Bounds on the volume of an inclusion in a body from a complex conductivity measurement

Abstract

We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of the electric field, current, and certain products of the electric field and current. All of these average values are computed from a single electrical impedance tomography measurement of the voltage and current on the boundary of the body. Additionally, the bounds are tight in the sense that at least one of the bounds gives the exact volume of the inclusion for certain geometries and boundary conditions.

Explicit examples of extremal quasiconvex quadratic forms that are not polyconvex

“…Q(Re(û(k) ⊗ k)) + Q(Im(û(k) ⊗ k)) ≥ 0, due to the quasiconvexity of Q. It is then clear by (23), that the equality holds if and only ifû(k) = 0 if k does not have one of the forms (l, l, l), (−l, l, l), (l, −l, l), (l, l, −l) and alsô u 1 (l, l, l) =û 2 (l, l, l) =û 3 (l, l, l), −û 1 (−l, l, l) =û 2 (−l, l, l) =û 3 (−l, l, l), u 1 (l, −l, l) = −û 2 (l, −l, l) =û 3 (l, −l, l), u 1 (l, l, −l) =û 2 (l, l, −l) = −û 3 (l, l, −l).…”

“…In the 1980's a lot of attention was focussed on bounding the effective tensors of composites, and quasiconvex functions played an important role in this development: see the books [1,5,13,21] and references therein. In the last few years it was realized [7,8, see also 15,9,23,10] that similar methods can be useful for bounding of the Dirichlet to Neumann map of inhomogeneous bodies (which for a two-phase body can be applied in an inverse manner to bound the volume of an inclusion from Dirichlet and Neumann boundary data). Clearly there is an interest in obtaining sharp bounds on the Dirchlet to Neumann map not just for affine data but for other Dirichlet boundary conditions as well.…”

Criteria for guaranteed breakdown in two-phase inhomogeneous bodies

“…Following the procedure of Thaler and Milton [52], we can use the positivity of the variance, g (α) · g (α) ≥ 0, for all c (α) ∈ R 2 , where g (α) is given by (3.5), to obtain the condition that the matrices S α given by (3.6) are positive semidefinite. Making the substitutions (4.21) and the symmetric matrices S α f (2) ;…”

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

“…We note that when the admittivities of the inclusion and the surrounding body are known, some bounds for |D| were derived by Thaler, Milton [18] using the splitting method, and by Kang, Kim, Lee, Li, Milton [13] using the translation method. This paper is organized as follows.…”