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Talenti's weighted Pólya–Szegő inequality extends to Sobolev functions on arbitrary Borel sets, producing Faber–Krahn bounds for the first eigenvalue of weighted p-Laplacians.

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2026-06-29 06:32 UTC pith:UNAHYBVD

load-bearing objection Extends Talenti's weighted Pólya-Szegő to zero-trace functions on arbitrary Borel sets and adds concrete new examples, but the trace preservation step on irregular domains needs checking. the 1 major comments →

arxiv 2605.29721 v1 pith:UNAHYBVD submitted 2026-05-28 math.AP math.SP

Spectral inequalities for weighted p-Laplacians via Talenti symmetrization

Giulio Bartoli , Giorgio Saracco This is my paper
classification math.AP math.SP
keywords weighted p-LaplacianTalenti symmetrizationPólya–Szegő inequalityFaber–Krahn inequalityDirichlet eigenvalueSobolev functionsrearrangement inequalitiesconvex cones
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that a rearrangement inequality previously limited to Lipschitz functions continues to hold when the functions are taken from the Sobolev space with zero trace on any Borel subset of the ambient domain. This extension immediately supplies a lower bound for the first (p,q)-eigenvalue of the weighted Dirichlet p-Laplacian by comparing any domain to its symmetrized counterpart. The same argument recovers the classical Euclidean and Gaussian cases while also covering previously untreated weights such as homogeneous functions on convex cones and certain log-concave perturbations. A reader would care because the result supplies a uniform method for obtaining spectral comparison inequalities once the measure is absolutely continuous with respect to Lebesgue measure.

Core claim

We prove that Talenti's weighted Pólya–Szegő inequality extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets Ω ⊂ X. This yields Faber–Krahn-type inequalities for the first (p,q)-eigenvalue of the weighted Dirichlet p-Laplacian. Several concrete families of measures are shown to fit the abstract setting, including the Euclidean and Gaussian cases together with new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations.

What carries the argument

Talenti symmetrization producing the weighted Pólya–Szegő inequality that compares the weighted gradient integrals of a function and its radially decreasing rearrangement.

Load-bearing premise

The underlying measure must be absolutely continuous with respect to Lebesgue measure on the open connected set X.

What would settle it

An explicit Sobolev function with zero boundary trace on a non-symmetric Borel set whose weighted gradient integral is strictly smaller than that of its rearrangement would falsify the claimed extension of the inequality.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The first (p,q)-eigenvalue on any admissible domain is at least as large as the eigenvalue on the symmetrized domain of the same measure.
  • The comparison holds uniformly for all measures that are absolutely continuous with respect to Lebesgue measure.
  • New explicit lower bounds follow for the first eigenvalue under homogeneous weights on convex cones.
  • Analogous bounds hold for anisotropic Gaussian measures and for log-concave perturbations of the Gaussian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks of the eigenvalue inequality on irregular domains could be used to test the sharpness of the symmetrized bound.
  • If similar rearrangement inequalities can be proved for other differential operators, the same abstract argument would immediately give Faber–Krahn results for those operators as well.
  • The framework suggests that shape-optimization problems for the first eigenvalue can be reduced to the radially symmetric case once the weight satisfies the absolute-continuity assumption.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper establishes that Talenti's weighted Pólya–Szegő inequality, originally for Lipschitz functions on the open connected set X, extends to functions in the weighted Sobolev space W^{1,p}_0(Ω, μ) with zero boundary trace, where Ω is an arbitrary Borel subset of X and μ is absolutely continuous with respect to Lebesgue measure. This extension is applied to obtain Faber–Krahn-type inequalities for the first (p,q)-eigenvalue of the weighted Dirichlet p-Laplacian. Several examples are presented, covering classical Euclidean and Gaussian settings as well as new cases for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave perturbations.

Significance. If the extension step is fully rigorous, the work supplies a general abstract framework for deriving spectral inequalities via symmetrization in weighted spaces, unifying and extending known results to irregular domains. The provision of multiple concrete examples, including novel ones, adds concrete value and testability to the abstract theorem.

major comments (1)
  1. [main extension argument] Proof of the main extension (likely the argument following the statement that Talenti's inequality extends to W^{1,p}_0(Ω,μ) for Borel Ω): the zero-trace condition on non-open Borel sets is typically understood in the capacity sense (quasi-everywhere). The manuscript must explicitly show that the weighted rearrangement u* inherits the zero-trace property while preserving the relevant integrals; absolute continuity of μ alone does not automatically guarantee control over the capacity of the support or the behavior on sets of positive capacity but zero Lebesgue measure. This step is load-bearing for the central claim.
minor comments (2)
  1. [abstract / introduction] The abstract states that μ is absolutely continuous w.r.t. Lebesgue on the open connected set X; clarify whether this absolute continuity is used only for the measure or also to identify the underlying space with Lebesgue measure in the rearrangement construction.
  2. [examples] In the examples section, the transition from the abstract theorem to the concrete cases (e.g., homogeneous weights in cones) should include a brief verification that the chosen weight satisfies the absolute-continuity hypothesis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the key technical point in the extension of Talenti's inequality. We address the major comment below.

read point-by-point responses

  1. Referee: [main extension argument] Proof of the main extension (likely the argument following the statement that Talenti's inequality extends to W^{1,p}_0(Ω,μ) for Borel Ω): the zero-trace condition on non-open Borel sets is typically understood in the capacity sense (quasi-everywhere). The manuscript must explicitly show that the weighted rearrangement u* inherits the zero-trace property while preserving the relevant integrals; absolute continuity of μ alone does not automatically guarantee control over the capacity of the support or the behavior on sets of positive capacity but zero Lebesgue measure. This step is load-bearing for the central claim.

    Authors: We agree that the zero-trace condition on Borel sets is understood in the capacity sense and that this requires explicit verification for the rearranged function u*. The current argument invokes absolute continuity of μ to transfer measure-theoretic properties of level sets, but does not contain a separate paragraph confirming that u* vanishes quasi-everywhere on the boundary of the symmetrized domain while preserving the integrals that enter the eigenvalue inequality. We will add a short lemma (or dedicated remark) in the proof of the main extension theorem that uses the definition of weighted capacity and the fact that μ ≪ ℒ^N to show that sets of capacity zero are preserved under the weighted rearrangement. This addition will make the load-bearing step fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: direct extension of external Talenti inequality via symmetrization

full rationale

The derivation rests on extending a pre-existing weighted Pólya–Szegő inequality (originally for Lipschitz functions) to Sobolev functions with zero trace on Borel sets, using Talenti symmetrization to obtain Faber–Krahn bounds. No step reduces a claimed prediction or eigenvalue bound to a fitted parameter, self-defined quantity, or self-citation chain; the central argument is a proof of extension under the stated absolute continuity of μ, which is independent of the target spectral inequalities. The provided abstract and reader summary contain no equations or claims that collapse by construction to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of Sobolev spaces, absolutely continuous measures, and the classical Talenti inequality; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption μ is absolutely continuous w.r.t. Lebesgue measure on open connected X ⊂ ℝ^N
    Stated in the first sentence of the abstract as the setting for the weighted p-Laplacian.
  • standard math Talenti's weighted Pólya–Szegő inequality holds for Lipschitz functions
    Invoked as the starting point that is being extended.

pith-pipeline@v0.9.1-grok · 5655 in / 1369 out tokens · 22926 ms · 2026-06-29T06:32:02.679135+00:00 · methodology

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Cite this review

Pith. "Pith review of Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization." pith.science (2026). https://pith.science/paper/UNAHYBVD

@misc{pith2026260529721,
  author       = {Pith},
  title        = {Pith review of: Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNAHYBVD}},
  note         = {Machine review of arXiv:2605.29721}
}
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read the original abstract

We consider the weighted $p$-Laplacian associated with a measure $\mu$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted P\'olya--Szeg\H{o} inequality -- originally established for Lipschitz functions on $X$ -- extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $\Omega\subset X$. This yields Faber--Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, including classical Euclidean and Gaussian cases alongside new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations.

discussion (0)

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