On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation.

Abstract

We study the mapping properties of boundary integral oper_x005F_x0002_ators arising when solving two-dimensional, time-harmonic waves scat_x005F_x0002_tered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood frequencies, continuity and coercivity results are derived to prove well_x005F_x0002_posedness of the associated first kind boundary integral equati