# Optimal estimates for the perfect conductivity problem with inclusions close to the boundary

## TL;DR

This paper derives optimal bounds for the electric field's blow-up rate when a convex perfect conductor is near the boundary of a domain, extending previous circular case results to general shapes and dimensions.

## Contribution

It introduces a new energy-based method to establish sharp gradient bounds for perfect conductors of arbitrary shape near boundaries, applicable in all dimensions.

## Key findings

- Established pointwise upper and lower bounds for the electric field gradient.
- Derived optimal blow-up rates for conductors with arbitrary shapes.
- Extended analysis from circular to general convex inclusions in all dimensions.

## Abstract

When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blow-up rates of electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earlier by Ammari,Kang,Lee,Lee and Lim(2007). From the view of methodology, the technique we develop in this paper is significantly different from the previous one restricted to the circular case, which allows us further investigate the general elliptic equations with divergence form.