log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator

Lei Qin

REVIEW 4 major objections 4 minor 1 cited by

log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator

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Pith's one-line read This paper proves that on a bounded C^2 convex domain, the first positive eigenfunction of the weighted p-Laplace operator is log-concave, and that the first eigenvalue obeys a Brunn-Minkowski-type inequality under Minkowski combinations…

desk verdict Plausible extension of log-concavity to the Gaussian p-Laplace for all p>1, but the Brunn-Minkowski proof leans on an unproved convex inf-convolution lemma and the manuscript needs a careful sign cleanup. read the letter →

arxiv 2411.16377 v1 pith:ACEYXVXI submitted 2024-11-25 math.AP

Lei Qin This is my paper · ORCID

classification math.AP MSC 35P3035J9252A40

keywords log-concavityweightedp-LaplaceoperatorBrunn-MinkowskiinequalityfirsteigenvalueGaussianmeasureconvexbodiesquasilinearellipticequationsconcavitymaximumprinciple

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 proves two convexity statements for the Gaussian-weighted $p$-Laplace eigenvalue problem on bounded convex domains with smooth boundary. The first is that the positive first eigenfunction $u$ of $\Delta_{p,\gamma}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)-\langle x,\nabla u\rangle |\nabla u|^{p-2}$ is log-concave: $w=-\ln u$ is convex in $\Omega$ for every $p>1$. The second is that the first eigenvalue $\lambda_{p,\gamma}$ is convex under Minkowski addition of $C^2$ convex bodies, meaning $\lambda_{p,\gamma}(\Omega_t)\le(1-t)\lambda_{p,\gamma}(\Omega_0)+t\lambda_{p,\gamma}(\Omega_1)$ for $\Omega_t=(1-t)\Omega_0+t\Omega_1$. Together these extend the classical log-concavity of Laplacian eigenfunctions and Brunn-Minkowski-type inequalities to a degenerate quasilinear weighted setting.

What carries the argument

The load-bearing objects are the weighted $p$-Laplace operator $\Delta_{p,\gamma}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)-\langle x,\nabla u\rangle |\nabla u|^{p-2}$, the log-transform $w=-\ln u$, and the inf-convolution of the convex functions $w_i$ on the Minkowski sum $\Omega_t=(1-t)\Omega_0+t\Omega_1$. On the eigenfunction side, the argument runs through a regularized variational problem whose solutions are smooth in the interior, to which a concavity maximum principle applies; uniform $C^{1,\beta}$ and $C^{2,\beta}$ estimates let the regularization converge to the weak solution and carry convexity of the regularized logarithm to $w$. On the eigenvalue side, the Brunn-Minkowski inequality is transported via inf-convolution: the paper defines $\tilde w(x)=\inf\{(1-t)w_0(x_0)+tw_1(x_1):x=(1-t)x_0+tx_1\}$, shows $\tilde w$ is $C^1$, obtains the comparison $\Delta_p\tilde w\le(1-t)\Delta_p w_0+t\Delta_p w_1$ in the appropriate sense, and converts this into a bound on the Rayleigh quotient of the trial function $e^{-\tilde w}$. The transfer relies on convex duality results from [22] and on comparison lemmas from [11] originally stated for strictly concave sup-convolutions, applied here to the convex $w_i$ after an implicit duality step.

What would settle it

Compute $\lambda_{p,\gamma}$ numerically for two concentric balls in $\mathbb R^2$ (or a ball and an ellipse) for several $p>1$, and check whether $\lambda_{p,\gamma}(\Omega_t)\le(1-t)\lambda_{p,\gamma}(\Omega_0)+t\lambda_{p,\gamma}(\Omega_1)$; a violation for any $C^2$ convex bodies refutes Theorem 1.2. For Theorem 1.1, solve (1.1) on a $C^2$ convex domain with a flat boundary segment and inspect the Hessian of $w=-\ln u$ at points where $\nabla u\neq0$; one negative eigenvalue of $D^2w$ refutes log-concavity.

Watch

Extended reading notes

Core claim

The central claim of the paper is that log-concavity and Brunn-Minkowski convexity survive the passage from the linear Laplacian to the weighted $p$-Laplace operator. On a bounded convex domain with $C^2$ boundary, the first positive eigenfunction of (1.1) has the property that $w=-\ln u$ is convex, so all positive superlevel sets of $u$ are convex. On the class of $C^2$ convex bodies, the first eigenvalue satisfies $\lambda_{p,\gamma}((1-t)\Omega_0+t\Omega_1)\le(1-t)\lambda_{p,\gamma}(\Omega_0)+t\lambda_{p,\gamma}(\Omega_1)$, a Brunn-Minkowski-type inequality for the eigenvalue map. The proof route is to establish existence, uniqueness, Hopf boundary behavior, and global $C^{1,\alpha}$ regularity for the weak solution; to prove log-concavity through a regularized problem and a concavity maximum principle; and then to derive the eigenvalue inequality by inf-convolution of the convex functions $w_i=-\ln u_i$ together with a comparison argument for the $p$-Laplacian of the inf-convolution.

Load-bearing premise

The second theorem rests on assuming without proof that the comparison lemmas for combining strictly concave functions still hold for the inf-convolutions of the convex functions $w_i=-\ln u_i$; if that transfer fails, the Brunn-Minkowski inequality does not follow.

Editorial extensions

If this is right

Every positive superlevel set $\{u>c\}$ of the first eigenfunction is convex whenever $\Omega$ is convex, so Brunn-Minkowski-type measure bounds apply to these level sets.
The eigenvalue map $\Omega\mapsto\lambda_{p,\gamma}(\Omega)$ is convex along Minkowski linear combinations, giving the nonlinear analogue of the Brunn-Minkowski inequalities already known for torsional rigidity and $p$-capacity.
For $p=2$, Theorems 1.1 and 1.2 recover the log-concavity and Brunn-Minkowski results for the Gaussian Laplacian (Ornstein-Uhlenbeck) eigenvalue problem recently obtained by other methods.
The comparison theory developed here (weak comparison principle, Hopf boundary lemma, uniqueness up to scale) supplies tools for further variational problems for the weighted $p$-Laplacian on Gaussian spaces.

Reading between the lines

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

A radial numerical test is immediately available: for concentric balls the Minkowski combination is a ball, so the inequality $\lambda_{p,\gamma}(B_R)\le(1-t)\lambda_{p,\gamma}(B_r)+t\lambda_{p,\gamma}(B_s)$ with $R=(1-t)r+ts$ can be checked from the one-dimensional radial eigenfunction problem; a violation would refute Theorem 1.2.
The same inf-convolution strategy should work for other convex weights $e^{-V}$ replacing the Gaussian density, since the drift term enters only through the first-order term; testing $V=|x|^4/4$ would separate Gaussian-specific properties from the general convex-geometric mechanism.
The approximating sequence $\Omega_k\uparrow\Omega$ in Theorem 1.1 is where strong convexity is dropped, so a $C^2$ convex domain with a flat boundary segment is the natural place to look for counterexamples if log-concavity does not survive without strong convexity.
Writing out the duality step between sup-convolutions and inf-convolutions explicitly would turn Section 5 into a general comparison theorem for convex solutions of (5.1), making the Brunn-Minkowski inequality independent of the regularity machinery.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: this is a seriously intended extension of Sakaguchi's log-concavity theorem to the Gaussian-weighted p-Laplace operator for every p>1, together with a Brunn-Minkowski inequality for the first eigenvalue. The log-concavity part is the stronger half and is probably right; the BM part is plausible but as written does not close.

What is genuinely new: the unweighted p-Laplace case is Sakaguchi, the weighted Ornstein-Uhlenbeck case is only p=2 (Colesanti–Francini–Livshyts–Salani), and the all-p weighted extension is not in the literature. The paper also correctly identifies that the lack of homogeneity prevents the usual weak BM statement, and it gives a full existence–uniqueness–regularity section, including Hopf and comparison lemmas adapted to the Gaussian weight. That is useful groundwork.

Where the soft spots are, in proportion:

1. The BM proof has a load-bearing gap. Lemmas 5.1 and 5.2 quoted from [11] are stated for strictly concave functions and sup-convolution. In Theorem 1.2 the paper applies them to the convex functions w_i = -ln u_i and uses inf-convolution. The remark (5.2) gestures at the convex-duality route, but the actual statement and proof of the inf-convolution analogue—the Hessian formula and the pointwise comparison for Delta_p tilde w—are not given. Without that comparison, the inequality (5.12) has no foundation. I think the analogue is likely true, but it needs to be derived, not asserted.

2. The manuscript is full of sign and transcription errors in exactly the places a referee would need to check. Theorem 1.2 defines Omega_t = (1-t)Omega_0 + Omega_1, dropping the factor t. In Proposition 3.3 the comparison inequality has the same integrand on both sides (u1 instead of u2 on the right). In Section 4, upsilon_epsilon is sometimes -ln u_epsilon and sometimes ln u_epsilon; Proposition 4.9 concludes "upsilon_epsilon = ln u_epsilon is concave" from a proof that works with the opposite sign. These are individually fixable, but they make line-by-line verification of Theorem 1.1 unnecessarily hard.

3. Minor: the convergence argument for F_epsilon and the boundary term in (5.14) are standard but sketched; a referee will want more details on the passage to the limit in the singular set C_t.

Who this is for: people working on convexity of solutions to degenerate elliptic equations and on Brunn–Minkowski inequalities for spectral quantities. They will get a clear statement of a natural open problem's likely answer, but they should not take the BM proof as finished.

My recommendation: send it to a serious referee. The result is important enough, and Theorem 1.1 is close to being solid. The referee should be asked to request a self-contained proof of the inf-convolution lemma and a full correction of the sign conventions before the paper is accepted.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the Gaussian-weighted p-Laplace eigenvalue problem (1.1) on bounded convex domains with C^2 boundary, with operator Δ_{p,γ}u = div(|∇u|^{p-2}∇u) - (x,∇u)|∇u|^{p-2}. After proving existence, positivity, uniqueness and C^{1,α} regularity (Theorem 3.1), the paper claims that the first positive eigenfunction is log-concave (Theorem 1.1) and that the first eigenvalue satisfies the Brunn-Minkowski-type inequality λ_{p,γ}(Ω_t) ≤ (1-t)λ_{p,γ}(Ω_0) + tλ_{p,γ}(Ω_1) for C^2 convex bodies (Theorem 1.2). The proof of Theorem 1.1 uses a regularized variational problem, interior C^{1,β}/C^{2,β} estimates, and Korevaar's concavity maximum principle; the proof of Theorem 1.2 uses log-concavity and the inf-convolution of the convex functions w_i = -ln u_i, following the strategy of Colesanti-Cuoghi-Salani.

Significance. If the proofs can be completed, Theorem 1.1 is a natural extension of known log-concavity results for p-Laplace eigenfunctions (Sakaguchi, Barles) to the Gaussian-weighted operator, and Theorem 1.2 would be a new convexity result for the weighted first eigenvalue in the class of convex bodies. The paper is honest about the main difficulty, namely that the equation is only weakly regular, and it attempts a regularization route rather than assuming higher regularity. It also correctly identifies that the non-homogeneity of the eigenvalue prevents the usual normalization in the Brunn-Minkowski inequality. The main value is in the two theorem statements and the methodological template; the manuscript does not contain reproducible code or machine-checked proofs, but that is not expected for this type of result.

major comments (4)

[Section 5 (Lemmas 5.1–5.2 and proof of Theorem 1.2)] The proof of Theorem 1.2 invokes, without proof, an inf-convolution analogue of Lemmas 5.1 and 5.2. The lemmas as quoted from [11] are stated for strictly concave functions and for sup-convolution, whereas the functions w_i = -ln u_i are convex and the construction (5.3)–(5.5) is an inf-convolution. The remark (5.2) records the convex-duality identity (-u)^* = (1-t)(-u_0)^* + t(-u_1)^*, but the paper does not derive the transformed Hessian formula D^2 w(z) = [(1-t)A^{-1} + tB^{-1}]^{-1} nor the p-Laplacian comparison Δ_p w(z) ≤ (1-t)Δ_p w_0(x) + tΔ_p w_1(y). Inequality (5.12) is precisely where this comparison is used, so Theorem 1.2 is not established as written. If the comparison fails for some p in (1,2), the argument collapses; at minimum a proof or a precise citation of the convex/inf-convolution version is required.
[Section 4 (Propositions 4.8–4.10 and Theorem 1.1)] The sign and definition of υ_ε are inconsistent across Section 4. Proposition 4.8 defines υ_ε = -ln u_ε and studies c_ε = υ_ε((1-t)x+ty) - (1-t)υ_ε(x) - tυ_ε(y) in order to prove convexity of -ln u_ε, while Proposition 4.9 states that υ_ε = ln u_ε is concave in Ω_ν and Proposition 4.10 and Theorem 1.1 conclude that υ = ln u is concave. Since concavity of ln u is equivalent to convexity of -ln u, the two conventions can be reconciled, but as written the equation in Proposition 4.8, with the coefficient displayed as (ε+|∇υ_ε|)^{(p-2)/2}, is not shown to be the equation satisfied by -ln u_ε under either sign convention, and the statement of Proposition 4.9 uses the opposite sign from Proposition 4.8. The limiting argument (4.17)–(4.20) and the application of Korevaar's principle cannot be checked line by line until this is fixed.
[Proposition 3.3] The comparison principle is not proved as written. After passing to the unweighted form, the proof states ∫ e^{-|x|^2/2}|∇u_1|^{p-2}∇u_1·∇ϕ ≤ ∫ e^{-|x|^2/2}|∇u_1|^{p-2}∇u_1·∇ϕ, with the same function u_1 on both sides; the subsequent test-function computation uses u_1 and u_2, indicating a typo, but the displayed inequality is tautological. Since Proposition 3.3 is used in the proof of Hopf's lemma (Proposition 3.5) and in the uniqueness argument in Theorem 3.1, the regularity and uniqueness part of Theorem 1.1 depends on a corrected proof.
[Proof of Theorem 1.1 (approximation by strongly convex domains)] In the proof of Theorem 1.1, the passage from strongly convex domains Ω_k to Ω relies on the assertion that 'the C^β(Ω)-estimate of the solution to (1.1) is independent of small smooth perturbation of the boundary ∂Ω'. This is not proved and is not automatic, since the C^β constants for solutions on varying domains may depend on the boundary geometry. A reference or explicit argument is needed to justify the uniform bound ‖u_k‖_{C^β(Ω_k)} ≤ c used before applying Arzelà-Ascoli.

minor comments (4)

[Theorem 1.2 statement] The displayed definition of Ω_t reads Ω_t = (1-t)Ω_0 + Ω_1 and is missing the factor t on Ω_1; it should be Ω_t = (1-t)Ω_0 + tΩ_1, as used throughout the proof.
[Equation (5.9) and definition of F_ε] In (5.9) and in the definition of F_ε, the second second-order term in each bracket should involve the unit vector n_i rather than n_{i,ε}; as printed, both occurrences are written with n_{i,ε}, so the ε-dependence does not cancel in the intended way.
[Proof of Theorem 1.2, after (5.14)] The text says 'by (5.7) and (5.7)' where the second reference should be (5.8), and in the surrounding sentences '∇w_{i,ε}' is sometimes written as '∇w_i,ε'; these notational slips should be corrected.
[Equation (3.2)] Equation (3.2) appears to have a typo on the right-hand side: the test function should be ψ, not ∇ψ, in the integral λ∫|u|^{p-2}u e^{-|x|^2/2}ψ dx; otherwise the weak formulation does not match (2.4).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained apart from ordinary external citations.

full rationale

Theorems 1.1 and 1.2 are derived from the hypotheses rather than assumed. Theorem 1.1 is obtained by regularizing the degenerate problem (4.1)-(4.3), applying Korevaar's concavity maximum principle from [17] to the smooth approximants, deriving boundary tangent-plane controls from [17, Lemma 2.4], and then passing to the limit through the uniform C^{1,β} estimates and the uniqueness result proved in Theorem 3.1. The limit passage is asserted and supported by the cited Tolksdorf/Di Benedetto regularity theory, which is external and does not contain the target log-concavity conclusion. Theorem 1.2 uses the log-concavity of w_i = -ln u_i just established, constructs the inf-convolution w̃, and applies Lemmas 5.1 and 5.2 quoted from the independent paper [11]. The inequality (5.12) yields a Rayleigh quotient bound for the trial function ũ = e^{-w̃}, and the variational definition (1.2) then gives λ_{p,γ}(Ω_t) ≤ (1-t)λ_0 + tλ_1. No parameter is fitted to the target eigenvalue, no conclusion is imported from the author's own prior work, and no cited result assumes the theorem being proved. The only caveat visible is a technical one, not a circular one: Lemmas 5.1 and 5.2 are stated for concave functions and sup-convolution, while the proof uses convex functions and inf-convolution via the conjugation remark (5.2); if the convex analogue of the p-Laplacian comparison fails, Theorem 1.2 would be incorrect, but that would be a correctness gap, not a reduction of the theorem to its own inputs. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants appear; the only tunable objects are the regularization parameter ε and the approximating domains Ω_k, both taken to limits rather than fit. The paper uses standard Sobolev and regularity theory, Korevaar's concavity principle, comparison and Hopf results adapted from Tolksdorf, and the inf-convolution lemmas of Colesanti, Cuoghi, and Salani. No new physical entities are introduced.

assumptions (5)

standard math Korevaar's concavity maximum principle ([17, Theorem 1.3]) applies to the regularized solutions u_ε in Ω_δ
Invoked in Proposition 4.8; requires smooth classical solutions and boundary support conditions, which the paper obtains via Proposition 4.7 and boundary estimates.
standard math Comparison principle and Hopf lemma for the weighted p-Laplace operator, as stated in Propositions 3.3 and 3.5
Adapted from Tolksdorf [24] and Gilbarg-Trudinger [15]; used for uniqueness and for the nonzero normal derivative at the boundary. The proof in Proposition 3.3 contains a typo with duplicated integrands.
standard math Regularity theorems of Di Benedetto [3], Tolksdorf [24], and Ladyzhenskaya-Ural'tseva [18] give C^{1,α} and C^{2+β} estimates for the degenerate equations
Used in Theorem 3.1 and Propositions 4.7 and 4.9 to pass from weak to classical solutions.
domain assumption Lemmas 5.1 and 5.2 from Colesanti-Cuoghi-Salani [11] extend to the inf-convolution of convex functions w_i
The paper applies these lemmas to the convex functions w_i=-ln u_i and the inf-convolution (5.5) without proving the convex analogue; this is the main external load-bearing premise of Theorem 1.2.
domain assumption Ω is a bounded convex domain with C^2 boundary, and ∂Ω is connected for uniqueness; p>1
These are the hypotheses of Theorems 1.1 and 1.2 and Theorem 3.1; they guarantee boundary regularity and Hopf-type behavior.

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Pith. "Pith review of log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator." pith.science (2026). https://pith.science/paper/ACEYXVXI

@misc{pith2026241116377,
  author       = {Pith},
  title        = {Pith review of: log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACEYXVXI}},
  note         = {Machine review of arXiv:2411.16377}
}

abstract

In this paper, we investigate the log-concavity property of the first eigenfunction to the weighted $p$-Laplace operator in class of bounded, convex and smooth domain. Moreover, we prove a Brunn-Minkowski-type inequality for the first eigenvalue to the weighted $p$-Laplace operator in the class of $C^2$ convex bodies in $\R^n$

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The strong log-concavity for first eigenfunction of the Ornstein-Uhlenbeck operator in the class of convex bodies
math.AP 2025-07 conditional novelty 6.0 of 10

The first Dirichlet eigenfunction of the Ornstein-Uhlenbeck operator on any bounded convex domain is strongly log-concave, and the equality case in the Brunn-Minkowski inequality for its principal frequency is charact...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

Colesanti, P

A. Colesanti, P. Cuoghi and P. Salani, Brunn-Minkowski inequalit ies for two functionals involving the p-Laplace operator, Appl. Anal. 85 (2006), no. 1-3, 45–66

work page 2006

[2] [1]

Garc ´ ıa Azorero and I

J. Garc ´ ıa Azorero and I. Peral Alonso, Existence and nonuniqu eness for the p-Laplacian: nonlinear eigen- values, Comm. Partial Diﬀerential Equations 12 (1987), no. 12, 1389–1430

work page 1987

[3] [2]

Barles, Remarks on uniqueness results of the ﬁrst eigenvalue of the p-Laplacian, Ann

G. Barles, Remarks on uniqueness results of the ﬁrst eigenvalue of the p-Laplacian, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), no. 1, 65–75

work page 1988

[4] [3]

Di Benedetto, C1+α local regularity of weak solutions of degenerate elliptic equations, N onlinear Anal

E. Di Benedetto, C1+α local regularity of weak solutions of degenerate elliptic equations, N onlinear Anal. 7 (1983), no. 8, 827–850

work page 1983

[5] [4]

Borell, Capacitary inequalities of the Brunn-Minkowski type, M ath

C. Borell, Capacitary inequalities of the Brunn-Minkowski type, M ath. Ann. 263 (1983), no. 2, 179–184

work page 1983

[6] [5]

Borell, Greenian potentials and concavity, Math

C. Borell, Greenian potentials and concavity, Math. Ann. 272 (1985), no. 1, 155–160

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