{"id":"6aa68171-ce5d-4985-8d75-ddff221d6c0d","arxiv_id":"2411.11181","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper introduces the logarithmic p-Laplacian and proves its first eigenvalue is the s to 0 derivative of the fractional p-Laplacian eigenvalue, with eigenfunction convergence, maximum principles, and a boundary Hardy inequality.","lead":"This paper defines a nonlinear logarithmic p-Laplacian as the derivative of the fractional p-Laplacian at order zero, and develops its Dirichlet theory. It shows the first eigenvalue of the new operator controls the small-order asymptotics of the fractional p-Laplacian and inherits a Faber-Krahn inequality.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The eigenvalue derivative is normalization-dependent: shifting the small-s normalization by κ C_N,p s² changes λ_L,p by κ without altering the s→0 limit, so sign and maximum-principle statements are convention-bound.","rationale":"The reader identified the normalization constant as the weakest assumption, and I agree that this is the most load-bearing concern. The paper's central link, λ_L,p(Ω) = d/ds|_{s=0} λ_s,p(Ω), is meaningful only after C_N,s,p is fixed; for p ≠ 2 there is no canonical constant, and a perturbation of the form κ C_N,p s² preserves the s→0 limit while shifting λ_L,p by κ. This directly affects the sign-based maximum and comparison principles, which are among the paper's advertised main results. The concern does not indicate an internal mathematical error: the proofs are coherent for the chosen operator, and Remark 2.5 explicitly discloses the convention dependence. Because the authors are transparent and the claims are correct under the stated convention, the conditional verdict remains appropriate rather than a rejection. The concrete test is analytic and would settle whether the shift is exactly as predicted; if performed, it would confirm the convention-dependence without undermining the paper's internal correctness.","tokens_in":52837,"tokens_out":26645,"duration_ms":270074,"concrete_test":"Recompute Theorem 1.1 and Theorem 1.3 with C̃_N,s,p = C_N,s,p + κ C_N,p s² for 0 < s < 1/2, extended smoothly to s ≥ 1/2. If the expansion gains κ|u|^{p−2}u and λ̃_L,p(Ω) = λ_L,p(Ω) + κ, while the s→0 limit of the operator remains the identity, then the normalization-dependence is confirmed: choose κ so that a fixed ball crosses λ_L,p = 0 and Theorem 7.5 flips.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For p ≠ 2 there is no Fourier symbol fixing C_N,s,p, and Remark 2.5 concedes that a different choice changes only the zero-order part of LΔp. This is not merely cosmetic: take C̃_N,s,p = C_N,s,p + κ C_N,p s² for small s and extend smoothly. The limit s→0+ of (−Δ̃_p)^s is still |u|^{p−2}u, but the first-order term in expansion (1.1) gains κ|u|^{p−2}u. Repeating the Rayleigh-quotient argument in Theorem 1.3 then gives λ̃_L,p(Ω) = λ_L,p(Ω) + κ, with the same first eigenfunctions. Hence Theorems 1.7 and 7.5, which tie the maximum principle to the sign of λ_L,p, have thresholds that move with κ; a domain whose eigenvalue is positive in one convention can be negative in another. The quantitative spectral asymptotics and the validity regime of the maximum principles are therefore not intrinsic properties of an invariant nonlinear operator, but properties of one selected normalization. The authors disclose this in Remark 2.5, so the argument is internally consistent, but the central claim must be read as convention-dependent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a nonlinear operator LΔp, called the logarithmic p-Laplacian, obtained as the s-derivative at 0 of a suitably normalized fractional p-Laplacian. Theorem 1.1 gives an explicit integral representation, and a variational framework is built on the space X_p^0(Ω). The main results are: a logarithmic boundary Hardy inequality for locally plump domains (Theorem 1.2), the spectral expansion λ_L,p = d/ds λ_s,p at s=0 with L^p-convergence of first eigenfunctions (Theorem 1.3), a Faber-Krahn inequality, L∞ bounds, and strong maximum/comparison principles whose validity is tied to the sign of λ_L,p. The proofs are detailed and combine pointwise estimates with Γ-convergence-type arguments.","tokens_in":53053,"tokens_out":14997,"duration_ms":209637,"significance":"If the proofs hold, this is the first systematic nonlinear analogue of the logarithmic Laplacian, and the eigenvalue-derivative result is a genuine theorem rather than a definition. The Hardy inequality for arbitrary p>0 on locally plump sets and the L∞-bounds are substantial technical contributions. Credit is due for the transparent disclosure in Remark 2.5 that the normalization constant is not fixed by a Fourier symbol for p≠2; nevertheless, this caveat limits the interpretation of the sign-dependent maximum principles.","major_comments":[{"comment":"The operator LΔp and the value λ_L,p(Ω) are normalization-dependent. Taking C̃_N,s,p = C_N,s,p + κ C_N,p s² for small s and extending smoothly gives a family whose s→0 limit is still |u|^{p−2}u but whose first-order term is LΔp + κ|u|^{p−2}u. Repeating the Rayleigh-quotient argument in Theorem 1.3 gives λ̃_L,p = λ_L,p + κ, with the same first eigenfunctions. Consequently the criterion λ_L,p(Ω)>0 that decides the maximum principle in Theorem 1.7 (and the threshold in Corollary 7.9) is not an intrinsic property of the nonlinear operator but depends on the arbitrary choice of CN,s,p. Since the authors explicitly concede this in Remark 2.5, the mathematics is internally consistent, but the abstract and introduction currently present LΔp as the object emerging from the formal derivative without this qualifier. I ask the authors to state the convention-dependence prominently and to formulate Theorem 1.7 with an explicit reference to the operator (1.2) under the Section 2.4 normalization.","section":"Section 2.4, Remark 2.5; Theorems 1.3 and 1.7"},{"comment":"Lemma 7.6 is used in an essential way to derive the lower bound in Lemma 7.7 and hence the small-volume maximum principle in Corollary 7.9, but its proof is attributed only to 'personal communication'. Although a proof is included in the text, the provenance statement is insufficient for a reader to verify independence and for the journal to assess novelty; please provide a proper reference or explain the source and the degree of overlap with the unpublished material from which it was taken.","section":"Section 7, Lemma 7.6 and Lemma 7.7"}],"minor_comments":[{"comment":"The symbol W_0^{s,p}(U) is defined twice with different meanings: first as the space of W^{s,p}(R^N)-functions with zero exterior data, and then as the C_c^∞-closure in the ∥·∥_{s,p,U}-norm. Please use distinct notation for these two spaces.","section":"Section 2.2"},{"comment":"The piecewise definition of CN,s,p is discontinuous at s=1/2 for p≠2 (it is continuous for p=2). Since the paper only analyzes the limits s→0 and s→1, this does not affect the proofs, but the authors should comment on whether the discontinuity is intentional or whether a continuous normalization is preferable.","section":"Section 2.4"},{"comment":"The abstract and the opening paragraph present LΔp as 'the' logarithmic p-Laplacian emerging from the formal derivative, without mentioning that for p≠2 the derivative depends on the choice of CN,s,p. A short qualifier such as 'for the normalization chosen in Section 2.4' would prevent misinterpretation.","section":"Abstract and Introduction"},{"comment":"In the proof, the integral operator defined at the start of Section 3 and the derivative operator defined in (1.2) are both denoted LΔp, which is confusing. Please use distinct symbols in the proof, e.g., LΔp for the integral operator and LΔp^deriv for the derivative.","section":"Proof of Theorem 1.1, Section 3"},{"comment":"The expression 'N + α p + p − N p− s1p2' is hard to parse; rewrite with parentheses and explicit exponents, e.g., N + α p + p − N p − s_1 p^2, and check the inequality.","section":"Proof of Lemma 7.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically substantial and the proofs appear correct; my main concern is the framing around the normalization constant. I would not reject on those grounds, but the revision should address the convention-dependence explicitly in the abstract and in the statement of the maximum-principle theorem. The technical content (Whitney decomposition, Γ-convergence, L∞ estimates) is of high quality."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a technically strong, honest paper. It generalizes the logarithmic Laplacian of Chen–Weth to p ∈ (1,∞), and the main spectral result (Theorem 1.3) is a genuine Γ-convergence/compactness argument, not a circular restatement. The representation (1.2), the continuity estimates, the logarithmic boundary Hardy inequality via Whitney cubes, and the maximum/comparison principles are all non-obvious and mostly well-proved.\n\nThe catch, and I think the stress-test is right: for p ≠ 2 the operator LΔp is not canonical. The normalization CN,s,p is chosen by hand to make the two limits behave like identity and p-Laplacian. Remark 2.5 openly says any other choice changes only the zero-order part. But that “only” matters: add κ CN,p s² to CN,s,p for small s and the first-order term in (1.1) gains κ|u|^{p−2}u, so λ_L,p shifts by κ and the eigenfunctions stay the same. The sign of λ_L,p, and therefore the validity threshold in Theorems 1.7 and 7.5, moves with κ. A domain positive under one convention is negative under another. So the quantitative spectral asymptotics and the maximum-principle theorems are statements about a chosen normalization, not about an invariant nonlinear operator. The authors disclose this, and the internal logic is consistent; it is a conditional, convention-relative result, not a falsity.\n\nMinor soft spot: Lemma 7.6 rests on personal communication and an unpublished master thesis. The proof is included and looks fine, but the provenance is thin. I would ask the authors to either verify it independently or cite the thesis properly.\n\nThe paper deserves a serious referee. It is the first systematic treatment of the p ≠ 2 case, solves several genuinely hard problems (Hardy inequality, compact embedding, eigenvalue asymptotics), and the main proofs are detailed enough to check. The normalization issue should be confronted head-on in the final version—ideally by stating in the abstract that the operator is defined relative to the chosen normalization, or by finding a natural normalization condition. Anyone working on small-order limits of fractional p-Laplacians or on nonlocal maximum principles will want to read it. I would accept it for peer review and would probably cite the Hardy inequality and the representation, with a caveat about the convention.","headline":"A technically strong and honest generalization of the logarithmic Laplacian to p ≠ 2, but for p ≠ 2 the operator is normalization-dependent, so the eigenvalue and maximum-principle thresholds are properties of a chosen convention, not of an intrinsic nonlinear operator.","tokens_in":53626,"tokens_out":2651,"would_cite":true,"duration_ms":26391,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35D30","35B50","35B51","35P30","35R11","46E35","35A23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Differentiating the fractional p-Laplacian at s=0 yields the logarithmic p-Laplacian, whose first Dirichlet eigenvalue is the derivative of the fractional one.","keywords":["logarithmic p-Laplacian","fractional p-Laplacian","Dirichlet eigenvalue","Faber-Krahn inequality","maximum principle","comparison principle","Hardy inequality","p-Lévy operators"],"falsifier":"Compute, for a fixed p≠2 and a bounded Lipschitz domain such as the unit ball, the limit as s→0+ of (λs,p(Ω)−1)/s using an independently justified normalization of the fractional p-Laplacian (e.g. the semigroup or Balakrishnan representation) and compare it with the variational infimum for λL,p(Ω) defined from (1.2); any disagreement beyond numerical precision would disprove Theorem 1.3.","tokens_in":52603,"feed_emoji":"","tokens_out":6737,"duration_ms":53518,"temperature":0.7,"pith_summary":"This paper introduces the logarithmic p-Laplacian LΔp, defined as the derivative at s=0 of the fractional p-Laplacian (−Δp)s, and shows it has an explicit nonlocal integral representation. The authors build a variational framework on a new space Xp0(Ω) and prove that the first Dirichlet eigenvalue of LΔp is the derivative at s=0 of the first eigenvalue of the fractional p-Laplacian, with Lp-convergence of the normalized first eigenfunctions. From this they deduce a Faber–Krahn inequality, strong maximum and comparison principles (which hold exactly when the first eigenvalue is positive), L∞ bounds for solutions, and a logarithmic boundary Hardy inequality. The work generalizes the linear logarithmic Laplacian (p=2) to all p∈(1,∞).","feed_headline":"Logarithmic p-Laplacian's first eigenvalue is the s=0 derivative","feed_subtitle":"This new nonlocal operator governs small-order limits, with Faber–Krahn and maximum principles.","key_machinery":"The operator LΔp itself, defined pointwise by (1.2) and realized variationally on the space Xp0(Ω) of functions with finite double integral of |u(x)−u(y)|p/|x−y|N over unit balls, together with the auxiliary function hΩ(x) that encodes the domain's boundary shape and the chosen normalization constant CN,p. The key identity is the scaling law λL,p(rΩ)=λL,p(Ω)−pln r, and the Γ-convergence-type Lemma 7.2 that passes the derivative of the fractional energy to EL,p.","core_discovery":"The central claim is that the formal derivative at s=0 of the fractional p-Laplacian computes an explicit operator LΔp given by (1.2), a nonlocal, nonlinear operator of logarithmic order in which the kernel |z|−N is integrable near zero but has a critical tail at infinity. On the space Xp0(Ω), the paper proves that the first Dirichlet eigenvalue of LΔp satisfies λL,p(Ω) = (d/ds)λs,p(Ω) at s=0, and the associated Lp-normalized positive eigenfunctions converge in Lp to the first eigenfunction of LΔp. This eigenvalue link is what powers the Faber–Krahn inequality and the characterization of maximum principles by the sign of λL,p(Ω).","pith_inferences":["Because any other normalization of the fractional p-Laplacian shifts LΔp by a multiple of the identity, the numerical value of λL,p(Ω) and the sign that triggers maximum principles are convention-dependent; a physically or probabilistically motivated normalization would need to be selected before applying these results to models.","The eigenvalue derivative formula suggests a quantitative way to compare the spectral gap of fractional problems at small s across different domains, potentially informing optimization of the fractional order s in image processing or population dynamics models.","One could test the stability of the Faber–Krahn inequality: the open question of strictness for non-balls is already noted in the paper, and a numerical check for p≠2 on a rectangle versus the ball of same area would indicate whether the inequality is generically strict.","The boundary Hardy inequality may extend to unbounded locally plump sets, which the paper's Whitney-decomposition proof already covers in principle, potentially simplifying proofs in related nonlocal problems."],"forward_implications":["For every bounded Lipschitz domain, the first Dirichlet eigenvalue of the fractional p-Laplacian has a first-order expansion λs,p(Ω)=1+sλL,p(Ω)+o(s) as s→0+, giving a new spectral quantity computed from the logarithmic operator.","The Faber–Krahn inequality for LΔp holds: among bounded Lipschitz sets of fixed volume, the ball minimizes the first eigenvalue.","The strong maximum principle for LΔp holds if and only if λL,p(Ω)>0; for large domains the eigenvalue becomes negative and the principle fails.","A small-volume maximum principle follows: for any bound on the coefficient c, all sufficiently small domains have positive first eigenvalue and thus satisfy the maximum principle.","The logarithmic boundary Hardy inequality (with ln+(1/δx) weight) holds for all p≥1 and locally plump sets, giving a characterization of the space Xp0(Ω)."],"supporting_citations":[{"why":"Introduces the linear logarithmic Laplacian (p=2) and its Fourier symbol, the model that the nonlinear operator generalizes.","marker":"[13]"},{"why":"Provides the p-Lévy operator framework and the properties of the space Xp0(Ω), including compact embeddings and Poincaré inequalities.","marker":"[27]"},{"why":"Supplies the Faber–Krahn inequality for the fractional p-Laplacian used to deduce the same inequality for LΔp.","marker":"[7]"},{"why":"Gives the ideas from which the Whitney-decomposition proof of the logarithmic boundary Hardy inequality is built.","marker":"[2]"},{"why":"Supplies the pointwise estimates on the map g(a)=|a|p−2a used in the proof of Theorem 1.1 and in later estimates.","marker":"[41]"},{"why":"Provides the discrete Picone inequality used to prove simplicity of the first eigenvalue of LΔp.","marker":"[6]"}],"fun_headline_variants":["Logarithmic p-Laplacian: s=0 derivative of fractional p-Laplacian","Faber–Krahn for logarithmic p-Laplacian from s=0 derivative","Maximum principle for logarithmic p-Laplacian tied to eigenvalue sign","First eigenvalue of fractional p-Laplacian's derivative at s=0"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the hand-selected normalization constant CN,s,p of the fractional p-Laplacian; for p≠2 there is no Fourier symbol to fix it, and any different choice that still has a derivative at s=0 changes the zero-order term of LΔp, so the operator, the value of λL,p(Ω), and the sign conditions for maximum principles are properties of one chosen convention rather than intrinsic invariants.","fun_headline_variants_meta":{"raw":{"variants":["Logarithmic p-Laplacian: s=0 derivative of fractional p-Laplacian","Faber–Krahn for logarithmic p-Laplacian from s=0 derivative","Maximum principle for logarithmic p-Laplacian tied to eigenvalue sign","First eigenvalue of fractional p-Laplacian's derivative at s=0"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000663,"raw_usage":{"total_tokens":3015,"prompt_tokens":918,"completion_tokens":2097,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":2013}},"tokens_in":534,"tokens_out":2097,"duration_ms":101815,"temperature":1.0,"reasoning_tokens":2013,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:50:13.965194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a fixed p≠2 and a bounded Lipschitz domain such as the unit ball, the limit as s→0+ of (λs,p(Ω)−1)/s using an independently justified normalization of the fractional p-Laplacian (e.g. the semigroup or Balakrishnan representation) and compare it with the variational infimum for λL,p(Ω) defined from (1.2); any disagreement beyond numerical precision would disprove Theorem 1.3.","supporting_citations":[{"cited_title":"Chen and T","cited_arxiv_id":null,"evidence_quote":"Introduces the linear logarithmic Laplacian (p=2) and its Fourier symbol, the model that the nonlinear operator generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the p-Lévy operator framework and the properties of the space Xp0(Ω), including compact embeddings and Poincaré inequalities."},{"cited_title":"Brasco, E","cited_arxiv_id":null,"evidence_quote":"Supplies the Faber–Krahn inequality for the fractional p-Laplacian used to deduce the same inequality for LΔp."},{"cited_title":"Lindgren","cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise estimates on the map g(a)=|a|p−2a used in the proof of Theorem 1.1 and in later estimates."},{"cited_title":"Brasco and G","cited_arxiv_id":null,"evidence_quote":"Provides the discrete Picone inequality used to prove simplicity of the first eigenvalue of LΔp."}],"review_version":1}