At the heart of their technique is the notion of a spectral germ for a class of operator pencils, which quantifies the leading order of frequency dispersion of waves in a heterogeneous medium near the bottom of the spectrum of the associated differential operator with periodic coefficients. In the case when \(\mu ^\varepsilon \) is the Lebesgue measure, operator-norm estimates for the Maxwell system have been obtained, using the spectral approach, in (for the “non-magnetic” case with no currents, as an application of the spectral germ technique introduced in the paper), (for magnetic field and induction in the presence of currents), (for the full system in the non-magnetic case), and (for the general Lebesgue measure case).Ī research programme similar to ours, although outside the context of thin structures and using a different analytical approach, has been pursued by Birman, Suslina, and subsequently by Suslina and her students, starting with. Equipped with this new machinery, in the present paper we set out to tackle a vector problem, in particular the system of Maxwell equations, which is of interest in applications to electromagnetism. \) The key technical tool for the proof of remainder estimates was a Poincaré-type inequality in an appropriate Sobolev space of quasiperiodic functions, conditioned by the fact that we deal with an arbitrary measure.
The strategy for the analysis of the family thus obtained was to use an asymptotic approximation for the solution in powers of \(\varepsilon ,\) carefully analyse the homogenisation corrector as a function of \(\varepsilon \) and \(\theta ,\) and obtain an estimate for the remainder that is uniform with respect to \(\theta. As a starting point of our approach, we considered the PDE family obtained from ( 1.1) by the Floquet transform (see ), in some sense replacing the macroscopic variable by an additional parameter \(\theta \) (“quasimomentum”), akin to the Fourier dual variable for PDE with constant coefficients. For arbitrarily small values of \(\varepsilon >0,\) we formulate and analyse the Maxwell system of equations of electromagnetism on \(\varepsilon \)-periodic sets \(S^\varepsilon \subset ^d,\) and the matrix-function A is uniformly positive definite.
0 kommentar(er)
