  In high-contrast composite materials, the electric field concentration is a
common phenomenon when two inclusions are close to touch. It is important from
an engineering point of view to study the dependence of the electric field on
the distance between two adjacent inclusions. In this paper, we derive the
upper and lower bounds of the gradient of solutions to the conductivity problem
where two perfectly conducting inclusions are located very close to each other.
To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two
folds: First, we weaken the smoothness of the inclusions from C^{2,\alpha} to
C^{1,\alpha}. To obtain an pointwise upper bound of the gradient, we follow an
iteration technique developed by Bao-Li-Li (ARMA 2015), who mainly deal with
the system of linear elasticity. However, when the inclusions are of C^{1,
\alpha}, we can not use W^{2,p} estimates for elliptic equations any more. In
order to overcome this new difficulty, we take advantage of De Giorgi-Nash
estimates and Campanato's approach to apply an adapted version of the iteration
technique with respect to the energy. A lower bound in the shortest line
between two inclusions is also obtained to show the optimality of the blow-up
rate. Second, when two inclusions are only convex but not strictly convex, we
prove that blow-up does not occur any more. Moreover, the establishment of the
relationship between the blow-up rate of the gradient and the order of the
convexity of the inclusions reveals the mechanism of such concentration
phenomenon.
