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

TL;DR

This paper provides new proofs and generalizations of gradient estimates in high-contrast composite materials with convex inclusions in dimensions three and higher, addressing previous open problems.

Contribution

It offers an alternative proof of existing estimates for convex inclusions of arbitrary shape and extends results to flatter boundary conditions near touching points.

Findings

Established gradient bounds for convex inclusions of arbitrary shape in dimensions n≥3.

Extended estimates to cases with flatter boundaries near touching points.

Solved an open problem for spherical inclusions with dimensions n≥4.

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 $\epsilon$, the distance between the inclusions, where $\epsilon$ 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 $n\geq 3$. This result solves the problem raised by \cite{W}, where the spherical inclusions with $n\geq 4$ is considered. Moreover, we also generalize the above results with flatter boundaries near touching points.