Composite media, almost touching disks and the maximum principle

TL;DR

This paper studies the behavior of solutions to elliptic equations in domains with two nearly touching disks with different conductivities, providing a new proof of a gradient bound using the maximum principle.

Contribution

It offers a novel proof of a gradient bound for elliptic equations in composite media with nearly touching disks, extending previous results by Li-Vogelius.

Findings

Established a new proof of the gradient bound.

Analyzed the impact of almost touching disks on elliptic equations.

Applied the maximum principle to composite media.

Abstract

We consider the setting of two disks in a domain in $\mathbb{R}^2$ which are almost touching and have finite and positive conductivities, giving rise to a divergence form elliptic equation with discontinuous coefficients. We use the maximum principle to give a new proof of a gradient bound of Li-Vogelius.