Published Paper
Inserted: 4 apr 2023
Last Updated: 12 sep 2025
Journal: Nonlinear Analysis
Volume: 239
Year: 2024
Doi: https://doi.org/10.1016/j.na.2023.113436
Abstract:
We describe the asymptotic behaviour of the minimal heterogeneous
$d$-capacity of a small set, which we assume to be a ball for simplicity, in a
fixed bounded open set $\Omega\subseteq \mathbb{R}^d$, with $d\geq2$. Two
parameters are involved: $\varepsilon$, the radius of the ball, and $\delta$,
the length scale of the heterogeneity of the medium. We prove that this
capacity behaves as $C\lvert\log \varepsilon\rvert^{1-d}$, where $C=C(\lambda)$ is an
explicit constant depending on the parameter
$\lambda:=\lim_{\varepsilon\to0}
\log \delta
/
\log\varepsilon
$.
We determine the $\Gamma$-limit of oscillating integral
functionals subjected to Dirichlet boundary conditions on periodically
perforated domains. Our first result is used to study the
behaviour of the functionals near the perforations which, in this instance, are balls of
radius $\varepsilon$. We prove that an additional strange term arises involving $C(\lambda)$.
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