Inverse conductivity problem with one measurement: Uniqueness of multi-layer structures

TL;DR

The paper tackles the inverse conductivity problem for general multi-layer inclusions using a single measurement by introducing Generalized Polarization Tensors (GPTs) for multi-layer media and deriving perturbation formulas. It develops a layer-potential framework to obtain integral representations and a far-field expansion that encodes geometric and material information. The authors establish uniqueness results for locating the multi-layer structure and for recovering the conductivity distribution in multi-layer concentric disks, first under high-order probing (explicit CGPTs) and then under low-order probing with a invertible measurement matrix. These results rely on explicit constructions of GPTs, generalized polarization matrices, and their invertibility properties. The work lays groundwork for one-measurement identification of layered inclusions with potential applications in imaging and cloaking design.

Abstract

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

Theorems & Definitions (31)

• Lemma 2.1
• proof : Proof
• Theorem 2.1
• proof : Proof
• Definition 3.1
• Theorem 3.1
• proof : Proof
• Theorem 3.2
• Lemma 4.1
• proof : Proof