Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modeling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator that are valid for sufficiently large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction and that are only assumed to be bounded (i.e., L ^∞) in the other spatial directions. This class of coefficients therefore includes coefficients modeling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibers (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly oscillatory coefficients and study the limiting behavior when the period of oscillations goes to zero. In particular, we bound the H ¹ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to the best of our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.