The Polyharmonic Dirichlet Problem and Path Counting

TL;DR

This paper introduces explicit solutions to the m-fold Laplace equation in half-space using boundary layer potentials, with coefficients linked to path counting problems, advancing understanding of higher-order Laplace problems.

Contribution

It provides explicit formulas for boundary layer potentials solving the m-fold Laplace equation, connecting solutions to combinatorial path counting.

Findings

Explicit formulas for boundary layer potentials

Coefficients determined by path counting

Solutions applicable to half-space Dirichlet problems

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.