The Calder\'{o}n inverse problem for isotropic quasilinear conductivities

TL;DR

This paper proves a global uniqueness result for the Calderón inverse problem in isotropic quasilinear conductivities, using higher order linearizations and complex geometric optics solutions to recover conductivity differentials.

Contribution

It introduces a novel approach combining higher order linearizations and CGO solutions to solve the inverse problem for quasilinear isotropic conductivities.

Findings

Established a uniqueness theorem for the inverse conductivity problem.

Reduced the recovery of conductivity differentials to a completeness property.

Used complex geometric optics solutions with concentrated amplitudes.

Abstract

We prove a global uniqueness result for the Calder\'{o}n inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension $n\ge 3$. Performing higher order linearizations of the nonlinear Dirichlet--to--Neumann map, we reduce the problem of the recovery of the differentials of the quasilinear conductivity, which are symmetric tensors, to a completeness property for certain anisotropic products of solutions to the linearized equation. The completeness property is established using complex geometric optics solutions to the linearized conductivity equation, whose amplitudes concentrate near suitable two dimensional planes.