Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

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 Ω. This problem is gov erned by two small parameters: ε, the size of the inclusion (which is not restrictive to assume to be a ball), and δ, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as C| log ε| −1. The coefficient C is ex plicitly 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 δ|/| log ε|.