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

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