RESUMEN
For a nonlocally perturbed
half-space we consider the scattering of time-harmonic acoustic
waves. A second kind boundary integral equation formulation is proposed
for the sound-soft case, based on a standard ansatz
as a combined single- and double-layer potential but replacing the usual
fundamental solution of the Helmholtz equation with an appropriate half-space
Green's function. Due to the unboundedness of the
surface, the integral operators are noncompact.
In contrast to the two-dimensional case, the integral operators are
also strongly singular, due to the slow decay at infinity of the
fundamental solution of the three-dimensional Helmholtz equation. In
the case when the surface is sufficiently smooth (Lyapunov) we show that the integral operators are
nevertheless bounded as operators on $L^2(\Gamma)$ and
on $L^2(\Gamma)\cap BC(\Gamma)$ and that the operators depend
continuously in norm on the wave number and on $\Gamma$. We further
show that for \emph{mild} roughness, i.e.,
a surface $\Gamma$ which does not differ too much from a plane, the
boundary integral equation is uniquely solvable in the space
$L^2(\Gamma)\cap BC(\Gamma)$ and the scattering
problem has a unique solution which satisfies a limiting absorption
principle in the case of real wave number