Optimal gradient estimates for the insulated conductivity problem with dimensions more than two

Abstract

In high-contrast composite materials, the electric (or stress) field may blow up in the narrow region between inclusions. The gradient of solutions depend on $\hat{I}\mu$, the distance between the inclusions, where $\hat{I}\mu$ approaches to $0$. By using the maximum principle techniques, we give another proof of the Dong-Li-Yang estimates \cite{DLY} for any convex inclusions of arbitrary shape with $n\ge 3$. This result solves the problem raised by \cite{W}, where the spherical inclusions with $n\ge 4$ is considered. Moreover, we also generalize the above results with flatter boundaries near touching points.

The proof in this paper is incorrect