{"id":"c7cb14b6-dcd7-4659-9280-4fa5073a61e0","arxiv_id":"2608.13443","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves log-concavity of the first p-eigenfunction with convex potentials, a sharp one-dimensional gap inequality for every p>1, and a higher-dimensional p=2 collapse dichotomy.","lead":"For the Dirichlet p-Laplacian on convex domains with convex potentials, this paper proves a sharp lower bound on the gap between the first two eigenvalues in one dimension and reveals a clean change of behavior at p=2 in higher dimensions. A generalist should read it because it takes a celebrated linear spectral-gap theorem and shows how it does, and does not, survive in a nonlinear PDE setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The sharp one-dimensional bound for all convex potentials rests on eq. (4.1), an unproved nonlinear comparison principle whose cited source [5] is a p=2 Robin problem.","rationale":"The reader's weakest_assumption is precisely the comparison principle (4.1). I agree. I checked the one-dimensional part of the paper. Part (i) for linear potentials is self-contained: scaling, evenness/concavity of µ̂, nodal decomposition, and the convexity estimate are all given. Part (ii), however, has no independent argument; it is exactly one paragraph invoking (4.1). The paper's own text signals missing support both by 'We omit the proof' (for the nonlinear Lavine identity) and by 'whose comparison argument carries over unchanged.' Because [5] treats p=2 with Robin data, the phrase 'carries over unchanged' is doing a large amount of work. If (4.1) fails for some p, the sharp one-dimensional theorem and the N=1 case of Theorem 1.7 would need revision. I found no comparably serious issue in the higher-dimensional estimates: the regularization and log-concavity proof is long but internally coherent, the weighted Poincaré argument is detailed, and the collapsing-domain dichotomy is backed by explicit constructions. The dependence on unpublished preprints [7] and [65] is a second weakness, but it is less central because those ingredients are numerical constants/identities that can be checked independently. Therefore I would keep the reader's CONDITIONAL verdict; no change.","tokens_in":33067,"tokens_out":10683,"duration_ms":108065,"concrete_test":"Re-derive eq. (4.1) from [5, Theorem 1.2] in the present p-Dirichlet setting, writing the comparison difference Γ_p(I_D,V)-Γ_p(I_D,ax) in terms of u_1,u_2 and tracking the term ∫_{I_D}(u_2 φ_p(u'_2)-u_1 φ_p(u'_1)) dx. If this term is not controlled (it is not identically zero for p≠2, as the paper states), eq. (4.1) does not follow. Independently, a numerical shooting computation for p=1.5 and V(x)=x^2 can check whether Γ_p(I_D,V) < min_a Γ_p(I_D,ax), which would falsify (4.1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 proves Theorem 1.9(i) for linear potentials via nodal decomposition and the concavity of µ̂. Theorem 1.9(ii), which is the sharp bound for every convex V, is then obtained in one step from eq. (4.1): there exists a ∈ R with Γ_p(I_D,V) ≥ Γ_p(I_D,ax). This is the only bridge from arbitrary convex potentials to the sharp constant. The paper describes (4.1) as the homogeneous Dirichlet analogue of [5, Theorem 1.2] and says the comparison argument 'carries over unchanged'; but [5] is a p=2 linear Schrödinger problem with Robin boundary conditions. No statement or proof is supplied for the p-Dirichlet case. The adaptation is not a formality: for p≠2 the equation is nonlinear, the ground-state transform does not linearize the second variation, and the boundary terms in the comparison do not cancel (the paper itself notes, in the Lavine identity discussion, that u_j φ_p(u'_j) is not a derivative and the extra term does not vanish for p≠2). The manuscript also says 'We omit the proof' for the key identity preceding that discussion. Thus Theorem 1.9(ii), the abstract's one-dimensional sharp gap for every convex potential, and the N=1 entry of Theorem 1.7 rest on an unverified theorem-level assertion. This is a missing-support objection, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fundamental gap of the Dirichlet p-Laplacian with convex potentials on bounded convex domains. For N>=2 it proves log-concavity of the positive first eigenfunction, establishes a dichotomy on collapsing convex domains (gap tends to 0 for 1<p<2, to 3 pi^2/D^2 for p=2, and diverges for p>2), and derives dimension-free gap estimates for p>=2, including an enhanced estimate for zero potential. It also proves existence and degeneration of diameter-normalized gap minimizers for p>2. For N=1 it claims the sharp inequality lambda_{2,p}-lambda_{1,p} >= (p-1)(2^p-1)(pi_p/D)^p for every convex potential, with equality exactly for constant potentials. The one-dimensional proof is self-contained for linear potentials, but the extension to arbitrary convex potentials rests on an unproved comparison principle stated in eq. (4.1).","tokens_in":33244,"tokens_out":19849,"duration_ms":192791,"significance":"If the results are correct, this is a substantial contribution: it provides the first systematic fundamental-gap theory for the nonlinear Dirichlet p-Laplacian with convex potentials, identifies p=2 as a critical exponent through collapsing-domain examples, and gives a sharp one-dimensional constant. The higher-dimensional machinery -- degenerate weighted Poincare inequalities, quantitative stability of the L^p-Poincare inequality, and the compactness argument for minimizers -- is carefully developed with explicit constants and appears to be new. The one-dimensional linear-potential calculation via the concave centered eigenvalue function is elegant and fully proved. The main obstacle is the unsupported comparison principle (4.1), which currently prevents the sharp one-dimensional claim for arbitrary convex potentials from being regarded as established.","major_comments":[{"comment":"Equation (4.1) is a theorem-level assertion that is neither proved nor stated precisely, yet it carries the entire extension from linear to arbitrary convex potentials in Theorem 1.9(ii). The text says this is the homogeneous Dirichlet analogue of [5, Theorem 1.2] and that the comparison argument 'carries over unchanged', but [5] treats the linear Schrodinger operator (p=2) with Robin boundary conditions. The adaptation to the nonlinear Dirichlet p-problem is not a formality: for p != 2 the eigenvalue equation is nonlinear, and the paper itself notes two paragraphs earlier that u_j phi_p(u'_j) is not a derivative and the extra term does not vanish for p != 2. Because Theorem 1.9(ii), and hence the abstract's sharp one-dimensional gap for every convex potential, is deduced from (4.1) in a single step, the proof is incomplete at a load-bearing point. The equality characterization 'precisely for constant potentials' in Theorem 1.9(ii) is also unsupported, since it uses the strictness assertion in (4.1). Please supply a full proof of (4.1), or a precise statement of the comparison result with a rigorous demonstration that the argument of [5] extends to the Dirichlet p-problem.","section":"Section 4, Eq. (4.1)"}],"minor_comments":[{"comment":"The nonlinear analogue of Lavine's identity is stated with the sentence 'We omit the proof.' Since this identity is not used in the subsequent argument, it should either be proved, cited to a complete reference, or removed; as written it introduces an unproved statement into the discussion around the main one-dimensional theorem.","section":"Section 4, paragraph before Lemma 1.8"},{"comment":"The expression '(p-1) 2 rho_p^{2-p}' in Theorem 1.5(i) is ambiguous and should be written as '(p-1)^2 rho_p^{2-p}'.","section":"Section 1, Eq. (1.5) and Theorem 1.5(i)"},{"comment":"Section 2 states 'throughout this section, let N>=2', but Theorem 1.1 is stated for N>=1; please indicate explicitly how the one-dimensional case is covered.","section":"Section 2, opening paragraph"},{"comment":"The remark that for p>2 the lower bound 'follows instead from Corollary 1.6 below' is imprecise, because Corollary 1.6(iii) yields the diverging lower bound only after combining (1.12) with the scaling of lambda_1 on the collapsing domains; please spell out the dependence on epsilon.","section":"Section 3.1, after Proposition 1.2"},{"comment":"There are numerous typographical and spacing errors, such as 'providedifferentforms' and missing spaces around displayed equations; a careful copyedit is needed before final publication.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the unproved comparison principle (4.1), cited to [5], a paper co-authored by the second author. I recommend requiring a complete proof or a precise, verifiable reference for this comparison before acceptance. The higher-dimensional parts appear carefully executed and would constitute a solid contribution on their own if the one-dimensional convex-potential claim were either fully proven or separated into a conditional statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short version: this is a real paper with real results, and the main advertised theorem for all convex potentials has a hole that needs filling. The hole is exactly the comparison principle in eq. (4.1). Everything else I checked is in good shape.\n\nWhat's new and solid: log-concavity of the first p-eigenfunction with convex potentials (Theorem 1.1) is proved via regularization and a two-point maximum principle; the proof is detailed and I didn't find a gap. The higher-dimensional dichotomy on collapsing domains (Proposition 1.2) is clean, and the dimension-free gap estimates for p≥2 (Theorems 1.4 and 1.5) are substantial. The one-dimensional case for linear potentials (Theorem 1.9(i)) is the most elegant part: the centered eigenvalue μ̂ is concave and even, and the nodal decomposition reduces the gap to a one-parameter inequality. That part is self-contained and correct as far as I can tell.\n\nThe soft spot: Theorem 1.9(ii) for arbitrary convex V is deduced in one line from (4.1), which asserts the gap is minimized over convex potentials by an affine potential. The paper says this is the Dirichlet analogue of [5, Theorem 1.2] and \"the comparison argument carries over unchanged.\" But [5] is a p=2 Robin problem. For p≠2 the equation is nonlinear, the ground-state transform doesn't linearize the second variation, and the paper itself points out that u φ_p(u') is not a derivative, so the boundary terms in Lavine's argument don't vanish. That means (4.1) is a theorem-level assertion, not a routine adaptation. It's not obviously false—I suspect it's true—but it's load-bearing and needs a proof. A referee should ask for one.\n\nAlso worth noting: the sharp constant c_p and the ground-state identity are quoted from two arXiv preprints ([7], [65]). That's a minor concern; the paper's own linear-potential and higher-dimensional arguments don't depend on those preprints in a way that would collapse, but it would help if the authors stated the precise results or verified them.\n\nWho's it for: anyone working on p-Laplacian eigenvalues, convex domains, or fundamental gaps. It deserves a serious referee and would likely be a strong paper after the comparison principle is supplied (or the theorem restricted to linear potentials, which would still be a meaningful contribution). My verdict: conditional accept if I were editor; the authors should be asked to provide a complete proof of (4.1) or to state the one-dimensional result without convex potentials.\n\nBest,\n\n[Your name]","headline":"A substantial nonlinear spectral geometry paper: log-concavity, higher-dimensional dichotomy, and the one-dimensional linear-potential bound are solid, but the advertised sharp bound for all convex potentials rests on an unproved comparison principle from a p=2 Robin paper.","tokens_in":33866,"tokens_out":3063,"would_cite":true,"duration_ms":30864,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P30","35J92","35P15","49R05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a sharp universal lower bound on the fundamental gap of the Dirichlet p-Laplacian in one dimension, and shows that in higher dimensions the exponent p=2 separates domains with collapsing gaps from domains with growing gaps.","keywords":["Dirichlet p-Laplacian","fundamental gap","convex potential","log-concavity","weighted Poincaré inequality","collapsing domains","sharp one-dimensional bound","p-Laplacian eigenvalues"],"falsifier":"Compute, analytically or numerically, the first two Dirichlet p-eigenvalues on $I=(-1/2,1/2)$ for $p=3$ and $V(x)=x^2$, and compare the gap to $\\min_a[\\lambda_{2,p}(I,ax)-\\lambda_{1,p}(I,ax)]$. If the quadratic-potential gap falls below the best linear-potential gap, the comparison principle in (4.1) is false and the one-dimensional convex-potential theorem would need reproof; if it does not, the key premise survives at least this test.","tokens_in":32722,"feed_emoji":"📐","tokens_out":6560,"duration_ms":62356,"temperature":0.7,"pith_summary":"This paper studies the gap between the first two eigenvalues of the Dirichlet p-Laplacian with a convex potential on a bounded convex domain. In one dimension it proves a sharp universal lower bound for every p>1: $\\lambda_{2,p}-\\lambda_{1,p} \\geq (p-1)(2^p-1)(\\pi_p/D)^p$, with equality exactly for constant potentials. In higher dimensions it shows that p=2 is a critical exponent: there are smooth convex collapsing domains whose gap tends to 0 for $1<p<2$, approaches $3\\pi^2/D^2$ for p=2, and diverges for p>2, while for $p\\geq2$ a dimension-free gap bound depending only on p and the diameter holds under convexity of the potential. These results matter because they extend a classical family of spectral-gap estimates from linear Schrödinger operators to a nonlinear, degenerate elliptic setting, and identify the exact exponent at which convexity still controls the gap.","feed_headline":"Flat potentials minimize the p-Laplacian fundamental gap","feed_subtitle":"Sharp one-dimensional bound, and a higher-dimensional dichotomy with p=2 as the critical case.","key_machinery":"The argument runs on three mechanisms. First, the log-concavity of the positive first eigenfunction, proved by a uniformly elliptic regularization and a two-point maximum principle, turns the measure $u_1^p\\,dx$ into a log-concave weight. Second, a quantitative remainder identity for $|\\xi|^p$---the function $C_p(\\xi,\\eta)=|\\xi|^p-|\\xi-\\eta|^p-p|\\xi-\\eta|^{p-2}(\\xi-\\eta)\\cdot\\eta$, bounded below by $c_p|\\eta|^p$---converts ground-state identities into stability estimates for the $L^p$ Poincaré inequality. Third, a degenerate weighted Poincaré inequality with weight $|\\nabla\\log u_1|^{p-2}$ supplies the $p>2$ dimension-free bound. In one dimension, the central object is the even concave function $\\hat\\mu(\\beta)=\\lambda_{1,p}((0,1),\\beta x)-\\beta/2$, whose concavity yields the sharp constant through a nodal decomposition of the second eigenfunction.","core_discovery":"For $N=1$, the paper claims that for every $p>1$ and every convex potential $V$ on an interval of length $D$, the fundamental gap satisfies $\\lambda_{2,p}(I_D,V)-\\lambda_{1,p}(I_D,V) \\geq (p-1)(2^p-1)(\\pi_p/D)^p$, with equality if and only if $V$ is constant. The proof uses a nodal decomposition of the second eigenfunction and the concavity of a centered first-eigenvalue function to reduce the problem to linear potentials. In dimensions $N\\geq2$, for $p\\geq2$ and convex potentials, the paper establishes a dimension-free gap lower bound $\\lambda_{2,p}-\\lambda_{1,p} \\geq c_p(p-1)(\\pi_p/D)^p$, and for zero potential an enhanced estimate of order $\\lambda_{1,p}^{(p-2)/p}D^{-2}$. On collapsing smooth convex domains it proves a dichotomy: the gap vanishes for $1<p<2$, stays of order $D^{-2}$ for $p=2$, and diverges like $\\varepsilon^{2-p}$ for $p>2$. The paper also proves log-concavity of the positive first eigenfunction for convex potentials, which is the geometric input that enables the weighted Poincaré arguments.","pith_inferences":["The one-dimensional proof's reliance on the unproved comparison principle in (4.1) is the structural weak point: if that principle fails, the sharp bound still holds for linear potentials but the extension to all convex potentials would require a different mechanism.","The higher-dimensional dichotomy suggests that for $1<p<2$ the first two eigenfunctions become asymptotically degenerate in thin domains, so nonlinear p-Laplacian diffusion may display anomalously slow spectral mixing compared with the linear case $p=2$; this is not explored in the paper.","The conjecture that $\\lim_{p\\downarrow2} G_{p,N}=3\\pi^2$, if true, would unify the sharp one-dimensional constant with the classical $p=2$ value and could be tested numerically by solving the eigenvalue problem on the collapsing family of Proposition 1.2 at p close to 2.","A direct numerical check of the comparison principle for $p=3$ and $V(x)=x^2$ on an interval would either secure or break the one-dimensional convex-potential theorem."],"forward_implications":["In one dimension, constant potentials minimize the fundamental gap among all convex potentials, with an explicit constant $(p-1)(2^p-1)(\\pi_p/D)^p$ that depends only on $p$ and the interval length.","For $p\\geq2$, every bounded convex domain with convex potential has a gap lower bound depending only on $p$ and the diameter, generalizing the classical dimension-free estimate for $p=2$.","For $1<p<2$, there exist convex domains of fixed diameter with arbitrarily small fundamental gap, so no positive dimension-free gap bound can hold in that range.","For $p>2$, diameter-normalized gap minimizers exist among bounded convex domains, and any family of such minimizers degenerates as $p\\downarrow2$, so $p=2$ is the only exponent where collapsing prevents attainment.","For zero potential and $p>2$, the gap grows at least like $\\lambda_{1,p}^{(p-2)/p}D^{-2}$, and along the collapsing domains it diverges as $\\varepsilon^{2-p}$."],"supporting_citations":[{"why":"Supplies the one-dimensional comparison principle invoked in (4.1), which reduces the gap for arbitrary convex potentials to the gap for a linear potential.","marker":"[5]"},{"why":"Provides the p=2 integral identity for linear potentials that the nonlinear proof replaces with a nodal decomposition.","marker":"[42]"},{"why":"Supplies the sharp weighted Poincaré inequality for log-concave measures used in Theorem 1.4 and Proposition 1.3.","marker":"[29]"},{"why":"Gives the sharp p=2 constant $3\\pi^2/D^2$ that anchors the higher-dimensional dichotomy and the $p\\downarrow2$ limit.","marker":"[4]"},{"why":"Obtains the zero-potential $L^p$-Poincaré stability estimate that the paper sharpens and extends to convex potentials.","marker":"[65]"},{"why":"Provides the exact ground-state identity with remainder $C_p(\\xi,\\eta)$ used in the stability estimates.","marker":"[7]"},{"why":"Proves log-concavity of the first eigenfunction for $V\\equiv0$, the result the paper adapts to convex potentials.","marker":"[54]"},{"why":"Gives the mountain-pass characterization of $\\lambda_{2,p}$ used to turn stability estimates into gap bounds.","marker":"[22]"},{"why":"Provides the quantitative sharpening of the p=2 gap bound that forces minimizers to degenerate as $p\\downarrow2$.","marker":"[3]"},{"why":"Supplies the inradius (Hersch–Protter) bound used to rule out collapse of minimizing sequences for $p>2$.","marker":"[14]"}],"fun_headline_variants":["Sharp 1D gap bound for p-Laplacian with convex potential","p=2 marks critical transition in p-Laplacian gaps","Log-concave eigenfunctions yield p-Laplacian gap estimates","Exact gap formula for 1D p-Laplacian revealed","Higher-D p-Laplacian gaps: vanish, constant, or diverge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire one-dimensional result for arbitrary convex potentials rests on an unproved comparison principle stated in (4.1): the gap for any convex potential on an interval is at least the gap for some linear potential on that interval. If that comparison fails, Theorem 1.9(ii) loses its support.","fun_headline_variants_meta":{"raw":{"variants":["Sharp 1D gap bound for p-Laplacian with convex potential","p=2 marks critical transition in p-Laplacian gaps","Log-concave eigenfunctions yield p-Laplacian gap estimates","Exact gap formula for 1D p-Laplacian revealed","Higher-D p-Laplacian gaps: vanish, constant, or diverge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1741,"prompt_tokens":1087,"completion_tokens":654,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":703,"completion_tokens_details":{"reasoning_tokens":561}},"tokens_in":703,"tokens_out":654,"duration_ms":6451,"temperature":1.0,"reasoning_tokens":561,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:53:00.610700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, analytically or numerically, the first two Dirichlet p-eigenvalues on $I=(-1/2,1/2)$ for $p=3$ and $V(x)=x^2$, and compare the gap to $\\min_a[\\lambda_{2,p}(I,ax)-\\lambda_{1,p}(I,ax)]$. If the quadratic-potential gap falls below the best linear-potential gap, the comparison principle in (4.1) is false and the one-dimensional convex-potential theorem would need reproof; if it does not, the key premise survives at least this test.","supporting_citations":[{"cited_title":"Andrews, J","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional comparison principle invoked in (4.1), which reduces the gap for arbitrary convex potentials to the gap for a linear potential."},{"cited_title":"Lavine, The eigenvalue gap for one-dimensional convex potentials,Proc","cited_arxiv_id":null,"evidence_quote":"Provides the p=2 integral identity for linear potentials that the nonlinear proof replaces with a nodal decomposition."},{"cited_title":"Ferone, C","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp weighted Poincaré inequality for log-concave measures used in Theorem 1.4 and Proposition 1.3."},{"cited_title":"Andrews and J","cited_arxiv_id":null,"evidence_quote":"Gives the sharp p=2 constant $3\\pi^2/D^2$ that anchors the higher-dimensional dichotomy and the $p\\downarrow2$ limit."},{"cited_title":"Stability of the $L^{p}$-Poincar\\'e inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps","cited_arxiv_id":"2602.05968","evidence_quote":"Obtains the zero-potential $L^p$-Poincaré stability estimate that the paper sharpens and extends to convex potentials."},{"cited_title":"Sharp remainder of the $L^{p}$-Poincar\\'e inequality for Baouendi-Grushin vector fields","cited_arxiv_id":"2507.01681","evidence_quote":"Provides the exact ground-state identity with remainder $C_p(\\xi,\\eta)$ used in the stability estimates."},{"cited_title":"Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirich- let problems,Ann","cited_arxiv_id":null,"evidence_quote":"Proves log-concavity of the first eigenfunction for $V\\equiv0$, the result the paper adapts to convex potentials."},{"cited_title":"Cuesta and H","cited_arxiv_id":null,"evidence_quote":"Gives the mountain-pass characterization of $\\lambda_{2,p}$ used to turn stability estimates into gap bounds."},{"cited_title":"Brasco, On principal frequencies and inradius in convex sets,Bruno Pini Math","cited_arxiv_id":null,"evidence_quote":"Supplies the inradius (Hersch–Protter) bound used to rule out collapse of minimizing sequences for $p>2$."}],"review_version":1}