Homogenization of Maxwell's equations in a split ring geometry (Q2786351)

The authors use the technique of homogenization to study the system NEWLINENEWLINE\[NEWLINE\operatorname{curl} E_\eta= i\omega\mu_0 H_\eta, \qquad \operatorname{curl} H_\eta= i\omega\varepsilon_\eta\varepsilon_0 E_\eta,NEWLINE\]NEWLINE NEWLINEwith positive real constants \(\omega\), \(\mu_0\) and \(\varepsilon_0\). For a constant \(\kappa\) they study the relative permitivity NEWLINENEWLINE\[NEWLINE\varepsilon_\eta= \begin{cases} 1 +i \frac{\kappa}{\eta^2} &\text{in }\Sigma_\eta,\\ 1&\text{in }\mathbb R^3\backslash \Sigma_\eta,\end{cases}NEWLINE\]NEWLINE NEWLINEwhere \(\Sigma_\eta\) is a family of so-called split rings. In particular, they rely on the two-scale convergence technique to deduce the weak two-scale limits of NEWLINENEWLINE\[NEWLINEE_\eta (x)\rightharpoonup E_0 (x, y), \qquad H_\eta\rightharpoonup H_0 (x, y), \qquad J_\eta= \eta\varepsilon_\eta E_\eta \rightharpoonup J_0 (x, y).NEWLINE\]NEWLINE NEWLINEThe cell problem for \(E_0\) and for the pair \((H_0, J_0 )\) is also considered.