Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ƒ∈C1,1(ℝ). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all Lp spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.