The Polyharmonic Dirichlet Problem and Path Counting

2013/05/22 by Thomas Hangelbroek, Hangelbroek, Thomas, Aaron Lauve +1
Computer Science · Mathematics · #05A15 #31B30 #35J40 #41A63 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Differential Equations and Numerical Methods #FOS: Mathematics #Spectral Theory in Mathematical Physics #math.AP #math.CA #math.CO #msc:05A15 #msc:31B30 #msc:35J40 #msc:41A63

paper · pdf · doi:10.48550/arxiv.1305.5063

34 pages

arxiv created 2013/05/22 · openalex publication_date 2013/05/22 · arxiv updated 2013/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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