A not-so-strange term coming from somewhere

Arghir Zarnescu; Giacomo Canevari; Kirill Cherednichenko

arxiv: 2511.20071 · v2 · submitted 2025-11-25 · 🧮 math.AP

A not-so-strange term coming from somewhere

Giacomo Canevari , Kirill Cherednichenko , Arghir Zarnescu This is my paper

Pith reviewed 2026-05-17 05:14 UTC · model grok-4.3

classification 🧮 math.AP MSC 35B2735J05

keywords homogenizationperforated domainsRobin boundary conditionsSteklov problemstrange termeffective reaction termspectral characterization

0 comments

The pith

When Robin boundary conditions on perforations are scaled with the inverse surface area, the homogenized Laplace equation gains an extra nonlinear zeroth-order term.

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

The paper studies Laplace's equation inside a domain riddled with small periodic holes that carry Robin boundary conditions. By scaling the Robin coefficient with the reciprocal of the holes' total surface area, the authors identify a regime in which the boundary reaction and the interior diffusion balance at the same order. The resulting effective equation includes a reaction term that depends nonlinearly on the Robin parameter, and this dependence is obtained by solving a Steklov-type eigenvalue problem in which the eigenvalue enters both the differential equation and the boundary condition. This construction continuously interpolates between the Neumann and Dirichlet homogenization problems and recovers the classical capacitary strange term when the coupling becomes strong.

Core claim

In the identified scaling regime the two-scale limit satisfies an effective equation that contains, besides the standard Laplacian, a zeroth-order term whose coefficient is a nonlinear function of the Robin parameter. This coefficient is determined by the principal eigenvalue and the corresponding eigenfunction of a Steklov-type spectral problem posed on the reference cell, where the spectral parameter appears simultaneously in the bulk equation and on the boundary. The resulting term varies continuously with the Robin parameter, recovering the Neumann problem at one end and the Dirichlet problem at the other, while reproducing the known strange term of capacitary type in the strong-coupling

What carries the argument

The Steklov-type spectral problem with the spectral parameter appearing both inside the equation and in the boundary condition, which supplies the nonlinear coefficient in the homogenized reaction term.

If this is right

The homogenized model can predict the behavior of solutions in periodically perforated media under intermediate Robin conditions.
The nonlinear dependence provides a continuous family of effective problems connecting the insulating and perfectly conducting limits.
In the limit of large Robin parameter the extra term coincides with the classical capacitary correction.
The spectral formulation allows numerical computation of the effective coefficient by solving a cell problem.

Where Pith is reading between the lines

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

This framework could be adapted to time-dependent diffusion or to nonlinear elliptic equations with similar boundary scalings.
The appearance of a Steklov problem suggests possible connections to other spectral homogenization results involving boundary oscillations.
Numerical experiments comparing the full perforated problem with the homogenized equation for finite Robin values would test the accuracy of the nonlinear correction.

Load-bearing premise

The Robin coefficient must be scaled exactly proportionally to the inverse of the total surface area of all perforations in order to place surface and bulk effects at the same order.

What would settle it

Compute the solution of the perforated problem numerically for a chosen Robin parameter in a large but finite number of periods, solve the proposed homogenized equation with the nonlinear term obtained from the cell spectral problem, and check whether their difference tends to zero as the period size goes to zero.

Figures

Figures reproduced from arXiv: 2511.20071 by Arghir Zarnescu, Giacomo Canevari, Kirill Cherednichenko.

read the original abstract

We consider Laplace's equation in a periodically perforated domain with Robin boundary conditions on the holes, where the Robin coefficient is scaled proportionally to the inverse total surface area of the performations. We identify a regime in which surface and bulk effects contribute at the same order and show that the homogenised equation contains an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterised via a Steklov-type spectral problem in which the spectral parameter appears both in the equation and in the boundary condition. The resulting term interpolates continuously between the Neumann and Dirichlet limits, recovering the classical capacitary strange term in the strong-coupling limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies a nonlinear zeroth-order term in the homogenized Laplace equation for Robin conditions scaled by inverse perforation surface area, defined through a Steklov problem with the spectral parameter appearing in both the interior equation and the boundary condition.

read the letter

The main point is that under this particular scaling of the Robin coefficient, surface and bulk effects balance at the same order, so the effective equation gains an extra nonlinear correction term that varies continuously with the parameter and recovers the usual capacitary term in the strong-coupling limit. The term is extracted from an auxiliary Steklov-type eigenvalue problem where the spectral parameter sits in both the PDE and the Robin boundary condition. That construction is the genuinely new piece here. The paper sets up the scaling and the regime cleanly and shows how the auxiliary problem interpolates between the Neumann and Dirichlet extremes, which is a neat way to unify those limits in one model. It should be useful for people who need effective descriptions of surface-driven transport in perforated media. The soft spot is exactly the one flagged in the stress-test note: the claim rests on the nonlinear map from Robin coefficient to the effective term being single-valued and continuous. If the spectral analysis in the paper does not establish a distinguished real eigenvalue across the full parameter range and prove the functional is well-behaved, the interpolation could have gaps. The abstract alone does not give the estimates or the spectral details, so that part needs careful checking in the proofs. This is aimed at readers working on homogenization in domains with small holes or on effective boundary conditions in materials science. Someone already familiar with strange terms and Steklov problems will see the value quickly. It is worth sending to a serious referee because the scaling choice and the auxiliary problem are interesting and the result, if the spectral part holds up, fills a gap between known limits.

Referee Report

1 major / 1 minor

Summary. The paper studies the homogenization of the Laplace equation in a periodically perforated domain with Robin boundary conditions on the holes. The Robin coefficient is scaled proportionally to the inverse total surface area of the perforations. The authors identify a regime where surface and bulk effects are of the same order and derive a homogenized equation that includes an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterized by a Steklov-type spectral problem in which the spectral parameter appears both in the PDE and in the boundary condition. The term provides a continuous interpolation between the Neumann and Dirichlet limits, recovering the classical capacitary strange term in the strong-coupling limit.

Significance. Should the mathematical details be fully rigorous, this result would contribute to the understanding of homogenization in domains with small perforations under intermediate boundary conditions. The nonlinear dependence and the use of a non-standard spectral problem to capture it represent a novel aspect that bridges the gap between weak and strong coupling regimes. This could be significant for applications in physics and engineering where boundary reactions are neither negligible nor dominant.

major comments (1)

The central claim that the homogenized equation contains a well-defined additional nonlinear zeroth-order term rests on the properties of the Steklov-type eigenvalue problem. Specifically, it is necessary to prove that this problem admits a distinguished real eigenvalue and that the associated map from the Robin coefficient to the effective term is single-valued and continuous for the relevant range of parameters. Without such analysis, the interpolation between Neumann and Dirichlet limits cannot be guaranteed to be rigorous.

minor comments (1)

There is a typographical error in the abstract: 'performations' should read 'perforations'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We appreciate the recognition of the novelty in using a non-standard spectral problem to capture the intermediate regime. Below we address the major comment point by point.

read point-by-point responses

Referee: The central claim that the homogenized equation contains a well-defined additional nonlinear zeroth-order term rests on the properties of the Steklov-type eigenvalue problem. Specifically, it is necessary to prove that this problem admits a distinguished real eigenvalue and that the associated map from the Robin coefficient to the effective term is single-valued and continuous for the relevant range of parameters. Without such analysis, the interpolation between Neumann and Dirichlet limits cannot be guaranteed to be rigorous.

Authors: We agree that a detailed analysis of the Steklov-type eigenvalue problem is crucial for rigor. In the current manuscript, we define the problem and state the existence of the principal eigenvalue based on standard spectral theory for Robin problems, but we acknowledge that the continuity and single-valuedness of the map require explicit verification. We will revise the paper by adding a lemma proving that for each Robin coefficient α > 0 in the scaling regime, there exists a unique positive real eigenvalue λ(α) which is simple, and that λ(α) depends continuously on α. This will be done using perturbation theory for self-adjoint operators and the variational characterization. With this addition, the nonlinear term is well-defined and the interpolation is rigorous. We believe this addresses the concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via auxiliary spectral problem

full rationale

The paper assumes a specific scaling of the Robin coefficient with inverse total surface area to balance surface and bulk contributions at the same order. It then derives a homogenized equation containing a nonlinear zeroth-order term, which is characterized (not fitted or redefined) by solving an auxiliary Steklov-type eigenvalue problem with the spectral parameter appearing in both the interior equation and the Robin boundary condition. This characterization is presented as an independent mathematical construction that interpolates between Neumann and Dirichlet limits. No step reduces the output to the input by construction, renames a known result, or relies on a load-bearing self-citation whose validity is internal to the paper. The result is self-contained against the stated spectral problem and the explicit scaling assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard elliptic PDE theory for existence of solutions and on the specific scaling choice that creates the balanced regime; no new physical entities are introduced.

axioms (2)

standard math Solutions to the Laplace equation with Robin boundary conditions exist and are unique in the periodically perforated domain.
Standard background result from elliptic boundary-value theory invoked for the original problem.
domain assumption The chosen scaling of the Robin coefficient with inverse total surface area produces a regime in which surface and bulk contributions are of the same order.
This scaling is the key modeling choice that enables the identification of the intermediate homogenized term.

pith-pipeline@v0.9.0 · 5404 in / 1413 out tokens · 39244 ms · 2026-05-17T05:14:24.163460+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The homogenised equation contains an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterised via a Steklov-type spectral problem in which the spectral parameter appears both in the equation and in the boundary condition.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

βκ*(β) = σn(n-2)β / (σn(n-2) + β) ... converges to the capacitary strange term ... as β→∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Auchmuty and Q

G. Auchmuty and Q. Han. Representations of solutions of Laplacian boundary value problems on exterior regions.Appl. Math. Opt., 69:21–45, 2014

work page 2014
[2]

T. P. Bennett, G. D’Alessandro, and K. R. Daly. Multiscale models of colloidal dispersion of particles in nematic liquid crystals.Phys. Rev. E, 90:062505, 2014. 34

work page 2014
[3]

Borisov, G

D. Borisov, G. Cardone, and Durante T. Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve.Proceedings of the Royal Society of Edinburgh, 146A:1115–1158, 2016

work page 2016
[4]

M. C. Calderer, A. DeSimone, D. Golovaty, and A. Panchenko. An effective model for nematic liquid crystal composites with ferromagnetic inclusions.SIAM Journal on Applied Mathematics, 74(2):237–262, 2014

work page 2014
[5]

Canevari and A

G. Canevari and A. Zarnescu. Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation.Mathematical Models and Methods in Applied Sciences, 30(02):309–342, 2020

work page 2020
[6]

Cherednichenko, P

K. Cherednichenko, P. Dondl, and F. Rösler. Norm-resolvent convergence in perforated domains. Asymptotic Analysis, 110(3–4):163–184, 2018

work page 2018
[7]

Cherednichenko, A

K. Cherednichenko, A. Kiselev, and L. Silva. Operator-norm resolvent asymptotic analysis of con- tinuous media with low-index inclusions.Mathematical Notes, 111:373–387, 2022

work page 2022
[8]

Cherednichenko, A

K. Cherednichenko, A. Kiselev, I. Velčić, and J. Žubrinić. Effective behaviour of critical-contrast pdes: micro-resonances, frequency conversion, and time dispersive properties. II.Communications in Mathematical Physics, 496:Article number 72, 72 pp., 2025

work page 2025
[9]

Cioranescu, P

D. Cioranescu, P. Donato, F. Murat, and E. Zuazua. Homogenisation and corrector for the wave equation in domains with small holes.Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e sé, 18(2):251–293, 1991

work page 1991
[10]

Cioranescu and F

D. Cioranescu and F. Murat. Un terme étrange venu d’ailleurs, I and II. InNonlinear Differential Equations and Their Applications, Collège de France Seminar, Vol. 60 and 70, Research Notes in Mathematics, pages 98–138, 154–178. Pitman, London, 1982–1983

work page 1982
[11]

Cioranescu and F

D. Cioranescu and F. Murat. A strange term coming from nowhere. InTopics in the Mathem- atical Modelling of Composite Materials, Progress in Nonlinear Differential Equations and Their Applications, Vol. 31, pages 45–93. Birkhäuser, Boston, 1997

work page 1997
[12]

Della Pietra, C

F. Della Pietra, C. Nitsch, R. Scala, and C. Trombetti. An optimization problem in thermal insulation with robin boundary conditions.Communications in Partial Differential Equations, 46(12):2288–2304, 2021

work page 2021
[13]

Girouard, A

A. Girouard, A. Henrot, and J. Lagacé. From Steklov to Neumann via homogenisation.Archive for Rational Mechanics and Analysis, 239:981–1023, 2021

work page 2021
[14]

Girouard and J

A. Girouard and J. Lagacé. Large Steklov eigenvalues via homogenisation on manifolds.Inventiones Mathematicae, 226:1011–1056, 2021

work page 2021
[15]

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik.Homogenization of Differential Operators and Integral Functionals. Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg, 1994

work page 1994
[16]

Kacimi and F

H. Kacimi and F. Murat. Estimation de l’erreur dans des problemes de Dirichlet ou apparait un terme étrange. InPartial Differential Equations and the Calculus of Variations: Essays in Honor of Ennio De Giorgi, volume 2, pages 661–296. Birkhäuser, Boston, 1989

work page 1989
[17]

S. Kaizu. The Robin problems on domains with many tiny holes.Proceedings of the Japan Academy, Series A, Mathematical Sciences, 61(2):39–42, 1985

work page 1985
[18]

Khrabustobskyi and M

A. Khrabustobskyi and M. Plum. Operator estimates for homogenization of the Robin Laplacian in a perforated domain.Journal of Differential Equations, 338:474–517, 2022

work page 2022
[19]

Khrabustovskyi and O

A. Khrabustovskyi and O. Post. Operator estimates for the crushed ice problem.Asymptotic Analysis, 110(3–4):137–161, 2018

work page 2018
[20]

E. H. Lieb and M. Loss.Analysis, volume 14 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. 35

work page 2001
[21]

V. A. Marchenko and E. Ya. Khruslov. Boundary-value problems with fine-grained boundary. Matematicheskii Sbornik (N.S.), 65(107)(3):458–472, 1964. [Russian]

work page 1964
[22]

V. A. Marchenko and E. Ya. Khruslov.Boundary Value Problems in Domains with a Fine-grained Boundary. Naukova Dumka, Kiev, 1974. [Russian]

work page 1974
[23]

Rauch and M

J. Rauch and M. Michael Taylor. Potential and scattering theory on wildly perturbed domains. Journal of Functional Analysis, 18(1):27–59, 1975

work page 1975
[24]

M. T. van Rossem, S. Wilks, P. R. Secor, M. Kaczmarek, and G D’Alessandro. Homogenization modelling of antibiotic diffusion and adsorption in viral liquid crystals.Royal Society Open Science, 10(1):221120, 2023

work page 2023
[25]

von Below and G

J. von Below and G. François. Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition.Bull. Belg. Math. Soc. Simon Stevin, 12:505–519, 2005

work page 2005
[26]

Wang and Q

X. Wang and Q. Du. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches.Journal of Mathematical Biology, 56(3):347–371, 2008. 36

work page 2008

[1] [1]

Auchmuty and Q

G. Auchmuty and Q. Han. Representations of solutions of Laplacian boundary value problems on exterior regions.Appl. Math. Opt., 69:21–45, 2014

work page 2014

[2] [2]

T. P. Bennett, G. D’Alessandro, and K. R. Daly. Multiscale models of colloidal dispersion of particles in nematic liquid crystals.Phys. Rev. E, 90:062505, 2014. 34

work page 2014

[3] [3]

Borisov, G

D. Borisov, G. Cardone, and Durante T. Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve.Proceedings of the Royal Society of Edinburgh, 146A:1115–1158, 2016

work page 2016

[4] [4]

M. C. Calderer, A. DeSimone, D. Golovaty, and A. Panchenko. An effective model for nematic liquid crystal composites with ferromagnetic inclusions.SIAM Journal on Applied Mathematics, 74(2):237–262, 2014

work page 2014

[5] [5]

Canevari and A

G. Canevari and A. Zarnescu. Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation.Mathematical Models and Methods in Applied Sciences, 30(02):309–342, 2020

work page 2020

[6] [6]

Cherednichenko, P

K. Cherednichenko, P. Dondl, and F. Rösler. Norm-resolvent convergence in perforated domains. Asymptotic Analysis, 110(3–4):163–184, 2018

work page 2018

[7] [7]

Cherednichenko, A

K. Cherednichenko, A. Kiselev, and L. Silva. Operator-norm resolvent asymptotic analysis of con- tinuous media with low-index inclusions.Mathematical Notes, 111:373–387, 2022

work page 2022

[8] [8]

Cherednichenko, A

K. Cherednichenko, A. Kiselev, I. Velčić, and J. Žubrinić. Effective behaviour of critical-contrast pdes: micro-resonances, frequency conversion, and time dispersive properties. II.Communications in Mathematical Physics, 496:Article number 72, 72 pp., 2025

work page 2025

[9] [9]

Cioranescu, P

D. Cioranescu, P. Donato, F. Murat, and E. Zuazua. Homogenisation and corrector for the wave equation in domains with small holes.Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e sé, 18(2):251–293, 1991

work page 1991

[10] [10]

Cioranescu and F

D. Cioranescu and F. Murat. Un terme étrange venu d’ailleurs, I and II. InNonlinear Differential Equations and Their Applications, Collège de France Seminar, Vol. 60 and 70, Research Notes in Mathematics, pages 98–138, 154–178. Pitman, London, 1982–1983

work page 1982

[11] [11]

Cioranescu and F

D. Cioranescu and F. Murat. A strange term coming from nowhere. InTopics in the Mathem- atical Modelling of Composite Materials, Progress in Nonlinear Differential Equations and Their Applications, Vol. 31, pages 45–93. Birkhäuser, Boston, 1997

work page 1997

[12] [12]

Della Pietra, C

F. Della Pietra, C. Nitsch, R. Scala, and C. Trombetti. An optimization problem in thermal insulation with robin boundary conditions.Communications in Partial Differential Equations, 46(12):2288–2304, 2021

work page 2021

[13] [13]

Girouard, A

A. Girouard, A. Henrot, and J. Lagacé. From Steklov to Neumann via homogenisation.Archive for Rational Mechanics and Analysis, 239:981–1023, 2021

work page 2021

[14] [14]

Girouard and J

A. Girouard and J. Lagacé. Large Steklov eigenvalues via homogenisation on manifolds.Inventiones Mathematicae, 226:1011–1056, 2021

work page 2021

[15] [15]

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik.Homogenization of Differential Operators and Integral Functionals. Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg, 1994

work page 1994

[16] [16]

Kacimi and F

H. Kacimi and F. Murat. Estimation de l’erreur dans des problemes de Dirichlet ou apparait un terme étrange. InPartial Differential Equations and the Calculus of Variations: Essays in Honor of Ennio De Giorgi, volume 2, pages 661–296. Birkhäuser, Boston, 1989

work page 1989

[17] [17]

S. Kaizu. The Robin problems on domains with many tiny holes.Proceedings of the Japan Academy, Series A, Mathematical Sciences, 61(2):39–42, 1985

work page 1985

[18] [18]

Khrabustobskyi and M

A. Khrabustobskyi and M. Plum. Operator estimates for homogenization of the Robin Laplacian in a perforated domain.Journal of Differential Equations, 338:474–517, 2022

work page 2022

[19] [19]

Khrabustovskyi and O

A. Khrabustovskyi and O. Post. Operator estimates for the crushed ice problem.Asymptotic Analysis, 110(3–4):137–161, 2018

work page 2018

[20] [20]

E. H. Lieb and M. Loss.Analysis, volume 14 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. 35

work page 2001

[21] [21]

V. A. Marchenko and E. Ya. Khruslov. Boundary-value problems with fine-grained boundary. Matematicheskii Sbornik (N.S.), 65(107)(3):458–472, 1964. [Russian]

work page 1964

[22] [22]

V. A. Marchenko and E. Ya. Khruslov.Boundary Value Problems in Domains with a Fine-grained Boundary. Naukova Dumka, Kiev, 1974. [Russian]

work page 1974

[23] [23]

Rauch and M

J. Rauch and M. Michael Taylor. Potential and scattering theory on wildly perturbed domains. Journal of Functional Analysis, 18(1):27–59, 1975

work page 1975

[24] [24]

M. T. van Rossem, S. Wilks, P. R. Secor, M. Kaczmarek, and G D’Alessandro. Homogenization modelling of antibiotic diffusion and adsorption in viral liquid crystals.Royal Society Open Science, 10(1):221120, 2023

work page 2023

[25] [25]

von Below and G

J. von Below and G. François. Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition.Bull. Belg. Math. Soc. Simon Stevin, 12:505–519, 2005

work page 2005

[26] [26]

Wang and Q

X. Wang and Q. Du. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches.Journal of Mathematical Biology, 56(3):347–371, 2008. 36

work page 2008