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Aylwin, R., Jerez-Hanckes, C., & Pinto, J. (2020). On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation. Integr. Equ. Oper. Theory, 92(2), 41 pp.
Abstract: We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered 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 wellposedness of the associated first kind boundary integral equations.