Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

Andrea Braides; Giuseppe Cosma Brusca

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arXiv · 2206.06093

Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

Andrea Braides·Giuseppe Cosma Brusca

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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 $C|\log\e|^{-1}$. 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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Andrea Braides, Giuseppe Cosma Brusca. 2022-06-13. Asymptotic behaviour of the capacity in two-dimensional heterogeneous media. https://arxiv.org/abs/2206.06093

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