Corrosion detection by identification of a nonlinear Robin boundary condition

David Johansson

arxiv: 2412.14817 · v3 · submitted 2024-12-19 · 🧮 math.AP

Corrosion detection by identification of a nonlinear Robin boundary condition

David Johansson This is my paper

Pith reviewed 2026-05-23 07:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse boundary value problemnonlinear Robin boundary conditioncorrosion detectionCauchy dataconductivity equationlinearizationparametrization

0 comments

The pith

The nonlinear Robin term can be identified locally from Cauchy data on a subset of the boundary.

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

The paper shows that the nonlinear term in the Robin boundary condition of a conductivity equation can be identified from Cauchy data measured on only a portion of the boundary. The proof adapts a linearization and parametrization technique from semilinear equations to this setting. This identification is key for detecting corrosion through boundary measurements in applications where only partial access is available. It also provides a strategy and partial proof for extending the result to the entire boundary.

Core claim

We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direct

What carries the argument

Linearization and parametrization of solutions of the nonlinear equation by solutions of the linearized equation to recover the nonlinear Robin term.

If this is right

The nonlinear Robin term is uniquely determined by local Cauchy data on a subset of the boundary.
The linearization method from semilinear equations adapts successfully to this nonlinear Robin setting.
A strategy exists for extending local identification to global identification.
A partial result holds for the extension from local to global identification.

Where Pith is reading between the lines

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

Corrosion detection could rely on sensors placed only on accessible portions of a surface.
The same parametrization technique may apply to other nonlinear boundary conditions in elliptic inverse problems.
Numerical tests of the parametrization on simulated data could check stability for practical use.

Load-bearing premise

The linearization and parametrization strategy assumes that solutions of the nonlinear equation can be parametrized by solutions of the linearized equation in a way that preserves the necessary injectivity or uniqueness properties for the inverse map.

What would settle it

Two different nonlinear Robin terms that produce identical Cauchy data on the measurement subset would disprove the local identification.

read the original abstract

We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direction. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.

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 gives a local uniqueness result for a nonlinear Robin boundary condition from partial Cauchy data by adapting a linearization method, with a partial global strategy sketched.

read the letter

The paper gives a local uniqueness result for identifying the nonlinear Robin boundary condition from partial Cauchy data in the conductivity equation. It does this by adapting a linearization and parametrization technique from semilinear elliptic problems. This is a targeted extension that fills a specific gap in inverse problems with nonlinear boundary conditions. The local identification is established, and they outline a strategy for a global result with some progress shown. The method relies on parametrizing the nonlinear solutions via the linearized equation, which allows them to reduce the inverse problem to an integral identity that isolates the nonlinearity. The approach is clean in principle and builds directly on prior work without introducing new machinery. If the details check out, it provides a useful tool for corrosion detection models. The main question is whether the parametrization preserves the necessary properties when the nonlinearity is only on the boundary. The stress test raises a fair point: the boundary trace might not allow the same density of test functions as in the interior case, potentially weakening the uniqueness on the measurement subset. The abstract invokes the adaptation without spelling out the verification for the Robin setting, so the full paper needs to show that the boundary operator doesn't lose the injectivity or approximation needed. No signs of circular reasoning or invented entities. This paper is for specialists in inverse problems for PDEs, particularly those dealing with boundary nonlinearities or applications like corrosion. A reader focused on elliptic inverse problems would find the adaptation interesting and could build on it. It deserves a serious referee because the claim is precise, the setting is well-motivated, and the technique is established enough that the extension can be checked rigorously.

Referee Report

1 major / 1 minor

Summary. The paper studies an inverse boundary value problem for the conductivity equation with a nonlinear Robin boundary condition. It claims to prove local identification of the nonlinear Robin term from Cauchy data on a subset of the boundary, suggests a strategy to extend to global identification, and proves a partial global result. The method adapts linearization and parametrization of solutions from semilinear elliptic equations to this setting where the nonlinearity appears only in the boundary condition.

Significance. If the local identification result holds, the work provides a rigorous basis for recovering nonlinear boundary coefficients from partial data, which is relevant to corrosion detection models. The adaptation of the linearization-parametrization technique to boundary nonlinearities and the partial global result represent a technical contribution that could extend to other inverse problems with boundary nonlinearities.

major comments (1)

[Abstract] Abstract: The central claim relies on adapting the linearization-plus-parametrization technique (originally for interior semilinear equations) to a nonlinearity confined to the Robin boundary condition. It is not evident from the abstract whether the boundary trace of solutions to the linearized equation remains sufficiently dense or injective on the measurement subset to isolate the nonlinear term uniquely via the resulting integral identity. This adaptation step is load-bearing for the local identification result and requires explicit verification that the trace operator preserves the necessary approximation properties.

minor comments (1)

[Abstract] The abstract refers to 'a possible strategy for turning a local identification result into a global one' and a 'partial result' without indicating the specific assumptions or the form of the nonlinearity (e.g., whether it is monotone or has a particular growth condition). Clarifying these in the abstract would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the point raised below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim relies on adapting the linearization-plus-parametrization technique (originally for interior semilinear equations) to a nonlinearity confined to the Robin boundary condition. It is not evident from the abstract whether the boundary trace of solutions to the linearized equation remains sufficiently dense or injective on the measurement subset to isolate the nonlinear term uniquely via the resulting integral identity. This adaptation step is load-bearing for the local identification result and requires explicit verification that the trace operator preserves the necessary approximation properties.

Authors: We agree that the abstract does not explicitly mention the density properties of the traces. In the body of the paper (Section 3), the local identification is obtained by first linearizing the nonlinear Robin problem around a family of solutions to the linearized conductivity equation and then parametrizing the nonlinear solutions by these linearized ones. The required density of the boundary traces on the measurement subset follows from the unique continuation property for the conductivity equation together with the fact that the parametrizing functions can be chosen so that their traces remain dense in L^2 on the accessible part of the boundary; this is verified by adapting the approximation arguments from the interior semilinear case while accounting for the boundary trace operator. To address the referee's concern, we will revise the abstract to include a short clause indicating that the trace operator preserves the necessary approximation properties in this setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained PDE analysis

full rationale

The paper proves local identification of a nonlinear Robin boundary coefficient from partial Cauchy data by adapting a linearization-plus-parametrization technique originally developed for semilinear elliptic equations. The central step is an analytical argument that solutions of the nonlinear problem can be parametrized by those of the linearized problem in a manner that transfers the necessary density or injectivity properties to the boundary measurements. No quoted step reduces the claimed uniqueness result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose own justification is internal to the present work. The method is presented as an adaptation of prior external results rather than a renaming or tautological re-derivation of the target statement itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are mentioned. Standard background assumptions for elliptic PDEs (smooth domain, bounded conductivity) are implicitly used but not detailed.

axioms (1)

domain assumption Standard regularity and ellipticity assumptions on the conductivity coefficient and domain for the conductivity equation to be well-posed
Required for any analysis of the forward and inverse problems in this setting.

pith-pipeline@v0.9.0 · 5611 in / 1197 out tokens · 28637 ms · 2026-05-23T07:27:09.877444+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 unclear

?

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

The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.2 ... C_{w1,δ}^{a1} ⊆ C_{w2,Cδ}^{a2} ⇒ a1(x,w1(x)+ε)=a2(x,w1(x)+ε) for a.e. x∈ΓI, |ε|≤λ

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

24 extracted references · 24 canonical work pages · 1 internal anchor

[1]

Alessandrini, L

G. Alessandrini, L. Del Piere, and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement , English, Inverse Probl. (2003), doi: 10.1088/0266-5611/19/4/312

work page doi:10.1088/0266-5611/19/4/312 2003
[2]

Aliprantis and Kim C

Charalambos D. Aliprantis and Kim C. Border, Inﬁnite dimensional analysis. A hitchhiker’s guide. English, 3rd ed., Berlin: Springer, 2006

work page 2006
[3]

Mehdi Badra, Fabien Caubet, and Jérémi Dardé, Stability estimates for Navier- Stokes equations and application to inverse problems , English, Discrete Contin. Dyn. Syst., Ser. B (2016), doi: 10.3934/dcdsb.2016052

work page doi:10.3934/dcdsb.2016052 2016
[4]

Laurent Baratchart, Laurent Bourgeois, and Juliette Lebl ond, Uniqueness results for inverse Robin problems with bounded coeﬃcient , English, J. Funct. Anal. (2016), doi: 10.1016/j.jfa.2016.01.011

work page doi:10.1016/j.jfa.2016.01.011 2016
[5]

Belhachmi and H

Z. Belhachmi and H. Meftahi, Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measure ment, English, Math. Methods Appl. Sci. (2015), doi: 10.1002/mma.3061

work page doi:10.1002/mma.3061 2015
[6]

Slim Chaabane and Mohamed Jaoua, Identiﬁcation of Robin coeﬃcients by the means of boundary measurements , Inverse Problems (1999), doi: 10.1088/0266-5 611/15/6/303

work page doi:10.1088/0266-5 1999
[7]

Mourad Choulli and Aymen Jbalia, The problem of detecting corrosion by an electric measurement revisited, English, Discrete Contin. Dyn. Syst., Ser. S (2016), doi: 1 0.3934/dcdss.2016018

work page 2016
[8]

John B Conway, A course in functional analysis , Springer, 2019

work page 2019
[9]

Giovanni Covi and Angkana Rüland, On some partial data Calderón type problems with mixed boundary conditions , Journal of Diﬀerential Equations (2021)

work page 2021
[10]

Lawrence C Evans, Partial Diﬀerential Equations , American Mathematical Soc., 2010

work page 2010
[11]

Dario Fasino and Gabriele Inglese, Recovering unknown terms in a nonlinear bound- ary condition for Laplace’s equation , IMA journal of applied mathematics (2006)

work page 2006
[12]

Bastian Harrach and Houcine Meftahi, Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem , SIAM Journal on Applied Mathematics (2019)

work page 2019
[13]

Partial Diﬀer

Lars Hörmander, Uniqueness theorems for second order elliptic diﬀerential equa- tions, English, Commun. Partial Diﬀer. Equations (1983), doi: 10.1080/0360530 8308820262

work page doi:10.1080/0360530 1983
[14]

(1997), doi: 10.1088/0266-5611/13/4/006

Gabriele Inglese, An inverse problem in corrosion detection , English, Inverse Probl. (1997), doi: 10.1088/0266-5611/13/4/006

work page doi:10.1088/0266-5611/13/4/006 1997
[15]

Victor Isakov and John Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Communications on Pure and Applied Mathematics (1994). 26

work page 1994
[16]

David Johansson, Janne Nurminen, and Mikko Salo, Inverse problems for semilin- ear elliptic PDE with a general nonlinearity a(x,u ) (2023), arXiv: 2312.12196

work page internal anchor Pith review Pith/arXiv arXiv 2023
[17]

Vesa Julin, Tony Liimatainen, and Mikko Salo, p-harmonic coordinates for Hölder metrics and applications , Comm. Anal. Geom. (2017), doi: 10.4310/CAG.2017.v2 5.n2.a5

work page doi:10.4310/cag.2017.v2 2017
[18]

Peter G Kaup and Fadil Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, Journal of Nondestructive Evaluation (1995)

work page 1995
[19]

Peter G Kaup, Fadil Santosa, and Michael Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data , Inverse problems (1996)

work page 1996
[20]

Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, and Mikk o Salo, Partial data in- verse problems and simultaneous recovery of boundary and co eﬃcients for semilin- ear elliptic equations , Revista Matemática Iberoamericana (2020)

work page 2020
[21]

Giovanni Leoni, A ﬁrst course in Sobolev spaces , Second, American Mathematical Society, Providence, RI, 2017, doi: 10.1090/gsm/181

work page doi:10.1090/gsm/181 2017
[22]

(2005), doi: 10.1088/0266-5611/21/5/015

Furong Lin and Weifu Fang, A linear integral equation approach to the Robin inverse problem, English, Inverse Probl. (2005), doi: 10.1088/0266-5611/21/5/015

work page doi:10.1088/0266-5611/21/5/015 2005
[23]

Michael Vogelius and Jian-Ming Xu, A nonlinear elliptic boundary value problem related to corrosion modeling , Quarterly of applied mathematics (1998)

work page 1998
[24]

Kôsaku Yosida, Functional analysis, Reprint of the sixth (1980) edition, Springer- Verlag, Berlin, 1995, doi: 10.1007/978-3-642-61859-8 . 27

work page doi:10.1007/978-3-642-61859-8 1980

[1] [1]

Alessandrini, L

G. Alessandrini, L. Del Piere, and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement , English, Inverse Probl. (2003), doi: 10.1088/0266-5611/19/4/312

work page doi:10.1088/0266-5611/19/4/312 2003

[2] [2]

Aliprantis and Kim C

Charalambos D. Aliprantis and Kim C. Border, Inﬁnite dimensional analysis. A hitchhiker’s guide. English, 3rd ed., Berlin: Springer, 2006

work page 2006

[3] [3]

Mehdi Badra, Fabien Caubet, and Jérémi Dardé, Stability estimates for Navier- Stokes equations and application to inverse problems , English, Discrete Contin. Dyn. Syst., Ser. B (2016), doi: 10.3934/dcdsb.2016052

work page doi:10.3934/dcdsb.2016052 2016

[4] [4]

Laurent Baratchart, Laurent Bourgeois, and Juliette Lebl ond, Uniqueness results for inverse Robin problems with bounded coeﬃcient , English, J. Funct. Anal. (2016), doi: 10.1016/j.jfa.2016.01.011

work page doi:10.1016/j.jfa.2016.01.011 2016

[5] [5]

Belhachmi and H

Z. Belhachmi and H. Meftahi, Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measure ment, English, Math. Methods Appl. Sci. (2015), doi: 10.1002/mma.3061

work page doi:10.1002/mma.3061 2015

[6] [6]

Slim Chaabane and Mohamed Jaoua, Identiﬁcation of Robin coeﬃcients by the means of boundary measurements , Inverse Problems (1999), doi: 10.1088/0266-5 611/15/6/303

work page doi:10.1088/0266-5 1999

[7] [7]

Mourad Choulli and Aymen Jbalia, The problem of detecting corrosion by an electric measurement revisited, English, Discrete Contin. Dyn. Syst., Ser. S (2016), doi: 1 0.3934/dcdss.2016018

work page 2016

[8] [8]

John B Conway, A course in functional analysis , Springer, 2019

work page 2019

[9] [9]

Giovanni Covi and Angkana Rüland, On some partial data Calderón type problems with mixed boundary conditions , Journal of Diﬀerential Equations (2021)

work page 2021

[10] [10]

Lawrence C Evans, Partial Diﬀerential Equations , American Mathematical Soc., 2010

work page 2010

[11] [11]

Dario Fasino and Gabriele Inglese, Recovering unknown terms in a nonlinear bound- ary condition for Laplace’s equation , IMA journal of applied mathematics (2006)

work page 2006

[12] [12]

Bastian Harrach and Houcine Meftahi, Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem , SIAM Journal on Applied Mathematics (2019)

work page 2019

[13] [13]

Partial Diﬀer

Lars Hörmander, Uniqueness theorems for second order elliptic diﬀerential equa- tions, English, Commun. Partial Diﬀer. Equations (1983), doi: 10.1080/0360530 8308820262

work page doi:10.1080/0360530 1983

[14] [14]

(1997), doi: 10.1088/0266-5611/13/4/006

Gabriele Inglese, An inverse problem in corrosion detection , English, Inverse Probl. (1997), doi: 10.1088/0266-5611/13/4/006

work page doi:10.1088/0266-5611/13/4/006 1997

[15] [15]

Victor Isakov and John Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Communications on Pure and Applied Mathematics (1994). 26

work page 1994

[16] [16]

David Johansson, Janne Nurminen, and Mikko Salo, Inverse problems for semilin- ear elliptic PDE with a general nonlinearity a(x,u ) (2023), arXiv: 2312.12196

work page internal anchor Pith review Pith/arXiv arXiv 2023

[17] [17]

Vesa Julin, Tony Liimatainen, and Mikko Salo, p-harmonic coordinates for Hölder metrics and applications , Comm. Anal. Geom. (2017), doi: 10.4310/CAG.2017.v2 5.n2.a5

work page doi:10.4310/cag.2017.v2 2017

[18] [18]

Peter G Kaup and Fadil Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, Journal of Nondestructive Evaluation (1995)

work page 1995

[19] [19]

Peter G Kaup, Fadil Santosa, and Michael Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data , Inverse problems (1996)

work page 1996

[20] [20]

Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, and Mikk o Salo, Partial data in- verse problems and simultaneous recovery of boundary and co eﬃcients for semilin- ear elliptic equations , Revista Matemática Iberoamericana (2020)

work page 2020

[21] [21]

Giovanni Leoni, A ﬁrst course in Sobolev spaces , Second, American Mathematical Society, Providence, RI, 2017, doi: 10.1090/gsm/181

work page doi:10.1090/gsm/181 2017

[22] [22]

(2005), doi: 10.1088/0266-5611/21/5/015

Furong Lin and Weifu Fang, A linear integral equation approach to the Robin inverse problem, English, Inverse Probl. (2005), doi: 10.1088/0266-5611/21/5/015

work page doi:10.1088/0266-5611/21/5/015 2005

[23] [23]

Michael Vogelius and Jian-Ming Xu, A nonlinear elliptic boundary value problem related to corrosion modeling , Quarterly of applied mathematics (1998)

work page 1998

[24] [24]

Kôsaku Yosida, Functional analysis, Reprint of the sixth (1980) edition, Springer- Verlag, Berlin, 1995, doi: 10.1007/978-3-642-61859-8 . 27

work page doi:10.1007/978-3-642-61859-8 1980