The polyharmonic Dirichlet problem and path counting

This article provides a solution to the m-fold Laplace equation in the half-space R-+(d) under certain Dirichlet conditions. The solution we present is a series of m boundary layer potentials. We give explicit formulas for these layer potentials as linear combinations of powers of the Laplacian applied to the Dirichlet data, with coefficients determined by certain path counting problems. (C) 2013 Elsevier Masson SAS. All rights reserved.