{"id":"cff37173-2926-4a07-8a75-2ade90c5f21a","arxiv_id":"2608.00199","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Gradient boundedness in spherical cones holds precisely when the first nontrivial cross-sectional Laplacian eigenvalue is ≥ N−1; weighted Lipschitz and second-order estimates hold for p-Laplace equations.","lead":"The paper proves a sharp spectral condition for boundedness of gradients of solutions to elliptic equations on spherical cone domains, and establishes weighted regularity for p-Laplace equations. The result separates the cone geometries where solutions are Lipschitz at the tip from those where gradients necessarily blow up.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nonlinear theorems import the C^{1,α} estimate (3.13) from an unpublished preprint, and the domain Σ_{1/2,1} may not satisfy the hypotheses needed.","rationale":"The reader's weakest_assumption correctly identifies the dependence on [3]'s unpublished estimate as the main risk. I agree that the linear theorem is solid and the p-Laplace theorems are conditional on (3.13). My stress-test sharpens this: even if [3] is correct for C^{1,β} domains, the specific domain Σ_{1/2,1} to which (3.13) is applied has edges at the spherical caps, so it may fall outside the hypotheses of [3]. This is a concrete gap in the written proof, not just a lack of independent verification. The suggested test—checking the applicability of [3]'s theorem to Σ_{1/2,1}—would settle whether the concern lands. If [3] turns out to cover such truncated cones, the nonlinear results stand without further issue; if not, the scaling argument must be repaired, e.g., by proving a local boundary estimate away from the edges and handling the caps as interior surfaces. Therefore the conditional verdict is appropriate and should remain: the linear theorem is strong, while the nonlinear theorems depend on an unverified and possibly inapplicable imported estimate.","tokens_in":28331,"tokens_out":13886,"duration_ms":137995,"concrete_test":"Examine arXiv:2601.07140 [3], Theorems 1.2–1.3, and list all hypotheses on the domain and boundary data. Check whether the truncated cone Σ_{1/2,1} (with edges at the spherical caps) satisfies those hypotheses, and whether the theorem yields the quantitative estimate (3.13) with constants depending only on N,p,δ,β,diam, Lipschitz characteristic, ∥∂Σ_{1/2,1}∥_{C^{1,β}}, ∥v_R∥_{W^{1,p}}, and ∥f_R∥_{L^{N+δ}}. If the theorem requires the full boundary to be C^{1,β} or imposes boundary conditions on the whole boundary, the estimate is not available, and Theorem 1.3's proof needs a local boundary regularity argument for domains with mixed conditions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 (λ1≥N−1 replaces convexity) is well supported: the proof reduces to the classical spectral expansions of Dauge/Kozlov–Maz'ya–Rossmann and the algebra ξ1≥1 ⟺ λ1≥N−1 is correct. The load-bearing weakness is in Theorems 1.3–1.4. The key step is (3.13), a quantitative C^{1,α} boundary estimate for the p-Laplace equation, imported from [3], an arXiv preprint by the first author with no independent verification. The scaling argument in §3 cannot close without it: inequalities (3.18)–(3.22) use (3.13) to control ∥R∇u0∥_{L∞} and [R^{1+α}∇u0]_{C^{0,α}} on each annulus, and the constants' independence of R is essential. There is a further, sharper problem: (3.13) is applied to v_R on Σ_{1/2,1}, but the full boundary of Σ_{1/2,1} has edges where the lateral boundary ∂Σ meets the spherical caps Σ∩∂B_R and Σ∩∂B_{R/2}. Under the paper's own Definition 2.1, Σ_{1/2,1} is not a C^{1,β} domain, and v_R satisfies no boundary condition on the caps. It is therefore unclear that [3]'s theorem applies as stated, even granting its internal correctness. The linear theorem, by contrast, does not use this estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies regularity at the vertex of spherical sectors Σ_D∩B_2 in R^N. For the Poisson problem with Dirichlet or Neumann boundary conditions, Theorem 1.1 shows that if f∈L^q and the first eigenvalue λ1(D) of the Laplace–Beltrami operator on the spherical cross-section D satisfies λ1(D)≥N−1, then weak solutions are globally Lipschitz (λ1=N−1) or C^{1,α} (λ1>N−1), with an explicit finite expansion (1.4). The proof uses the spectral decomposition of Dauge and Kozlov–Maz'ya–Rossmann, together with the exact equivalence ξ1≥1 ⟺ λ1≥N−1. For p-Laplace equations, Theorems 1.3–1.4 claim weighted gradient bounds, weighted Hölder continuity of |x|^{1+α}∇u, and W^{1,2} regularity of the stress field, via a scaling argument that relies on a quantitative C^{1,α} estimate imported from the first author's preprint [3]. The paper also cites Maz'ya's example to argue sharpness of the spectral threshold.","tokens_in":28671,"tokens_out":20772,"duration_ms":211462,"significance":"The linear theorem is a clean and attractive result: it replaces convexity by the spectral condition λ1(D)≥N−1, and the connection to the leading singular exponent ξ1 is exact and parameter-free. The proof is grounded in classical spectral theory and appears essentially sound. If correct, this is likely to be useful for regularity theory in non-smooth domains. The nonlinear results are more conditional: their proof depends on an unpublished estimate, and as written applies it to a domain with corners. These issues must be resolved before the p-Laplace claims can be accepted, but they do not undermine the linear part.","major_comments":[{"comment":"The estimate (3.13) is invoked for v_R on Σ_{1/2,1}. Under Definition 2.1, Σ_{1/2,1} is not a C^{1,β} domain: its boundary has edges where the lateral boundary ∂Σ meets the spherical caps |x|=1 and |x|=1/2. Moreover, v_R satisfies a homogeneous boundary condition only on ∂Σ∩(B_1\\B_{1/2}); on the caps it satisfies no boundary condition at all. Thus [3, Theorems 1.2–1.3] cannot be applied as stated. This estimate is the sole mechanism that yields the R-independent C^{1,α} bounds (3.18)–(3.22), and hence the conclusions (1.11)–(1.12). The proof must be repaired, for example by a localization that treats the caps as interior surfaces, or by a genuine regularity result for the mixed boundary value problem on the truncated cone. As written, Theorems 1.3 and 1.4 are not proven.","section":"§3, Eq. (3.13)"},{"comment":"The central quantitative ingredient (3.13) is taken from the first author's arXiv preprint [3], with no statement of the theorem's hypotheses and no proof. The scaling argument cannot close without this estimate. The authors should either include a complete proof of the imported estimate in an appendix, or state its precise hypotheses and verify them for the scaled solutions v_R. In particular, the issue of the artificial caps and the absence of boundary conditions on them is not addressed. Without access to the proof or a clear verification, the p-Laplace theorems rest on an unverified external result.","section":"§3, dependency on [3]"},{"comment":"The assertion that ∂Σ_{r,R} is of class C^{1,β} is inconsistent with Definition 2.1. The boundary has dihedral-type edges at ∂Σ∩∂B_r and ∂Σ∩∂B_R. This is not merely a wording problem; it is exactly the reason why the application of (3.13) fails. The remark should be corrected to say 'piecewise C^{1,β}' or 'C^{1,β} away from the edges'.","section":"Remark 2.2"}],"minor_comments":[{"comment":"'Weighted global lipschitzianity' should be 'weighted Lipschitz regularity' for clarity.","section":"Abstract"},{"comment":"The Neumann/Dirichlet alternatives are typeset awkwardly; please clarify the summation index and the exponent notation.","section":"Eq. (1.4)"},{"comment":"The notation B_{7/4R} is ambiguous; write B_{7R/4}.","section":"Eq. (3.4)"},{"comment":"There is a typo in the header: 'POL V ARA' should be 'POLVARA'.","section":"Author header"},{"comment":"Citing [3] for the C^{1,α} regularity of v_{ε,k} on Σ_{t/4,3t/2} is not necessary; standard interior regularity (e.g., Lieberman [36]) suffices and avoids the same domain-regularity issue.","section":"Lemma 3.2, Step 1"}],"recommendation":"major_revision","confidential_remarks":"The linear part of the paper is attractive and likely correct, and it deserves to be published. The nonlinear part, however, depends on an unpublished preprint and contains a concrete gap in the application of the imported estimate. I would advise the editor that acceptance should be conditioned on the authors either providing a self-contained proof or a precise, verified statement of the needed C^{1,α} estimate, and on fixing the application to the truncated cone. Consider whether the journal is comfortable with a central dependence on an as-yet-unpublished preprint."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The linear half of this paper is genuinely useful and essentially correct; the nonlinear half is a conditional addition that currently rests on an unpublished preprint and an application that likely doesn't meet that preprint's hypotheses.\n\nWhat's new: the explicit identification of λ1(D) ≥ N−1 as the threshold separating Lipschitz from non-Lipschitz behavior at conical vertices for the Poisson problem, for both Dirichlet and Neumann conditions. As far as I know that statement hasn't been written down before; it sharpens the classical Kondrat'ev/Maz'ya/Dauge expansion into a clean spectral condition. The proof via the eigenfunction expansion is transparent and correct: the equivalence ξ1≥1 ⟺ λ1≥N−1 is exact, and the reduction to Dauge's Theorem 6.4 checks out. The remarks about when equality actually gives C^1 are a nice touch. This part deserves a serious referee.\n\nSoft spots: The p-Laplace theorems (1.3–1.4) depend on the quantitative C^{1,α} boundary estimate (3.13), imported from the first author's arXiv preprint [3]. No independent verification of [3] is supplied, so the conclusions are conditional on that result. More concretely, the proof applies (3.13) to v_R on the annular sector Σ_{1/2,1}. That domain has edges where the lateral boundary meets the spherical caps, and v_R satisfies no boundary condition on the caps. Under the paper's own Definition 2.1, Σ_{1/2,1} is not C^{1,β}, so it is not clear that [3]'s theorem applies as stated. This is a real gap, not a technicality. The linear theorem does not use this step, so it is unaffected. There are also minor typos in the displayed estimates in (3.16)–(3.17), but they don't change the argument.\n\nSummary: This paper is worth engaging for the linear threshold and for the interesting conjecture connecting λ1(D) to other symmetry problems. The nonlinear results are plausible but should be treated as provisional until [3] is published or the estimate is verified for the actual domain. I'd send this to peer review—the linear theorem justifies that. For my own work I'd cite the linear part, not the p-Laplace part.","headline":"The linear threshold result is solid and new; the p-Laplace theorems are conditional on an unverified boundary-regularity estimate applied to a domain that likely doesn't satisfy its hypotheses.","tokens_in":29217,"tokens_out":2975,"would_cite":true,"duration_ms":32171,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35J60","35J92","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a single spectral quantity—the first nontrivial eigenvalue of a spherical domain—decides whether solutions of elliptic equations on cones have bounded gradients.","keywords":["elliptic regularity","conical singularities","spherical sectors","p-Laplacian","gradient bounds","spectral threshold","Laplace–Beltrami eigenvalues","second-order regularity"],"falsifier":"Take a smooth nonconvex spherical domain D with λ1(D) = N−1, solve the Poisson problem with a smooth right-hand side, and check whether the gradient remains bounded at the vertex; the paper predicts Lipschitz but generally not C¹ behavior. A single example with unbounded gradient at λ1(D) = N−1 would disprove the linear theorem, while an example at λ1(D) < N−1 with bounded gradient would disprove sharpness.","tokens_in":28183,"feed_emoji":"📐","tokens_out":4516,"duration_ms":49371,"temperature":0.7,"pith_summary":"The paper studies solutions of Poisson and p-Laplace equations in a spherical sector—a cone cut out by a domain D on the unit sphere—and asks when their gradients stay bounded at the cone tip. Its central result is that for the linear Poisson problem, bounded gradients are governed by one number: the first nontrivial eigenvalue λ1(D) of the Laplace–Beltrami operator on D. If λ1(D) = N−1 every weak solution is Lipschitz; if λ1(D) > N−1 solutions are C^{1,α} with zero gradient at the vertex; if λ1(D) < N−1 a counterexample shows gradients can blow up. Thus the spectral condition replaces convexity, which had been the standard sufficient hypothesis. For p-Laplace equations the paper proves weighted global Lipschitz and second-order regularity under a separate scaling argument.","feed_headline":"One eigenvalue decides gradient regularity on cones","feed_subtitle":"In nonconvex cones, a single spectral number replaces convexity as the sharp condition for bounded gradients.","key_machinery":"The spectral decomposition of the Laplacian in the cone writes a solution as a regular part plus finitely many singular harmonic modes P_i = r^{ξ_i}a_i(θ), where the exponents satisfy ξ(ξ+N−2)=λ_i(D). The threshold λ1(D)=N−1 becomes ξ1=1, exactly the exponent separating Lipschitz from non-Lipschitz radial behavior. For the nonlinear theorems, the load is carried by scale invariance: rescaling solutions to unit-width annuli and applying boundary regularity estimates, the paper controls |x|∇u uniformly across all scales.","core_discovery":"The paper establishes that the regularity of solutions at the tip of a cone is not determined by convexity of the spherical cross-section but by the position of λ1(D) relative to N−1. The mechanism is an explicit spectral decomposition: near the vertex a solution behaves like a finite sum of harmonic functions r^{ξ_i}Y_i(θ), with exponents ξ_i computed from the eigenvalues λ_i(D); the smallest exponent ξ1 crosses 1 exactly when λ1(D) crosses N−1. Lipschitz regularity holds when ξ1 ≥ 1, and C^{1,α} with vanishing gradient when ξ1 > 1. For the p-Laplacian, the paper shows via scaling that |x|∇u is bounded and |x|^{1+α}∇u is Hölder, and that the stress field has weighted square-integrability.","pith_inferences":["The sharp threshold suggests a broader principle: conical-domain regularity and symmetry results known for convex cross-sections should first be tested at λ1(D) ≥ N−1; the paper explicitly conjectures such an extension for symmetry problems.","One can numerically probe the equality case λ1(D) = N−1 on, say, a spherical lune on S², solving the Poisson equation with smooth data and checking whether the gradient is bounded yet not C¹; the paper's Remark 4.2 predicts such exceptional domains exist.","The nonlinear theorems would become fully self-contained if the imported boundary-regularity estimate were independently proved; until then, the linear threshold theorem is the robust core of the paper.","Because the argument uses only radial scaling and spectral expansion, the same threshold may transfer to other operators or to manifolds with conic singularities, but that is an extrapolation beyond the paper."],"forward_implications":["If the linear theorem is correct, all weak solutions of the Poisson problem on spherical sectors with λ1(D) ≥ N−1 are globally Lipschitz, even when the cross-section is nonconvex.","The threshold is sharp: below N−1, unbounded gradients can occur even for smooth bounded data.","When λ1(D) > N−1, solutions are differentiable at the vertex with gradient zero, so the singularity of the cone is fully smoothed out.","For p-Laplace equations, the weighted gradient bound |x|∇u ∈ L∞ holds for every p > 1 and for both Dirichlet and Neumann conditions, with a similar weighted second-order estimate for the stress field.","The spectral condition connects to known symmetry results, suggesting that convexity in problems like the critical exponent equation and isoperimetric inequalities may be replaceable by λ1(D) ≥ N−1."],"fun_headline_variants":["Cone gradient regularity: one eigenvalue rules","For cones, λ1≥N−1 gives Lipschitz gradient","Cone smoothness decided by a single eigenvalue","No convexity? Eigenvalue pinpoints cone regularity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The nonlinear half of the paper depends on a quantitative C^{1,α} boundary-regularity estimate for p-Laplace problems that is taken from another article and is not proved or checked here; the linear threshold theorem, by contrast, rests on classical spectral expansions.","fun_headline_variants_meta":{"raw":{"variants":["Cone gradient regularity: one eigenvalue rules","For cones, λ1≥N−1 gives Lipschitz gradient","Cone smoothness decided by a single eigenvalue","No convexity? Eigenvalue pinpoints cone regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000129,"raw_usage":{"total_tokens":949,"prompt_tokens":728,"completion_tokens":221,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":157}},"tokens_in":472,"tokens_out":221,"duration_ms":3086,"temperature":1.0,"reasoning_tokens":157,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T01:03:07.092930+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth nonconvex spherical domain D with λ1(D) = N−1, solve the Poisson problem with a smooth right-hand side, and check whether the gradient remains bounded at the vertex; the paper predicts Lipschitz but generally not C¹ behavior. A single example with unbounded gradient at λ1(D) = N−1 would disprove the linear theorem, while an example at λ1(D) < N−1 with bounded gradient would disprove sharpness.","supporting_citations":[],"review_version":1}