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 Rd when m ≥ 2 and d ≥ 3. We establish pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order ɛ−1/m+β with β=[−(d+m−3)+(d+m−3)2+4(d−2)]/(2m)∈(0,1/m) as the distance ɛ 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.