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Spatial non-locality of the Maxwell system on periodic structures

Kirill Cherednichenko, Serena D'Onofrio

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 $μ^\varepsilon,$ such that ${\rm supp}(μ^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed periodic measure $μ,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, dμ^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system.

Spatial non-locality of the Maxwell system on periodic structures

Abstract

For we analyse the Maxwell system of equations of electromagnetism on -periodic sets Assuming that a family of Borel measures such that is obtained by -contraction of a fixed periodic measure and for right-hand sides we prove order-sharp norm-resolvent convergence estimates for the solutions of the system.

Paper Structure

This paper contains 11 sections, 17 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces of quasiperiodic functions
  3. Floquet transform
  4. An analogue of Helmholtz decomposition
  5. Poincaré-type inequality
  6. Electric displacement $D^\varepsilon$
  7. Asymptotic approximation of $\mathfrak{D}^\varepsilon_\theta$
  8. Estimate for $R^\varepsilon_\theta$
  9. Conclusion of the proof of Theorem \ref{['main_thm_gen']}
  10. Magnetic induction $B^\varepsilon$
  11. Comparison to the formal expansion approach

Key Result

Proposition 3.1

For each $\varepsilon>0$ the following unitary equivalence between the resolvent of $\mathcal{A}^\varepsilon$ and the direct integral of the resolvents of $\mathcal{A}_{\varepsilon\theta},$$\theta\in \varepsilon^{-1}Q',$ holds: where $e_{\varepsilon\theta},$$\overline{e}_{\varepsilon\theta}$ represent the operators of multiplication by $e_{\varepsilon\theta},$$\overline{e}_{\varepsilon\theta},$ r

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof : Sketch of proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Lemma 5.1
  • ...and 23 more