Existence, uniqueness and explicit bounds for acoustic scattering by an infinite Lipschitz boundary with an impedance condition

Abstract. We study a boundary value problem for the Helmholtz equation with an impedance boundary condition, in two and three dimensions, modelling the scattering of time harmonic acoustic waves by an unbounded rough surface. Via analysis of an equivalent variational formulation we prove this problem to be well-posed when: i) the boundary has the strong local Lipschitz property and the frequency is small; ii) the rough surface is the graph of a bounded Lipschitz function (with arbitrary frequency). An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms, are explicit in terms of the wavenumber k, the geometry of the scatterer and the parameters describing the surface impedance.