Optimal gradient estimates for conductivity problems with imperfect low-conductivity interfaces

This paper studies field concentration between two nearly touching conductors separated by imperfect low-conductivity interfaces, modeled by Robin boundary conditions. It is known that for any sufficiently small interfacial bonding parameter $\gamma > 0$, the gradient remains uniformly bounded with respect to the separation distance $\varepsilon$. In contrast, for the perfect bonding case ($\gamma = 0$, corresponding to the perfect conductivity problem), the gradient may blow up as $\varepsilon \to 0$ at a rate depending on the dimension. In this work, we establish optimal pointwise gradient estimates that explicitly depend on both $\gamma$ and $\varepsilon$ in the regime where these parameters are small. These estimates provide a unified framework that encompasses both the previously known bounded case ($\gamma > 0$) and the singular blow-up scenario ($\gamma = 0$), thus furnishing a complete and continuous characterization of the gradient behavior throughout the transition in $\gamma$. The key technical achievement is the derivation of new regularity results for elliptic equations as $\gamma\to0$, along with a case dichotomy based on the relative sizes of $\gamma$ and a distance function $\delta(x')$. Our results hold for strictly relatively convex conductors in all dimensions $n \geq 2$.