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

TL;DR

This paper derives precise gradient estimates for the insulated conductivity problem involving general m-convex inclusions in higher dimensions, revealing how the singularity depends on eigenvalues of an associated elliptic operator.

Contribution

It provides new pointwise upper bounds on the gradient for m-convex inclusions, extending understanding of singular behavior in insulated conductivity models.

Findings

Gradient bounds depend on the first non-zero eigenvalue of an elliptic operator.

Singular behavior characterized for inclusions with m-convex shapes, including curvilinear cubes.

Estimates are shown to be sharp for axisymmetric insulators.

Abstract

In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized $m$-convex inclusions under these two cases with $m\geq2$, 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.