@article {Ashton20140884,
author = {Ashton, A. C. L.},
title = {Laplace{\textquoteright}s equation on convex polyhedra via the unified method},
volume = {471},
number = {2176},
year = {2015},
doi = {10.1098/rspa.2014.0884},
publisher = {The Royal Society},
abstract = {We provide a new method to study the classical Dirichlet problem for Laplace{\textquoteright}s equation on a convex polyhedron. This new approach was motivated by Fokas{\textquoteright} 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{\textendash}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.},
issn = {1364-5021},
URL = {http://rspa.royalsocietypublishing.org/content/471/2176/20140884},
eprint = {http://rspa.royalsocietypublishing.org/content/471/2176/20140884.full.pdf},
journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences}
}