Optimal estimate of electromagnetic field concentration between two nearly-touching inclusions in the quasi-static regime

We investigate the electromagnetic field concentration between two nearly-touching inclusions that possess high-contrast electric permittivities in the quasi-static regime. By using layer potential techniques and asymptotic analysis in the low-frequency regime, we derive low-frequency expansions that provide integral representations for the solutions of the Maxwell equations. For the leading-order term ${\text{bE}}_0$ of the asymptotic expansion of the electric field, we prove that it has the blow up order of ${\epsilon }^{-1}\mathrm{\mid }\ln \epsilon {\mathrm{\mid }}^{-1}$ within the radial geometry, where $\epsilon$ signifies the asymptotic distance between the inclusions. By delicate analysis of the integral operators involved, we further prove the boundedness of the first-order term ${\text{bE}}_1$. We also conduct extensive numerical experiments which not only corroborate the theoretical findings but also provide more discoveries on the field concentration in the general geometric setup. Our study provides the first treatment in the literature on field concentration between nearly-touching material inclusions for the full Maxwell system.