# Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

created by braidesa on 13 Jun 2022
modified on 22 Jun 2022

[BibTeX]

Preprint

Inserted: 13 jun 2022
Last Updated: 22 jun 2022

Year: 2022

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 $C\ \log\varepsilon\ ^{-1}$. 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 $\log\delta / \log\varepsilon$.