QUARTERLY OF APPLIED MATHEMATICS VOLUME LXV, NUMBER 2 JUNE 2007, PAGES 339–355 S 0033-569X(07)01034-1 Article electronically published on January 16, 2007 ESTIMATES FOR THE ELECTRIC FIELD IN THE PRESENCE OF ADJACENT PERFECTLY CONDUCTING SPHERES By HABIB AMMARI (Centre de Math´ ematiques Appliqu´ ees, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France ), GEORGE DASSIOS (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom), HYEONBAE KANG (Department of Mathematical Sciences and RIM, Seoul National University, Seoul 151-747, Korea ), and MIKYOUNG LIM (Centre de Math´ ematiques Appliqu´ ees, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France ) Abstract. In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero. 1. Introduction. Frequently in two phase composites, inclusions are located very closely and may even touch; see [3]. It is therefore natural and important to find out if the electric field in the presence of closely spaced inclusions can be arbitrarily large or not. The purpose of this paper is to deal with the problem in three dimensions and show that, unlike the two dimensional case, the electric field is bounded regardless of the distance between the two inclusions. In the conductivity model, the electric field is given by ∇u, where u is the solution to ⎧ ⎨ ⎩ ∇· ( χ(R d \ B 1 ∪ B 2 )+ k 1 χ(B 1 )+ k 2 χ(B 2 ) ) ∇u =0 in R d (d =2, 3), u(x) − H(x)= O(|x| 1−d ) as |x|→ +∞. (1.1) Here H is a given harmonic function in R d such that H(0) = 0, B 1 and B 2 represent the inclusions, k 1 and k 2 are their conductivities, and χ(E) denotes the indicator function Received April 30, 2006. 2000 Mathematics Subject Classification. Primary 35J25; Secondary 73C40. Key words and phrases. Electric field, gradient estimates, composite materials. E-mail address : ammari@cmapx.polytechnique.fr E-mail address : G.Dassios@damtp.cam.ac.uk E-mail address : hkang@math.snu.ac.kr E-mail address : mklim@cmapx.polytechnique.fr c 2007 Brown University Reverts to public domain 28 years from publication 339 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf