Spatial non-locality of the Maxwell system on periodic structures

TL;DR

This paper investigates the behavior of the Maxwell equations on periodic structures, providing precise estimates for how solutions converge as the structure's scale shrinks, revealing spatial non-locality effects.

Contribution

It establishes order-sharp norm-resolvent convergence estimates for Maxwell systems on epsilon-periodic sets, advancing understanding of electromagnetic behavior in periodic media.

Findings

Proves sharp convergence estimates for Maxwell solutions on periodic structures.

Shows the impact of spatial non-locality in electromagnetic systems.

Analyzes the effect of epsilon-contraction of measures on solution behavior.

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.