Alexey A. Pozharskii | Searcharxiv

arXiv2011

Homogenisation for a statinory Maxwell system

We study homogenization problem for the stationary Maxwell system. It is supposed that the magnetic permeability and the dielectric permittivity locally close to fast-oscillating (with respect to some small parameter) periodic functions which can change the form on rather big distances. An asymptotic behavior to solutions of the Maxwell system outside of its spectrum is obtained. We also describe asymptotic behavior of resolvent with control of the remainder in terms of some appropriate operator norms.

math-ph↗

arXiv2009

Averaging in scattering problems

We consider the scattering that is described by the equation $(-Δ_x + q(x,\frac{x}ε) - E)ψ= f(x), ψ= ψ(x,ε) \in \C, x \in \R^d, ε> 0, E > 0,$ where $q(x,y)$ is a periodic function of $y$, $q$ and $f$ have compact supports with respect to $x$. We are interested in the solution satisfying the radiation condition at infinity and describe the asymptotic behavior of the solution as $ε\to 0$. In addition, we find the asymptotic behavior of the scattering amplitude of the plain wave. Either of them (the solution and the amplitude) in the leading orders are described by the averaged equation with the potential $$\hat{q}(x) = \frac{1}{|Ω|}\int_Ωq(x,y)dy.$$

math-ph↗