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

Abstract

AbstractFor arbitrarily small values of $$\varepsilon >0,$$ ε > 0 , we formulate and analyse the Maxwell system of equations of electromagnetism on $$\varepsilon $$ ε -periodic sets $$S^\varepsilon \subset {{\mathbb {R}}}^3.$$ S ε ⊂ R 3 . Assuming that a family of Borel measures $$\mu ^\varepsilon ,$$ μ ε , such that $$\mathrm{supp}(\mu ^\varepsilon )=S^\varepsilon ,$$ supp ( μ ε ) = S ε , is obtained by $$\varepsilon $$ ε -contraction of a fixed 1-periodic measure $$\mu ,$$ μ , and for right-hand sides $$f^\varepsilon \in L^2({\mathbb {R}}^3, d\mu ^\varepsilon ),$$ f ε ∈ L 2 ( R 3 , d μ ε ) , we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic “singular structures”, when $$\mu $$ 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 $$L^2$$ 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.