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Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds

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Pith's one-line read For the Trudinger equation, a relative p-Faber–Krahn inequality is equivalent to volume doubling plus a sub-Gaussian upper estimate for subsolutions.

desk verdict A solid extension of Grigor'yan's FK/heat-kernel equivalence to Trudinger's equation; the new converse direction is cleanly proved, with only minor citation-level caveats in Section 4. read the letter →

arxiv 2608.07691 v1 pith:HNR4SDLZ submitted 2026-08-07 math.AP math.DG

Philipp Sürig This is my paper · ORCID
classification math.APmath.DG MSC 35K5558J3535K92
keywords TrudingerequationdoublynonlinearparabolicRiemannianmanifoldrelativeFaber–Krahninequalityvolumedoublingsub-Gaussianupperestimatep-Laplacianlong-timedecay
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

The paper studies the Trudinger equation $\partial_t u = \Delta_p u^{1/(p-1)}$ on geodesically complete, non-compact Riemannian manifolds and asks how the behavior of its subsolutions is tied to the geometry. Its central claim is a characterization: a relative $p$-Faber–Krahn inequality (a quantitative lower bound on Dirichlet energy in terms of the volume of the support) holds if and only if the manifold satisfies volume doubling and every non-negative bounded weak subsolution obeys a sub-Gaussian upper estimate, meaning decay $\exp(-c(d(x,A)/t^{1/p})^{p/(p-1)})$ away from the initial support. This is the nonlinear, $p>1$ counterpart of the classical $p=2$ heat-equation characterization, and it turns a purely geometric isoperimetric-type inequality into a quantitative diffusion statement. The paper also proves an improved long-time upper bound under a uniform $p$-Faber–Krahn inequality, showing that the shape of the Faber–Krahn function $\Lambda_p$ controls the large-time decay profile. If correct, these results give a geometric criterion for sub-Gaussian decay of Trudinger subsolutions and a method for bounding the first $p$-eigenvalue of geodesic balls.

What carries the argument

The load-bearing object is the first eigenfunction of the $p$-Laplacian on a precompact domain $D$: a non-negative $\varphi\in W^{1,p}_0(D)$ with $-\Delta_p \varphi = \lambda_{1,p}(D) \varphi^{p-1}$ and $\int_D \varphi^p = 1$. Its zero extension gives a separated weak subsolution $v(x,t)=e^{-\lambda_{1,p}(D)t}\tilde\varphi(x)^{p-1}$ of the Trudinger equation, and applying $(\mathrm{UE}_p)$ to $v$ converts the exponential decay rate into a lower bound on $\lambda_{1,p}(D)$; comparing this with the volume-doubling ball lower bound $\mu(B(x_D,r)) \ge c(r/R)^N \mu(B(x_0,R))$ yields the required Faber–Krahn inequality. In the long-time half, the engine is the Caccioppoli-type inequality for powers $u^{\sigma/p}$, together with the monotonicity of the $L^\sigma$ norms of solutions; these turn the uniform Faber–Krahn inequality into a differential inequality for $\Phi(t)=\int_M u^\sigma$ whose solution is controlled by the integral $\gamma$ defined by $t = \int_0^{\gamma(t)} dv/(\Lambda_p(v)v)$. The reverse doubling estimate $\mu(B(x,Ar)) \ge c A^{N_0} \mu(B(x,r))$ is used to handle the case where a ball is nearly filled by the support.

What would settle it

A concrete test would be to compute, on a geodesically complete non-compact manifold known to satisfy volume doubling and the sub-Gaussian estimate (for example, a manifold with polynomial volume growth), the first $p$-eigenvalue $\lambda_{1,p}(D)$ for a nested family of balls and check the lower bound $\lambda_{1,p}(D) \ge c R^{-p}(\mu(B(x_0,R))/\mu(D))^{p/N}$; a sequence of domains violating this bound would disprove the equivalence in Theorem 1.4.

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Extended reading notes

Core claim

On a geodesically complete non-compact Riemannian manifold, for any $p>1$, the relative $p$-Faber–Krahn inequality is equivalent to volume doubling together with the sub-Gaussian upper estimate for non-negative bounded weak subsolutions of the Trudinger equation. The forward half $(\mathrm{FK}_p) \Rightarrow (\mathrm{VD})$ is proved here by an iteration argument; the half $(\mathrm{FK}_p) \Rightarrow (\mathrm{UE}_p)$ is taken from the author's previous Theorem 1.1 in [21]. The reverse half $(\mathrm{VD})+(\mathrm{UE}_p) \Rightarrow (\mathrm{FK}_p)$ is proved by feeding the separated solution $v(x,t)=e^{-\lambda_{1,p}(D)t}\varphi(x)^{p-1}$ into the sub-Gaussian estimate, where $\varphi$ is the first $p$-Laplacian eigenfunction on a precompact domain $D$; this forces the lower bound $\lambda_{1,p}(D) \ge c R^{-p}(\mu(B_R)/\mu(D))^{p/N}$, which is exactly the relative Faber–Krahn inequality. Under a uniform Faber–Krahn inequality with function $\Lambda_p$, the paper derives a two-sided correspondence between the long-time decay of $L^\sigma$ norms and the integral $\gamma(t)=\int_0^t dv/(\Lambda_p(v)v)$, yielding Theorem 1.7 as an improvement over the earlier $t^{-n/p}$ bound in cases where $\Lambda_p$ decays logarithmically.

Load-bearing premise

The full equivalence rests on a previously published theorem, cited as [21], that the relative $p$-Faber–Krahn inequality alone forces the sub-Gaussian upper estimate; the paper does not reprove that direction, so the characterization stands or falls with the soundness of that borrowed result.

Editorial extensions

If this is right

  • A manifold with the relative $p$-Faber–Krahn inequality must satisfy volume doubling and must give the sub-Gaussian upper estimate for every non-negative bounded weak subsolution of Trudinger's equation.
  • A manifold with volume doubling and the sub-Gaussian upper estimate for one exponent $p_0$ automatically satisfies the relative $p_0$-Faber–Krahn inequality, and then the same sub-Gaussian estimate holds for every $p>p_0$ (Corollary 1.5).
  • Under a uniform $p$-Faber–Krahn inequality with $\Lambda_p(v) \ge c v^{-p/n}$, every non-negative bounded solution satisfies the long-time bound (1.5); when $\Lambda_p(v) \simeq (\log v)^{-\alpha}$, the time factor is $t^{-n/(p\sigma)}\exp(c\sigma t^{1/(\alpha+1)})$ rather than $t^{-n/p}$.
  • Conversely, from an upper estimate of the form (4.44) one recovers a lower bound $\lambda_{1,p}(\Omega) \ge c\Lambda_p(C\mu(\Omega))$ for every precompact open set, so the long-time shape of the estimate and the Faber–Krahn function determine each other.

Reading between the lines

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

  • The proof of Theorem 3.6 does not use the full nonlinear structure beyond homogeneity and the eigenvalue equation, so the same equivalence may hold for other doubly nonlinear parabolic equations with a separated exponential solution; testing this on porous-medium-type variants would be a natural extension.
  • Theorem 4.3 suggests a sharp transition in long-time decay: polynomial $\Lambda_p$ gives $t^{-n/p}$, while logarithmic $\Lambda_p$ degrades the time factor to $t^{-n/(p\sigma)}$ with an extra stretched-exponential in $t^{1/(\alpha+1)}$; the paper does not prove matching lower bounds, so whether these rates are sharp remains open.
  • Corollary 1.5 could be used as a bootstrap in proving $L^1$–$L^\infty$ smoothing: on a doubling manifold, establishing the sub-Gaussian estimate at a single exponent automatically upgrades it to all larger exponents, which may simplify regularity arguments for Trudinger-type flows.
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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

2 major / 3 minor

Summary. The paper studies the Trudinger equation ∂t u = Δp u^{1/(p-1)} on geodesically complete non-compact Riemannian manifolds and proves an equivalence between a relative p-Faber-Krahn inequality and the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions. Theorem 1.4 states (FK_p) ⇔ (VD)+(UE_p). The forward direction (FK_p)⇒(VD) is proved in Proposition 3.1; the implication (FK_p)⇒(UE_p) is cited from the author's earlier Theorem 1.1 in [21]. The reverse direction (VD)+(UE_p)⇒(FK_p) is proved in Theorem 3.6 by constructing an exponentially decaying subsolution from the first p-Laplacian eigenfunction and applying the upper estimate at a density point. The paper also derives an improved long-time upper estimate under a uniform Faber-Krahn inequality (Theorem 4.3, Theorem 1.7) and a converse statement (Proposition 4.7).

Significance. If correct, Theorem 1.4 is a nonlinear counterpart of Grigor'yan's characterization for the heat equation and is a valuable contribution to nonlinear potential theory on manifolds. The novel direction (FK_p)⇐(VD)+(UE_p) is proved with standard tools and the dependence on previous results is clearly disclosed. The improved upper estimate in Theorem 1.7 and the converse Proposition 4.7 are useful additions. The paper is generally well written and the main proofs are presented in sufficient detail, apart from the specific gaps noted below.

major comments (2)
  1. [Section 4, Lemma 4.2] The proof of Lemma 4.2 applies Lemma 2.3, which is stated for solutions of (2.8), to a subsolution u. Specifically, after the Hölder inequality the proof replaces ∫M u by ∫M u0 'using also Lemma 2.3'. Since u is only assumed to be a non-negative bounded subsolution, the monotonicity of the L1 norm is not covered by Lemma 2.3 as stated, nor by the Caccioppoli inequality of Lemma 2.2, which is restricted to σ ≥ p/(p−1). The author should either prove that t ↦ ∥u(·,t)∥_{L1(M)} is non-increasing for subsolutions (for example, by a cut-off argument in the weak formulation (2.11)) or extend Lemma 2.3 to subsolutions. This is necessary for the proof of (4.33) and hence for Theorem 4.3 and the stated result Theorem 1.7.
  2. [Section 3, Proposition 3.1] In the iteration leading to the volume doubling property, the factor μ(B(x,r/2^m))^{(1/(1+ν))^m} is dropped when sending m→∞. Because μ(B(x,r/2^m)) tends to zero as m→∞, the convergence of this factor to 1 is not automatic. On a Riemannian manifold it follows from the local volume comparability μ(B(x,ρ))≍ρ^n for small ρ, which makes the exponent (1/(1+ν))^m tend to zero while the measure decays at most polynomially, but this justification is omitted. Without it the limiting step in the derivation of (VD) is incomplete; please add a short argument or a reference.
minor comments (3)
  1. [Definition 1.1] The set D={w>0} is not necessarily open for a general w∈W^{1,p}_0(B). Since the proofs in Section 3 work with precompact open sets D, it would be helpful to state explicitly that the relative Faber-Krahn inequality is assumed for all such D, or to add a remark explaining that the general case follows by approximation.
  2. [Lemma 3.5] The approximation of the test function ψ by bounded functions and the passage to the limit ε→0 are only sketched; adding a few more details would improve readability and remove any doubt about the validity of the limiting argument in (3.24).
  3. [Throughout] There are minor typographical and convention issues, including inconsistent capitalization in the title and abstract, and the reuse of symbols c and C without comment in statements such as Theorem 3.6 and Proposition 4.7. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.4 is assembled from first-principles arguments and disclosed published results, with the one imported direction explicitly attributed to [21].

full rationale

After walking the derivation chain, I find no circularity. Theorem 1.4 splits into three parts: (FK_p) implies (VD), proved in Proposition 3.1 by applying (FK_p) to the cone function w=(r-d(x,·))_+ and iterating; (FK_p) implies (UE_p), imported from the author's prior Theorem 1.1 in [21] and explicitly disclosed in Remark 3.7; and (FK_p) is implied by (VD)+(UE_p), proved in Theorem 3.6 by constructing the subsolution v=e^{-lambda t} phi^{p-1} from the p-Laplacian eigenfunction and applying (UE_p). Each new derivation is first-principles given the stated hypotheses, and the cited inputs [21], [9], [11], and [10] are published, parameter-free statements whose assumptions do not include Theorem 1.4. The use of Lemma 2.3 from [9] in Lemma 4.2 is stated for a solution while Lemma 4.2 concerns a subsolution, but for nonnegative subsolutions the needed L^1 monotonicity follows by integrating the differential inequality, so this is at most a minor presentation gap rather than circularity. The secondary Theorem 4.3 adds a uniform-Faber-Krahn long-time estimate; its proof again invokes Caccioppoli and L^sigma-decay lemmas from prior work, but no fitted parameter is renamed as a prediction and no equation is taken as its own conclusion. The paper is self-contained against external benchmarks in the sense that its new implications are proven from stated assumptions, and its only imported implication is openly identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The constants c, C, nu, N entering FK_p, VD and UE_p are hypotheses of the theorems, not numbers fitted to any data. The exponent sigma in Theorem 1.7 and Theorem 4.3 is a free parameter of the statement (sigma >= p/(p-1)), and the Faber-Krahn function Lambda_p is given data; none is tuned to make the derivation work. No new physical or mathematical entities are postulated; the constructed subsolution v=e^{-lambda t} phi^{p-1} is assembled from the existing eigenfunction phi and involves no new object with independent degrees of freedom.

assumptions (7)
  • domain assumption Caccioppoli-type inequality (Lemma 2.2, adapted from Lemma 2.6 of [11])
    Used to obtain the L^sigma differential inequality (4.35) driving Lemma 4.2; not proved in the present text.
  • domain assumption Monotone decay of L^sigma norms for solutions (Lemma 2.3, from Lemma 2.9 of [9])
    Used to control the integral of u_0 in Lemma 4.2.
  • domain assumption (FK_p) implies (UE_p), Theorem 1.1 of [21]
    Supplies one half of the equivalence in Theorem 1.4; the paper proves only the converse.
  • domain assumption Lemma 5.1 of [21] converts L^sigma decay into pointwise estimates with sub-Gaussian tail
    Bridges Lemma 4.2 to the L-infinity pointwise bound in Theorem 4.3.
  • standard math Sobolev inequality on precompact domains (3.19) with exponent kappa>1
    Used in Lemma 3.4 to bootstrap L^q regularity of the p-eigenfunction to L-infinity.
  • standard math Compact embedding W^{1,p}_0(D) into L^p(D) for precompact D and existence of the p-Laplacian eigenfunction
    Underlies Lemma 3.3.
  • standard math Reverse volume doubling (3.30) follows from (VD) with constants c, N_0 (see [10])
    Used in the second case of Theorem 3.6's proof.

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Pith. "Pith review of Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds." pith.science (2026). https://pith.science/paper/HNR4SDLZ

@misc{pith2026260807691,
  author       = {Pith},
  title        = {Pith review of: Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNR4SDLZ}},
  note         = {Machine review of arXiv:2608.07691}
}
abstract

We consider on Riemannian manifolds the Trudinger equation \begin{equation*}\partial _{t}u=\Delta _{p}u^{\frac{1}{p-1}},\end{equation*} where $p>1$. We prove that a relative $p$--Faber--Krahn inequality is equivalent to the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions of Trudinger's equation. We also derive an improved long-time upper estimate under a uniform $p$--Faber--Krahn inequality.

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Works this paper leans on

24 extracted references · 14 canonical work pages

  1. [21]

    P. S¨ urig. Sharp sub-Gaussian upper bounds for subsolutions of Trudinger’s equation on Riemannian manifolds.Nonlinear Analysis, 249:113641, 2024

    2024

  2. [1]

    B¨ ogelein, F

    V. B¨ ogelein, F. Duzaar, P. Marcellini, and C. Scheven. Doubly nonlinear equations of porous medium type.Archive for Rational Mechanics and Analysis, 229(2):503–545, 2018

    work page 2018

  3. [2]

    P. Buser. A note on the isoperimetric constant.Ann. Sci. Ecole Norm. Sup., 15:213–230, 1982

    1982

  4. [3]

    G. Carron. In´ egalit´ es isop´ erim´ etriques de Faber-Krahn et cons´ equences. InActes de la table ronde de g´ eom´ etrie diff´ erentielle (Luminy, 1992)”, Collection SMF S´ eminaires et Congres, volume 1, pages 205–232, 1996

    work page 1992

  5. [4]

    Coulhon and D

    T. Coulhon and D. Hauer. Regularisation effects of nonlinear semigroups.arXiv preprint arXiv:1604.08737, 2016

    arXiv 2016

  6. [5]

    Coulhon and L

    T. Coulhon and L. Saloff-Coste. Isop´ erim´ etrie pour les groupes et les vari´ et´ es.Revista Matem´ atica Iberoamericana, 9(2):293–314, 1993

    work page 1993

  7. [6]

    Doubling measures, Poincar\'e inequalities and parabolic Harnack inequalities for a doubly nonlinear equation

    T. Elenius and J. Kinnunen. Doubling measures, Poincar´ e inequalities and parabolic Harnack inequalities for a doubly nonlinear equation.arXiv preprint arXiv:2605.27665, 2026

    work page Pith review arXiv 2026

  8. [7]

    Grigor’yan

    A. Grigor’yan. The heat equation on non-compact Riemannian manifolds.Math. USSR Sb., 72:47–77, 1992

    1992

Show all 24 references
  1. [8]

    Grigor’yan

    A. Grigor’yan. Heat kernel upper bounds on a complete non-compact manifold.Revista Matem´ atica Iberoamericana, 10(2):395–452, 1994

    1994

  2. [9]

    Grigor’yan and P

    A. Grigor’yan and P. S¨ urig. Sharp propagation rate for Leibenson’s equation on Rie- mannian manifolds.Ann. Scuola Norm. Super. Pisa, 2024

    2024

  3. [10]

    Grigor’yan and J

    A. Grigor’yan and J. Hu. Upper bounds of heat kernels on doubling spaces.Mosc. Math. J, 14(3):505–563, 2014

    2014

  4. [11]

    Grigor’yan and P

    A. Grigor’yan and P. S¨ urig. Finite propagation speed for Leibenson’s equation on Rie- mannian manifolds.Comm. Anal. Geom, 32(9):2467–2504, 2024

    2024

  5. [12]

    K. Ishige. On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation.SIAM Journal on Mathematical Analysis, 27(5):1235–1260, 1996. 16

    1996

  6. [13]

    A. V. Ivanov. Regularity for doubly nonlinear parabolic equations.Journal of Mathe- matical Sciences, 83(1):22–37, 1997

    1997

  7. [14]

    Leibenson

    L. Leibenson. General problem of the movement of a compressible fluid in a porous medium. Izv Akad. Nauk SSSR.Geography and Geophysics, 9:7–10, 1945

    1945

  8. [15]

    Leibenson

    L. Leibenson. Turbulent movement of gas in a porous medium.Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz, 9:3–6, 1945

    1945

  9. [16]

    Lindgren and P

    E. Lindgren and P. Lindqvist. On a comparison principle for Trudinger’s equation. Advances in Calculus of Variations, 15(3):401–415, 2022

    2022

  10. [17]

    Lindqvist

    P. Lindqvist. On the equation div |∇u|p−2∇u −λ|u| p−2u= 0.Proceedings of the American Mathematical Society, 109(1):157–164, 1990

    1990

  11. [18]

    Lindqvist

    P. Lindqvist. A nonlinear eigenvalue problem.Topics in mathematical analysis, 3:175– 203, 2008

    2008

  12. [19]

    Lindqvist, M

    P. Lindqvist, M. Parviainen, and J. Siltakoski. Lipschitz continuity and equivalence of positive viscosity and weak solutions to Trudinger’s equation: P. Lindqvist et al. Mathematische Annalen, 394(3):59, 2026

    2026

  13. [20]

    Saloff-Coste.Aspects of Sobolev-type inequalities

    L. Saloff-Coste.Aspects of Sobolev-type inequalities. LMS Lecture Notes Series, vol. 289. Cambridge Univ. Press, 2002

    2002

  14. [22]

    P. S¨ urig. Existence results for Leibenson’s equation on Riemannian manifolds.arXiv preprint arXiv:2601.20640, 2026

    2026 arXiv

  15. [23]

    P. S¨ urig. Long time upper bounds for solutions of Leibenson’s equation on Riemannian manifolds.arXiv preprint arXiv:2604.23227, 2026

    2026 arXiv

  16. [24]

    N. S. Trudinger. Pointwise estimates and quasilinear parabolic equations.Communica- tions on Pure and Applied Mathematics, 21(3):205–226, 1968. Universit¨ at Bielefeld, Fakult¨ at f¨ ur Mathematik, Postfach 100131, D-33501, Bielefeld, Ger- many philipp.suerig@uni-bielefeld.de 17

    1968

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