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Fundamental Gaps for the Dirichlet \(p\)-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.13443 v1 pith:YDIQM2DJ submitted 2026-08-13 math.AP

Rui Chen , Daniel Hauer This is my paper · ORCID
classification math.AP MSC 35P3035J9235P1549R05
keywords Dirichletp-Laplacianfundamentalgapconvexpotentiallog-concavityweightedPoincaréinequalitycollapsingdomainssharpone-dimensionalboundeigenvalues
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 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 $12, 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\varepsilon^{2-p}$.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

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).

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 (1)
  1. [Section 4, Eq. (4.1)] 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.
minor comments (5)
  1. [Section 4, paragraph before Lemma 1.8] 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.
  2. [Section 1, Eq. (1.5) and Theorem 1.5(i)] 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}'.
  3. [Section 2, opening paragraph] 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.
  4. [Section 3.1, after Proposition 1.2] 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.
  5. [Throughout] There are numerous typographical and spacing errors, such as 'providedifferentforms' and missing spaces around displayed equations; a careful copyedit is needed before final publication.

Circularity Check

1 steps flagged · score 5.0 of 10

Sharp one-dimensional bound for every convex potential rests on eq. (4.1), a load-bearing comparison principle imported by self-citation from a p=2 Robin problem.

  1. self citation load bearing [Section 4, eq. (4.1) and its use in the proof of Theorem 1.9(ii)]
    "We first recall the following one-dimensional comparison principle. If V is convex on I_D, then there exists an affine function ℓ(x)=ax+b such that Γ_p(I_D,V)≥Γ_p(I_D,ℓ)=Γ_p(I_D,ax), (4.1) with strict inequality unless V is affine. This is the homogeneous Dirichlet analogue of [5, Theorem 1.2], whose comparison argument carries over unchanged."

    The reduction is explicit: Theorem 1.9(ii) is obtained by applying (4.1) to pass from V to ax, and then invoking part (i). Eq. (4.1) itself is not proved in the paper; it is asserted as the homogeneous Dirichlet analogue of [5, Theorem 1.2], whose comparison argument 'carries over unchanged.' [5] is co-authored by the present second author and concerns a linear p=2 Schrödinger operator with Robin boundary conditions. No statement or proof is supplied for the nonlinear Dirichlet p-Laplacian, and the paper's own discussion records that u_j φ_p(u_j') is not a derivative for p≠2, so Lavine's p=2 argument does not directly extend.

full rationale

The core of the paper is self-contained: Section 2 proves log-concavity by regularization and two-point maximum principle; Section 3 derives the collapsing-domain dichotomy, the degenerate weighted Poincaré inequality, Theorem 1.5 and the minimizer existence directly; and Theorem 1.9(i) for linear potentials is proved from the concavity of μ̂ and nodal decomposition. The circular hinge is Theorem 1.9(ii): the extension from linear to arbitrary convex potentials is exactly eq. (4.1), which the paper states without proof as a transfer of [5, Theorem 1.2] from a self-cited p=2 Robin setting. The paper itself notes the p≠2 obstruction to Lavine's argument, so the unproved transfer is load-bearing, not formal. The explicit 'We omit the proof' for the nonlinear Lavine identity is flagged, but that identity is not used in the main argument. Higher-dimensional results and the linear one-dimensional case are independent, so score 5 rather than 6+.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All external inputs are listed here. The paper itself contributes the proofs of log-concavity, the degenerate weighted Poincare inequality, the stability estimates, the collapse dichotomy, and the linear-potential one-dimensional calculation. The unproved comparison principle and the two preprint dependencies are the main uncharged entries.

assumptions (6)
  • domain assumption Ground-state identity from [7, Cor 3.5]: E_V(w) - lambda_{1,p} integral |w|^p = integral C_p(grad w, u1 grad(w/u1)) dx
    Used in Theorem 1.4 proofs; the cited paper is an unreviewed 2025 preprint.
  • domain assumption Sharp remainder constant c_p for the convexity gap (eq. (1.4)), with c_p = (p-1) rho_p^{2-p} for p>2, from [65, Lemma 2.7]
    Gives the explicit constant in Theorem 1.5(i); [65] is an unreviewed 2026 preprint.
  • ad hoc to paper One-dimensional comparison principle: for convex V on I_D there is an affine ell such that Gamma_p(I_D,V) >= Gamma_p(I_D,ell) (eq. (4.1))
    Asserted to carry over unchanged from [5, Theorem 1.2], which treats a linear (p=2) Robin problem; no proof is given. Load-bearing for Theorem 1.9(ii).
  • standard math Sharp weighted Poincare inequality for log-concave weights (Lemma 3.5), from [29, Theorem 1.1]
    Published peer-reviewed result; provides the dimension-free Poincare constant.
  • standard math Continuity of variational p-eigenvalues in p (cited [24])
    Used in the p down to 2 limiting argument of Theorem 1.7.
  • standard math Mountain-pass characterization of lambda_{2,p} (eq. (1.10)), from [22, Proposition 15]
    Connects the second eigenvalue to paths in the Lp-sphere; used throughout Section 3.3.

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Pith. "Pith review of Fundamental Gaps for the Dirichlet \(p\)-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy." pith.science (2026). https://pith.science/paper/YDIQM2DJ

@misc{pith2026260813443,
  author       = {Pith},
  title        = {Pith review of: Fundamental Gaps for the Dirichlet \(p\)-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDIQM2DJ}},
  note         = {Machine review of arXiv:2608.13443}
}
abstract

We study fundamental gaps for the Dirichlet \(p\)-Laplacian on bounded convex domains with convex potentials. We prove log-concavity of the positive first eigenfunction by a regularization and two-point maximum principle. For \(N\geq2\), we identify a sharp transition at \(p=2\) through collapsing smooth convex domains: the gap vanishes for \(1<p<2\), remains of order \(D^{-2}\) for \(p=2\), and diverges for \(p>2\). For \(p\geq2\) and convex potentials, we first establish a degenerate weighted Poincar\'e inequality, which yields quantitative stability estimates for the \(L^p\)-Poincar\'e inequality and, in turn, dimension-free bounds for the fundamental gap; for zero potential, we further obtain an enhanced gap estimate involving both the first eigenvalue and the diameter. We also prove existence of diameter-normalized gap minimizers for \(p>2\) and show that they degenerate as \(p\downarrow2\). Finally, for $N=1,$ we prove the sharp inequality \[ \lambda_{2,p}(I_D,V)-\lambda_{1,p}(I_D,V) \geq (p-1)(2^p-1)\left(\frac{\pi_p}{D}\right)^p \] for every \(p>1\) and every convex potential, with equality precisely for constant potentials.

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