Gradient estimates for the insulated conductivity problem with partially flat inclusions

TL;DR

This paper investigates the insulated conductivity problem with partially flat inclusions, showing that unlike strictly convex cases, the gradient of solutions remains bounded and does not blow up under any uniform background fields.

Contribution

It establishes that partially flat inclusions prevent gradient blow-up in the insulated conductivity problem, extending understanding beyond strictly convex geometries.

Findings

Gradient of solutions remains bounded with partially flat inclusions

Blow-up does not occur under any uniform background fields

Contrasts with blow-up behavior in strictly convex inclusions

Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between inclusions approaches 0. The optimal blow-up rate was proved in [10] and was achieved in the presence of a uniform background gradient field. In this paper, we demonstrate that when the inclusions are partially flat, the gradient of solutions does not blow up under any uniform background fields.