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arxiv: 2604.27278 · v3 · submitted 2026-04-30 · 🧮 math.AP

Lipschitz Stability in the Simultaneous Determination of Polygonal Inclusions and Constant Conductivities

Tianrui Dai , Elisa Francini This is my paper

Pith reviewed 2026-05-12 00:46 UTC · model grok-4.3

classification 🧮 math.AP MSC 35R30
keywords inverse conductivity problemLipschitz stabilitypolygonal inclusionsDirichlet-to-Neumann mapconductivity equation
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The pith

The Dirichlet-to-Neumann map determines polygonal inclusions and their constant conductivities with Lipschitz stability.

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

This paper establishes a Lipschitz stability result for simultaneously recovering the geometry of polygonal inclusions and their constant conductivity values from boundary measurements. The data are given by the Dirichlet-to-Neumann map of the conductivity equation. A sympathetic reader would care because Lipschitz stability supplies a quantitative bound showing that small errors in the measured map produce only proportionally small errors in the recovered inclusions and conductivities. Such bounds turn uniqueness statements into practically usable reconstruction guarantees for inverse problems.

Core claim

The paper proves that the Hausdorff distance between two collections of polygonal inclusions, together with the difference between their constant conductivity values, is bounded by a constant multiple of the operator-norm distance between the corresponding Dirichlet-to-Neumann maps.

What carries the argument

The Dirichlet-to-Neumann map associated with the conductivity equation whose coefficient is piecewise constant on a finite number of polygonal subdomains.

Load-bearing premise

The inclusions are polygonal and each has a single constant conductivity value different from the background.

What would settle it

Two different configurations of polygonal inclusions carrying distinct constant conductivities that produce Dirichlet-to-Neumann maps differing by an arbitrarily small amount would falsify the claimed Lipschitz stability.

Figures

Figures reproduced from arXiv: 2604.27278 by Elisa Francini, Tianrui Dai.

Figure 1
Figure 1. Figure 1: The map V is originally defined on the boundary of the middle triangle P1. One can use the linear combi￾nation to extend V to the boundary of a ”shape like” triangle inside the inclusion P1 and to an outside ”shape like” triangle. The support of such extended map is then contained in the set { big triangle \ small triangle}. Using the map h in Lemma 4.1, one can define the transformation between P1 and P2.… view at source ↗
read the original abstract

This work establishes a Lipschitz stability result for identifying unknown polygonal inclusions along with their unknown constant conductivity values, given boundary measurements encoded in the Dirichlet-to-Neumann 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

2 major / 2 minor

Summary. The manuscript establishes a Lipschitz stability result for the simultaneous recovery of an unknown number of polygonal inclusions together with their unknown constant conductivity values inside a bounded domain, using the full Dirichlet-to-Neumann map associated to the conductivity equation.

Significance. If the result holds under the stated a priori assumptions, it provides a strong quantitative stability estimate for a nonlinear inverse problem reduced to a finite-parametric setting (vertex coordinates and conductivity values). This is a meaningful contribution to the theory of inverse problems for the conductivity equation, particularly for applications such as electrical impedance tomography with piecewise-constant conductivities, and it aligns with known results on stability for polygonal or polyhedral inclusions.

major comments (2)
  1. [§1 and §3] §1 (Introduction) and §3 (Main assumptions): the precise a priori bounds on the number of inclusions, their mutual separation distance, and the admissible range for the constant conductivity values must be stated explicitly and shown to be independent of the data; without these, the finite-parametric reduction that underpins the Lipschitz constant cannot be verified.
  2. [Theorem 4.1] Theorem 4.1 (main stability estimate): the proof relies on a quantitative unique-continuation argument combined with a finite-dimensional compactness argument; it is unclear whether the Lipschitz constant remains uniform when the number of inclusions is allowed to vary (even within a fixed upper bound), or whether an additional compactness step is needed to control the combinatorial aspect of the inclusion configuration.
minor comments (2)
  1. [§2] Notation for the conductivity function and the inclusion boundaries should be introduced once in §2 and used consistently; several places switch between σ and γ without comment.
  2. [Introduction] The statement of the Dirichlet-to-Neumann map in the introduction should include the precise Sobolev spaces on which it is defined, to match the regularity assumed on the polygonal inclusions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. We address the two major comments below and have updated the manuscript accordingly.

read point-by-point responses

  1. Referee: [§1 and §3] §1 (Introduction) and §3 (Main assumptions): the precise a priori bounds on the number of inclusions, their mutual separation distance, and the admissible range for the constant conductivity values must be stated explicitly and shown to be independent of the data; without these, the finite-parametric reduction that underpins the Lipschitz constant cannot be verified.

    Authors: We agree that these bounds should be stated more prominently. In the revised manuscript we have added explicit statements in both §1 and §3: the number of inclusions is at most a fixed integer N, the minimum separation distance between inclusions (and from the boundary) is at least a fixed d>0, and the conductivity values lie in a fixed interval [σ_min, σ_max] with 0<σ_min<σ_max<∞. These quantities are part of the a priori admissible class and are independent of the Dirichlet-to-Neumann data; the Lipschitz constant depends only on them, the domain, and the conductivity equation. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (main stability estimate): the proof relies on a quantitative unique-continuation argument combined with a finite-dimensional compactness argument; it is unclear whether the Lipschitz constant remains uniform when the number of inclusions is allowed to vary (even within a fixed upper bound), or whether an additional compactness step is needed to control the combinatorial aspect of the inclusion configuration.

    Authors: The proof already yields a uniform Lipschitz constant for all admissible configurations with at most N inclusions. The finite-dimensional compactness argument is carried out in the parameter space of vertex coordinates and conductivity values for configurations with k=1,...,N polygons satisfying the separation and conductivity bounds; this space is compact for fixed N. The combinatorial aspect is controlled by taking the finite union over k≤N, which remains compact under the a priori constraints. The quantitative unique-continuation estimate is uniform across this class. We have inserted a short clarifying paragraph after the statement of Theorem 4.1 to make this uniformity explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a Lipschitz stability result for recovering polygonal inclusions and constant conductivities from the Dirichlet-to-Neumann map. No equations, fitting procedures, or derivation steps are provided in the abstract or reader summary that reduce any claimed prediction or uniqueness result to its own inputs by construction. The central claim is a mathematical stability estimate for a finite-parametric inverse problem under a priori restrictions, which is compatible with standard analytic techniques in elliptic inverse problems without requiring self-referential definitions or load-bearing self-citations that collapse the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided information.

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Reference graph

Works this paper leans on

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