{"id":"66ae0c81-1cae-428a-8ca2-b98ed1060431","arxiv_id":"1909.00159","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a p-curl system in a bounded convex domain, weak solutions exist uniquely and their curl lies in L∞, with the bound ‖curl u‖_{L∞} ≤ C‖f‖_{L^{3,1}}^{1/(p-1)}.","lead":"This math paper proves that the curl of a solution to a nonlinear Maxwell-type system is bounded in convex three-dimensional domains, under a mild integrability assumption on the data. It is relevant to researchers studying regularity for degenerate elliptic equations and for models of currents in type-II superconductors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Identity (2.5)–(2.6), the core pointwise computation behind (2.4), is false: the div(A×B) sign is reversed and the replacement for ω·curlcurlω is not a valid vector identity, so the L∞ bound is not proved as written.","rationale":"The reader correctly identifies a dependence on external estimates from [23] and [25]/[28], but the more serious problem is an internal algebraic error in the proof's key pointwise computation. Equation (2.6) is asserted as a direct computation yet fails for a simple divergence-free vector field even in the non-degenerate case p=2. Since (2.4) and all subsequent bounds derive from this computation, the manuscript as written does not establish Theorem 1.2. The theorem may still be true and repairable by a different argument, but the current proof contains a central invalid step. Therefore the appropriate verdict is REJECT of the paper in its present form, with the concrete check above serving as a quick way to confirm or refute the flaw.","tokens_in":7384,"tokens_out":13727,"duration_ms":115377,"concrete_test":"Recompute (2.5)–(2.6) in the simplest case p=2, taking Ω to be a convex domain containing the point (1,1,0), with local vector field ω=(0,0,xy) and f=curlω=(x,−y,0). Evaluate both sides of (2.6); equality fails by 2(x²+y²). Then either supply the corrected vector identity and re-derive (2.4), checking that the boundary term has the required sign, or conclude that the level-set argument in Section 2 does not prove Theorem 1.2. This single symbolic check isolates whether the main estimate is grounded.","verdict_should_be":"REJECT","load_bearing_attack":"The central estimate (1.3) is obtained from the level-set inequality (2.4), which rests entirely on (2.5)–(2.6). Applying div(A×B)=curlA·B−A·curlB with A=G(|ω|)ω and B=curlω gives div(Gω×curlω)=f·curlω − Gω·curlcurlω, so −f·curlω = −div(Gω×curlω) − Gω·curlcurlω; the signs in (2.5) are reversed. The subsequent identity G(|ω|)ω·curlcurlω = −div(G(|ω|)∇|ω||ω|)+G(|ω|)|∇|ω||²+G′(|ω|)|∇|ω||²|ω| is also not true for general divergence-free vector fields. For a concrete counterexample, take p=2, a small ball, and in the quadrant x,y>0 set ω=(0,0,xy) and f=curlω=(x,−y,0). Then divω=0 and div f=0. Direct computation gives −f·curlω = −(x²+y²), div(ω×curlω)=x²+y², and −div(∇|ω||ω|)+|∇|ω||²=0, so the right side of (2.6) is x²+y², not −(x²+y²). Thus inequality (2.4), the convexity boundary control, and the final L∞ estimate are not established by the proof as written. This is an internal algebraic gap, not merely an imported-citation issue.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the steady-state p-curl system (1.1) with the tangential boundary condition u × ν = 0 on a bounded convex domain in R^3. Theorem 1.2 claims existence, uniqueness, and the global L∞ estimate (1.3) for curl u in terms of the L^{3,1} norm of the divergence-free forcing f. The proof is organized as an existence argument by minimization, followed by a three-step regularization: smooth data, smooth domains, and general f. The central step is a level-set inequality (2.4) for |curl u|, which is derived from the pointwise identities (2.5)-(2.6), integrated with the aid of estimates quoted from [23] and a convexity boundary sign from [15].","tokens_in":7732,"tokens_out":9852,"duration_ms":124174,"significance":"If Theorem 1.2 were proved, the result would be a meaningful analogue for curl-type systems of the Cianchi-Maz'ya global gradient boundedness for p-Laplacian systems, with direct relevance to steady-state Bean critical-state models of type-II superconductors. The paper is clearly organized, and the variational existence part and the use of the convexity boundary sign are standard. However, the proof of the L∞ estimate rests on pointwise vector identities that are not correct, so the central claim of the paper is not established as written.","major_comments":[{"comment":"The vector identity in (2.5) has the wrong sign. Applying div(A×B)=curlA·B−A·curlB with A=G(|ω|)ω and B=curlω, and using curl(G(|ω|)ω)=f from (1.1), gives div(G(|ω|)ω×curlω)=f·curlω−G(|ω|)ω·curlcurlω, hence f·curlω=div(G(|ω|)ω×curlω)+G(|ω|)ω·curlcurlω. The manuscript writes −f·curlω on the left-hand side. Since (2.6) and then the level-set inequality (2.4) are obtained from (2.5), this sign error invalidates the derivation of the main estimate.","section":"Section 2, Eq. (2.5)"},{"comment":"The identity G(|ω|)ω·curlcurlω = −div(G(|ω|)∇|ω||ω|)+G(|ω|)|∇|ω||²+G′(|ω|)|∇|ω||²|ω| is not true for general divergence-free vector fields. For a concrete counterexample, take p=2, so G=1, and let ω=(y,z,0) in a small ball centered at (0,1,0). Then divω=0, |ω|=√(y²+z²), and at the center ∇|ω|=(0,1,0), so |∇|ω||²=1 and div(∇|ω||ω|)=0. A direct computation gives curlω=(−1,0,−1) and curlcurlω=(0,0,1), so the left side of the claimed identity is ω·curlcurlω=0, while the right side is 0−0+1=1. Thus (2.6) is false as a pointwise identity, and the subsequent inequality (2.4) is not established.","section":"Section 2, Eq. (2.6)"},{"comment":"The estimates imported from [23] are not shown to apply to the quantity |curl u| appearing in (2.12). As printed, the denominators in (2.13) and (2.14) contain |∇|∇ω||, not |∇|ω||, and the reference concerns a different limiting Ginzburg-Landau system. No argument is given that the level-set estimates transfer to the present p-curl setting, so even if (2.5)-(2.6) were corrected, the integration step from (2.12) to (2.15) would still lack support.","section":"Section 2, Eqs. (2.13)-(2.14)"},{"comment":"The regularity input is attributed inconsistently: the introduction credits [28] for C^{1,α} regularity of weak solutions, while the proof cites the main theorem in [25]. Moreover, the assertion that classical regularity gives u∈C³ on {|curl u|>t} is not derived; for a degenerate quasilinear curl system this is a nontrivial claim and needs either a proof or a precise reference. Since the level-set computation and the boundary integrations in (2.4) require this smoothness, this is a load-bearing gap.","section":"Section 2, Step 1"}],"minor_comments":[{"comment":"The abstract contains a typo: \"cur l\" should be \"curl\".","section":"Abstract"},{"comment":"Scalar products are written ambiguously as juxtapositions, e.g., \"f curl ω\" and \"−f curlω\"; these should be written as f·curlω for clarity and to avoid confusion in the sign-sensitive identities.","section":"Throughout Section 2"},{"comment":"Even if the quoted result is relevant, the displayed estimates use |∇|∇ω|| in the denominators, while the expression in (2.12) has |∇|ω||; the notation should be aligned if those estimates are to be substituted.","section":"Section 2, Eqs. (2.13)-(2.14)"},{"comment":"When approximating the domain by smooth domains Ω_m containing Ω, the manuscript does not explain how the forcing f, a priori defined only on Ω, is extended to Ω_m; this needs to be stated explicitly.","section":"Section 2, Step 2"},{"comment":"There is a stray comma after the closing brace of the third integral in (2.17); it should be removed.","section":"Section 2, Eq. (2.17)"}],"recommendation":"reject","confidential_remarks":"The central L∞ estimate rests entirely on the sign-reversed identity (2.5) and the false identity (2.6). These are not local typographical issues: the derivation of (2.4) collapses, and the main theorem is not proved as written. I see no simple correction within the scope of the manuscript, and therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper claims a global L∞ bound for the curl of weak solutions to a p-curl system in convex domains. That is a natural result, parallel to Cianchi–Maz'ya for p-Laplacian systems, and the existence/uniqueness part of Theorem 1.2 is fine. But the proof of the L∞ estimate collapses: the pointwise identity (2.5)–(2.6) is false.\n\nThe authors state that div(Gω×curlω) = f·curlω − Gω·curlcurlω, which is correct, so −f·curlω = −div(Gω×curlω) − Gω·curlcurlω. The sign on the divergence in (2.5) is reversed. The subsequent replacement for Gω·curlcurlω is also not a valid identity for divergence-free vector fields. A concrete check: take p=2, ω=(0,0,xy), f=curlω=(x,−y,0). Then −f·curlω = −(x²+y²), while the right-hand side of (2.6) evaluates to +(x²+y²). So the sign error is real, not a typo. Since (2.4) is built on (2.6), the convexity boundary argument and both cases p≥2 and p<2 rest on this faulty computation. The final estimate (1.3) is not proved.\n\nThere are secondary issues. Step 1 cites [25] for the C^{1,α} regularity that the introduction attributes to [28]; that mismatch needs fixing. The estimates quoted from [23], (2.13)–(2.14), are taken as black boxes, and there are notation slips (|∇|∇ω|| vs |∇|ω||). But these are minor compared to the failed identity.\n\nWhat the paper does well: it identifies a real gap in the literature, sets up the variational framework cleanly, and the convexity boundary estimate from [15] is standard. The existence part via the convex functional is correct. If the authors can replace (2.5)–(2.6) with a correct identity, the rest of the argument might be salvageable. As written, though, the main theorem is unsupported.\n\nI would not cite this paper in its current form. If it comes to me for review, I would reject, with a clear explanation of the sign error. A desk reject seems appropriate; the flaw is elementary and central.","headline":"The main L∞ estimate is unproved: the core identity (2.5)–(2.6) has a sign error and is false for divergence-free vector fields, so the paper's central theorem collapses as written.","tokens_in":8280,"tokens_out":7846,"would_cite":false,"duration_ms":71689,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","46E40","35Q61","82D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for a bounded convex domain $\\Omega\\subset\\mathbb{R}^3$, the p-curl system with divergence-free data in $L^{3,1}$ has a unique weak solution whose curl is globally bounded, with explicit dependence on the data's norm.","keywords":["p-curl system","curl operator","convex domain","Lorentz space","L-infinity estimate","weak solution","Bean critical-state model","type-II superconductor"],"falsifier":"Fix $p=2$, take the unit cube as $\\Omega$ and a fixed smooth divergence-free $f$ with finite $L^{3,1}$ norm; approximate the cube by smooth convex domains and compute $\\max|\\mathrm{curl}\\,u_m|$ for the corresponding solutions. If this maximum grows without bound as the approximation approaches the cube while $\\|f\\|_{L^{3,1}}$ stays fixed, the theorem's $L^\\infty$ estimate would be false.","tokens_in":7171,"feed_emoji":"🧲","tokens_out":11032,"duration_ms":92987,"temperature":0.7,"pith_summary":"This paper studies the semilinear p-curl system $\\mathrm{curl}(|\\mathrm{curl}\\, u|^{p-2}\\,\\mathrm{curl}\\, u)=f$ on a bounded convex domain $\\Omega\\subset\\mathbb{R}^3$, with $\\mathrm{div}\\,u=0$ and $u\\times\\nu=0$ on the boundary. The main result is that when the data $f$ is divergence-free and lies in the Lorentz space $L^{3,1}(\\Omega)$, there is a unique weak solution and its current density $\\mathrm{curl}\\,u$ is globally bounded, with $\\|\\mathrm{curl}\\,u\\|_{L^\\infty(\\Omega)}\\le C\\|f\\|_{L^{3,1}(\\Omega)}^{1/(p-1)}$. A sympathetic reader should care because $\\mathrm{curl}\\,u$ is the total current density in the Bean critical-state model for type-II superconductors, so the theorem turns a physically expected statement into a provable estimate on nonsmooth convex geometries, and it places the p-curl system on the same footing as the p-Laplacian in terms of derivative boundedness.","feed_headline":"Curl proven bounded for p-curl system in convex domains","feed_subtitle":"The estimate controls superconducting current density from Lorentz-space data, covering nonsmooth convex shapes.","key_machinery":"The mechanism is a level-set integration scheme over the superlevel sets $\\{|\\mathrm{curl}\\,u|>t\\}$. Writing $\\omega=\\mathrm{curl}\\,u$ and $G(s)=s^{p-2}$, the paper combines the identity $\\mathrm{div}(A\\times B)=\\mathrm{curl}\\,A\\cdot B-A\\cdot\\mathrm{curl}\\,B$ with a curl-curl expansion to obtain the differential identity $-\\mathrm{div}(G(|\\omega|)\\omega\\times\\mathrm{curl}\\,\\omega)+\\mathrm{div}(G(|\\omega|)\\nabla|\\omega||\\omega|)=(p-1)|\\omega|^{p-2}|\\nabla|\\omega||^2-f\\cdot\\mathrm{curl}\\,\\omega$. Integrating this over $\\{|\\omega|>t\\}$ and applying Cauchy's inequality produces the level-set inequality (2.4), whose boundary term is controlled by convexity: the second fundamental form of $\\partial\\Omega$ contributes $B(\\omega_\\tau,\\omega_\\tau)\\le0$ on the boundary, where $\\omega_\\tau$ is the tangential part of $\\omega$. Coarea-type estimates from [23] then bound the integrated terms by $\\|f\\|_{L^{3,1}(\\Omega)}$, leaving a polynomial inequality in the superlevel parameter $T$ that forces $\\|\\omega\\|_{L^\\infty(\\Omega)}$ to be finite and satisfy the stated power bound.","core_discovery":"The paper's central claim is Theorem 1.2: for a bounded convex domain $\\Omega\\subset\\mathbb{R}^3$, every divergence-free $f\\in L^{3,1}(\\Omega)$ gives a unique weak solution $u\\in W^p_t(\\Omega,\\mathrm{div}\\,0)$ to the system, and the bound (1.3) holds: $\\|\\mathrm{curl}\\,u\\|_{L^\\infty(\\Omega)}\\le C\\|f\\|_{L^{3,1}(\\Omega)}^{1/(p-1)}$ with $C$ depending only on $p$ and $\\Omega$. The proof first establishes the estimate under the extra assumptions of smooth boundary and smooth data, then removes both assumptions by approximation: smooth convex domains $\\Omega_m$ converging to $\\Omega$ in the Hausdorff distance, and smooth divergence-free approximations of $f$ in $L^{3,1}$. The $L^\\infty$ bound is obtained not by pointwise comparison but by controlling the distribution of the superlevel sets of $|\\mathrm{curl}\\,u|$ through a differential inequality in the level parameter $t$, which yields a polynomial bound and then the desired $L^\\infty$ estimate in both cases $p\\ge2$ and $p<2$.","pith_inferences":["The proof uses convexity only to make the boundary term non-positive; a natural test is whether a non-convex Lipschitz domain with a reentrant corner admits a divergence-free $f\\in L^{3,1}$ for which $\\mathrm{curl}\\,u$ becomes unbounded, which would show the convexity assumption is sharp.","Because the level-set estimates force a polynomial inequality in $T$, one could try to extract a modulus of continuity or Hölder exponent for $\\mathrm{curl}\\,u$ from the same inequalities; the paper does not state such a regularity result.","The Lorentz space $L^{3,1}$ is slightly smaller than $L^3$; a plausible extension is to test whether the same method fails at the endpoint $L^3$, or whether a weak-type $L^{3,\\infty}$ datum still yields some unbounded but locally integrable current."],"forward_implications":["The current density $\\mathrm{curl}\\,u$ is bounded up to the boundary in every bounded convex domain, including polyhedral and other nonsmooth convex shapes, for every $p\\in(1,\\infty)$.","The estimate is quantitative: the $L^\\infty$ norm of the current grows at most like the $(p-1)$-th root of the $L^{3,1}$ norm of the data, so small data gives controllable currents.","Uniqueness follows from strict convexity of the energy $\\int_\\Omega(\\frac1p|\\mathrm{curl}\\,u|^p-f\\cdot u)\\,dx$, so the solution map from divergence-free data to current density is well defined.","The result transfers the p-Laplacian phenomenon of global derivative boundedness to the curl setting, reinforcing the elliptic character of the p-curl system."],"supporting_citations":[{"why":"Supplies the boundary regularity result used in Step 1: weak solutions are smooth up to the boundary when the domain and data are smooth.","marker":"[25]"},{"why":"Supplies the level-set integral estimates (2.13) and (2.14) that turn superlevel-set integrations into bounds controlled by the L^{3,1} norm of f.","marker":"[23]"},{"why":"Supplies the boundary identity involving the second fundamental form, used to show the boundary term is non-positive on convex domains.","marker":"[15]"},{"why":"Provides smooth convex domains that approximate the original domain in the Hausdorff distance, used to remove the smoothness assumption on the boundary.","marker":"[22]"},{"why":"Provides the variational existence and uniqueness theorem for the strictly convex energy functional defining weak solutions.","marker":"[19]"},{"why":"Provides the divergence-free smooth approximation of the data in the Lorentz space, used to remove the smoothness assumption on f.","marker":"[12]"},{"why":"Supplies the limiting step that passes from smooth approximated data to the general divergence-free L^{3,1} data.","marker":"[11]"}],"fun_headline_variants":["Bounded curl for p-curl systems on convex shapes","Lorentz-space data yields L∞ curl bound on convex domains","Curl controlled for p-curl systems on nonsmooth convex domains","Global curl boundedness for p-curl systems in convex domains","From L^{3,1} to L∞: curl bounded for p-curl systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on two imported results: that weak solutions are smooth up to the boundary on smooth domains, and that the level-set estimates from [23] hold verbatim for $|\\mathrm{curl}\\,u|$; if either is not available, the claimed $L^\\infty$ bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Bounded curl for p-curl systems on convex shapes","Lorentz-space data yields L∞ curl bound on convex domains","Curl controlled for p-curl systems on nonsmooth convex domains","Global curl boundedness for p-curl systems in convex domains","From L^{3,1} to L∞: curl bounded for p-curl systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000382,"raw_usage":{"total_tokens":1968,"prompt_tokens":831,"completion_tokens":1137,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":1044}},"tokens_in":447,"tokens_out":1137,"duration_ms":11097,"temperature":1.0,"reasoning_tokens":1044,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:59:55.365633+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $p=2$, take the unit cube as $\\Omega$ and a fixed smooth divergence-free $f$ with finite $L^{3,1}$ norm; approximate the cube by smooth convex domains and compute $\\max|\\mathrm{curl}\\,u_m|$ for the corresponding solutions. If this maximum grows without bound as the approximation approaches the cube while $\\|f\\|_{L^{3,1}}$ stays fixed, the theorem's $L^\\infty$ estimate would be false.","supporting_citations":[{"cited_title":"Yin, On a singular limit problem for nonlinear Maxw ell’s equations, J","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary regularity result used in Step 1: weak solutions are smooth up to the boundary when the domain and data are smooth."},{"cited_title":"Xiang, L∞ estimate for a limiting form of Ginzburg-Landau systems in c onvex domains, J","cited_arxiv_id":null,"evidence_quote":"Supplies the level-set integral estimates (2.13) and (2.14) that turn superlevel-set integrations into bounds controlled by the L^{3,1} norm of f."},{"cited_title":"Grisvard, Elliptic problems in nonsmooth domains, P itman Advanced Pub","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary identity involving the second fundamental form, used to show the boundary term is non-positive on convex domains."},{"cited_title":"Verchota, Layer potentials and regularity for the Di richlet problem for Laplace’s equation in Lipschitz domains, J","cited_arxiv_id":null,"evidence_quote":"Provides smooth convex domains that approximate the original domain in the Hausdorff distance, used to remove the smoothness assumption on the boundary."},{"cited_title":"Struwe, Variational Methods, 3rd edition, Springer -Verlag, Berlin, 2006","cited_arxiv_id":null,"evidence_quote":"Provides the variational existence and uniqueness theorem for the strictly convex energy functional defining weak solutions."},{"cited_title":"Costabel, A remark on the regularity of solutions of M axwell’s equations on Lipschitz domains, Math","cited_arxiv_id":null,"evidence_quote":"Provides the divergence-free smooth approximation of the data in the Lorentz space, used to remove the smoothness assumption on f."},{"cited_title":"Cianchi, V.G","cited_arxiv_id":null,"evidence_quote":"Supplies the limiting step that passes from smooth approximated data to the general divergence-free L^{3,1} data."}],"review_version":1}