Spatial non-locality of the Maxwell system on periodic structures

Abstract

For $\varepsilon>0,$ we analyse the Maxwell system of equations of electromagnetism on $\varepsilon$-periodic sets $S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures $\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed periodic measure $\mu,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system.