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)

中文翻译：

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n 维球上 Neumann-Poisson 问题的格林函数

摘要 我们提供了泊松方程格林函数的初等推导，其中包含任意维度球的诺依曼边界数据。令人惊讶的是，直到最近，这个格林函数才知道最多三个维度，并且缺乏本科 PDE 课程级别的显式构造（即使在低维度）。如果从脑电图 (EEG) 方程的格林函数（右侧偶极子的泊松方程）推导出泊松方程的格林函数，这种情况就会改变