Gradient estimates for the insulated conductivity problem with partially flat inclusions

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.