Gradient estimates for the insulated conductivity problem: the case of $m$-convex inclusions

Abstract

We consider an insulated conductivity model with two neighboring inclusions of $m$-convex shapes in ${\mathbb{R}}^d$ when $m\ge 2$ and $d\ge 3$. We establish the pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order ${\epsilon }^{-1/m+\beta }$ with $\beta =[-(d+m-3)+\sqrt{(d+m-3{)}^2+4(d-2)}]/(2m)\in (0,1/m)$, as the distance $\epsilon$ between these two insulators tends to zero. In particular, the optimality of the blow-up rate is also demonstrated for a class of axisymmetric $m$-convex inclusions.

Exposition improved