An integral equation method for a boundary value problem arising in unsteady water wave problems

Abstract

In this paper we consider the 2D Dirichlet boundary value problem for Laplace's equation in a nonlocally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragḿen-Lindel̈of arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalization of the Fredholm alternative in Arens et al. [2]. This then leads to an existence proof for the boundary value problem. © 2006 Rocky Mountain Mathematics Consortium.