  When a convex perfectly conducting inclusion is closely spaced to the
boundary of the matrix domain, a bigger convex domain containing the inclusion,
the electric field can be arbitrary large. We establish both the pointwise
upper bound and the lower bound of the gradient estimate for this perfect
conductivity problem by using the energy method. These results give the optimal
blow-up rates of electric field for conductors with arbitrary shape and in all
dimensions. A particular case when a circular inclusion is close to the
boundary of a circular matrix domain in dimension two is studied earlier by
Ammari,Kang,Lee,Lee and Lim(2007). From the view of methodology, the technique
we develop in this paper is significantly different from the previous one
restricted to the circular case, which allows us further investigate the
general elliptic equations with divergence form.
