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Abstract
For arbitrarily small values of ε>0, we formulate and analyse the Maxwell system of equations of electromagnetism on εperiodic sets Sε⊂R3. Assuming that a family of Borel measures με, such that supp(με)=Sε, is obtained by εcontraction of a fixed 1periodic measure μ, and for righthand sides fε∈L2(R3,dμε), we prove ordersharp normresolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic “singular structures”, when μ is supported by lowerdimensional 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 Poincarétype 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 language  English 

Article number  67 
Number of pages  43 
Journal  Calculus of Variations and Partial Differential Equations 
Volume  61 
Issue number  2 
DOIs  
Publication status  Published  7 Feb 2022 
Bibliographical note
Funding Information:We are grateful to Igor Velčić for fruitful discussions, which motivated our version of the Poincaré inequality of Sect. . KC is grateful for the support of the Engineering and Physical Sciences Research Council: Grant EP/L018802/2 “Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory”.
ASJC Scopus subject areas
 Analysis
 Applied Mathematics
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Dive into the research topics of 'Operatornorm homogenisation estimates for the system of Maxwell equations on periodic singular structures'. Together they form a unique fingerprint.Projects
 2 Finished

Newton Mobility Grant : Homogenisation of Degenerate Equations and Scattering for New Materials
1/02/17 → 31/01/19
Project: Research council

Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory
Engineering and Physical Sciences Research Council
23/07/14 → 22/06/19
Project: Research council