Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set $\mathrm{\Omega }$. 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 modelled by oscillating coefficients. We show that this capacity behaves as $C\mathrm{\mid }\log \text{e}{\mathrm{\mid }}^{-1}$. 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 $\mathrm{\mid }\log \delta \mathrm{\mid }\mathrm{/}\mathrm{\mid }\log \epsilon \mathrm{\mid }$.