In this paper, we are concerned with the gradient estimate of the electric field due to two nearly touching dielectric inclusions, which is a central topic in the theory of composite materials. We derive accurate quantitative characterisations of the
gradient fields in the transverse electromagnetic case within the quasi-static regime, which clearly indicate the optimal blowup rate or non-blowup of the gradient fields in different scenarios. There are mainly two novelties of our study. First, the sizes of the two material inclusions may be of different scales. Second, we consider our study in the quasi-static regime, whereas most of the existing studies are concerned with the static case.
We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the ge
ometry of the conductive rod. In particular, we accurately characterise the localisation of the gradient field (i.e. the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer-potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod.The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderons inverse inclusion problem in the literature.
