A simple proof for the insulated conductivity problem and application to flat boundaries

In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $\epsilon$ between the inclusions, which tends to $0$. The purpose of this paper is to provide a simple proof of optimal pointwise estimates for the insulated conductivity problem in any dimension, including the case of flat inclusions. Our approach is based on two fundamental tools: the maximum principle and the Hopf lemma. A key feature of this method is that it avoids the flattening techniques commonly used in the literature, such as those in \citet{dong2021optimal,dong2022gradient}, which require transforming the narrow region into an n-dimensional cuboid. We show that the solution of the insulated problem is $\alpha$-order ($\alpha\in[0,1)$) polynomial growth for $n\geq2$ near the origin. Moreover, when the boundaries near the origin are flat, we prove that the gradient of the solution remains uniformly bounded.