Green’s Function for the Neumann–Poisson Problem on n-Dimensional Balls

Abstract

We provide an elementary derivation of the Green function for Poisson’s equation with Neumann boundary data on balls of arbitrary dimension. Surprisingly, until very recently this Green function was only known in dimensions up to three, and an explicit construction (even in low dimensions) on the level of an undergraduate PDE class was lacking. This changes if one derives the Green function for Poisson’s equation from the Green function for the electroencephalography (EEG) equation (Poisson’s equation with dipole right-hand side)

• MSC: Primary 35J08
• Secondary 35C05
• 35E05
• 35J25

Acknowledgments

This work was supported by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung10.13039/501100005306 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXS 2044—390685587, Mathematics Münster: Dynamics—Geometry—Structure.