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arxiv: 2605.05021 · v1 · submitted 2026-05-06 · 🧮 math.AP

Reconstruction of non-self-adjoint anisotropic and complex inclusions in the Calder\'on problem

Henrik Garde , David Johansson , Thanasis Zacharopoulos This is my paper

Pith reviewed 2026-05-08 16:30 UTC · model grok-4.3

classification 🧮 math.AP
keywords monotonicity methodCalderón probleminclusion detectionnon-self-adjoint perturbationsanisotropic conductivitypartial datainverse problemsunique continuation
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The pith

Monotonicity methods detect inclusions in the partial data anisotropic Calderón problem under general non-self-adjoint perturbations.

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

The paper generalizes the monotonicity method to identify inclusions when the background conductivity and permittivity can include very general non-self-adjoint terms. The forward model treats both anisotropic real conductivity and anisotropic permittivity together and the results apply in every dimension two and higher. A reader would care because the approach keeps the detection procedure intact while dropping the global self-adjointness requirement that limited earlier work. Only local definiteness near the inclusion boundaries and unique continuation from the self-adjoint part of the background are needed.

Core claim

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calderón problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension d ≥ 2. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general L^∞ non-self-adjoint perturbations. We only require unique continuation based on

What carries the argument

The monotonicity method, extended by replacing global self-adjointness with a reachability condition through a self-adjoint background set plus a local definiteness condition near inclusion boundaries.

If this is right

  • Inclusions remain detectable even when non-self-adjoint perturbations are arbitrary away from their boundaries.
  • The method continues to work with only partial boundary measurements in dimensions d ≥ 2.
  • Unique continuation is required only for the self-adjoint part of the background conductivity.
  • The same monotonicity argument applies to both conductivity and permittivity inclusions simultaneously.

Where Pith is reading between the lines

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

  • The local character of the self-adjoint requirement may let the method apply to heterogeneous media where global symmetry fails.
  • Numerical tests with deliberately non-self-adjoint complex admittivities could check whether the monotonicity signature survives in practice.
  • Similar localization of self-adjointness might extend the approach to other inverse problems that currently demand full self-adjoint coefficients.

Load-bearing premise

Inclusion boundaries must be reachable from the domain boundary along a path where the background conductivity remains self-adjoint, together with a definiteness condition holding near those boundaries.

What would settle it

An explicit example of an inclusion that satisfies the reachability and definiteness conditions yet produces no detectable monotonicity signature when the non-self-adjoint perturbation is present.

read the original abstract

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calder\'on problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension $d \geq 2$. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general $L^\infty$ non-self-adjoint perturbations. We only require unique continuation based on the self-adjoint part of the background conductivity, thus making the methods compatible with generic unique continuation results.

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 / 2 minor

Summary. The manuscript generalizes recent results on the monotonicity method for inclusion detection in the partial-data anisotropic Calderón problem to very general non-self-adjoint perturbations. It introduces a forward model incorporating both anisotropic real conductivity and anisotropic permittivity, with results claimed to hold in any dimension d ≥ 2. The key hypotheses are that inclusion boundaries are reachable from the domain boundary via a set on which the background conductivity is self-adjoint, a definiteness condition holds near the inclusion boundaries, general L^∞ non-self-adjoint perturbations are permitted away from those boundaries, and unique continuation is required only for the self-adjoint part of the background conductivity.

Significance. If the central claims hold, the work meaningfully extends the applicability of monotonicity-based reconstruction to complex, lossy, and non-self-adjoint media without imposing smallness conditions on the perturbations or coercivity on the full operator. The restriction of unique continuation to the self-adjoint background component on a connecting set is a strength, as it aligns with generic unique-continuation theorems and avoids stronger global assumptions. This broadens the method's relevance for applications such as electrical impedance tomography in heterogeneous or complex materials.

minor comments (2)
  1. The introduction would benefit from a concise table or paragraph explicitly contrasting the new assumptions (definiteness only near boundaries, unique continuation only on the self-adjoint part) with those in the cited prior works on the monotonicity method.
  2. Notation for the anisotropic conductivity and permittivity tensors could be clarified in §2 by adding a short remark on how the complex-valued case reduces to the real anisotropic case when the imaginary part vanishes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The referee summary correctly identifies the main contributions and assumptions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generalizes the monotonicity method for partial-data anisotropic Calderón problems to non-self-adjoint perturbations by introducing a forward model with anisotropic conductivity and permittivity, plus explicit assumptions on boundary-reaching self-adjoint sets, definiteness near inclusions, and unique continuation restricted to the self-adjoint background part. These extensions are stated independently of the target reconstruction result and rely on standard external unique-continuation theorems rather than self-referential definitions or fitted inputs. No load-bearing step reduces by construction to a prior self-citation, ansatz, or renamed empirical pattern; the central claims remain externally verifiable against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The reconstruction claim rests on three domain assumptions stated in the abstract: reachability of inclusion boundaries through self-adjoint background regions, a definiteness condition near those boundaries, and unique continuation only for the self-adjoint part of the background conductivity. No free parameters or invented entities are introduced.

axioms (3)
  • domain assumption Inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint
    Explicitly required for the monotonicity method to apply in the non-self-adjoint setting.
  • domain assumption A definiteness condition holds near the inclusion boundaries
    Needed to ensure the monotonicity argument works near the inclusions.
  • domain assumption Unique continuation holds based only on the self-adjoint part of the background conductivity
    Allows compatibility with generic unique-continuation theorems while permitting non-self-adjoint perturbations away from boundaries.

pith-pipeline@v0.9.0 · 5425 in / 1548 out tokens · 43222 ms · 2026-05-08T16:30:31.115356+00:00 · methodology

discussion (0)

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Reference graph

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