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Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

Kirill Cherednichenko, Serena D'Onofrio

Abstract

For arbitrarily small values of $\varepsilon>0,$ we formulate and 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 1-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. Our analysis includes the case of periodic "singular structures", when $μ$ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for $L^2$ functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method.

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

Abstract

For arbitrarily small values of we formulate and 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 1-periodic measure and for right-hand sides we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method.

Paper Structure

This paper contains 15 sections, 17 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces of quasiperiodic functions
  3. Floquet transform
  4. Quasiperiodic Helmholtz decomposition
  5. Asymptotic approximation for $u_\theta^\varepsilon$
  6. Proof of Theorem \ref{['main_thm']}: asymptotic estimate for $R_\theta^\varepsilon$ as $\varepsilon\to0$
  7. Conclusion of proof: convergence estimate for $z_\theta^\varepsilon$
  8. Estimates for the electric field and displacement
  9. Further developments of the method
  10. Curls of zero for a measure supported by a plane
  11. Approximation in $H^1_{\mathop{\mathrm{curl}}\nolimits,\kappa}(Q, d\mu)$ by smooth functions
  12. Poincaré inequality
  13. The norm of the transversal curl is the norm of the tangential gradient
  14. "Scalar" Poincaré inequality for a single plane
  15. Connectivity argument

Key Result

Proposition 3.2

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

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof : Sketch of the proof
  • Proposition 4.1
  • proof
  • Remark 4.3
  • Proposition 5.1
  • proof
  • ...and 26 more