Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

TL;DR

This paper analyzes how the minimal inhomogeneous two-capacity of small sets in a plane behaves asymptotically, depending on the size of the inclusion and the periodic inhomogeneity, with explicit formulas derived.

Contribution

It provides an explicit asymptotic formula for the capacity in heterogeneous media considering two small parameters, extending previous understanding of capacity behavior.

Findings

Capacity behaves as C/|log ε| with explicit C

C depends on the minimum of oscillating coefficient and homogenized matrix

The asymptotic coefficient involves a harmonic mean based on the ratio of logs

Abstract

We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set $\Omega$. This problem is governed by two small parameters: $\varepsilon$, the size of the inclusion (which is not restrictive to assume to be a ball), and $\delta$, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as $C|\log\e|^{-1}$. The coefficient $C$ is explicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behaviour of $|\log\delta|/|\log\varepsilon|$.