Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

Abstract

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form D = {(x, z) ∈ Rn × R: z > f(x)} where f: Rn → R is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example, acoustic scattering problems, problems involving elastic waves and problems in potential theory, have been reformulated as second kind integral equations u + Ku = v in the space BC of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator A = I + K under consideration, with an emphasis on the function space setting BC. Firstly, under which conditions is A a Fredholm operator, and, secondly, when is the finite section method applicable to A?. © 2008 Rocky Mountain Mathematics Consortium.