Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions

TL;DR

This paper derives gradient estimates for perfect and insulated conductivity problems with multiple inclusions, showing how the gradients behave as inclusions approach each other in bounded domains.

Contribution

It provides new gradient bounds for both perfect and insulated conductivity problems with multiple inclusions, especially near close inclusions.

Findings

Gradient estimates for perfect conductivity problems.

Upper bounds for insulated conductivity problems.

Behavior of solutions as inclusions approach each other.

Abstract

In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in $\mathbb{R}^n, n\ge 2$. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions.