Calculus of Variations and Geometric Measure Theory

G. C. Brusca

Homogenization in perforated domains at the critical scale

created by brusca on 04 Apr 2023
modified on 15 Nov 2023


Accepted Paper

Inserted: 4 apr 2023
Last Updated: 15 nov 2023

Journal: Nonlinear Analysis
Year: 2023

ArXiv: 2304.01123 PDF


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
$. 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 $C(\lambda)$.