Optimal estimates for the electric field in two dimensions

Abstract

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 in [H. Ammari, H. Kang, M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005) 277-286], is that they give an optimal information about the blow-up of the electric field as the conductivities of the inclusions degenerate. (C) 2007 Elsevier Masson SAS. All rights reserved.