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. We discover a \textit{new} dichotomy for the field concentration depending on the bonding parameter $\gamma$. Specifically, we show that the gradient of solution is uniformly bounded independent of $\varepsilon$ (the distance between two inclusions) when $\gamma$ is sufficiently small. However, the gradient may blow up when $\gamma$ is large. Moreover, we identify the threshold of $\gamma$ and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.