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

Kirill Cherednichenko, Serena D'Onofrio

Research output: Contribution to journalArticlepeer-review

Abstract

For arbitrarily small values of ε>0, we formulate and analyse the Maxwell system of equations of electromagnetism on ε-periodic sets Sε⊂R3. Assuming that a family of Borel measures με, such that supp(με)=Sε, is obtained by ε-contraction of a fixed 1-periodic measure μ, and for right-hand sides fε∈L2(R3,dμε), 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 L2 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.
Original languageEnglish
Article number67
Number of pages43
JournalCalculus of Variations and Partial Differential Equations
Volume61
Issue number2
DOIs
Publication statusPublished - 7 Feb 2022

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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