Asymptotics for the concentrated field between closely located hard inclusions in all dimensions

When hard inclusions are frequently spaced very closely, the elec- tric field, which is the gradient of the solution to the perfect conductivity equation, may be arbitrarily large as the distance between two inclusions goes to zero. In this paper, our objectives are two-fold: First, we extend the as- ymptotic expansions of [26] to the higher dimensions greater than three by capturing the blow-up factors in all dimensions, which consist of some certain integrals of the solutions to the case when two inclusions are touching; second, our results answer the optimality of the blow-up rate for any m; n ≥ 2, where m and n are the parameters of convexity and dimension, respectively, which is only partially solved in [29].