Optimal boundary gradient estimates for the insulated conductivity problem

In this paper we study the boundary gradient estimate of the solution to the insulated conductivity problem with the Neumann boundary data when a convex insulating inclusion approaches the boundary of the matrix domain. The gradient of solutions may blow up as the distance between the inclusion and the boundary, denoted as $\varepsilon$, approaches to zero. The blow up rate was previously known to be sharp in dimension $n=2$ (see Ammari et al.\cite{AKLLL}). However, the sharp rates in dimensions $n\geq3$ are still unknown. In this paper, we solve this problem by establishing upper and lower bounds on the gradient and prove that the optimal blow up rates of the gradient are always of order $ε^{-1/2}$ for general strictly convex inclusions in dimensions $n\geq3$. Several new difficulties are overcome and the impact of the boundary data on the gradient is specified. This result highlights a significant difference in blow-up rates compared to the interior estimates in recent works (\cites{LY,Weinkove,DLY,DLY2,LZ}), where the optimal rate is $ε^{-1/2+β(n)}$, with $β(n)\in(0,1/2)$ varying with dimension $n$. Furthermore, we demonstrate that the gradient does not blow up for the corresponding Dirichlet boundary problem.