Let \(U \subseteq \mathbb{R}^n\) be an open, bounded set with \(C^1\) boundary. The function

\[G(x, y) = \phi(y - x) - \phi^x(y) \quad (x, y \in U, x \neq y)\]

is called Green’s function for \(U\) of \(\Delta\), with

\[ \begin{cases} \Delta \phi^x = 0 & \text{in } U\\ \phi^x = \phi(y - x) & \text{on } \partial U, \end{cases} \]

where \(\phi\) denotes the fundamental solution of the Laplacian.

Remarks
  • The Green’s function is symmetric in both arguments [1, 2.2 Theorem 13].

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References Link to heading

  1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.