Partial data stability for the inverse fractional conductivity problem

Antti Kujanp\"a\"a; Giovanni Covi; Jesse Railo

REVIEW 2 major objections 7 minor 36 references

Partial data stability for the inverse fractional conductivity problem

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Partial exterior measurements for the fractional conductivity problem determine the interior conductivity with logarithmic stability, and with log-log stability when the conductivities are not known outside the domain.

desk verdict First partial-data stability estimates for the fractional conductivity problem, but the proof of the log-log theorem uses a regularity hypothesis the stated assumptions don't imply. read the letter →

arxiv 2505.18567 v1 pith:XQQM6IQG submitted 2025-05-24 math.AP

Giovanni Covi , Antti Kujanpää , Jesse Railo This is my paper · ORCID

classification math.AP MSC 35R3026A3342B37

keywords fractionalLaplacianinverseconductivityproblempartialdatalogarithmicstabilityLiouvillereductionnonlocalestimatesDirichlet-to-Neumannmap

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

This paper proves that for the fractional, nonlocal analogue of the classical conductivity inverse problem, measurements taken only on bounded pieces of the exterior domain still control the conductivity inside the domain, with at most logarithmic loss of accuracy. If the two conductivities are known to agree in the whole exterior, the interior $H^s(\Omega)$ difference is bounded by a power of the logarithm of the measurement error (Theorem 1.1). If their difference merely has compact support and they agree on the measurement set, the same conclusion holds in $L^p$ with a log-log modulus (Theorem 1.2). This matters because inverse imaging problems are typically unstable, and quantitative stability rates tell users how small measurement errors must be to resolve features at a given precision.

What carries the argument

The fractional Liouville reduction: multiplying a solution of the conductivity equation by $\gamma^{1/2}$ turns it into a solution of $(-\Delta)^s v + q v = 0$ with $q = -(-\Delta)^s(\gamma^{1/2}-1)/\gamma^{1/2}$, and conjugates the conductivity Dirichlet-to-Neumann map into the Schr\"odinger one through multiplication by $\gamma^{\pm 1/2}$. The argument also relies on a regularity lemma showing that $\gamma^{\pm 1/2}$ maps $\widetilde H^s$ spaces homeomorphically, on a known logarithmic stability theorem for fractional Schr\"odinger equations with partial exterior data, and, for the log-log case, on a proposition that bounds a compactly supported function in terms of its fractional Laplacian outside the set.

What would settle it

Verify whether the assumed $H^{2s+\epsilon,n/s}(\mathbb{R}^n)$ condition implies the weighted-space hypothesis $H^{(2s+\epsilon)/\theta_0,\theta_0 n/s}$ used in the cited comparison proposition for every allowed $\theta_0$; a counterexample for small $\epsilon$ would expose a gap in the proof of Theorem 1.2, and a pair of conductivities achieving a slower-than-logarithmic rate would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

The central discovery is that partial exterior data suffice for quantitative stability in the fractional conductivity problem, not just uniqueness. After applying the fractional Liouville reduction, which rewrites the conductivity equation as a fractional Schr\"odinger equation with reduced potential $q = -(-\Delta)^s(\gamma^{1/2}-1)/\gamma^{1/2}$, the paper proves that the operator-norm difference of the conductivity Dirichlet-to-Neumann maps dominates the difference of the reduced potentials. An elliptic estimate then converts the potential difference into the $H^s(\Omega)$ difference of the conductivities themselves. The resulting moduli of continuity are explicit, and all constants depend only on the geometry, the measurement sets, the ellipticity bound, and the a priori smoothness bound, not on the individual conductivities.

Load-bearing premise

The proof requires the background deviations $m_i = \gamma_i^{1/2}-1$ to lie in the mixed-norm Sobolev space $H^{2s+\epsilon,n/s}(\mathbb{R}^n)$ with a uniform bound, and the reduced Schr\"odinger operators to avoid a zero Dirichlet eigenvalue; if either fails, the chain of estimates is not established.

Editorial extensions

If this is right

Partial exterior measurements are enough: when the conductivities agree a priori in the whole exterior, the interior $H^s(\Omega)$ conductivity error is bounded by $C|\log \delta|^{-\sigma}$, where $\delta$ is the operator-norm measurement error.
Without exterior agreement, compact support of the conductivity difference still gives control in $L^p(\mathbb{R}^n)$ for $1 \le p < 2n/(n-2s)$, at a log-log rate.
The constants in both estimates depend only on $s,n,\Omega$, the measurement sets, the ellipticity lower bound, and the a priori smoothness bound, so the stability is uniform over the admissible class.
This improves the earlier complete-data stability result by allowing the measurements to be taken only on bounded open subsets of the exterior, rather than on the whole exterior.
The nonlocal nature of the operator is used essentially: the proof controls the difference in a larger set by the fractional Laplacian of the difference in an exterior set, a step with no local analogue.

Reading between the lines

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

The log-log rate in Theorem 1.2 likely reflects the additional uncertainty from not knowing the conductivities outside $\Omega$; an analogous rate is known to be optimal in the local inverse conductivity problem, and one may conjecture a similar optimality here.
A direct check of whether the assumed $H^{2s+\epsilon,n/s}(\mathbb{R}^n)$ condition implies the weighted-space hypothesis used in the cited comparison proposition is a natural next step; the paper does not spell out that implication, despite relying on it.
The same reduction should yield partial-data stability for other nonlocal inverse problems, such as fractional magnetic Schr\"odinger equations, whenever a comparable comparison lemma for the relevant Dirichlet-to-Neumann maps is available.
Numerical experiments on simple conductivity families could test whether the logarithmic modulus is qualitatively sharp or merely an artifact of the proof technique.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a genuine step forward. It proves the first partial-data stability estimates for the fractional conductivity equation, with a logarithmic modulus under exterior agreement and log-log under compact support. The Liouville reduction chain is standard and mostly clearly written. The new Lemma 3.3 on multiplication preserving \tilde H^s is useful, and the reliance on RS20 and CRTZ24 is legitimate — those are independent published results. I agree with the reader's overall assessment: the main ideas are right, and the gaps are repairable.

Now the soft spots. In Proposition 3.7, the proof applies Theorem 3.1 (RS20) but never checks the condition that zero is not a Dirichlet eigenvalue for (−Δ)^s+q_j. That is a genuine hypothesis of the theorem, and the paper just asserts the reduced potentials satisfy the bounds. For a fixed conductivity this may be generic, but the theorem as stated covers all gamma in the class, so the proof is incomplete. This should be fixable by adding the non-eigenvalue assumption to the hypotheses or by an approximation argument.

The larger problem is in Theorem 1.2. The proof uses CRTZ24, Prop. 5.1 (restated as Prop. 4.1), which requires m_i ∈ H^{(2s+ε)/θ_0, θ_0 n/s} for some s/n<θ_0<1. The theorem only assumes m_i ∈ H^{2s+ε, n/s}. The stress-test note is right: the scaling exponents give (s+ε)/θ_0 > s+ε, so H^{2s+ε,n/s} is strictly larger than the needed space; there is no embedding in the right direction. The sentence “Repeating the argument in the proof of Proposition 3.7” only controls q_j, not m_i in the stronger space. Since this proposition is the only bridge from conductivity data to Schrödinger data in that theorem, the log-log estimate is not established under the stated hypotheses. The likely fix is to strengthen assumption (iii) or prove a variant of the comparison estimate under the weaker condition, but as written the proof has a real hole.

Minor: Lemma 3.4 uses an inequality ‖φψ‖ ≤ ‖φ‖ ‖ψ‖ in H^s, which fails for s

Who is this for? Researchers in fractional inverse problems, especially stability. It deserves a serious referee — the claims are new and plausible, and the gaps are identifiable and likely fixable. I recommend engaging with it, but the revision needs to address the non-eigenvalue condition and, more importantly, the regularity mismatch in Theorem 1.2.

Referee Report

2 major / 7 minor

Summary. The paper studies the inverse fractional conductivity problem with partial exterior measurements. The authors prove two stability estimates: a logarithmic bound for conductivities that agree a priori in the whole exterior, and a log-log bound when the difference has compact support and measurements are taken on a single exterior set. The proofs use the fractional Liouville reduction to relate the conductivity DN map to the fractional Schrödinger DN map, then invoke known stability results of Rüland–Salo and of Covi–Railo–Tyni–Zimmermann. The paper also contains a self-contained proof that multiplication by gamma^{1/2} and gamma^{-1/2} preserves the space tilde H^s(U).

Significance. If the stated estimates hold, these are the first partial-data stability results for the fractional conductivity problem with the expected logarithmic and log-log rates, extending the full-data result of CRTZ24. The reduction strategy is elegant and Lemma 3.3 is a useful independent technical contribution. However, the proof of Theorem 1.2 relies on a regularity hypothesis in a quoted proposition that is not implied by the theorem's assumptions, and the non-eigenvalue condition required by the main Schrödinger stability theorem is never checked. Both issues are repairable, but as written the main claims are not fully established.

major comments (2)

[Section 3, Proposition 3.7 and Theorem 3.1] The proof of Theorem 1.2 invokes Proposition 4.1, quoted from CRTZ24 Proposition 5.1, to bound the Schrödinger DN-map difference in terms of the conductivity DN-map difference. Proposition 4.1 requires the background deviations to satisfy m_i in H^{(2s+epsilon)/theta_0, theta_0 n/s}(R^n) for some s/n < theta_0 < 1, whereas Theorem 1.2 assumes only m_i in H^{2s+epsilon, n/s}(R^n). These spaces are not nested in the needed direction: on the level of homogeneous norms, H^{a,p} scales as lambda^{-n/p+a} and H^{b,q} as lambda^{-n/q+b}, and here b - n/q = (s+epsilon)/theta_0 is larger than a - n/p = s+epsilon, so no embedding H^{2s+epsilon,n/s} into H^{(2s+epsilon)/theta_0, theta_0 n/s} exists. The sentence 'Repeating the argument in the proof of Proposition 3.7' only yields the H^{delta, n/(2s)} bound on q_j, not the required Sobolev regularity of the background deviations m_i. Since Proposition 4.1 is the only bridge from the conductivity data to the Schrödinger-potential data in this proof, the log-log stability estimate is not established under the stated hypotheses. This gap is repairable by either strengthening assumption (iii) of Theorem 1.2 to the CRTZ24 condition or by proving the comparison estimate under the weaker H^{2s+epsilon,n/s} regularity.
[Section 3, Proposition 3.7] Theorem 3.1, which is the key input for Proposition 3.7, requires that zero is not a Dirichlet eigenvalue of the exterior value problem (-Delta)^s u + q_j u = 0 in Omega with u = 0 in Omega^e, for j = 1, 2. Proposition 3.7 states that 'It suffices to show that the conditions of Theorem 3.1 are met' and then verifies only the H^{delta,n/(2s)} bound on q_j. The non-eigenvalue condition is never stated, checked, or referenced for the reduced potentials q_j = -((-Delta)^s m_j)/gamma_j^{1/2}. Without this condition the DN maps Lambda_{q_j} are not known to be well-defined, and the stability estimate of Theorem 3.1 does not apply. Since the paper aims to use Theorem 3.1 as a black box, the missing verification is a load-bearing gap, though it may be filled by a perturbation argument or by imposing a smallness condition on the potentials.

minor comments (7)

[Abstract] In the abstract, 'shaper' should be 'sharper'.
[Introduction, page 1] The word 'furhter' should be 'further'.
[Section 2, Theorem 3.1 statement (page 7)] The typo 'non-emtpy' should be 'non-empty'.
[Section 4, proof of Theorem 1.2 (page 13)] The typo 'becuse' should be 'because'.
[Section 3, proof of Lemma 3.4, equation (5)] The estimate ||phi_k^pm psi - gamma^{pm 1/2} psi||_{H^s} <= ||phi_k^pm - gamma^{pm 1/2} chi||_{H^s} ||psi||_{H^s} is not a direct consequence of the H^s multiplication because H^s is not an algebra. Since psi is fixed, smooth, and compactly supported, multiplication by psi is a bounded operator on H^s, so the argument survives with a constant depending on psi. The authors should state this to avoid giving the impression that the displayed inequality holds as written.
[Section 4, Theorem 1.2 condition (i)] The assumption supp(gamma_1 - gamma_2) = Sigma is an equality; if the authors intend only that the support is contained in Sigma, the condition should be stated as supp(...) subset Sigma to avoid measure-theoretic pedantry.
[Section 4, proof of Theorem 1.2 (parameter p)] The proof works with p = 2n/(n-2s') for s' < s; the connection to an arbitrary p in the range stated in the theorem should be made explicit by choosing s' such that p < 2n/(n-2s') and then adjusting the constants.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the conductivity-to-Schrodinger reduction and stability estimates are built from independent published lemmas, not from the theorem being proved.

full rationale

The derivation chain is: convert the conductivity DN-map difference into a fractional Schrodinger DN-map difference via the Liouville reduction (Lemma 3.6), using the multiplication homeomorphism Lemmas 3.2 and 3.3, which are proved from the stated hypotheses and from [RZ24, Lemma 3.9]; apply the external stability theorem [RS20, Theorem 1.2] (stated as Theorem 3.1) to the reduced potentials q1, q2; and finally transfer the Schrodinger-potential estimate back to the conductivity difference through the elliptic estimate in Lemma 3.8. Each step is either proved in the paper or quotes a published result whose hypotheses do not include the target estimate. Theorem 1.2 additionally invokes [CRTZ24, Proposition 5.1] (restated as Proposition 4.1) and [GRSU20, Proposition 6.1] as black boxes; these are independent, peer-reviewed results, and their use is not circular even though CRTZ24 shares authors with the present paper. The regularity-gap concern raised in review (that the stated assumption m_i in H^{2s+epsilon,n/s} need not imply the stronger hypothesis of Proposition 4.1) is a potential correctness gap in the proof, not a circularity: the conclusion is not equivalent to the assumptions by construction, and no fitted parameter is renamed as a prediction. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters and no invented entities: the paper is a proof-based analysis. It relies on standard functional analysis, the Liouville reduction, and prior stability theorems (RS20, CRTZ24, GRSU20), plus the regularity and spectral assumptions documented above.

assumptions (4)

standard math Bessel potential space embeddings, including H^{delta,n/s}(Omega) subset H^{delta,n/(2s)}(Omega) and Triebel-Lizorkin identifications
Used in Proposition 3.7 to show q_j belongs to H^{delta,n/(2s)}, a hypothesis of Theorem 3.1.
standard math Kato-Ponce fractional product inequality
Used in Proposition 3.7 to bound the product m_j/(m_j+1) times (-Delta)^s m_j.
domain assumption Zero is not a Dirichlet eigenvalue of (-Delta)^s + q_i for the reduced Schrodinger operators
Required by Lemma 2.1 and Theorem 3.1 for the Schrodinger DN map to be well-defined; omitted from the theorem statements but used in Lemma 3.6 and Proposition 3.7.
domain assumption Background deviation regularity m_i in H^{2s+epsilon,n/s}(R^n) with uniform bound C1
The main a priori smoothness and decay condition on the conductivities, used throughout for the Liouville reduction, multiplication operator bounds, and the DN-map comparison.

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Cite this review

Pith. "Pith review of Partial data stability for the inverse fractional conductivity problem." pith.science (2026). https://pith.science/paper/XQQM6IQG

@misc{pith2026250518567,
  author       = {Pith},
  title        = {Pith review of: Partial data stability for the inverse fractional conductivity problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQQM6IQG}},
  note         = {Machine review of arXiv:2505.18567}
}

read the original abstract

The classical Calder\'on problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity problem on bounded smooth domains. Using the fractional Liouville reduction, we obtain a log-log stability estimate when the conductivities a priori agree in the measurement set and their difference has compact support. In the case in which the conductivities are assumed to agree a priori in the whole exterior of the domain, we obtain a shaper logarithmic stability estimate.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Stable determination of conductivity by boundary measurements

Giovanni Alessandrini. Stable determination of conductivity by boundary measurements. Appl. Anal. , 27(1-3):153--172, 1988

work page 1988

[2] [2]

Calder\' o n's inverse conductivity problem in the plane

Kari Astala and Lassi P\" a iv\" a rinta. Calder\' o n's inverse conductivity problem in the plane. Ann. of Math. (2) , 163(1):265--299, 2006

work page 2006

[3] [3]

Calder\' o n's inverse problem for anisotropic conductivity in the plane

Kari Astala, Lassi P\" a iv\" a rinta, and Matti Lassas. Calder\' o n's inverse problem for anisotropic conductivity in the plane. Comm. Partial Differential Equations , 30(1-3):207--224, 2005

work page 2005

[4] [4]

Lipschitz stability for the inverse conductivity problem

Giovanni Alessandrini and Sergio Vessella. Lipschitz stability for the inverse conductivity problem. Adv. in Appl. Math. , 35(2):207--241, 2005

work page 2005

[5] [5]

Calder\' o n

Alberto-P. Calder\' o n. On an inverse boundary value problem. In Seminar on N umerical A nalysis and its A pplications to C ontinuum P hysics ( R io de J aneiro, 1980) , pages 65--73. Soc. Brasil. Mat., Rio de Janeiro, 1980

work page 1980

[6] [6]

Stability estimates for the calderón problem with partial data

Pedro Caro, David Dos Santos Ferreira , and Alberto Ruiz. Stability estimates for the calderón problem with partial data. Journal of Differential Equations , 260(3):2457--2489, 2016

work page 2016

[7] [7]

A reduction of the fractional C alder\'on problem to the local C alder\'on problem by means of the C affarelli- S ilvestre extension, 2023

Giovanni Covi, Tuhin Ghosh, Angkana Rüland, and Gunther Uhlmann. A reduction of the fractional C alder\'on problem to the local C alder\'on problem by means of the C affarelli- S ilvestre extension, 2023. arXiv:2305.04227

arXiv 2023

[8] [8]

u land. The C alder\' o n problem for the fractional S chr\

Mihajlo Ceki\' c , Yi-Hsuan Lin, and Angkana R\" u land. The C alder\' o n problem for the fractional S chr\" o dinger equation with drift. Calc. Var. Partial Differential Equations , 59(3):Paper No. 91, 46, 2020

work page 2020