Gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

TL;DR

This paper investigates the gradient blow-up behavior in the insulated conductivity problem with inclusions in higher dimensions, improving the upper bound of the blow-up rate from the known sharp rate in 2D.

Contribution

It extends the understanding of gradient estimates in the insulated conductivity problem to dimensions three and higher, refining the upper bound of the blow-up rate.

Findings

Upper bound for gradient blow-up rate in n ≥ 3 is improved to 0(0 0 0 0 + eta) for some 0 > 0.

The result advances the theoretical understanding of singular behavior in high-dimensional conductivity problems.

The paper establishes that the blow-up rate is not as severe as previously thought in higher dimensions.

Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. An upper bound for the blow up rate was proved to be of order $\varepsilon^{-1/2}$. The upper bound was known to be sharp in dimension $n = 2$. However, whether this upper bound is sharp in dimension $n \ge 3$ has remained open. In this paper, we improve the upper bound in dimension $n \ge 3$ to be of order $\varepsilon^{-1/2 + \beta}$, for some $\beta > 0$.