solutions used throughout.","marker":"[22]"}],"fun_headline_variants":["p-Laplace on cylinders: local rate ℓ^{-1/p} for all p>1","Blow-up data no barrier: p-Laplace convergence rate holds for 1<p<2","Cylinder p-Laplace solutions: same rate, any boundary data","Local convergence for p<2: p-Laplace on long cylinders at rate ℓ^{-1/p}","Cross-sectional limit: p-Laplace large solutions on long cylinders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["p-Laplace on cylinders: local rate ℓ^{-1/p} for all p>1","Blow-up data no barrier: p-Laplace convergence rate holds for 1<p<2","Cylinder p-Laplace solutions: same rate, any boundary data","Local convergence for p<2: p-Laplace on long cylinders at rate ℓ^{-1/p}","Cross-sectional limit: p-Laplace large solutions on long cylinders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":3015,"prompt_tokens":878,"completion_tokens":2137,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2040}},"tokens_in":494,"tokens_out":2137,"duration_ms":17236,"temperature":1.0,"reasoning_tokens":2040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:43:33.491934+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\ell\\to\\infty$.","supporting_citations":[{"cited_title":"Chowdhury and N","cited_arxiv_id":null,"evidence_quote":"Supplies the identification of the limiting blow-up solution with a cross-sectional solution in Theorem 4.2; the current paper defers to it for $2\\le p<\\infty$ and says the $1<p<2$ case follows similarly."},{"cited_title":"D ´ ıaz and R","cited_arxiv_id":null,"evidence_quote":"Provides the comparison principle (Proposition 2.2) used to convert one-dimensional large-solution bounds into pointwise bounds for $u_\\ell$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the Keller-Osserman condition that ensures the one-dimensional comparison function in Proposition 2.3 exists."},{"cited_title":"Osserman","cited_arxiv_id":null,"evidence_quote":"Together with [18], the origin of the Keller-Osserman condition and the existence theory for boundary blow-up solutions used throughout."}],"review_version":1}