Asymptotic behavior of the capacity in two-dimensional heterogeneous media

We describe the asymptotic behavior of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set Ω. This problem is governed by two small parameters: ", the size of the inclusion (which is not restrictive to assume to be a ball), and ı, the period of the inhomogeneity modeled by oscillating coefficients. We show that this capacity behaves as C jlog "j-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 jlog ıj=jlog "j.