Gradient estimates for electric fields with multi-scale inclusions in the quasi-static regime

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.