Asymptotic behavior of the capacity in two-dimensional heterogeneous media
Andrea Braides, Giuseppe Cosma Brusca · Rendiconti Lincei Matematica e Applicazioni · 2023
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: \varepsilon , 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\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 behavior of |{\log\delta}|/|{\log\varepsilon}| .