The conductivity problem with imperfect bonding interfaces and finite internal conductivities

We study the field concentration phenomenon between two closely spaced inclusions with imperfect bonding interfaces of low conductivity type. The inclusions are assumed to have finite conductivities. The problem is governed by a system of elliptic equations coupled with Robin-type boundary conditions. While it is known that finite-conductivity inclusions with ideal interfaces yield bounded gradients, in this paper we show that with imperfect bonding interfaces, the gradient of the solution may blow up as $\epsilon$ (the distance between two inclusions) tends to zero when the bonding parameter $\gamma$ is large, while it is uniformly bounded independently of $\epsilon$ when the bonding parameter $\gamma$ is small. Moreover, we identify the threshold of $\gamma$ and the optimal blow-up rates under certain symmetry assumptions. Compared to the case when the inclusions are perfect conductors, we find a novel logarithmic blow-up phenomenon at the critical value of $\gamma$.