Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators (Analysis in Theory and Applications, 2022)
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. When the distance of inclusions, denoted by $ε$, goes to 0, the gradient of solutions may blow up. When two inclusions are strictly convex, it was known that an upper bound of the blow-up rate is of order $ε^{-1/2}$ for $n = 2$, and is of order $ε^{-1/2 + β}$ for some $β > 0$ when dimension $n \ge 3$. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.
Keywords: Conductivity problem; harmonic functions; maximum principle; gradient estimates.