Green function to the biharmonic equation on $n$-dimensional sphere

TL;DR

This paper derives explicit Green functions for the biharmonic equation on n-dimensional spheres, expanding the mathematical tools available for solving such PDEs in spherical geometries.

Contribution

The paper provides explicit series representations of Green functions for the biharmonic equation on n-dimensional spheres, including special cases with specific parameters.

Findings

Explicit Green functions expressed as Gegenbauer polynomial series.

Representation formulas for particular parameter sets.

Enhanced understanding of biharmonic problems on spherical domains.

Abstract

Homogeneous and inhomogeneous biharmonic equation are considered on the $n$-dimensional unit sphere. The Green function is given as a series of Gegenbauer polynomials. In the paper, explicit representations of the Green function are found for some sets of parameters.