Optimal Estimates for the Electric Field in Two-Dimensions

Abstract

The purpose of this paper is to set out optimal gradient estimates for solutions to the isotropic conductivity problem in the presence of adjacent conductivity inclusions as the distance between the inclusions goes to zero and their conductivities degenerate. This difficult question arises in the study of composite media. Frequently in composites, the inclusions are very closely spaced and may even touch. It is quite important from a practical point of view to know whether the electric field (the gradient of the potential) can be arbitrarily large as the inclusions get closer to each other or to the boundary of the background medium. In this paper, we establish both upper and lower bounds on the electric field in the case where two circular conductivity inclusions are very close but not touching. We also obtain such bounds when a circular inclusion is very close to the boundary of a circular domain which contains the inclusion. The novelty of these estimates, which improve and make complete our earlier results published in Math. Ann., is that they give an optimal information about the blow-up of the electric field as the conductivities of the inclusions degenerate.