Gradient estimates for the conductivity problem with imperfect bonding interfaces

We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. A previous conjecture suggested that the gradient of solutions remains bounded regardless of $\varepsilon$, the distance between two inclusions. In this article, we establish gradient estimates, indicating that the conjecture is true only when the bonding parameter $\gamma$ is sufficiently small, and the gradient could blow up when $\gamma$ is large and the boundary data is not aligned with shortest line connecting the two inclusions. Moreover, we derive the optimal blow-up rates under certain symmetry assumptions.