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Partial data Calder\'{o}n problem for quasilinear conductivities in dimension 2

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that in two dimensions, boundary measurements on any non-empty open portion of the boundary uniquely determine a quasilinear conductivity γ(x,u,∇u).

desk verdict The paper proves a genuinely new partial-data uniqueness result for 2D quasilinear conductivities with full gradient dependence, but the main theorem as stated is not established because the proof silently assumes real-valued coefficients and tensors. read the letter →

arxiv 2607.17126 v1 pith:ZAYBHVBK submitted 2026-07-19 math.AP

Ruirui Wu , Ting Zhou This is my paper
classification math.AP MSC 35R3035J6035J25
keywords quasilinearconductivitypartialdataCalderónprobleminverseboundaryvaluecomplexgeometricopticshigher-orderlinearizationtwodimensionsuniquenessstationaryphase
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 proves a uniqueness theorem for the partial-data Calderón problem with quasilinear conductivity in two dimensions. It shows that if two conductivities agree to infinite order on the boundary and their partial Dirichlet-to-Neumann maps match on any non-empty open boundary set, then the conductivities are identical everywhere. The proof uses higher-order linearization, which turns the problem into a completeness statement for symmetric tensors, and constructs complex geometric optics solutions that vanish on the inaccessible boundary. The construction introduces improved remainder estimates because derivatives of solutions appear in the integral identities. A sympathetic reader would care because this removes the gradient-direction restriction present in earlier partial-data results.

What carries the argument

The load-bearing machinery is a family of CGO solutions of the form u = γ₀^{-1/2}(e^{Φ/h}(a+ha₀+r₁) + e^{Φ̄/h}(ā+hā₀+r̄₁) + e^{φ/h}r₂), where Φ is a holomorphic phase that is purely real on the inaccessible boundary portion Γ₀ and Morse in the interior, and a is holomorphic and purely imaginary on Γ₀. These solutions vanish on Γ₀, so they can be used in partial-data measurements. The new ingredient is an improved H² remainder estimate ∥r₁∥_{H²}=o(1/h), needed because derivatives of the solutions appear in the integral identities from higher-order linearization. Together with choices of phases (e.g., Θ₁=λf, Θ₂=Φ-λf, Θ₃=-Φ̄) that have or lack critical points, the phases generate products of so

What would settle it

Compute the H² norm of the remainder r₁ for a specific Morse phase Φ (e.g., Φ(z)=z²) in the unit disk with Γ₀ a boundary arc, and check whether ∥r₁∥_{H²} is indeed o(1/h) as claimed; if it is only O(h^{-1}|log h|), the leading-order phase extraction in §3 would acquire additional terms. Alternatively, construct two complex-valued quasilinear conductivities agreeing to infinite order on the boundary that are distinct only in their imaginary parts and test whether the partial Dirichlet-to-Neumann maps could coincide; if they can, the theorem as stated fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for isotropic quasilinear conductivities γ: Ω×C×Cⁿ→C satisfying the stated holomorphicity and normalization assumptions, agreement to infinite order on ∂Ω plus equality of the partial Dirichlet-to-Neumann maps on any non-empty open Γ⊂∂Ω forces γ₁=γ₂ on all of Ω×C×Cⁿ. The core discovery is that uniqueness reduces to Proposition 1.2, a completeness result: if a symmetric tensor T of rank m vanishes to infinite order on the boundary and the integral identity (1.1) holds for all solutions of a linear conductivity equation, then T must vanish identically. The proof realizes this by selecting phase combinations so that stationary and nonstationary phase arguments

Load-bearing premise

The proof assumes at the extraction step (end of §3 and §5.1) that the conductivities and tensor components are real-valued, although Theorem 1.1 states complex-valued γ; if complex-valued conductivities can differ only in their imaginary parts while producing the same partial data, the argument as written would not detect the difference.

Editorial extensions

If this is right

  • Any non-empty open portion of the boundary determines the quasilinear conductivity uniquely in two dimensions, with no restriction on the direction of ∇u.
  • The result extends full-data uniqueness for quasilinear conductivities to partial data and improves earlier partial-data results that fixed the gradient direction.
  • The completeness result provides a general tool: a symmetric tensor vanishing on the boundary is zero if it satisfies the integral identity with all solutions; this may apply to other nonlinear inverse problems.
  • The phase-selection technique formalizes how combinations of phases create specific patterns in products of solutions, enabling both stationary and nonstationary phase arguments to recover tensor components.

Reading between the lines

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

  • The proof appears to require that the conductivities and tensor components be real-valued at the steps where real and imaginary parts are separated (e.g., §3: 'since we assume the coefficients are real valued'); if the theorem is meant for complex-valued conductivities as stated, that gap would need to be closed. This is an inference from the text, not a claim the paper makes.
  • The improved remainder estimate ∥r₁∥_{H²}=o(1/h) is justified by a short argument; if a more detailed proof reveals a logarithmic loss, the phase cancellations in the m=1 recovery might still work but with weaker rates, possibly limiting the range of h where the extraction is valid.
  • The phase choices with λf and Φ±kλf suggest a combinatorial recipe (choosing ⌊m/2⌋+1 sets) that could be transplanted to other partial-data inverse problems for quasilinear elliptic equations in two dimensions.
  • If the real-valued gap is repaired, one would predict the same uniqueness for complex-valued quasilinear conductivities; a direct way to test this is to attempt to recover a complex conductivity in a numerical simulation of the m=1 case.
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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

4 major / 5 minor

Summary. The paper claims a partial-data uniqueness theorem for the two-dimensional quasilinear conductivity equation div(γ(x,u,∇u)∇u)=0. Under assumptions that γ is holomorphic in the solution and gradient variables, that γ(x,τ,0)=1, and that two conductivities agree to infinite order on the boundary, equality of the partial Dirichlet-to-Neumann maps on an arbitrary non-empty open boundary subset Γ is claimed to imply γ1=γ2 in Ω×C×C^n. The proof combines higher-order linearization with the CGO solutions of [IUY10, GT11] that vanish on the complementary boundary portion. The main completeness result, Proposition 1.2, asserts that an integral identity involving products of solutions forces a smooth symmetric tensor T^{j_1...j_m} to vanish. The cases m=1 and m=2 are developed in detail; the case m≥3 is delegated to the authors' earlier full-data paper [LW23].

Significance. If the result is correct, it is a substantial advance: it removes the fixed-gradient-direction restriction in the previous partial-data quasilinear result [KKU23] and extends the two-dimensional full-data quasilinear uniqueness of [LW23] to partial boundary measurements. The strategy of using improved H² remainder estimates for CGO solutions, combined with phase choices that produce specific stationary/non-stationary patterns, is natural and potentially influential. The paper's strengths include a concrete construction of CGO solutions adapted to the nonlinear higher-order linearization and explicit phase selections for the low-order tensor recoveries. However, the current manuscript is not yet at the level of a published proof: the main theorem overstates its hypothesis by omitting a real-valuedness assumption that the proof uses in an essential way, several load-bearing estimates and reductions are asserted rather than proved, and the general-m case is not written out.

major comments (4)
  1. [§3, end; §4 (4.4)–(4.7); §5.1] The statements of Theorem 1.1 and Proposition 1.2 allow complex-valued γ and T, but the recovery step uses an unstated real-valuedness assumption. In §3 the authors obtain T^1+iT^2=0 and conclude T^1=T^2=0 'since we assume the coefficients are real valued.' For complex T^1,T^2 this implication is false: T^1+iT^2=0 gives only T^1=-iT^2. Similarly, in §4, combining (4.4) and (4.7) leaves one free complex parameter unless realness is assumed, and §5.1 again invokes 'all the entrances are real.' Thus Proposition 1.2 and Theorem 1.1 are not established in the stated complex-valued generality. Either prove the complex case or add an explicit real-valued hypothesis throughout the statement.
  2. [§5, including §5.2] The m≥3 case is delegated rather than proved. Section 5 begins 'Same as in section 4 of [LW23]' and §5.2 says 'with some technical details omitted.' Equations (5.3)–(5.4) and the linear independence of the resulting system are asserted by reference to [LW23], a preprint treating the full-data problem. Since Proposition 1.2 must hold for every m and is the load-bearing completeness step for Theorem 1.1, this is not a cosmetic omission. Please provide the induction, the derivation of the displayed coefficient system, and a self-contained invertibility argument, or state and prove a precise lemma.
  3. [Lemma 2.2] The proof of the H² remainder estimate ||r1||_{H²}=o(1/h) is a one-sentence assertion: 'By the form of ... we see that the same estimates hold with a loss of h² for any derivatives of order less than or equal to 2.' This estimate is used to control leading h^{-2} terms, for example in (3.18) and (3.21), so it carries real weight. The L² estimate alone does not directly yield the claimed H² bound with the required h-dependence; a rigorous oscillatory-integral or integration-by-parts argument is needed. Please supply the details.
  4. [§1, Theorem 1.1] The reduction of Theorem 1.1 to Proposition 1.2 is asserted but not demonstrated. Higher-order linearization of the D-N map should yield integral identities of the form (1.1) for the Taylor coefficients of γ1-γ2; one must also use the infinite-order boundary agreement and the partial-data hypothesis to control boundary terms. Since Proposition 1.2 is the core completeness statement, the paper should either give the linearization argument explicitly or cite a theorem that applies verbatim to this setting, rather than stating 'the proof reduces' without further explanation.
minor comments (5)
  1. [§2, Eq. (2.2)] The displayed CGO solution contains two identical summands: e^{Φ/h}(a+ha_0+r_1) appears twice. Presumably the second should involve e^{\bar Φ/h}(\bar a+h\bar a_0+\bar r_1). Please correct.
  2. [§3, around p. 18] The phrase 'Since ψ(p)≠0' uses an undefined point p; it should be z_0. Also, the constant C_{z_0} appearing in stationary-phase evaluations is never defined; please specify that it is the standard stationary-phase coefficient depending on the Hessian of the phase at z_0.
  3. [§5.2, Eqs. (5.3)–(5.4)] The notation T_{1⋯1} and T_{2⋯2} should indicate the length of the multi-index explicitly, especially for general m; likewise the coefficients C_j(s) depend on m, so C_j^{(m)}(s) would be clearer.
  4. [References] The bibliography entry for [LW23] appears as 'L W23' with an inserted space; check all internal citation labels and ensure the preprint reference includes a stable identifier.
  5. [Throughout] There are several small typographical and formatting issues: missing parentheses around exponentials in a few displays (e.g., Lemma 2.5), inconsistent use of 'real valued' vs. 'real-valued', and the statement of condition (i) in the introduction uses (x,z)∈Ω×C while z should be in C^n. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a CGO/higher-order-linearization proof; the same-author citation [LW23] is independent technical support and the real-valuedness gap is a correctness issue, not a circular one.

full rationale

I checked the claimed derivation chain against the paper's own equations. Theorem 1.1 is reduced to the completeness statement Proposition 1.2, and Proposition 1.2 is proved from the integral identity (1.1) by higher-order linearization, CGO solutions, and stationary/nonstationary phase asymptotics. The recovered tensor equations — e.g. ∂̄(T1+iT2)−(∂̄γ0/γ0)(T1+iT2)=0 in §3, T11+2iT12−T22=0 and T11+T22=0 in §4, and the m=3 combinations in §5.1 — are consequences of the assumed identity, not restatements of the input by construction. The only same-author citation that is load-bearing is [LW23] in §5, where the m≥3 coefficient algebra is imported: 'Same as in section 4 of [L W23]' and 'by the same proof in section 4 of [L W23]'. Under hard rule 4 this is independent support: [LW23] is a full-data result whose coefficient system is a parameter-free algebraic lemma that does not assume the partial-data equality or the target conclusion, so the citation does not make the derivation circular. The paper also openly omits technical details ('with some technical details omitted', §5), which is a completeness/correctness issue, not a circularity. Finally, the proof repeatedly uses real-valuedness ('since we assume the coefficients are real valued', end of §3; 'since we also assume all the entrances are real', §5.1), which is a hypothesis gap relative to the stated complex-valued Theorem 1.1 and Proposition 1.2, but this is not a fitted-input or definitional circularity. I find no circular step.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's contribution is the phase/product-expansion analysis and the improved remainder estimates; it inherits CGO constructions from [GT11, IUY10], the m≥3 coefficient algebra from [LW23], and an unflagged real-valuedness assumption. No fundamentally new entities are introduced.

free parameters (1)
  • λ (phase coefficient) = generic λ outside a finite excluded set (e.g., image of 2Φ'/f')
    Chosen by hand to force Θ1, Θ2, and higher phase combinations to have no critical points in Ω; without this, the non-stationary phase arguments fail.
assumptions (6)
  • domain assumption Local well-posedness of (1.1) for small boundary data from [CFK+21, Appendix B].
    Used to define Λ_γ^Γ for f∈B_δ(∂Ω); quoted, not proved.
  • domain assumption Equality of partial DN maps implies the integral identity (1.1) via higher-order linearization.
    The paper states 'The proof of Theorem 1.1 reduces to...' but does not show the reduction; it is standard from [CFK+21, LW23].
  • standard math CGO solution machinery from [GT11]/[IUY10]: existence of Morse holomorphic Φ purely real on Γ0 with critical point at arbitrary z0, Φ(z0)≠0, and amplitudes a purely imaginary on Γ0 with prescribed zeros; Carleman estimates give the stated remainder bounds.
    Section 2 relies on [GT11] and [IUY10] for the phase construction and Lemma 2.3; these are not reproved.
  • domain assumption Elliptic uniqueness: if ∇·(γ0∇w)=0 in Ω and w=0 on ∂Ω, then w=0.
    Requires γ0 to be real and elliptic (positive), which is not stated in Proposition 1.2; used after (3.4) and for T^1+iT^2.
  • domain assumption The m≥3 coefficient system (5.3)–(5.4) is linearly independent, as asserted by reference to [LW23, Section 4].
    Section 5.2 says 'by the same argument as in the above paper'; no proof is given here.
  • ad hoc to paper Real-valuedness of conductivities and tensor components T^{...}.
    Introduced only in the proof ('we assume the coefficients are real valued') and absent from Theorem 1.1 and Proposition 1.2; needed to conclude T1=T2=0 from T1+iT2=0.

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Pith. "Pith review of Partial data Calder\'{o}n problem for quasilinear conductivities in dimension 2." pith.science (2026). https://pith.science/paper/ZAYBHVBK

@misc{pith2026260717126,
  author       = {Pith},
  title        = {Pith review of: Partial data Calder\'on problem for quasilinear conductivities in dimension 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZAYBHVBK}},
  note         = {Machine review of arXiv:2607.17126}
}
read the original abstract

In this paper, we prove a uniqueness result for the partial data Calder\'{o}n problem with quasilinear conductivity in two dimensions. The proof is based on higher-order linearization and the use of CGO solutions in dimension two that vanish on part of the boundary. Since derivatives of the solutions appear in the integral identity, we need improved remainder estimates, for which we introduce a modification in the choice of the phase, analogous to limiting Carleman weights. We also analyze how combinations of phases produce specific patterns in the products of solutions, which allows us to apply both stationary and nonstationary phase arguments to recover the conductivity.

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Pith tools

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