The Polyharmonic Dirichlet Problem and Path Counting

Abstract

The purpose of this article is to provide a solution to the $m$-fold Laplace equation in the half space $R_+^d$ under certain Dirichlet conditions. The solutions we present are 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.