Scattering by a Bounded Highly Oscillating Periodic Medium and the Effect of Boundary Correctors
Fioralba Cakoni, Bojan B. Guzina, Shari Moskow, Tayler Pangburn · SIAM Journal on Applied Mathematics · 2019
We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size $\epsilon$. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the $O(\epsilon)$ boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the “bulk” corrector---the so-called periodic drift, which is found to emerge only at $O(\epsilon^2)$. We illustrate the analysis by examples in one and two spatial dimensions.