Spatial non-locality of the Maxwell system on periodic structures

Abstract

For $\epsilon >0,$ we analyse the Maxwell system of equations of electromagnetism on $\epsilon$-periodic sets $S^{\epsilon }\subset {\mathbb{R}}^3.$ Assuming that a family of Borel measures $\hat{I}{¼}^{\epsilon },$ such that $\mathrm{supp}(\hat{I}{¼}^{\epsilon })=S^{\epsilon },$ is obtained by $\epsilon$-contraction of a fixed periodic measure $\hat{I}¼,$ and for right-hand sides $f^{\epsilon }\in L^2({\mathbb{R}}^3,d\hat{I}{¼}^{\epsilon }),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system.

15 pages