Calculus of Variations and Geometric Measure Theory

A. Braides - G. C. Brusca

Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

created by braidesa on 13 Jun 2022
modified on 10 Dec 2023


Published Paper

Inserted: 13 jun 2022
Last Updated: 10 dec 2023

Journal: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
Volume: 34
Pages: 383-399
Year: 2023
Doi: 10.4171/RLM/1011


We describe the asymptotic behaviour 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 modelled by oscillating coefficients. We show that this capacity behaves as $C\
^{-1}$. 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 $