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

In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in R-d with d >= 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 >= 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 Sd-2. Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.