Blowup Analysis for the Perfect Conductivity Problem with convex but not strictly convex inclusions.

In the perfect conductivity problem, it is interesting to study whether the electric field can become arbitrarily large or not, in a narrow region between two adjacent perfectly conducting inclusions. In this paper, we show that the relative convexity of two adjacent inclusions plays a key role in the blowup analysis of the electric field and find some new phenomena. By energy method, we prove the boundedness of the gradient of the solution if two adjacent inclusions fail to be locally relatively strictly convex, namely, if the top and bottom boundaries of the narrow region are partially flat. The boundary estimates when an inclusion with partially flat boundary is close to the flat matrix boundary and estimates for the general elliptic equation of divergence form are also established in all dimensions.