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The Dirichlet Problem For the Logarithmic p-Laplacian

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Pith's one-line read Differentiating the fractional p-Laplacian at s=0 yields the logarithmic p-Laplacian, whose first Dirichlet eigenvalue is the derivative of the fractional one.

desk verdict 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. read the letter →

arxiv 2411.11181 v3 pith:IXRR3MXT submitted 2024-11-17 math.AP

Bart{l}omiej Dyda , Sven Jarohs , Firoj Sk This is my paper · ORCID
classification math.AP MSC 35D3035B5035B5135P3035R1146E3535A23
keywords logarithmicp-LaplacianfractionalDirichleteigenvalueFaber-KrahninequalitymaximumprinciplecomparisonHardyp-Lévyoperators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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,∞).

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 2.4, Remark 2.5; Theorems 1.3 and 1.7] 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.
  2. [Section 7, Lemma 7.6 and Lemma 7.7] 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.
minor comments (5)
  1. [Section 2.2] 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.
  2. [Section 2.4] 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.
  3. [Abstract and Introduction] 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.
  4. [Proof of Theorem 1.1, Section 3] 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.
  5. [Proof of Lemma 7.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the eigenvalue-derivative theorem is a genuine Gamma-convergence proof and the normalization convention is explicitly disclosed.

full rationale

The paper's central claim, Theorem 1.3, is not circular. The first eigenvalue λ_L,p is defined independently in (7.2) as the infimum of the energy EL,p, while the proof that it equals d/ds at s=0 of λ_s,p uses an upper bound from admissible test functions, a uniform boundedness argument for the fractional eigenfunctions, and a compactness/convergence lemma (Lemma 7.2) that is proved in the paper. This is a genuine Γ-convergence argument, not a restatement of definitions. The Faber-Krahn corollary imports the fractional Faber-Krahn inequality from [7] and passes to the derivative, so it does not assume its own conclusion. The boundary Hardy inequality is proved by a self-contained Whitney decomposition argument, not by citing the space X_p^0. The only normalization freedom is stated in Remark 2.5: for p≠2 the constant CN,s,p has no Fourier-symbol justification and "any other choice ... only changes the zero order part of the logarithmic p-Laplacian." This makes the quantitative operator and the sign threshold in the maximum principle convention-dependent, but the paper explicitly discloses this and does not claim normalization-invariance; convention dependence is a limitation, not a circular reduction. Self-citations such as Lemma 2.3 from [34] are minor technical inequalities and are not load-bearing. No step in the derivation chain reduces by construction to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on one hand-chosen normalization constant, on external functional-analytic facts about p-Levy spaces, on the fractional Faber-Krahn inequality, and on one supporting lemma whose proof is only available through personal communication. No parameter is fitted to data; the main 'free parameter' is the convention for the normalization of the fractional p-Laplacian.

free parameters (1)
  • Normalization constant CN,s,p for the fractional p-Laplacian = CN,p = p Γ(N/2)/(2 π^{N/2}); derivative ρN(p) = 2 ln 2 − γ + (p/2) ψ(N/2)
    The constant is chosen by hand in Section 2.4 so that the s to 0 and s to 1 limits behave like the identity and the p-Laplacian. For p≠2 there is no Fourier symbol to determine it, and Remark 2.5 notes that any other choice changes the zero-order part of LΔp and hence the value and sign of λ_L,p.
assumptions (5)
  • domain assumption Known properties of the space X_p^0(Ω): reflexivity, compact embedding into Lp(Ω), and density of C_c^∞(Ω), taken from [27].
    The variational framework and the eigenvalue compactness arguments rely on these external results (Propositions 4.1, 4.2, and 4.3).
  • domain assumption Faber-Krahn inequality for the fractional p-Laplacian, from [7, Theorem 3.5].
    Corollary 1.4 passes this inequality to the limit s to 0; any gap in the fractional result would propagate.
  • domain assumption Local plumpness is sufficient for the Whitney cube conditions in Theorem 5.1, proved in Proposition 5.9.
    The logarithmic boundary Hardy inequality is built on this geometric condition, and bounded Lipschitz sets are locally plump.
  • domain assumption Bounds for hΩ in Lemma 7.6, credited to personal communication and the master's thesis [37].
    These bounds drive the lower bound on λ_L,p and the small-volume maximum principle in Corollary 7.9.
  • standard math Discrete Picone inequality from [6, Proposition 4.2].
    Used in Theorem 7.1 to prove that the first eigenfunction is simple, up to scalar multiples.
invented entities (1)
  • Logarithmic p-Laplace operator LΔp independent evidence
    purpose: Central object of the paper, defined as the derivative at s=0 of the normalized fractional p-Laplacian and used for a variational Dirichlet theory.
    It is derived from a limiting expansion rather than postulated, and it carries testable consequences such as λ_L,p being the s-derivative of λ_s,p and the scaling λ_L,p(rΩ)=λ_L,p(Ω)−p ln r.

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Pith. "Pith review of The Dirichlet Problem For the Logarithmic p-Laplacian." pith.science (2026). https://pith.science/paper/IXRR3MXT

@misc{pith2026241111181,
  author       = {Pith},
  title        = {Pith review of: The Dirichlet Problem For the Logarithmic p-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXRR3MXT}},
  note         = {Machine review of arXiv:2411.11181}
}
abstract

We introduce and study the logarithmic $p$-Laplacian $L_{\Delta_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-\Delta_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator. We present a variational framework to study the Dirichlet problems involving the $L_{\Delta_p}$ in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional $p$-Laplacian and the logarithmic $p$-Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of $L_{\Delta_p}$. We discuss maximum and comparison principles for $L_{\Delta_p}$ in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of $L_{\Delta_p}$. In addition, we prove that the first Dirichlet eigenfunction of $L_{\Delta_p}$ is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian.

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