Inverse problems for semilinear elliptic equations with low regularity

David Johansson; Janne Nurminen; Mikko Salo

arxiv: 2505.05278 · v3 · submitted 2025-05-08 · 🧮 math.AP

Inverse problems for semilinear elliptic equations with low regularity

David Johansson , Janne Nurminen , Mikko Salo This is my paper

Pith reviewed 2026-05-22 16:17 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse problemssemilinear elliptic equationslow regularityunique determinationboundary measurementsgauge invariancenonlinearity recovery

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The pith

Boundary measurements determine a general nonlinearity a(x,u) up to gauge near a fixed solution for low-regularity semilinear elliptic equations.

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

The paper proves that a general nonlinearity a(x,u) is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. This extends earlier results on smoother coefficients to low-regularity settings where the coefficients may be rough or discontinuous. A sympathetic reader would care because many applied models in physics and engineering involve semilinear equations with irregular media, and boundary data is the only accessible information. If the claim holds, inverse problem techniques become viable in a wider class of physically realistic but mathematically challenging regimes.

Core claim

The authors show that a general nonlinearity a(x,u) is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in their recent smoother paper.

What carries the argument

Reduction to the previous smoother result, relying on the linearized operator around the fixed solution remaining invertible.

If this is right

Nonlinear terms become recoverable in media with discontinuous or low-regularity coefficients.
Gauge equivalence classes of nonlinearities are distinguishable by boundary data alone.
Inverse problems for semilinear equations extend to rougher physical settings without losing uniqueness.
The result applies to models with limited smoothness, such as certain reaction-diffusion systems.

Where Pith is reading between the lines

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

The method may adapt to other nonlinear PDE classes where linearization around a known solution is possible.
If the fixed solution can be varied, global uniqueness results might follow under extra assumptions on the domain.
Numerical tests with manufactured low-regularity nonlinearities and simulated boundary data could verify the recovery procedure.

Load-bearing premise

The fixed solution satisfies the semilinear equation with the low-regularity coefficients and the linearized operator around it is invertible.

What would settle it

Two distinct nonlinearities a1(x,u) and a2(x,u) that produce identical boundary measurements for the same semilinear equation around the same fixed solution would disprove the uniqueness claim.

read the original abstract

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in our recent paper (Johansson, Nurminen, Salo; ArXiv preprint 2312.12196).

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

This paper extends the authors' prior uniqueness results for determining nonlinearities in semilinear elliptic equations to lower regularity settings via linearization around a fixed solution.

read the letter

The main thing to know is that the paper supplies low-regularity versions of uniqueness theorems for a general nonlinearity a(x,u) from boundary measurements of the semilinear equation, possibly up to gauge and near a fixed solution. It is framed explicitly as a counterpart to their earlier arXiv 2312.12196 work, now in weaker function spaces that better match applications with rough coefficients. The low-regularity uniqueness statement itself is new and not covered in the cited prior literature. They carry the main ideas over by linearizing around a known solution u0 that satisfies the equation with the given coefficients. This is a solid technical move if the details hold, and it keeps the result relevant without requiring extra smoothness assumptions that would limit use. The abstract states the claim cleanly and avoids overreaching by tying everything to the previous paper. The reduction approach is efficient and shows they thought through how to adapt the smoother case. The soft spot sits in that same reduction. The linearized operator needs to support the density or unique continuation steps from the earlier work, yet the abstract gives no separate argument confirming invertibility or the necessary properties when coefficients drop to L^infty or W^{1,p} with p below the dimension. If this is simply inherited without re-checking the assumptions at low regularity, the transfer could fail for some admissible nonlinearities. That is the part a referee would need to examine line by line. This is written for people already working on inverse problems for nonlinear PDEs who track regularity questions. Readers who know the 2312.12196 paper will see the value quickest as a direct extension. It is the sort of incremental improvement the subfield follows. The work is grounded enough in its own terms to deserve peer review rather than a desk rejection, mainly so reviewers can verify the low-regularity linearization closes properly.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes low-regularity uniqueness results for inverse problems associated to semilinear elliptic equations. It shows that a general nonlinearity a(x,u) is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution u0 from boundary measurements, as a low-regularity counterpart to the authors' earlier work in arXiv:2312.12196.

Significance. If the central reduction is valid, the results would meaningfully extend inverse-problem theory for nonlinear elliptic equations to coefficient classes with minimal regularity (e.g., L^∞ or W^{1,p} with p < n). This is valuable because many applications involve data or coefficients that are not smooth, and the linearization-around-a-fixed-solution strategy is a natural way to transfer uniqueness from the smoother setting.

major comments (1)

[Proof of the main theorem (reduction step)] The proof reduces the low-regularity uniqueness statement to the smoother result of arXiv:2312.12196 by linearizing the semilinear equation around the fixed solution u0. For this reduction to be valid, the linearized operator L = Δ + ∂_u a(x,u0) must remain invertible (or permit the requisite unique-continuation/density arguments) in the low-regularity function spaces. The manuscript states that u0 satisfies the equation with the given coefficients but supplies no independent verification or quantitative condition ensuring invertibility when the coefficients lie only in the low-regularity class. This assumption is load-bearing for the neighborhood-uniqueness claim.

minor comments (1)

[Abstract and §1] The abstract and introduction would benefit from an explicit statement of the precise function spaces in which the coefficients a(x,u) and the fixed solution u0 are taken, as well as the precise notion of gauge freedom.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying a key point in the reduction step of the main theorem. We address this comment in detail below.

read point-by-point responses

Referee: The proof reduces the low-regularity uniqueness statement to the smoother result of arXiv:2312.12196 by linearizing the semilinear equation around the fixed solution u0. For this reduction to be valid, the linearized operator L = Δ + ∂_u a(x,u0) must remain invertible (or permit the requisite unique-continuation/density arguments) in the low-regularity function spaces. The manuscript states that u0 satisfies the equation with the given coefficients but supplies no independent verification or quantitative condition ensuring invertibility when the coefficients lie only in the low-regularity class. This assumption is load-bearing for the neighborhood-uniqueness claim.

Authors: We agree that the invertibility of the linearized operator L in the low-regularity spaces is essential for the validity of the reduction and that the manuscript would benefit from an explicit statement on this point. In the revised version we will insert a short lemma (placed immediately before the main reduction argument) that verifies the required properties. Under the standing assumptions that a(x,u) is Lipschitz continuous in the second variable with L^∞ coefficients and that u0 belongs to the natural solution space W^{1,p} (p < n), standard elliptic theory for operators with bounded measurable coefficients yields both the unique-continuation property and the invertibility of L on the appropriate Sobolev spaces. We will also record a quantitative smallness condition on the neighborhood of u0 that guarantees the operator remains invertible. This addition clarifies the reduction without altering the overall strategy or the low-regularity hypotheses. revision: yes

Circularity Check

1 steps flagged

Low-regularity uniqueness reduces to authors' prior smoother result via linearization whose invertibility is not re-verified

specific steps

self citation load bearing [Abstract]
"The main theorems are low regularity counterparts of the results in our recent paper (Johansson, Nurminen, Salo; ArXiv preprint 2312.12196)."

The uniqueness statement for general a(x,u) in low regularity is obtained by reducing to the cited prior theorem after linearization around u0. The reduction presupposes that L = Δ + ∂u a(x,u0) allows the same arguments as in the smoother case, yet the manuscript supplies no separate proof or quantitative condition guaranteeing this invertibility when coefficients are merely low-regularity (e.g., L^∞ or W^{1,p}). The central claim therefore inherits its validity from the self-cited result.

full rationale

The manuscript explicitly frames its main theorems as low-regularity counterparts of the authors' own prior work (arXiv:2312.12196). The derivation proceeds by linearizing the semilinear equation around a fixed solution u0 and invoking the smoother uniqueness result. While this adaptation supplies new content for the low-regularity setting, the key step—that the linearized operator remains invertible or permits the required density/unique-continuation arguments—receives no independent verification in the low-regularity class. This creates moderate self-citation dependence without reducing the entire claim to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard elliptic theory and the validity of the authors' earlier higher-regularity theorems; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption The linearised operator at the fixed solution remains invertible under the lowered regularity assumptions.
Required for the linearisation step that reduces the nonlinear inverse problem to a linear one.

pith-pipeline@v0.9.0 · 5573 in / 1269 out tokens · 33714 ms · 2026-05-22T16:17:25.762390+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We show that a general nonlinearity a(x,u) is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

The method is based on first linearization and a Runge approximation argument.

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

14 extracted references · 14 canonical work pages · 1 internal anchor

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Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$

arXiv: 2312.12196 [math.AP] . [KKU23] Y. Kian, K. Krupchyk, and G. Uhlmann. “ Partial data i nverse problems for quasilinear conductivity equations”. In: Math. Ann. 385.3-4 (2023), pp. 1611–1638. [KLU18] Y. Kurylev, M. Lassas, and G. Uhlmann. “ Inverse prob lems for Lorentzian manifolds and non-linear hyperbolic equations”. In: Invent. Math. 212.3 (2018),...

work page internal anchor Pith review Pith/arXiv arXiv 2023
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An inverse problem for the minimal sur face equation in the presence of a Riemannian metric

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Functional analysis and partial diﬀ erential equations. II

[Bro62] F. E. Browder. “ Functional analysis and partial diﬀ erential equations. II”. In: Math. Ann. 145 (1962), pp. 81–226. [BTW20] E. Blåsten, L. Tzou, and J.-N. W ang. “ Uniqueness for the inverse boundary value problem with singular potential s in 2D”. In: Math. Z. 295.3-4 (2020), pp. 1521–1535. [CFKKU21] C. I. Cârstea, A. Feizmohammadi, Y. Kian, K. K...

work page 1962

[2] [2]

Rema rks on the anisotropic Calderón problem

[CFO23] C. Cârstea, A. Feizmohammadi, and L. Oksanen. “ Rema rks on the anisotropic Calderón problem”. In: Proc. Amer. Math. Soc. 151.10 (2023), pp. 4461–4473. [Cha90] S. Chanillo. “ A problem in electrical prospection a nd an n-dimensional Borg-Levinson theorem”. In: Proc. Amer. Math. Soc. 108.3 (1990), pp. 761–767. [CLLO24] C. I. Cârstea, M. Lassas, T. ...

work page 2023

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[CLT24] C

arXiv: 2310.14268 [math.AP] . [CLT24] C. I. Cârstea, T. Liimatainen, and L. Tzou. The Calderón problem on Riemannian surfaces and of minimal surfaces

work page arXiv

[4] [4]

Det ermining an unbounded potential from Cauchy data in admissible geometries

arXiv: 2406.16944 [math.AP] . [DKS13] D. Dos Santos Ferreira, C. E. Kenig, and M. Salo. “ Det ermining an unbounded potential from Cauchy data in admissible geometries”. In: Comm. Partial Diﬀerential Equations 38.1 (2013), pp. 50–68. [FKO23] A. Feizmohammadi, Y. Kian, and L. Oksanen. “ Rigidit y of inverse problems for nonlinear elliptic equations on mani...

work page arXiv 2013

[5] [5]

Global uniqueness for a t wo-dimensional semilinear elliptic inverse problem

Lecture Notes in Mathematics. Positivity preserving and nonlinear higher o rder elliptic equations in bounded domains. Springer- Verlag, Berlin, 2010, pp. xviii+423. [GT01] D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001 , pp. xiv+5...

work page 2010

[6] [6]

Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$

arXiv: 2312.12196 [math.AP] . [KKU23] Y. Kian, K. Krupchyk, and G. Uhlmann. “ Partial data i nverse problems for quasilinear conductivity equations”. In: Math. Ann. 385.3-4 (2023), pp. 1611–1638. [KLU18] Y. Kurylev, M. Lassas, and G. Uhlmann. “ Inverse prob lems for Lorentzian manifolds and non-linear hyperbolic equations”. In: Invent. Math. 212.3 (2018),...

work page internal anchor Pith review Pith/arXiv arXiv 2023

[7] [7]

Springer, Berlin, 1992, pp. 434–441. [NS25] J. Nurminen and S. K. Sahoo. An inverse problem for a nonlinear biharmonic operator

work page 1992

[8] [8]

An inverse problem for the minimal su rface equation

arXiv: 2504.06624 [math.AP] . [Nur23a] J. Nurminen. “ An inverse problem for the minimal su rface equation”. In: Nonlinear Anal. 227 (2023), Paper No. 113163,

work page arXiv 2023

[9] [9]

Determining an unbounded potential for an elliptic equation with a power type nonlinearity

[Nur23b] J. Nurminen. “ Determining an unbounded potential for an elliptic equation with a power type nonlinearity”. In : J. Math. Anal. Appl. 523.1 (2023), Paper No. 126962,

work page 2023

[10] [10]

An inverse problem for the minimal sur face equation in the presence of a Riemannian metric

17 [Nur24] J. Nurminen. “ An inverse problem for the minimal sur face equation in the presence of a Riemannian metric”. In: Nonlinearity 37.9 (2024), Paper No. 095029,

work page 2024

[11] [11]

Inverse problems for semilinear elliptic PDE with measurements at a single point

[ST23] M. Salo and L. Tzou. “ Inverse problems for semilinear elliptic PDE with measurements at a single point”. In: Proc. Amer. Math. Soc. 151.5 (2023), pp. 2023–2030. [Sun05] Z. Sun. “ Conjectures in inverse boundary value prob lems for quasilinear elliptic equations”. In: Cubo 7.3 (2005), pp. 65–73. [Sun10] Z. Sun. “ An inverse boundary-value problem f...

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xxiii+755

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Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983, p

Mathematik und ihre Anwendungen in Physik und Tech nik [Mathematics and its Applications in Physics and Technolog y]. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983, p

work page 1983

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Electrical impedance tomography and C alderón’s problem

[Uhl09] G. Uhlmann. “ Electrical impedance tomography and C alderón’s problem”. In: Inverse Problems 25.12 (2009), pp. 123011,

work page 2009