Asymptotic behavior of the capacity in two-dimensional heterogeneous media

Abstract

We describe the asymptotic behavior 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: $\epsilon$, the size of the inclusion (which is not restrictive to assume to be a ball), and $\delta$, the period of the inhomogeneity modeled by oscillating coefficients. We show that this capacity behaves as $C|\log \epsilon {|}^{-1}$. 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 behavior of $|\log \delta |/|\log \epsilon |$.