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

Chandler-Wilde, S. N. and Lindner, M. (2008) Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method. Journal of Integral Equations and Applications, 20 (1). pp. 13-48. ISSN 0897-3962

## Abstract/Summary

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ 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 $A$?

Item Type: Article Yes Faculty of Science > School of Mathematical and Physical Sciences > Department of Mathematics and Statistics 1614 Rocky Mountain Mathematics Consortium