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 Rⁿ. 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.