  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.

Analysis of PDEs