Order-sharp norm-resolvent homogenisation estimates for Maxwell equations on periodic singular structures: the case of non-zero current and the general system

2020/07/09 by Cherednichenko, Kirill, D'Onofrio, Serena
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2007.04836

Abstract

For arbitrarily small values of ε>0, we formulate and analyse the Maxwell system of equations of electromagnetism on ε-periodic sets S^ε⊂\mathbb R³. Assuming that a family of Borel measures μ^ε, such that \rm supp(μ^ε)=S^ε, is obtained by ε-contraction of a fixed 1-periodic measure μ, and for right-hand sides f^ε∈ L²(\mathbb R³, dμ^ε), we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. In the resent work we address the case of non-zero current density in the Maxwell system and complete the analysis of the general setup including non-constant permittivity and permeability coefficients.

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