---
tag: 2F6AJ1DM11
kind: definition
name: Green's function for the Laplacian
msc: [31A15, 31B10, 35J05]
defines: M36SD56MWQ
uses: [0EWMDD6C14, 0DEB0JMN0W, 05D5EGMFYV, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation
---

## Definition

Let $\Omega$ be a bounded [[domain|0EWMDD6C14]] in $\C$ with [[boundary|05D5EGMFYV]] $\partial\Omega$, and let $w \in \Omega$.
A *Green's function of $\Omega$ with pole at $w$* is a function $g(\cdot, w) \colon \Omega \setminus \set{w} \to \R$ such that:

1. $g(\cdot, w)$ is [[harmonic|0DEB0JMN0W]] on $\Omega \setminus \set{w}$.
2. The function $z \mapsto g(z, w) + \log\abs{z - w}$ extends to a harmonic function on a neighbourhood of $w$.
3. $g(z, w) \to 0$ as $z \to \zeta$, for every $\zeta \in \partial\Omega$.

## Remarks

If such a function exists, it is unique, by the maximum principle applied to the difference of two of them.
It exists whenever the [[Dirichlet problem|8D56GQ4AG8]] on $\Omega$ is solvable for continuous boundary data: take $g(z, w) = h(z) - \log\abs{z - w}$, where $h$ is harmonic with boundary values $\log\abs{\zeta - w}$.
Example: for the unit disc, $g(z, w) = \log\abs{\dfrac{1 - \bar w z}{z - w}}$.
Here $\log\abs{1 - \bar w z}$ is harmonic on the disc because $1 - \bar w z \ne 0$ there, and on $\abs{z} = 1$ one has $\abs{1 - \bar w z} = \abs{\bar z - \bar w} = \abs{z - w}$, so $g = 0$ on the boundary.
Non-example: for $w \ne 0$, the function $z \mapsto -\log\abs{z - w}$ on the unit disc satisfies clauses 1 and 2, but at the boundary point $\zeta = -w/\abs{w}$ its limit is $-\log(1 + \abs{w}) \ne 0$, so clause 3 fails.
In the sense of distributions, $\Delta_z g(z, w) = -2\pi\delta_w$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The Green's function is positive on $\Omega \setminus \set{w}$ and symmetric: $g(z, w) = g(w, z)$.
In $\R^n$ with $n \ge 3$, the analogous definition uses the singularity $\abs{x - y}^{2 - n}$ in place of $-\log\abs{z - w}$, up to normalizing constants.