Uniqueness estimates for the general complex conductivity equation and their applications to inverse problems

TL;DR

This paper establishes quantitative uniqueness estimates for solutions to the complex conductivity equation under Lipschitz conditions and applies these results to inverse problems involving size estimation of inclusions with anisotropic complex admittivity.

Contribution

It proves three-ball inequalities for the complex conductivity equation with Lipschitz coefficients and explores their application to inverse boundary value problems.

Findings

Three-ball inequalities hold under Lipschitz assumptions.

Quantitative estimates aid in inverse problems for anisotropic complex admittivity.

New Carleman estimates are developed for complex conductivity equations.

Abstract

The aim of the paper is twofold. Firstly, we would like to derive quantitative uniqueness estimates for solutions of the general complex conductivity equation. It is still unknown whether the \emph{strong} unique continuation property holds for such equations. Nonetheless, in this paper, we show that the unique continuation property in the form of three-ball inequalities is satisfied for the complex conductivity equation under only Lipschitz assumption on the leading coefficients. The derivation of such estimates relies on a delicate Carleman estimate. Secondly, we study the problem of estimating the size of an inclusion embedded inside of a conductive body with anisotropic complex admittivity by one boundary measurement. The study of such inverse problem is motivated by practical problems.