Exterior boundary-value problems for the Helmholtz equation with a generalized impedance boundary condition

An exterior boundary-value problem for the Helmholtz equation is considered, the boundary condition being of the (generally nonlocal) form ∂u/∂v + Bu = g, B a bounded linear operator in the Hilbert space L2(Γ), Γ denoting the boundary of the exterior domain, and g given in L2(Γ). It is also required that B satisfy a certain dissipativeness condition. The basic existence, uniqueness, and continuous-dependence results are stated. Various techniques for construction of the solution are described, which are extensions of the methods known for the Neumann or “classical” impedance boundary-value problems. It is argued that such an exterior problem is the appropriate one to study when modeling the more difficult interior-exterior interface problem corresponding to the scattering of time-harmonic acoustic waves by a penetrable body.