title: Laplace's equation on convex polyhedra via the unified method.
abstract: We provide a new method to study the classical Dirichlet problem for Laplace's equation on a convex polyhedron. This new approach was motivated by Fokas' unified method for boundary value problems. The central object in this approach is the global relation: an integral equation which couples the known boundary data and the unknown boundary values. This integral equation depends holomorphically on two complex parameters, and the resulting analysis takes place on a Banach space of complex analytic functions closely related to the classical Paley-Wiener space. We write the global relation in the form of an operator equation and prove that the relevant operator is bounded below using some novel integral identities. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realization of the fundamental principle of Ehrenpreis.
authors.affiliation: Department of Applied Mathematics and Theoretical Physics , University of Cambridge , Cambridge CB3 0WA, UK.
authors.forename: A C L
authors.initials: AC
authors.lastname: Ashton
keywords: Laplace; Paley–Wiener; boundary value problem; unified method