Optimal estimates for transmission problems including relative conductivities with different signs

We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different signs, the gradient and higher order derivatives are bounded independent of $\varepsilon$, the distance between the inclusions. We also show that for general smooth strictly convex inclusions, when one inclusion is an insulator and the other one is a perfect conductor, the derivatives of any order is bounded independent of $\varepsilon$ in any dimensions $n \ge 2$.