Elliptic PDEs with constant coefficients on convex polyhedra via the unified method

We provide a new method to study the classical Dirichlet problem for constant coefficient second order elliptic PDEs on convex polyhedrons. Our approach is heavily motivated by Fokas' unified method for boundary value problems, and can be interpreted as the Fourier analogue to the classical boundary integral equations. 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 show that the analysis can be reduced to the case of Laplace's equation, from which the more general problem turns out to be a compact perturbation. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realisation of the fundamental principle of Ehrenpreis for all constant coefficient elliptic PDEs on convex polyhedra.