Laplace's equation on convex polyhedra via the unified method

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.