In this talk, I will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In the first part, I will present some results on the insulated conductivity problem when the current-electric field relation is linear (jointly with Hongjie Dong and Yanyan Li). We obtained an optimal gradient estimate in terms of the distance between inclusions. This solved one of the major open problems in this area. In the second part, I will present a recent work on the insulated conductivity problem when the current-electric field relation is the power law (jointly with Hongjie Dong and Hanye Zhu). We identify the optimal blow-up rate for the gradient in 2D, and prove some gradient estimates in any dimensions.
