Calculus of Variations and Geometric Measure Theory
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A. Braides - G. C. Brusca

Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

created by braidesa on 13 Jun 2022
modified on 22 Jun 2022

[BibTeX]

Preprint

Inserted: 13 jun 2022
Last Updated: 22 jun 2022

Year: 2022

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 $\Omega$. 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 $C\
\log\varepsilon\
^{-1}$. 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 behaviour of $
\log\delta
/
\log\varepsilon
$.


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