The perfect conductivity problem with arbitrary vanishing orders and non-trivial topology

TL;DR

This paper extends gradient estimates for harmonic functions in narrow regions between conductors with complex boundary behaviors, providing sharp bounds even with non-trivial topology and arbitrary vanishing orders.

Contribution

It generalizes previous estimates to more complex geometries and boundary conditions, establishing new sharp bounds for the perfect conductivity problem.

Findings

Extended gradient estimates to arbitrary vanishing orders.

Handled globally defined narrow regions with complex topology.

Proved the sharpness of the estimates in relation to conductor distance.

Abstract

The perfect conductivity problem concerns optimal bounds for the magnitude of an electric field in the presence of almost touching perfect conductors. This reduces to obtaining gradient estimates for harmonic functions with Dirichlet boundary conditions in the narrow region between the conductors. In this paper we extend estimates of Bao-Li-Yin to deal with the case when the boundaries of the conductors are given by graphs with arbitrary vanishing orders. Our estimates allow us to deal with globally defined narrow regions with possibly non-trivial topology. We also prove the sharpness of our estimates in terms of the distance between the perfect conductors. The precise optimality statement we give is new even in the setting of Bao-Li-Yin.