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 with . We establish the pointwise upper bounds on the gradient of the solution for the generalized m‐convex inclusions under these two cases with , 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 . Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.