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 mu ^varepsilon , such that mathrm{supp}(mu ^varepsilon )=S^varepsilon , is obtained by varepsilon -contraction of a fixed 1-periodic measure mu , and for right-hand sides f^varepsilon in L^2({mathbb {R}}^3, dmu ^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 mu 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.
Highlights
The operator-theoretic perspective on partial differential equations (PDE) with multiple scales has proved effective for obtaining sharp convergence results for problems of periodic homogenisation, see e.g. [13,14,23,54,63,66,70] for related developments in the “wholespace” setting, i.e. when the spatial domain is invariant with respect to shifts by the elements of a periodic lattice in Rd, d ≥ 2
As a starting point of our approach, we considered the PDE family obtained from (1.1) by the Floquet transform, in some sense replacing the macroscopic variable by an additional parameter θ (“quasimomentum”), akin to the Fourier dual variable for PDE with constant coefficients
We develop a new tool for proving the estimates, namely asymptotic expansions that are uniform in the quasimomentum, see Sect
Summary
The operator-theoretic perspective on partial differential equations (PDE) with multiple scales has proved effective for obtaining sharp convergence results for problems of periodic homogenisation, see e.g. [13,14,23,54,63,66,70] for related developments in the “wholespace” setting, i.e. when the spatial domain is invariant with respect to shifts by the elements of a periodic lattice in Rd , d ≥ 2. The operator-theoretic perspective on partial differential equations (PDE) with multiple scales has proved effective for obtaining sharp convergence results for problems of periodic homogenisation, see e.g. [13,14,23,54,63,66,70] for related developments in the “wholespace” setting, i.e. when the spatial domain is invariant with respect to shifts by the elements of a periodic lattice in Rd , d ≥ 2. The techniques developed in the above works have highlighted a variety of different new ways to interpret the process homogenisation, e.g. via the singular-value decomposition of operator resolvents or by extending the classical perturbation series to PDE families that.
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