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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 arbitrarily 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 blowup rates of an 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 et al. [J. Math. Pures Appl. (9), 88 (2007), pp. 307--324]. 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.

Optimal Estimates for the Perfect Conductivity Problem with Inclusions Close to the Boundary