Gradient estimates for the insulated conductivity problem with inclusions of the general $m$-convex shapes

Abstract

In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in ${\mathbb{R}}^d$ with $d\ge 3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized $m$-convex inclusions under these two cases with $m\ge 2$, which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non-zero eigenvalue of an elliptic operator of divergence form on ${\mathbb{S}}^{d-2}$. Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.

arXiv commentExposition improved and some computational details added