Estimating the volume fraction of an inclusion in a body from boundary measurements in two-dimensional conductivity problem

Abstract

In this thesis we consider the inverse problems of bounding the volume fraction of an inclusion embedded in a body from two sets of electrical boundary measurements. The inclusion and the body are two dimensional domains with constant conductivities, which are supposed to be known. Recently, it was derived the upper and lower bounds of the volume fraction of the inclusion in terms of two boundary measurements by Kang, Kim, and Milton [1]. We first derive simple bounds of the volume fraction using classical variational principles and null-Lagrangians and then review the results in [1]. We demonstrate some numerical examples to see the validity of the bounds.