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

TL;DR

This paper investigates how the shape of adjacent inclusions affects electric field blowup in perfect conductivity problems, revealing that non-strict convexity can prevent unbounded field intensities.

Contribution

It demonstrates that relative convexity of inclusions is crucial for electric field blowup analysis and provides boundedness results when inclusions are not strictly convex.

Findings

Bounded electric field gradients when inclusions are not locally strictly convex.

Boundary estimates for flat inclusions near flat boundaries.

Extensions of estimates to general elliptic divergence form equations.

Abstract

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.