Finite-order GPT and CGPT identifiability for multiple inclusions

Determine whether finitely many generalized polarization tensors or complex contracted generalized polarization tensors determine the geometric and material parameters of a prescribed family of multiple planar conductivity inclusions, including the parameters of the lemniscatic domain, the coefficients of the conformal map, and the componentwise conductivities.

Background

The paper develops Faber--Walsh field expansions and Faber--Walsh polarization tensors for multiple disjoint inclusions with independently varying conductivities. It establishes a triangular relation between the Faber--Walsh polarization tensors and the complex contracted generalized polarization tensors, thereby providing an alternative coordinate system for inverse problems.

The unresolved issue is whether finite measurement data, represented by finitely many GPTs or CGPTs, suffice to identify both geometry and material parameters within a prescribed family of multiple inclusions. The parameters specifically mentioned include the lemniscatic-domain data, conformal-map coefficients, and the conductivity assigned to each component.