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REVIEW 2 major objections 4 minor 34 references

Finsler $p$-Laplacian in domains becoming unbounded

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Pith's one-line read This paper proves that, on cylinders whose length tends to infinity, solutions of the Finsler p-Laplace equation converge to the cross-sectional solution at a polynomial rate, with matching asymptotics for energies and first eigenvalues.

desk verdict First treatment of the Finsler p-Laplacian on long cylinders, but Theorem 1.4's stated convergence rates are p-th-power too fast because the proof bounds an L^p integral, not the L^p norm. read the letter →

arxiv 2505.22329 v1 pith:AIYHBNYZ submitted 2025-05-28 math.AP

Luca Esposito , Lorenzo Lamberti , Dattatreya N. N. , Prosenjit Roy This is my paper
classification math.AP MSC 35Pxx35B4047J1049R0535P15
keywords Finslerp-LaplaciananisotropiccylindricaldomainsasymptoticbehaviourfirsteigenvalueMinkowskinormconvergencerateDirichletproblem
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 sets out to show that the Finsler p-Laplacian—the anisotropic operator built from a convex, positively homogeneous gauge $H$—behaves on very long cylinders exactly as the much simpler cross-sectional problem predicts. As the cylinder length $\ell$ tends to infinity, the solutions on the bounded cylinders converge to the solution on the fixed cross-section, with explicit polynomial rates in the general case and exponential rates for a subclass of gauges. The paper also proves that the energy per unit length converges to the cross-sectional energy with an $O(1/\ell)$ correction, and that the first eigenvalue converges to the cross-sectional eigenvalue at rate $O(1/\ell)$, with the sharp rate $1/\ell^p$ for a specific family of gauges. Because these operators include the Laplacian, the p-Laplacian and the pseudo-p-Laplacian as special cases, the results unify several asymptotics known separately for those equations.

What carries the argument

The carrier of the argument is the monotone vector field $\Phi(z)=H^{p-1}(z)\nabla H(z)$ defining the Finsler p-Laplacian, together with the inequalities A1–A2 that control it. The proofs use cut-off functions in the long variable $X_1$ to localize near the middle of the cylinder, the strip Poincaré inequality, and Jensen averaging over $X_1$ for the energy comparison. For the eigenvalue lower bound, the paper proves a Finsler version of Picone's identity; for the exponential rate, it applies a hole-filling iteration so that the excess gradient on an inner slab is bounded by a fraction of the excess on a slightly larger slab.

What would settle it

Check whether assumptions A1–A2 hold for a polyhedral gauge such as the $\ell^\infty$ norm, whose flat facets make strong monotonicity not automatic; if they fail there, the general polynomial rates cannot cover all gauges the paper admits. A complementary check is to compute $\lambda_1^\ell-\mu_\infty$ numerically for $\tilde H_q$ with $q<p$ and compare the gap with the claimed $1/\ell^p$ order.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the asymptotic behaviour on cylinders $\Omega_\ell=\ell\omega_1\times\omega_2$ is governed entirely by the cross-sectional problem on $\omega_2$, provided the gauge $H$ satisfies the monotonicity assumptions A1–A2. For the Dirichlet problem with right-hand side depending only on $X_2$, Theorem 1.4 gives $\|\nabla(u_\ell-u_\infty)\|_{L^p(\Omega_{\ell/2})}\le C_1\ell^{m-p/(p-1)}$ for $p\ge 2$ and $\le C_2\ell^{m-p^2/(2-p)}$ for $1<p<2$, where $u_\infty$ is the cross-sectional solution extended constantly in the $X_1$ variables. Theorem 1.5 shows $J_\ell(u_\ell)/|\ell\omega_1|\to J_\infty(u_\infty)$ with a $C/\ell$ correction, and Theorem 1.6 shows $\mu_\infty\le\lambda_1^\ell\le\mu_\infty+C/\ell$ for the first eigenvalues. For gauges of the form $\tilde H_q=(F^q+G^q)^{1/q}$ the paper identifies the sharper rate $1/\ell^p$ for the eigenvalue gap, and for $\tilde H_p$ with $p\ge 2$ it proves exponential convergence of the solutions.

Load-bearing premise

The load-bearing premise is that the convex, positively homogeneous function $H$ used to build the operator satisfies the strong monotonicity and Lipschitz-type inequalities A1–A2; if those fail, the polynomial rates and the lower eigenvalue bound do not follow from the stated hypotheses.

Editorial extensions

If this is right

  • In a long cylinder, the solution of the Dirichlet problem for any admissible gauge $H$ becomes, away from the ends, essentially a function of the cross-sectional variables only, with the error controlled by the stated power of $\ell$.
  • The total energy on the finite cylinder, divided by the cylinder length, approaches the cross-sectional energy, with the finite-length correction no larger than $C/\ell$.
  • The first eigenvalue of the Finsler p-Laplacian with Dirichlet conditions is bounded below by the cross-sectional eigenvalue and above by $\mu_\infty+C/\ell$.
  • For gauges of the form $\tilde H_q$ with $q\le p$ the eigenvalue gap is at least $C_1/\ell^p$, and for $q\ge p$ it is at most $C_2/\ell^p$, so the rate is optimal for this family.
  • The known cylinder asymptotics for the Laplacian, the p-Laplacian, the pseudo-p-Laplacian and constant-coefficient elliptic operators are recovered as special cases of one set of estimates.

Reading between the lines

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

  • Because Theorem 1.4 gives polynomial rates for all admissible gauges but exponential rates only for a separable subclass, the natural next question is whether the polynomial exponents are sharp; the difference between $p/(p-1)$ and $p^2/(2-p)$ suggests the true rate may depend on the smoothness of the gauge.
  • The monotonicity assumptions A1–A2 are the only place where regularity of the gauge matters; polyhedral crystalline energies that arise in Wulff-shape problems may fail them, so extending the estimates to nonsmooth or flat-faceted gauges would require an approximation argument.
  • The exact eigenvalue rate $1/\ell^p$ for $\tilde H_q$ points to a general heuristic: the rate is set by the lowest-order term in the gauge decomposition, so mixed gauges with different powers in different coordinate blocks should inherit the slower of the two rates.
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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 / 4 minor

Summary. The paper studies the Finsler (anisotropic) p-Laplacian on cylindrical domains Ωℓ = ℓω1 × ω2 whose length ℓ tends to infinity, with Dirichlet data depending only on the cross-sectional variables. Under structural monotonicity assumptions A1 and A2 on the operator Φ(z)=H^{p−1}(z)∇H(z), it claims polynomial convergence rates for solutions to the cross-sectional solution (Theorem 1.4), exponential rates for a subclass H̃p (Theorem 3.4), convergence of the normalized energy (Theorem 1.5), and convergence of the first eigenvalue with rate O(1/ℓ) (Theorem 1.6), plus an optimal rate O(1/ℓ^p) for a subclass H̃q (Theorem 5.1). The methods are variational: cut-off functions, Poincaré inequalities on strips, and Picone's identity.

Significance. The intended results, if fully established, would unify several previously known asymptotic statements for the Laplacian, the p-Laplacian, and the pseudo-p-Laplacian in long cylinders, with explicit rates and constants. The paper also contains useful technical ingredients, notably the hole-filling argument for the exponential rate and a version of Picone's identity for the Finsler p-Laplacian. However, the central convergence-rate theorem contains an exponent error that invalidates the stated rates, and the lower eigenvalue bound depends on an unproved regularity assertion. The paper is not acceptable in its current form, but the underlying approach is promising and the issues appear fixable.

major comments (2)
  1. [Section 3.1, Theorem 1.4, Eq. (16)] The proof bounds the integral Y = ∫_{Ω_{ℓ/2}} |∇(u_ℓ−u_∞)|^p dx by C ℓ^{m − p/(p−1)} for p ≥ 2 and by C ℓ^{m − p^2/(2−p)} for 1 < p < 2. The theorem, however, states bounds on the L^p norm of ∇(u_ℓ−u_∞). Taking the p-th root gives ∥∇(u_ℓ−u_∞)∥_{L^p(Ω_{ℓ/2})} ≤ C ℓ^{m/p − 1/(p−1)} for p ≥ 2 and ≤ C ℓ^{m/p − p/(2−p)} for 1 < p < 2, not the displayed exponents in (4) and (5). This is not a cosmetic issue: for example, with m = 3 and p = 2 the theorem's exponent is 1, so it does not even assert convergence, while the corrected bound grows as ℓ^{1/2}. Because |Ω_{ℓ/2}| ≈ ℓ^m, unnormalized L^p convergence fails whenever m > p/(p−1). The statement of Theorem 1.4 must be corrected, either by normalizing by |Ω_{ℓ/2}|^{1/p} or by restricting m, and the proof must be reconciled with the stated rates.
  2. [Section 5.1, lower bound of Theorem 1.6] The lower bound λ_ℓ^1 ≥ μ_∞ is proved by applying Picone's identity with test function φ^p/u_∞^{p−1}. The proof contains the assertion 'Since u_∞ > 0 and u_∞ ∈ C^1', but no reference or proof is given. Under the hypotheses of the paper, where H is only assumed convex and C^1 away from 0, C^1 regularity of the first eigenfunction of the Finsler p-Laplacian is not automatic and is not established. Without this regularity, the quotient φ^p/u_∞^{p−1} need not be an admissible test function in W^{1,p}_0(Ω_ℓ). This is a load-bearing gap in the proof of Theorem 1.6. The authors should either prove or cite the required C^{1,α} (or at least C^1) regularity for positive eigenfunctions under assumptions A1–A2, or add explicit hypotheses on H (such as stronger smoothness and uniform convexity) under which such regularity holds and verify that the examples satisfy them.
minor comments (4)
  1. [Section 3, Lemma 3.1] In the proof of Lemma 3.1 the constant is written as C = C_1 |ω_1|^{a p} with an undefined exponent 'a'; this should be C_1 |ω_1|^{(p−1)/p} (or a correctly defined quantity).
  2. [Assumptions A1–A2] The assumptions A1 and A2 are stated as hypotheses, but the paper does not discuss which of the examples in Section 1 (q-norms, H_{A,q}, H_{q,P,Λ}) satisfy them. For general convex, positively homogeneous, C^1 functions H, strong monotonicity of the form (A1) is not automatic. A short verification for the listed examples, or a reference, would make the scope of the theorems clearer.
  3. [Section 5.2, Theorem 5.1] The statement of Theorem 5.1 splits into q ≤ p and q ≥ p; for q = p the two inequalities combine to give the optimal rate, but this is not explicitly said. The phrase 'for q≤p ... and for q≥p ...' could be clarified to state that for q = p both the lower and upper bounds hold.
  4. [General presentation] There are several typos, e.g., 'any any' in Assumption A2, 'phenomenons' in the introduction, and an inconsistent use of ℓ in the proof of Theorem 3.4 where |∇X1 ρ_ℓ| ≤ c is written rather than a bound involving 1/ℓ. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the results follow from explicit structural assumptions A1-A2 and auxiliary lemmas proved in the paper, with no fitted parameters and no load-bearing self-citation chain.

full rationale

The paper's main results are conditional theorems: given the explicit monotonicity and Lipschitz-type conditions A1-A2 on the Finsler gauge H, the authors prove the convergence and eigenvalue estimates from the weak formulation and energy inequalities. The constants in the rates depend on H only through the structural constants in A1-A2 and the data f, omega_1, omega_2; nothing is fitted to the output. The auxiliary results used in the core derivation, such as the strip Poincare inequality (Lemma 2.7) and Picone's identity (Lemma 2.6), are either proved in the paper or proved there in the needed form, even when a classical reference is also cited. Earlier work by the same authors is mentioned only in the literature survey and is not used to justify the main theorems. The special case H = H_q is an example illustrating the general framework, not a target built into the estimates. The reader's concern about whether every admissible H satisfies A1-A2 is a hypothesis-checking issue, not circularity, and the skeptic's point about the displayed exponent in Theorem 1.4 concerns the consistency of the theorem statement with the proof's estimate on the L^p integral rather than any circular dependence of the result on its own conclusion. The derivation is therefore self-contained relative to its stated assumptions and earns a circularity score of 0.

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

No fitted or hand-tuned numerical parameters appear. The constants c1, c2, c3, theta_1, theta_2, and mu_infinity are structural data of the problem, not degrees of freedom adjusted to make the conclusion hold. No new physical entities, particles, forces, or dimensions are introduced; H_tilde_q and F, G are examples of admissible H, not independent postulates.

assumptions (6)
  • domain assumption A1/A2: the operator Phi(z) = H^{p-1}(z) grad H(z) is strongly monotone with the stated p-dependent constants, plus the Lipschitz-type bound A2 for 1 < p < 2.
    Stated in Section 1 and used in Lemma 3.1, Lemma 3.2, and Theorem 1.4. The paper does not characterize which gauges H satisfy it beyond examples.
  • domain assumption H is a convex, positively 1-homogeneous, C^1(R^n \ {0}) gauge with theta_1 |z| <= H(z) <= theta_2 |z|.
    Definition in Section 1; used in Lemma 2.1 and throughout for H-Holder, subadditivity, and bounded derivatives.
  • domain assumption u_infinity is a positive C^1 first eigenfunction of the cross-sectional problem.
    Assumed in Section 5.1 to justify the Picone test function phi^p / u_infinity^{p-1}; regularity and strict positivity for Finsler p-Laplacian eigenfunctions are not proved or cited.
  • domain assumption f is in L^{p/(p-1)}(omega_2), omega_1 is open bounded convex with 0 in omega_1, omega_2 is bounded open, and Dirichlet boundary data hold on Omega_ell.
    Problem setup in Section 2.1; used for existence, uniqueness, and all estimates.
  • standard math Poincare inequality on strips (Lemma 2.7) and the weak Picone identity (Lemma 2.6) hold as stated.
    Both are proved or referenced in Section 2; they are load-bearing in Theorems 1.4 and 1.6.
  • domain assumption For the exponential and exact-rate results, H belongs to the subclass H_tilde_q in (17) with F, G satisfying monotonicity A3, and q <= p or q >= p in Theorem 5.1.
    Restricts the results; the exponential argument uses the separated structure (18), and the sharp eigenvalue rate uses the split H^p >= F^p + G^p or H^p <= F^p + G^p.

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Pith. "Pith review of Finsler $p$-Laplacian in domains becoming unbounded." pith.science (2026). https://pith.science/paper/AIYHBNYZ

@misc{pith2026250522329,
  author       = {Pith},
  title        = {Pith review of: Finsler $p$-Laplacian in domains becoming unbounded},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIYHBNYZ}},
  note         = {Machine review of arXiv:2505.22329}
}
abstract

We study the asymptotic behavior of sequences of solutions, energies functionals, and the first eigenvalues associated with the Finsler $p$-Laplace operator, also known as the anisotropic $p$-Laplace operator on a sequence of bounded cylinders whose length tends to infinity. We prove that the solutions on the bounded cylinders converge to the solution on the cross-section, with a polynomial rate of convergence in the general case and exponential convergence in some special cases. We show that energies on finite cylinders, with the multiplication of a scaling factor, converge to the energy on the cross-section. Finally, we investigate the convergence of the first eigenvalue and, for a specific subclass, we provide the optimal convergence rate.

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Works this paper leans on

34 extracted references · 32 canonical work pages

  1. [1]

    Roy, and V

    Adimurthi, P. Roy, and V . Sahu. Fractional boundary hardy inequality for the critical cases, preprint, https://doi.org/10.48550/arxiv.2308.11956

    work page doi:10.48550/arxiv.2308.11956

  2. [2]

    Arbel and J

    E. Arbel and J. W. Cahn. A method for the absolute measurement of anisotropic surface free energies. Surface Science, 66(1):14–24, 1977

    work page 1977

  3. [3]

    Bao, S.-S

    D. Bao, S.-S. Chern, and Z. Shen. An introduction to Riemann-Finsler geometry, volume 200 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2000

    work page 2000

  4. [4]

    Bellettini and M

    G. Bellettini and M. Paolini. Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J., 25(3):537–566, 1996

    work page 1996

  5. [5]

    Belloni, V

    M. Belloni, V . Ferone, and B. Kawohl. Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z. Angew. Math. Phys., 54(5):771–783, 2003. Special issue dedicated to Lawrence E. Payne

    work page 2003

  6. [6]

    Ben and H.-S

    A.-Q. Ben and H.-S. Dou. Simulation and stability study of the flow around a cylinder in infinite domain. Procedia Engineering, 126:68–72, 2015. Frontiers in Fluid Mechanics Research

    2015

  7. [7]

    Buzzegoli and K

    M. Buzzegoli and K. Tuchin. Chiral magnetic e ffect in a cylindrical domain. Phys. Rev. D, 108:056008, Sep 2023

    work page 2023

  8. [8]

    M. Chipot. Asymptotic behaviour of operators sum of p-Laplacians. Discrete Contin. Dyn. Syst. Ser. S , 17(2):917–929, 2024

    work page 2024

Show all 34 references
  1. [9]

    Chipot, J

    M. Chipot, J. D ´avila, and M. del Pino. On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains. J. Fixed Point Theory Appl., 19(1):205–213, 2017

    2017

  2. [10]

    Chipot, A

    M. Chipot, A. Mojsic, and P. Roy. On some variational problems set on domains tending to infinity. Discrete Contin. Dyn. Syst., 36(7):3603–3621, 2016

    2016

  3. [11]

    Chipot and A

    M. Chipot and A. Rougirel. On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Commun. Contemp. Math., 4(1):15–44, 2002

    2002

  4. [12]

    Chipot and A

    M. Chipot and A. Rougirel. On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded. Trans. Amer. Math. Soc., 360(7):3579–3602, 2008

    2008

  5. [13]

    Chipot, P

    M. Chipot, P. Roy, and I. Shafrir. Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity. Asymptot. Anal., 85(3-4):199–227, 2013. 18

    2013

  6. [14]

    Chipot and Y

    M. Chipot and Y . Xie. On the asymptotic behaviour of the p-Laplace equation in cylinders becoming un- bounded. In Nonlinear partial differential equations and their applications, volume 20 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 16–27. GakkB otosho, Tokyo, 2004

    2004

  7. [15]

    Chipot and K

    M. Chipot and K. Yeressian. Exponential rates of convergence by an iteration technique.C. R. Math. Acad. Sci. Paris, 346(1-2):21–26, 2008

    2008

  8. [16]

    Chipot and K

    M. Chipot and K. Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete Contin. Dyn. Syst., 33(11-12):4875–4890, 2013

    2013

  9. [17]

    Chowdhury and N

    I. Chowdhury and N. N. Dattatreya. Boundary blow-up solutions of second order quasilinear equation on infinite cylinders

  10. [18]

    U. Clarenz. The Wul ff shape minimizes an anisotropic Willmore functional. Interfaces Free Bound., 6(3):351– 359, 2004

    2004

  11. [19]

    Cozzi, A

    M. Cozzi, A. Farina, and E. Valdinoci. Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations. Comm. Math. Phys., 331(1):189–214, 2014

    2014

  12. [20]

    Esposito, N

    L. Esposito, N. Fusco, and C. Trombetti. A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4(4):619–651, 2005

    2005

  13. [21]

    Esposito, P

    L. Esposito, P. Roy, and F. Sk. On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded. Asymptot. Anal., 123(1-2):79–94, 2021

    2021

  14. [22]

    Y . Giga. Surface evolution equations , volume 99 of Monographs in Mathematics . Birkh ¨auser Verlag, Basel,

  15. [23]

    P. Jana. Anisotropic p-Laplace equations on long cylindrical domain. Opuscula Math., 44(2):249–265, 2024

    2024

  16. [24]

    J. Jaro ˇs. Picone’s identity for a Finsler p-Laplacian and comparison of nonlinear elliptic equations. Math. Bohem., 139(3):535–552, 2014

    2014

  17. [25]

    Mezei and O

    I.-I. Mezei and O. Vas. Existence results for some Dirichlet problems involving Finsler-Laplacian operator. Acta Math. Hungar., 157(1):39–53, 2019

    2019

  18. [26]

    D. Mugnai. A limit problem for degenerate quasilinear variational inequalities in cylinders. In Recent trends in nonlinear partial differential equations. I. Evolution problems, volume 594 ofContemp. Math., pages 281–293. Amer. Math. Soc., Providence, RI, 2013

    2013

  19. [27]

    Novaga and E

    M. Novaga and E. Paolini. A computational approach to fractures in crystal growth. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., 10(1):47–56, 1999

    1999

  20. [28]

    L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1):259–268, 1992

    1992

  21. [29]

    Z. Shen. Lectures on Finsler geometry. World Scientific Publishing Co., Singapore, 2001

    2001

  22. [30]

    Taylor, J

    J. Taylor, J. Cahn, and C. Handwerker. Overview no. 98 i—geometric models of crystal growth. Acta Metal- lurgica et Materialia, 40(7):1443–1474, 1992

    1992

  23. [31]

    J. E. Taylor. Crystalline variational problems. Bull. Amer. Math. Soc., 84(4):568–588, 1978

    1978

  24. [32]

    G. Wul ff. Zur frage der geschwindigkeit des wachsthums und der aufl ¨osung der krystallfl¨achen. Zeitschrift f¨ ur Kristallographie - Crystalline Materials, 34:449 – 530, 1901

    1901

  25. [33]

    C. Xia. On a class of anisotropic problems. PhD thesis, Universit¨at Freiburg im Breisgau, 2012. 19 L. Esposito (corresponding author) Dipartimento di Matematica, Universit`a degli Studi di Salerno, Via Giov anniPaolo II 132, Fisciano 84084, Italy Email address: luesposi@unisa...

    2012

  26. [2006]

    A level set approach

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