Uniqueness Estimates for the General Complex Conductivity Equation and Their Applications to Inverse Problems

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 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.

The enclosure method for a generalized anisotropic complex conductivity equation

“…One motivation for the study of conductivity equations with complex conductivity comes from models of the biological tissues. See for example [2,4,5,23]. Moreover, anisotropic nature of some biological tissues such as bones and muscles is also important.…”

“…Moreover, anisotropic nature of some biological tissues such as bones and muscles is also important. Another motivation, which is directly associated to our mathematical model, arises from the modeling of the propagation of electromagnetic waves in a so called chiral or continuous chiroptical medium [2]. Chirality and chiral phenomena associated with electromagnetic waves have recently been covered in many papers, for example [1,6,13,15,17,22].…”

“…Here , μ are complex positive semi-definite matrices. If we additionally assume (i) μ is a real multiple of identity matrix, (ii) = ρe −iθ , μ = ρ e iθ with real positive definite matrices ρ, ρ, and (iii) both | ρ|, θ 1, then the source-free time-harmonic Maxwell equations with DBF constitutive relations leads to the following approximation equation [2]…”

“…Moreover, due to the assumptions (i)-(iii), we may neglect the imaginary part of ξ. See [2] for more details.…”

Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients

“…Having established the Carleman estimate, we then can apply the ideas in [6] to prove a three-region inequality and those in [4] to prove a three-ball inequality across the interface. With the help of the three-ball inequality, we can study the size estimate problem for the complex conductivity equation following the ideas in [3]. We will present these quantitative uniqueness results and the application to the size estimate in the forthcoming paper.…”