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arxiv: 2604.26458 · v1 · submitted 2026-04-29 · 🧮 math.AP

The local Calder\'on problem and the determination at the boundary of a complex anisotropic admittivity

Jessica Crosse , Romina Gaburro This is my paper

Pith reviewed 2026-05-07 11:02 UTC · model grok-4.3

classification 🧮 math.AP
keywords Calderón problemlocal Dirichlet-to-Neumann mapanisotropic admittivitystability estimatesinverse boundary problemcomplex conductivityboundary determinationHölder stability
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The pith

The local Dirichlet-to-Neumann map determines the boundary values of a parametrically structured complex anisotropic admittivity together with all derivatives.

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

The paper addresses the local inverse problem of recovering an anisotropic complex admittivity inside a domain from the Dirichlet-to-Neumann map prescribed only on a nonempty open portion of the boundary. It restricts attention to admittivities that can be written as the evaluation of a known one-parameter family of complex-symmetric matrices at an unknown scalar function. Under this structural assumption the authors prove that the local map controls the admittivity and every derivative of the admittivity on the accessible boundary portion. The resulting estimates are of Lipschitz type for the function itself and of Hölder type for its derivatives of arbitrary order. These boundary stability statements constitute the central contribution.

Core claim

Assuming the complex anisotropic admittivity takes the form σ(·) = A(·, a(·)), where the one-parameter family of complex-symmetric matrices A(·, t) is known in advance and only the scalar function a is unknown, the local Dirichlet-to-Neumann map on a boundary portion Σ determines the values of σ and all its derivatives on Σ. The determination is stable: Lipschitz continuity holds for σ itself and Hölder continuity holds for derivatives of any order, with the constants depending only on the local map.

What carries the argument

the representation σ = A(·, a(·)) with known matrix family A and unknown scalar a, which reduces boundary recovery to determining the scalar function a from the local map

If this is right

  • The admittivity σ is uniquely determined on Σ by the local map.
  • Every derivative of σ on Σ is likewise uniquely determined.
  • Quantitative error bounds for the boundary values follow directly from the distance between two local maps.
  • The estimates remain valid for complex-valued anisotropic coefficients in dimensions n ≥ 3.

Where Pith is reading between the lines

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

  • The boundary stability may serve as the first step in a two-stage reconstruction that combines local data with interior unique-continuation arguments.
  • Knowing the explicit form of A converts the original matrix-valued inverse problem into a scalar one at the boundary, which could simplify numerical algorithms.
  • Consistency checks on the recovered scalar a across overlapping boundary patches could be used to validate the parametric assumption in practice.

Load-bearing premise

The admittivity must be exactly of the form A(·, a(·)) for a known one-parameter family A of complex-symmetric matrices and an unknown scalar function a.

What would settle it

Two distinct scalar functions a1 and a2 whose corresponding admittivities produce identical local Dirichlet-to-Neumann maps on Σ but differ at some point of Σ would violate the claimed stability.

read the original abstract

We address Calder\'on's problem of stably determining the anisotropic complex admittivity $\sigma$ in a domain $\Omega\subset\mathbb{R}^n$, with $n\geq3$, representing a conducting medium, in terms of a Dirichlet-to-Neumann map locally prescribed on a non-empty portion $\Sigma$ of the boundary of $\Omega$, $\partial\Omega$. $\sigma$ is assumed to be of type $\sigma(\cdot)=A(\cdot,a(\cdot))$ in $\Omega$, where the one-parameter family of complex-symmetric matrices $[\lambda^{-1},\:\lambda]\ni t\mapsto A(\cdot,\: t)$ is assumed to be a-priori known and the scalar function $a$ is unknown. We establish Lipschitz and H\"older stability estimates at the boundary for $\sigma$ and its derivatives of arbitrary order on $\Sigma$, respectively, in terms of the local map.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript addresses the local Calderón problem of recovering a complex anisotropic admittivity σ in Ω ⊂ ℝ^n (n ≥ 3) from the local Dirichlet-to-Neumann map prescribed on a nonempty open portion Σ ⊂ ∂Ω. The admittivity is restricted to the parametrized form σ(·) = A(·, a(·)), where the one-parameter family t ↦ A(·, t) of complex-symmetric matrices is known a priori and only the scalar function a is unknown. Under suitable regularity assumptions on A and a, the authors derive Lipschitz stability for σ and Hölder stability for all derivatives of σ on Σ, expressed in terms of the local DN map.

Significance. If the estimates hold, the work supplies quantitative stability for boundary recovery in a reduced class of anisotropic complex admittivities, which is relevant to applications such as electrical impedance tomography with frequency-dependent or complex conductivities. The reduction to a scalar unknown via the known family A enables the stability analysis; the manuscript provides explicit Lipschitz and Hölder constants depending on the a-priori data.

minor comments (3)
  1. [§2] §2, Assumption (A1): the precise regularity class required on the family A(·, t) (e.g., C^∞ or analytic in t) should be stated explicitly, as it directly controls the Hölder exponents for higher derivatives of σ.
  2. [Theorem 1.1] Theorem 1.1: the dependence of the stability constants on the diameter of Σ and on the lower bound of the ellipticity constant of A should be made explicit rather than absorbed into generic constants.
  3. [§4] The proof of the higher-order derivative estimates relies on repeated differentiation of the DN map; a brief outline of the induction step or reference to the relevant lemma would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee's description accurately reflects the main results on Lipschitz stability for the admittivity and Hölder stability for its derivatives at the boundary, derived from the local Dirichlet-to-Neumann map under the given structural assumption on the family A.

Circularity Check

0 steps flagged

Minor self-citation present but not load-bearing; central stability estimates are independently derived

full rationale

The derivation begins from the explicit assumption that σ = A(·, a(·)) with the one-parameter family A known a priori, then uses the local DN map to obtain Lipschitz stability for σ and Hölder stability for all derivatives of a on Σ. This is a direct stability proof under stated regularity, not a reduction of the target quantities to fitted parameters or prior self-citations. Any self-citations are peripheral and do not carry the central claims, yielding only a minor score of 2 with no steps meeting the strict quotation-and-reduction criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the structural assumption on σ is the main modeling choice.

pith-pipeline@v0.9.0 · 5450 in / 1026 out tokens · 32866 ms · 2026-05-07T11:02:59.712219+00:00 · methodology

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Reference graph

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